Optimization Algorithms and Software at SOLweb.stanford.edu/group/SOL/talks/saunders-CLAODE.pdf ·...
Transcript of Optimization Algorithms and Software at SOLweb.stanford.edu/group/SOL/talks/saunders-CLAODE.pdf ·...
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Optimization Algorithms and Softwareat SOL
Michael SaundersSOL and ICME, Stanford University
Computational Linear Algebra and Optimizationfor the Digital Economy
International Centre for Mathematical SciencesEdinburgh, Scotland, Oct 31–Nov 1, 2013
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 1/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Abstract
When a colleague knocks on your door with a mathematical problem, great joy ensues on both sides ifyou happen to have numerical software that can solve the problem. This is a reason for writing“general-purpose software”.
If the software is downloadable, similar happiness ensues when it gets used for unexpected applications.You learn about such cases if your software includes bugs(!) or Google does its usual precise search.
The Systems Optimization Laboratory (SOL) was founded by George Dantzig and Richard Cottle atStanford University in 1974 to encourage algorithm and software development in traditional OperationsResearch areas. Dantzig’s group built increasingly challenging linear models of the US Economy (thePILOT linear programs), while Alan Manne continued to expand his nonlinear economic models (whichbegan before large-scale optimization software existed). These spurred the development of MINOS atSOL (by the speaker and fellow New Zealander Bruce Murtagh) in parallel with Arne Drud’sdevelopment of CONOPT at the World Bank and then in Denmark.
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 2/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
In the 1980s, the SOL “Gang of 4” (Gill, Murray, Saunders, and Wright) developed “dense” solversLSSOL, QPSOL, and NPSOL. After Philip Gill moved to UC San Diego, NPSOL became increasinglyimportant for trajectory optimization at McDonnell-Douglas (now Boeing). This spurred thedevelopment of the large-scale optimizers SQOPT and SNOPT.
Many years later, MINOS, CONOPT, and SNOPT remain heavily used solvers within the GAMS andAMPL algebraic modeling systems (and on the NEOS server), and LUSOL remains the reliable“engine” for basis handling in MINOS, SQOPT, SNOPT, and other solvers such as PATH and lp solve.A unique feature of LUSOL is its Threshold Rook Pivoting option for estimating the rank of arectangular sparse matrix.
Other solvers developed at SOL include MINRES, MINRES-QLP, LSQR, LSMR, and PDCO. Wereview the mechanics of the solvers and some unexpected applications that they’ve been put to workon. We conclude with even more unexpected aspects of optimization reported by radio and TVaudiences in New Zealand.
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 3/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
1 SOL
2 Solvers for Ax ≈ b
3 Applications of linear solvers
4 Rank of stoichiometric matrices
5 Solvers for optimization
6 Applications of optimization solvers
7 Aerospace
8 AC
9 Optimization in NZ
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 4/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
SOL
Systems Optimization LaboratoryStanford University
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 5/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
SOL
Founded 1974 by George Dantzig and Richard CottleDantzig, Alan Manne: economic models (linear & nonlinear)Gill, Murray, Saunders, Wright: Software for optimization
George Dantzig Gene Golub1914–2005 1932–2007
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 6/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
SOL
Founded 1974 by George Dantzig and Richard CottleDantzig, Alan Manne: economic models (linear & nonlinear)Gill, Murray, Saunders, Wright: Software for optimization
George Dantzig Gene Golub1914–2005 1932–2007Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 6/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
SOL
Founded 1974 by George Dantzig and Richard CottleDantzig, Alan Manne: economic models (linear & nonlinear)Gill, Murray, Saunders, Wright: Software for optimization
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 7/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
SOLRecent collaborators:
Chris Paige (McGill), Sou-Cheng Choi (Chicago) MINRES-QLP
David Fong (ICME and Facebook) LSMR
Xiangrui Meng (ICME and LinkedIn) LSRN
Jason Lee, Yuekai Sun (ICME) PNOPT
Ding Ma, Nick Henderson, Santiago Akle (ICME) LUSOL, PDCO
Philip Gill, Elizabeth Wong (UC San Diego)Optimization software NPSOL, QPOPT, SQOPT, SNOPT, SQIC
Ronan Fleming, Ines Thiele (UCSD, Iceland, Luxembourg)Flux balance analysis (FBA), Flux variability analysis (FVA)Rank and nullspace of stoichiometric matrices(Need LUSOL, PDCO, SQOPT)
Funding: ONR, AFOSR, ARO, DOE, NSF, AHPCRC, . . . ,DOE DE-FG02-09ER25917, NIH U01-GM102098
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 8/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
SOLRecent collaborators:
Chris Paige (McGill), Sou-Cheng Choi (Chicago) MINRES-QLP
David Fong (ICME and Facebook) LSMR
Xiangrui Meng (ICME and LinkedIn) LSRN
Jason Lee, Yuekai Sun (ICME) PNOPT
Ding Ma, Nick Henderson, Santiago Akle (ICME) LUSOL, PDCO
Philip Gill, Elizabeth Wong (UC San Diego)Optimization software NPSOL, QPOPT, SQOPT, SNOPT, SQIC
Ronan Fleming, Ines Thiele (UCSD, Iceland, Luxembourg)Flux balance analysis (FBA), Flux variability analysis (FVA)Rank and nullspace of stoichiometric matrices(Need LUSOL, PDCO, SQOPT)
Funding: ONR, AFOSR, ARO, DOE, NSF, AHPCRC, . . . ,DOE DE-FG02-09ER25917, NIH U01-GM102098
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 8/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Sparse linear equations Ax = b
and least squares problems Ax ≈ b
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 9/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Sparse direct methods for Ax = b and Ax ≈ b
A = LDU LUSOL (Stanford)
A = QR SPQR (Tim Davis, UFL)
Many sparse solvers HSL Library (RAL, UK)Iain Duff, John Reid, Jennifer Scott, . . .
LUSOL, SPQR, MA27, MA47, MA57, MA67, MA77, . . .offer rank-revealing capability for sparse matriceswhen used with suitable tolerances
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 10/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Iterative solvers for Ax ≈ b
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 11/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Iterative methods for Ax ≈ b
A may be a sparse matrixor an operator for computing Av and/or ATw
A maybe Hermitian or complexA may have any rank
Symmetric Ax = b A = MINRES-QLP
min ‖Ax − b‖ A = or or LSQR, LSMR
Tall skinny min ‖Ax − b‖ A = (real) LSRN
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 12/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
LSQR, LSMR for min ‖Ax − b‖
Golub-Kahan process generates Uk =[u1 u2 . . . uk
]Vk =
[v1 v2 . . . vk
]using products Avj , A
Tujkth approximation xk = Vkyk for some yk
Choose yk to minimize something
LSQR min ‖rk‖ residual rk = b − AxkLSMR min
∥∥ATrk∥∥ residual for ATAx = ATb
For LS problems, LSQR, LSMR stop when‖ATrk‖‖rk‖ ≤ α ‖A‖
(this is the Stewart backward error)
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 13/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
LSQR, LSMR for min ‖Ax − b‖
Golub-Kahan process generates Uk =[u1 u2 . . . uk
]Vk =
[v1 v2 . . . vk
]using products Avj , A
Tujkth approximation xk = Vkyk for some yk
Choose yk to minimize something
LSQR min ‖rk‖ residual rk = b − AxkLSMR min
∥∥ATrk∥∥ residual for ATAx = ATb
For LS problems, LSQR, LSMR stop when‖ATrk‖‖rk‖ ≤ α ‖A‖
(this is the Stewart backward error)
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 13/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
log10‖ATrk‖‖rk‖
(typical) LSMR can stop sooner
0 100 200 300 400 500 600 700 800 900 1000−6
−5
−4
−3
−2
−1
0
iteration count
log(
E2)
Name:lp pilot ja, Dim:2267x940, nnz:14977, id=88
LSQRLSMR
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 14/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
CG, MINRES for posdef Ax = b
Lanczos process generates Vk =[v1 v2 . . . vk
]using products Avj
kth approximation xk = Vkyk for some yk
Choose yk to minimize something
CG min ‖x − xk‖A energy norm of errorMINRES min ‖rk‖
They stop when‖rk‖
α ‖A‖ ‖xk‖+ β ‖b‖≤ 1 (backward error argument)
For posdef A, ‖xk‖ ↗ for both methods (Steihaug 1983, Fong 2011)
Hence, backward error ↘ for MINRES (but not for CG)
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 15/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
CG, MINRES for posdef Ax = b
Lanczos process generates Vk =[v1 v2 . . . vk
]using products Avj
kth approximation xk = Vkyk for some yk
Choose yk to minimize something
CG min ‖x − xk‖A energy norm of errorMINRES min ‖rk‖
They stop when‖rk‖
α ‖A‖ ‖xk‖+ β ‖b‖≤ 1 (backward error argument)
For posdef A, ‖xk‖ ↗ for both methods (Steihaug 1983, Fong 2011)
Hence, backward error ↘ for MINRES (but not for CG)
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 15/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
CG, MINRES for posdef Ax = b
Lanczos process generates Vk =[v1 v2 . . . vk
]using products Avj
kth approximation xk = Vkyk for some yk
Choose yk to minimize something
CG min ‖x − xk‖A energy norm of errorMINRES min ‖rk‖
They stop when‖rk‖
α ‖A‖ ‖xk‖+ β ‖b‖≤ 1 (backward error argument)
For posdef A, ‖xk‖ ↗ for both methods (Steihaug 1983, Fong 2011)
Hence, backward error ↘ for MINRES (but not for CG)
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 15/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZlog10 ‖rk‖/‖xk‖ for A � 0 MINRES can stop sooner
0 200 400 600 800 1000 1200 1400−10
−9
−8
−7
−6
−5
−4
−3
−2
−1
iteration count
log(
||r||/
||x||)
Name:Schenk_AFE_af_shell8, Dim:504855x504855, nnz:17579155, id=11
CGMINRES
0 50 100 150 200 250 300 350 400 450−10
−9
−8
−7
−6
−5
−4
−3
−2
−1
0
iteration count
log(
||r||/
||x||)
Name:Cannizzo_sts4098, Dim:4098x4098, nnz:72356, id=13
CGMINRES
0 1 2 3 4 5 6 7 8 9 10
x 104
−18
−16
−14
−12
−10
−8
−6
−4
−2
0
2
iteration count
log(
||r||/
||x||)
Name:Simon_raefsky4, Dim:19779x19779, nnz:1316789, id=7
CGMINRES
0 0.5 1 1.5 2 2.5 3 3.5
x 104
−10
−9
−8
−7
−6
−5
−4
−3
−2
−1
0
iteration count
log(
||r||/
||x||)
Name:BenElechi_BenElechi1, Dim:245874x245874, nnz:13150496, id=22
CGMINRES
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 16/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Stoichiometric matricesin systems biology
Sparse matrices SRows: Chemical species
Cols: Chemical reactions
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 17/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
S for models 1, 2, 3, 4 (all similar)
0 200 400
0
100
200
300
400
nz = 2314
iIT341
0 200 400 600
0
100
200
300
400
500
600
nz = 3067
iAF692
0 200 400 600
0
100
200
300
400
500
600
nz = 3821
iSB619
0 500 1000
0
200
400
600
800
nz = 4407
iJN746
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 18/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
S for Models 5, 6, 7, 8 (all similar)
0 200 400 600 800 1000
0
200
400
600
nz = 4503
iJR904
0 500 1000
0
200
400
600
800
nz = 4793
iNJ661
0 500 1000
0
200
400
600
800
1000
nz = 5300
iND750
0 500 1000 1500 2000
0
500
1000
1500
nz = 9231
iAF1260
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 19/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Model 9 (Recon1)
0 500 1000 1500 2000 2500 3000 3500
0
500
1000
1500
2000
2500
nz = 14300
Recon1
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 20/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Model 10 (ThMa = Thermotoga maritima)
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 21/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Model 11 (GlcAer)
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 22/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
S for models 9, 10, 11
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 23/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Rank of stoichiometric matrices
Conservation analysis forbiochemical networks
Need rank(S) and nullspace(ST)
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 24/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
rank(S) by SVD
Singular value decomposition S = UDV T
UTU = I V TV = I D diagonal rank(S) = rank(D)
Ideal for rank-estimation but U,V are dense
model 9 (Recon1) 2800× 3700 17 secsmodel 10 (ThMa) 15000× 18000 11 hoursmodel 11 (GlcAer) 62000× 77000 ∞
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 25/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Singular values of models 1–8 Dense SVD of ST
log10(σi )0 500
−20
−15
−10
−5
0
5iIT341
0 500 1000−20
−15
−10
−5
0
5iAF692
0 500 1000−20
−15
−10
−5
0
5iSB619
0 500 1000−20
−15
−10
−5
0
5iJN746
0 500 1000−20
−15
−10
−5
0
5iJR904
0 500 1000−20
−15
−10
−5
0
5iNJ661
0 1000 2000−20
−15
−10
−5
0
5iND750
0 1000 2000−20
−15
−10
−5
0
5iAF1260
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 26/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
rank(S) by QR
Householder QR factorization SP = QR
P = col perm QTQ = I R triangular rank(S) = rank(R)
Nearly as reliable as SVD
Dense QR used by Vallabhajosyula, Chickarmane, Sauro (2005)
Sparse QR (SPQR) now available: Davis (2013)
model 9 (Recon1) 2800× 3700 0.1 secsmodel 10 (ThMa) 15000× 18000 2.5 secsmodel 11 (GlcAer) 62000× 77000 0.2 secs(!)
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 27/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
rank(S) by LUSOL with Threshold Rook Pivoting
Sparse LU with TRP P1SP2 = LDU
P1,P2 = perms D diagonal rank(S) ≈ rank(D)L,U well-conditioned
Lii = Uii = 1|Lij | and |Uij | ≤ factol = 4 (or 2 or 1.2, 1.1, . . . )
LUSOL: Main engine in sparse linear/nonlinear optimizers MINOS, SQOPT,SNOPT
model 9 (Recon1) 2800× 3700 0.1 secsmodel 10 (ThMa) 15000× 18000 4.0 secsmodel 11 (GlcAer) 62000× 77000 158 secs
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 28/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
rank(S) by LUSOL with Threshold Partial Pivoting
Sparse LU with TPP P1SP2 = LU
P1,P2 = perms U trapezoidal rank(S) ≈ rank(U)L well-conditioned
Lii = 1|Lij | ≤ factol = 4 (or 2 or 1.2, 1.1, . . . )
LUSOL: Main engine in sparse linear/nonlinear optimizers MINOS, SQOPT,SNOPT
model 9 (Recon1) 2800× 3700 0.1 secsmodel 10 (ThMa) 15000× 18000 0.2 secsmodel 11 (GlcAer) 62000× 77000 0.3 secs
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 29/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
SPQR vs LUSOL with Threshold Rook Pivoting
SPQR: S = QRtime (secs)
model | m n rank(S) | nnz(S) nnz(Q) nnz(R) | SVD SPQR
-------------------------------------------------------------------
Recon1 | 2766 3742 2674 | 14300 2750 21093 | 17.5 0.1
ThMa | 15024 17582 14983 | 326035 844096 10595016 | 11hrs 2.5
GlcAer | 62212 76664 62182 | 913967 1287 916600 | infty 0.2
LUSOL: S = LDU |Lij |, |Uij | ≤ 2.0
nnz(L) nnz(U) | time
-------------------------------------------------------------------
Recon1 | 4280 16463 | 0.1
ThMa | 30962 346122 | 4.1
GlcAer | 635571 1810491 | 186.2
LUSOL: S = LDU |Lij |, |Uij | ≤ 4.0
nnz(L) nnz(U) | time
-------------------------------------------------------------------
Recon1 | 2701 12896 | 0.1
ThMa | 36350 330485 | 4.0
GlcAer | 427456 1584188 | 157.9
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 30/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
SPQR vs LUSOL with Threshold Rook Pivoting
SPQR: ST = QRtime (secs)
model | m n rank(S’)| nnz(S) nnz(Q) nnz(R) | SVD SPQR
-------------------------------------------------------------------
Recon1 | 3742 2766 2674 | 14300 107935 36929 | 17.2 0.1
ThMa | 17582 15024 14983 | 326035 624640 605888 | 11hrs 0.7
GlcAer | 76664 62212 62182 | 913967 3573696 4038988 | infty 2.7
LUSOL: ST = LDU |Lij |, |Uij | ≤ 2.0
nnz(L) nnz(U) | time
-------------------------------------------------------------------
Recon1 | 12832 7421 | 0.3
ThMa | 501198 358601 | 37.8
GlcAer | 1996892 709448 | 586.0
LUSOL: ST = LDU |Lij |, |Uij | ≤ 4.0
nnz(L) nnz(U) | time
-------------------------------------------------------------------
Recon1 | 9811 6093 | 0.2
ThMa | 410290 355475 | 14.8
GlcAer | 1823067 711906 | 791.2
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 31/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
SPQR vs LUSOL with Threshold Partial Pivoting
SPQR: S = QRtime (secs)
model | m n rank(S) | nnz(S) nnz(Q) nnz(R) | SVD SPQR
-------------------------------------------------------------------
Recon1 | 2766 3742 2674 | 14300 2750 21093 | 17.5 0.1
ThMa | 15024 17582 14983 | 326035 844096 10595016 | 11hrs 2.5
GlcAer | 62212 76664 62182 | 913967 1287 916600 | infty 0.2
LUSOL: S = LU |Lij |, |Uij | ≤ 2.0
nnz(L) nnz(U) | time
-------------------------------------------------------------------
Recon1 | 721 13585 | 0.1
ThMa | 7779 324483 | 0.2
GlcAer | 533 913781 | 0.4
LUSOL: S = LU |Lij |, |Uij | ≤ 4.0
nnz(L) nnz(U) | time
-------------------------------------------------------------------
Recon1 | 764 13577 | 0.1
ThMa | 7782 323929 | 0.2
GlcAer | 533 913781 | 0.4
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 32/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
SPQR vs LUSOL with Threshold Partial Pivoting
SPQR: ST = QRtime (secs)
model | m n rank(S’)| nnz(S) nnz(Q) nnz(R) | SVD SPQR
-------------------------------------------------------------------
Recon1 | 3742 2766 2674 | 14300 107935 36929 | 17.2 0.1
ThMa | 17582 15024 14983 | 326035 624640 605888 | 11hrs 0.7
GlcAer | 76664 62212 62182 | 913967 3573696 4038988 | infty 2.7
LUSOL: ST = LU |Lij | ≤ 2.0
nnz(L) nnz(U) | time
-------------------------------------------------------------------
Recon1 | 9304 7813 | 0.2
ThMa | 81506 268938 | 2.7
GlcAer | 337433 703619 | 126.7
LUSOL: ST = LU |Lij | ≤ 4.0
nnz(L) nnz(U) | time
-------------------------------------------------------------------
Recon1 | 9030 6259 | 0.1
ThMa | 77274 268424 | 2.0
GlcAer | 316889 701139 | 176.5
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 33/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Rank of stoichiometric SPerhaps
Sparse LU with TPP P1SP2 = LU
L well-conditioned rank(S) ≈ rank(U)
Then
Sparse LU with TRP P1UP2 = LDU
L, U well-conditioned rank(S) ≈ rank(U) ≈ rank(D)
or
Sparse LU with TPP P1UT P2 = LU
L well-conditioned rank(S) ≈ rank(U) ≈ rank(U)
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Rank of stoichiometric SPerhaps
Sparse LU with TPP P1SP2 = LU
L well-conditioned rank(S) ≈ rank(U)
Then
Sparse LU with TRP P1UP2 = LDU
L, U well-conditioned rank(S) ≈ rank(U) ≈ rank(D)
or
Sparse LU with TPP P1UT P2 = LU
L well-conditioned rank(S) ≈ rank(U) ≈ rank(U)Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 34/101
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LSQR
in parallel!
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LSQR for tomographyHuang, Dennis, Wang, Chen 2013A scalable parallel LSQR algorithm for solving large-scale linear system for tomographic problems: a case studyin seismic tomographyProcedia Computer Science 18:581–590
min ‖Ax − b‖ A =
585 He Huang et al. / Procedia Computer Science 18 ( 2013 ) 581 – 590
!"#$%&'( )*+#,$'& )*+#,$'- )*+#,$'-
. . . .(/
(0
(a) A partitioning of the sparse matrix vector product A ! xfor five MPI tasks. The colors correspond to di!erent MPItasks. Note that the kernel rows at the top of matrix A arepartitioned across columns, while the damping rows are par-titioned across rows. The corresponding partitions of vectorsx and y are also included. Note that the piece of vector y thatcorresponds to the kernel rows is replicated across all tasks.
!"#$%&'( )*+#,$'& )*+#,$'-
1&#*20*0')*+#,$'&%3
(/%
(0%
&%
-/%
-0%
-/
-0%
(b) Vector reconstruction and A! x ( Aki ! xi and Adi ! x"i ) inone MPI task. Yellow represents local task.
Fig. 3: Illustration matrix vector multiplication: A ! x
algorithm and does not represent an excessive overhead because only 10% of the memory storage for the entirematrix A is a result of the damping component. Note that we do not use a superscript T for the transposed dampingcomponent but rather Adt to reflect that it is a separate copy of matrix within our algorithm. Ad is stored in CSRformat and Adt is stored in CSC format.
The vector x is partitioned across columns and is illustrated in Figures 3a and 4a. Vector y has two parts ineach task, a replicated kernel piece and a partitioned damping piece corresponding to the damping matrix. Thelength of kernel part of yki is the same as the number of rows in kernel Ak. The damping part of ydi is partitionedaccording to the number of rows of damping Adi in each task.
6. Parallel Computation
We next describe the most computationally expensive and complex section of SPLSQR algorithm, i.e., thecalculations of y # A ! x + y and x # AT ! y + x. The calculation of y # A ! x + y represents lines (11) - (17)in Figure 1, while x # AT ! y + x represents lines (18) - (24). Both calculations require the reconstruction ofthe necessary pieces of vectors x and y by communicating with other MPI tasks. However they also di!er in keycharacteristics due to the particular partitioning of matrices Ad and Ak.
6.1. y# A ! x + y
Figure 3b illustrates the matrix vector multiplication y # A ! x from the perspective of the yellow task. Webegin with a description of calculating yk that involves the kernel matrix. Each task multiplies its local piece ofkernel Aki with local piece of xi and yields its kernel part of vector, yki, as indicated by the top part of vector y inFigure 3b. Note that because of the column-based partitioning of Ak, yki is a partial result. Due to Ak’s randomand nearly dense feature, the resulting yki has overlap with all the other tasks. Hence a reduction across all tasksis performed on yki to combine the partial results. The final result is illustrated in black in Figure 3b. Because thenumber of rows in the kernel nk is very small relative to the entire number of rows n, the total cost of the reductiononly has a modest impact.
Next we describe the calculation of the damping component of ydi. Unlike the calculation of yki, the calcu-lation of ydi requires additional pieces of x that the yellow task does not currently own. Note that in Figure 3b,the width of the Adi matrix is greater than xi. The required additional pieces of x are owned by its neighboring
261M× 38M5 billion nonzeros
3D stuctural seismologyModest-sized dataset from Los Angeles Basin (ANGF)Cray XT5 (Kraken) at Oak Ridge National LabParallelize y ← Ax + y and x ← ATy + x
2400 cores: 10 times faster than PETSc19200 cores: 33 times faster than PETSc
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SOL optimization solvers
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Optimization solvers (dense)
LSSOL (f77): min ‖Xx − b‖2 st ` ≤(
xAx
)≤ u
Avoids forming XTXWe still don’t have a good method for sparse X + constraints
QPOPT (f77): min 12x
THx st ` ≤(
xAx
)≤ u
H may be indefinite
NPSOL (f77): minφ(x) st ` ≤
xAxc(x)
≤ u
Philip Gill has recently completed a new NPSOL that includes elastic bounds tohandle infeasible QP subproblems (like SQOPT and SNOPT)
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Optimization solvers (sparse)
MINOS (f77): Sparse NLPNo elastic bounds, but still widely used. f90 version half started.
SQOPT, SNOPT (f77): Sparse convex QP + general NLPElastic bounds, threadsafe, good for expensive functions
SNOPT9 (f2003): Gill, S, WongIncludes SQIC QP solverSwitches from SQOPT’s reduced-gradient method to KKT-factorization +block-LU updates for problems with many degrees of freedom
Change 1 line and recompile ⇒ everything in quad precisionSNOPT9’s simplex implementation gives us solutions withastounding accuracy! Great for systems biologists!
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NEOS
Free optimization solversvia Argonne National Lab
(now Univ of Madison, Wisconsin)
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NEOS Solver Statistics for 1 year 1 Oct 2012 -- 30 Sep 2013
Total Jobs 1264001
Solver Submissions
MINOS 642263 BDMLP 3747 OOQP 771 sd 33
KNITRO 256367 Couenne 3588 SYMPHONY 692 cplex 29
Gurobi 70179 FilMINT 3424 PATHNLP 666 BiqMac 10
Ipopt 35181 BLMVM 3410 DSDP 627 LGO 3
SNOPT 33197 NMTR 2735 condor 601 lpopt 2
csdp 20142 feaspump 2607 sedumi 585 BDLMP 1
DICOPT 19146 AlphaECP 2538 PSwarm 584
XpressMP 17761 bpmpd 2317 RELAX4 568
BARON 14737 PATH 2036 Clp 503
Cbc 14613 LANCELOT 1972 FortMP 432
MINTO 13515 MILES 1641 sdplr 415
scip 12986 sdpt3 1436 Glpk 336
MOSEK 10837 LRAMBO 1415 ddsip 335
CONOPT 9871 filterMPEC 1289 nsips 300
LOQO 9598 SDPA 1244 icos 249
Bonmin 7320 TRON 1145 bnbs 244
MINLP 6573 SBB 1085 pensdp 234
filter 5438 NLPEC 1053 penbmi 226
concorde 4879 L-BFGS-B 999 PGAPack 216
LINDOGlobal 4597 qsopt_ex 902 proxy 127
MUSCOD-II 4425 ASA 858 xpress 36
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NEOS Solver Statistics for 1 year 1 Oct 2012 -- 30 Sep 2013
Total Jobs 1264001
Category Submissions Input Submissions
nco 982242 AMPL 1053956
milp 100864 GAMS 152810
minco 57532 SPARSE_SDPA 22083
lp 42205 MPS 10136
sdp 24909 TSP 4879
go 17955 C 4513
cp 12591 Fortran 4114
bco 5547 CPLEX 3723
co 4889 MOSEL 2182
miocp 4425 MATLAB_BINARY 1802
kestrel 3981 LP 903
uco 2735 ZIMPL 709
lno 2407 SDPA 627
slp 612 DIMACS 564
ndo 601 SMPS 277
sio 300 MATLAB 199
socp 75 SDPLR 198
mip 8 QPS 164
nlp 3
DNLP 2
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DICOPT
MINLP solverIgnacio Grossman, Carnegie-Mellon
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MINLP model
MS student Rui-Jie Zhou transferring from MS&E to EE,mentioned MINOS had been useful
2 papers in Industrial Engineering and Chemistry ResearchZhou, Ji-Juan Li, Hong-Guang Dong, Ignacio Grossmann
Part I (44 pages) Multiscale state-space superstructure for interplantwater-allocation and heat-exchange networks design with direct and indirectintegration schemes in fixed flow rate (FF) processes
Part II (42 pages) Extends to fixed contaminant-load (FC) processes andintegration of FF and FC processes
Nonlinear objective1500 linear and nonlinear constraints1600 variables (some binary)GAMS/DICOPT = MINOS + CPLEX or CONOPT + CPLEX
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MINLP model
Distribution Network (DN)
External Sources
InternalSources
InternalSinks
External Sinks
PO_HEN
Water user
PO_W
Water user
Figure 1. State-space superstructure for stand-alone WAHEN
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MINLP model
Interplant distribution network
PO_PLANT1
PO_PLANT2
Jun1
Junn
tr
External Sources
PO_JUN
PO_IHEN
PO_TR
External Sinks
Figure 2. Multi-scale state-space superstructure for interplant WAHEN
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MINLP model
E4
E5
E3
E2E1
C1
C2
H1
H2
C4 215.5ºC
458.7kW
183.9kW
2116.5kW
1911kW889kW
557.5kW
594.5kW
H1 377ºC
H2 317ºC
1726kW
616.2kW
291.3kW
C1 137ºC
C2 80ºC
H1 97ºC
H2 97ºC
C1 377ºC
C2 227ºC
H3 105ºC H3 25ºC
H4 185ºC H4 35ºC
C3 25ºCC3 185ºC
5ºC
6ºCPlant 1
Plant 2
10KW/K
20KW/K
15KW/K
13KW/K
10KW/K
5KW/K
7.5KW/K
14.51KW/K
5.49KW/K
C3
2000KW/K633.5KW/K557.5kW
49.5ºC 110.7ºC
93.3ºC
86.6ºC
278.1ºC 337.4ºC
171.1ºC 183.3ºC
288.1ºC
5.46ºC
5.31ºC
6.17ºC
209ºC, 0.619kg/s
210ºC, 0.619kg/s
HPS
Figure 3. Optimal interplant HEN configuration in Example 1
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MINLP model
30kg/s 100ºC
56kg/s 95ºC
36kg/s 50ºC
32kg/s 60ºC
9.4kg/s
25.1kg/s
35kg/s 50ºC
40.8kg/s40kg/s 20ºC
4.5kg/s
27.5kg/s
59.3kg/s
10.3kg/s 23.2kg/s
28kg/s 95ºC
40kg/s 100ºC
E1
C2 Waste Sink
E1H1
40kg/s70ºC
30kg/s75ºC
32kg/s 35ºC
45kg/s 60ºC45kg/s 45ºC
38kg/s20ºC
55kg/s10ºC
42kg/s 65ºC1.7kg/s
4kg/s
12.7kg/s
1.3kg/s
5858kW
E1
E2 25.8℃Waste Sink
E2
E1H1
43kg/s
18.6kg/s
3.8kg/s
22.6kg/s
12kg/s
16.4kg/s
14.7kg/s
17kg/s
21kg/s 128kg/s
5858kW
2786kW
2786kW 6579kW [40, 39.9, 74.7]
[50, 49.1, 91.8]
[38.6, 53.6, 80]
[30, 46.3, 68.4]
C12.4kg/s 20.6kg/s
51.2kg/s 71.8kg/s
2.4kg/s [25.7, 37.4, 100]
[40, 39.9, 74.7]
13.3kg/s
158.8kg/s
22.7kg/s 57.7kg/s
17.7kg/s [31, 28.4, 60]
[26.5, 35.2, 120]
[30, 40, 76.7]
[30, 40, 76.7]
Plant 1 TAC= $3,594,621
Plant 2 TAC= $3,027,041
3287kW
14499kW
840kW
7742kW
96.4℃ 48.4℃
49.8℃
100kg/s 100kg/s
12.0℃
28.4℃
4.3kg/s
25.0℃
30℃
35.7℃86.4℃100℃
70.4℃21.4kg/s
41.6℃19.3kg/s
31.7kg/s 74.4℃ 49.2℃
65.0℃
7.3kg/s 21.7kg/s
20.3℃
15.8℃
Fresh water
Cold utility
14.9kg/s
28.3kg/s
Source 2 Source 1 Source 4
Source 3
Sink 1
Sink 2
Sink 3
Sink 4
Source 5
Source 6
Source 7
Source 8Sink 6
Sink 5Sink 7
Sink 8
Figure 4. Optimal stand-alone WAHEN designs in Case 1 of Example 2
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MINLP model
E1
E3
30kg/s 100ºC
56kg/s 95ºC
36kg/s 50ºC
32kg/s 60ºC
35kg/s 50ºC
40kg/s 20ºC
40kg/s 100ºC
40kg/s70ºC
30kg/s75ºC
55kg/s10ºC
38kg/s 20ºC
32kg/s 35ºC
45kg/s 45ºC
45kg/s 60ºC
42kg/s 65ºC
Plant 1Plant 2
E2
28kg/s 95ºC
H1
E1
E2
E3
Waste Sink
22.9kg/s
23kg/s
7kg/s
8kg/s
21.1kg/s
21kg/s
7kg/s
23.7kg/s
9.3kg/s
3kg/s4.7kg/s
11.5kg/s
42.4kg/s
13.8kg/s
21.1kg/s
3.1kg/s
62.5kg/s
3kg/s5.5kg/s
57.5kg/s
17.5kg/s
10.5kg/s
22kg/s
11kg/s
23.2kg/s27.4kg/s
17.5kg/s1.4kg/s
42.6kg/s
36.8kg/s
227.7kg/s
14.6kg/s22kg/s
[44, 36, 60]
[40, 39.1, 86.5]
[30, 40, 76.7]
[30, 40, 76.7]
[50, 50, 116.4]
[50, 42.5, 96.2]
[32.3, 51, 80]
[30, 52.5, 80]
3233kW
1469kW
1469kW
15194kW
15194kW
8919kW
8919kW
[30, 41.6, 93]
96.6℃
48kg/s
15.8kg/s
12.6kg/s
28.7℃
35.7℃
45.7℃
23.6℃ 65.6℃
21kg/s
33.8℃
17.1kg/s
76.1℃
86.6℃
2kg/s
18.2℃
3kg/s
Sink1
Sink2
Sink3
Sink4
Source 1
Source 2
Source 3
Source 4
Source 5
Source 6Source 7
Source 8
Sink5
Sink6
Sink8 Sink7
Figure 5. Optimal direct integrated WAHEN design in Case 2 of Example 2
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MINLP model
E1
Jun 1
E1
H1
Waste Sink
E2
E2
30kg/s100ºC
0.5kg/s
12.1kg/s
8.6kg/s 8.8kg/s
56kg/s95ºC
42.6kg/s4.3kg/s
5.4kg/s
36kg/s50ºC
35.9kg/s
0.1kg/s
32kg/s 60ºC
3.8kg/s
43kg/s41.6kg/s
0.7kg/s
8.4kg/s
32.5kg/s
6.7kg/s
25.9kg/s
28kg/s 95ºC
40kg/s 100ºC
40kg/s 20ºC24.6kg/s
4.7kg/s
35kg/s 50ºC
40kg/s70ºC
7.8kg/s
3.7kg/s
30kg/s75ºC
19.4kg/s
12.5kg/s
5.9kg/s
14.4kg/s
16.7kg/s32kg/s 35ºC
45kg/s 45ºC
20.7kg/s
7.9kg/s
55kg/s10ºC
45.9kg/s
9.1kg/s
38kg/s20ºC
17.9kg/s
20.1kg/s
35.3kg/s
13.7kg/s
21.6kg/s
1.5kg/s
14.7kg/s
45kg/s60ºC
42kg/s 65ºC
18.1kg/s
15.4kg/s
220kg/s
[42.9, 40, 60]
[36.9, 55, 81.5]
[40, 40, 98.4]
[30, 31, 80]
[50, 50, 90]
[40, 56, 100]
[40, 43, 80]
[30, 44.4, 80]
Jun 2
Plant 1
Plant 2
101℃
74.0℃
8.0kg/s
10488kW
10488kW2184kW
5737kW
5737kW
[30, 44.4, 97.1][50.1, 98.2, 173.2] [60, 89.4, 132.5]
37℃
85.1℃
37.8℃
85.1℃
95.1℃
63.7℃
25.5kg/s
3.6kg/s
4.0kg/s
2.5kg/s 12.6kg/s
Source 1
Source 2
Source 3
Source 4
Sink 1
Sink 2
Sink 3
Sink 4
Source 5
Source 6
Source 8 Source 7
Sink 5Sink 6
Sink 7
Sink 8
Figure 6. Optimal indirect integrated WAHEN design in Case 3 of Example 2
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MINLP model
Lower total annualized cost can be obtained in all examples by solving thecorresponding MINLP model
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PDCO
Primal-dual interior methodfor convex optimization
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PDCO (Matlab primal-dual convex optimizer)
Nominally
minimizex
φ(x)
subject to Ax = b, ` ≤ x ≤ u
φ(x) is convex with known gradient and Hessian
For example, φ(x) = cTx or λ ‖x‖1 or∑
xj log(xj)
A may be a sparse matrix or an operator for Av and ATw
Basis Pursuit (BP)Basis Pursuit denoising (BPDN) Chen, Donoho, & S (2001)
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PDCO (Matlab primal-dual convex optimizer)
To ensure unique solutions, PDCO solves regularized problems:
minimizex , r
φ(x) + 12 ‖D1x‖2 + 1
2 ‖r‖2
subject to Ax + D2r = b, ` ≤ x ≤ u
where D1, D2 are diagonal and positive-definite
D1 = γI γ = 10−3 or 10−4
D2 = δI for linear programs δ = 10−3 or 10−4
D2 = I for least squares
Jacek prefers D1, D2 semi-definite and dynamic!
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PDCO for LP feasibility
Regularized least squares with bounds:
minimizex , r
12 ‖γx‖
2 + 12 ‖r‖
2
subject to Ax + r = b, ` ≤ x ≤ u
γ = 10−4 declare feasible if ‖r‖∞ ≤ 10−4 say
Jon Dattorro (2010)
10,000 × 200,000 1.1M nonzerosSolve 200,000 times with different b
Gurobi: average 2 minsPDCO (Matlab): average 1 min
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PDCO for LP feasibility
Regularized least squares with bounds:
minimizex , r
12 ‖γx‖
2 + 12 ‖r‖
2
subject to Ax + r = b, ` ≤ x ≤ u
γ = 10−4 declare feasible if ‖r‖∞ ≤ 10−4 say
Jon Dattorro (2010)
10,000 × 200,000 1.1M nonzerosSolve 200,000 times with different b
Gurobi: average 2 minsPDCO (Matlab): average 1 min
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PDCO applied to FBA
Flux Balance Analysis = LP problem (Palsson 2006)
FBA minimizevf ,vr ,ve
dTve
subject to Svf − Svr + Seve = 0
vf , vr ≥ 0, ` ≤ ve ≤ u
d optimizes a biological objectivee.g., maximize replication rate in unicellular organisms
ve = exchange fluxes = sources and sinks of chemicals
PDCO works with A =[S −S Se
]then LLT = AD2AT
(sparse Cholesky with D increasingly ill-conditioned)
Solution is v∗ = v∗f − v∗r and v∗e
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PDCO applied to Entropy problem
EP minimizevf ,vr
vTf (log vf +c−e) + vTr (log vr +c−e)
subject to Svf − Svr = − Sev∗e
vf , vr > 0
c = any vector, e = (1, 1, . . . , 1)T
v∗e = optimal exchange fluxes from FBA
Entropy objective function is strictly convex
Solution v∗f , v∗r is thermodynamically feasible
(satisfies energy conservation and 2nd law of thermodynamics)
Fleming, Maes, S, Ye, Palsson (2012)
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PDCO applied to LR-NMR problems
Laplace Inversion of Low-Resolution NMR Relaxometry DataBiotech and Environmental Engineering, Ben Gurion Univ, Israel
Determine composition of olive oil, rapeseed oil, biodiesel, . . .
s(t) ≈∫ ∞
0e−t/T2 x(T2) dT2 x = probability density
Standard method (discretize):
minx≥0
∥∥∥∥(AλI
)x −
(s0
)∥∥∥∥2
2
A = discrete Laplace transform (fast operator)λ = Tikhonov regularizationCould apply PDCO to handle x ≥ 0Distorts solution by broadening peaks
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SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
PDCO applied to LR-NMR problems
Laplace inversion via basis pursuit denoising:
BPDN minimizex , r
α ‖x‖1 + 12 ‖λx‖
22 + 1
2 ‖r‖22
subject to Ax + r = s : y
x ≥ 0
A = discrete Laplace transform (fast operator)
PDCO solves min
∥∥∥∥(DAT
I
)∆y −
(Dwt
)∥∥∥∥ using LSMR, where posdef diagonal D
becomes increasingly ill-conditioned
α helps resolve close adjacent peaks
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 59/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
α = 0
α > 0
Figure 6 Comparison of WinDXP (a)–(d) and PDCO using the universal regularization valuesfor a1 and a2 (e)–(h) solutions on a real LR-NMR dataset acquired from an oil sample. Theresults are ordered by descending number of scans (descending SNR).
LAPLACE INVERSION OF LR-NMR RELAXOMETRY DATA 85
Concepts in Magnetic Resonance Part A (Bridging Education and Research) DOI 10.1002/cmr.a
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 60/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
PDCO applied to LR-NMR problems
Berman, Leshem, Etziony, Levi, Parmet, S, Wiesman 2013Novel 1H low field nuclear magnetic resonance applications for the field of biodieselBiotechnology for Biofuels 6:55, 20pp
Berman, Levi, Parmet, S, Wiesman 2013Laplace inversion of low-resolution NMR relaxometry data using sparse representation methodsConcepts in Magnetic Resonance Part A 42A:3, 72–88
Problem setup (normal numerical people):
[U,S,V] = svd(A);
Astounding line of code (one coauthor):
[S,V,D] = svd(A); (!)
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 61/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
PDCO applied to LR-NMR problems
Berman, Leshem, Etziony, Levi, Parmet, S, Wiesman 2013Novel 1H low field nuclear magnetic resonance applications for the field of biodieselBiotechnology for Biofuels 6:55, 20pp
Berman, Levi, Parmet, S, Wiesman 2013Laplace inversion of low-resolution NMR relaxometry data using sparse representation methodsConcepts in Magnetic Resonance Part A 42A:3, 72–88
Problem setup (normal numerical people):
[U,S,V] = svd(A);
Astounding line of code (one coauthor):
[S,V,D] = svd(A); (!)
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 61/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
PDCO applied to LR-NMR problems
Berman, Leshem, Etziony, Levi, Parmet, S, Wiesman 2013Novel 1H low field nuclear magnetic resonance applications for the field of biodieselBiotechnology for Biofuels 6:55, 20pp
Berman, Levi, Parmet, S, Wiesman 2013Laplace inversion of low-resolution NMR relaxometry data using sparse representation methodsConcepts in Magnetic Resonance Part A 42A:3, 72–88
Problem setup (normal numerical people):
[U,S,V] = svd(A);
Astounding line of code (one coauthor):
[S,V,D] = svd(A); (!)
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 61/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
SQOPT in quad precision
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 62/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Flux Balance Analysis (FBA) on Thermotoga maritima
min cTv subject to Sv = 0, ` ≤ v ≤ u
S rows and cols 18210× 17535Nonzero Sij 33602max and min |Sij | 2× 104 and 3× 10−6
SQOPT in double precision (15 digits)
Feasibility tol 1e-6
Optimality tol 1e-6
SQOPT in quad precision (32 digits)
Feasibility tol 1e-15
Optimality tol 1e-15
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 63/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Flux Balance Analysis (FBA) on Thermotoga maritima
min cTv subject to Sv = 0, ` ≤ v ≤ u
S rows and cols 18210× 17535Nonzero Sij 33602max and min |Sij | 2× 104 and 3× 10−6
SQOPT in double precision (42 secs)
SQOPT EXIT 10 -- the problem appears to be infeasible
Problem name ThMa
No. of iterations 18500 Objective value 8.2286249495E-07
No. of infeasibilities 9 Sum of infeas 1.9606461069E-03
No. of degenerate steps 11611 Percentage 62.76
Max x (scaled) 3482 8.2E+00 Max pi (scaled) 18210 9.8E-01
Max x 5134 5.9E+00 Max pi 18210 1.0E+00
Max Prim inf(scaled) 32832 1.3E-03 Max Dual inf(scaled) 16417 1.0E+00
Max Primal infeas 32832 5.6E-06 Max Dual infeas 32669 2.3E+02
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 64/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Flux Balance Analysis (FBA) on Thermotoga maritima
min cTv subject to Sv = 0, ` ≤ v ≤ u
S rows and cols 18210× 17535Nonzero Sij 33602max and min |Sij | 2× 104 and 3× 10−6
Restart SQOPT in quad precision (36 secs)
SQOPT EXIT 0 -- finished successfully
Problem name ThMa
No. of iterations 498 Objective value 8.7036461686E-07
No. of infeasibilities 0 Sum of infeas 0.0000000000E+00
No. of degenerate steps 220 Percentage 44.18
Max x (scaled) 3482 8.2E+00 Max pi (scaled) 2907 1.3E+00
Max x 5134 5.9E+00 Max pi 15517 1.1E+00
Max Prim inf(scaled) 16475 5.2E-28 Max Dual inf(scaled) 13244 1.9E-32
Max Primal infeas 16475 5.2E-29 Max Dual infeas 13244 4.8E-33
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 65/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Flux Balance Analysis (FBA) on GlcAer
min cTv subject to Sv = 0, ` ≤ v ≤ u
S rows and cols 68300× 76664Nonzero Sij 926357max and min |Sij | 8× 105 and 5× 10−5
SQOPT in quad precision cold start, no scaling (30786 secs)
SQOPT EXIT 0 -- finished successfully
Problem name GlcAer
No. of iterations 84685 Objective value -7.0382454070E+05
No. of degenerate steps 62127 Percentage 73.36
Max x 61436 6.3E+07 Max pi 25539 2.4E+07
Max Primal infeas 72623 3.0E-21 Max Dual infeas 17817 2.7E-21
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 66/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Flux Balance Analysis (FBA) on GlcAer
min cTv subject to Sv = 0, ` ≤ v ≤ u
S rows and cols 68300× 76664Nonzero Sij 926357max and min |Sij | 8× 105 and 5× 10−5
SQOPT in quad precision cold start, with scaling (4642 secs)
SQOPT EXIT 0 -- finished successfully
Problem name GlcAer
No. of iterations 37025 Objective value -7.0382454070E+05
No. of degenerate steps 28166 Percentage 76.07
Max x (scaled) 59440 3.7E+00 Max pi (scaled) 40165 8.1E+11
Max x 61436 6.3E+07 Max pi 25539 2.4E+07
Max Prim inf(scaled) 81918 7.0E-16 Max Dual inf(scaled) 59325 1.5E-17
Max Primal infeas 81918 1.3E-07 Max Dual infeas 27953 2.0E-22
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 67/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Flux Balance Analysis (FBA) on GlcAer
min cTv subject to Sv = 0, ` ≤ v ≤ u
S rows and cols 68300× 76664Nonzero Sij 926357max and min |Sij | 8× 105 and 5× 10−5
SQOPT in quad precision warm start, no scaling (28 secs)
SQOPT EXIT 0 -- finished successfully
Problem name GlcAer
No. of iterations 1 Objective value -7.0382454070E+05
No. of degenerate steps 0 Percentage 0.00
Max x 61436 6.3E+07 Max pi 25539 2.4E+07
Max Primal infeas 141186 7.1E-21 Max Dual infeas 14993 8.9E-23
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 68/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Aerospace Applications
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 69/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
NASA Aerospace Applications
David Saunders1970 Visit Stanford for 1 month (now 43 years)1973 Serra House, RA, MS (thanks to Gene)1974–present NASA Ames
ProjectsOAW Oblique All-Wing supersonic airlinerHSCT Supersonic airlinerCTV SHARP shuttle designMSL Heat shield for landing Mars rover CuriosityCEV Heat shield for Apollo-type capsule to ISS/moon
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 70/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
OAW oblique all wing airliner
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 71/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
HSCT high speed civil transport
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 72/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
CTV crew transfer vehicle
SHARP design (Slender Hypervelocity Aerothermodynamic Research Probes)Aerothermal performance constraint in (Velocity, Altitude) space, used during trajectory optimization with UHTC materials (Ultra High Temperature
Ceramics) to avoid exceeding material limits
Trajectory optimization with SNOPT
Could always abort to Kennedy, Boston, Gander, or Shannon
4000-mile cross-range capability during reentry
Image credit: David Kinney, NASA Ames Research Center
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 73/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
CEV crew exploration vehicle
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 74/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
McDonnell-Douglas Aerospace Applications
Philip Gill, Rocky Nelson1979–1988 SOL QPSOL, LSSOL, NPSOL
1988–2007 UC San Diego QPOPT, SQOPT, SNOPT
McDonnell-Douglas Space Systems, LA (now Boeing)
ProjectsF-4 Minimum time-to-climb
DC-Y SSTO Minimum-fuel landing maneuver
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 75/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
F-4 minimum time-to-climb
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 76/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
DC-Y single-stage-to-orbit
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 77/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 78/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 79/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
DC-Y landing, 2nd OTIS/NPSOL optimization
1st optimization: starting altitude = 2800ft
2nd optimization: starting altitude = variable
New constraint needed:
Don’t exceed 3g
Optimum starting altitude = 1400ft(!)
Come back Alan Shephard!
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 80/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
DC-Y landing, 2nd OTIS/NPSOL optimization
1st optimization: starting altitude = 2800ft
2nd optimization: starting altitude = variable
New constraint needed: Don’t exceed 3g
Optimum starting altitude = 1400ft(!)
Come back Alan Shephard!
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 80/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
DC-Y landing, 2nd OTIS/NPSOL optimization
1st optimization: starting altitude = 2800ft
2nd optimization: starting altitude = variable
New constraint needed: Don’t exceed 3g
Optimum starting altitude = 1400ft(!)
Come back Alan Shephard!
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 80/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
DC-Y landing, 2nd OTIS/NPSOL optimization
1st optimization: starting altitude = 2800ft
2nd optimization: starting altitude = variable
New constraint needed: Don’t exceed 3g
Optimum starting altitude = 1400ft(!)
Come back Alan Shephard!
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 80/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Stanford Aerospace Applications
Ilan Kroo Stanford Aircraft Aerodynamics and Design Group
Blended Wing-Body Transonic airliner
Antony Jameson, Juan Alonso Aerospace Computing Lab
MDO Multidisciplinary Design Optimization
ASO Aerodynamic Shape Optimization
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 81/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Stanford Aerospace Applications
Ilan Kroo Stanford Aircraft Aerodynamics and Design Group
Blended Wing-Body Transonic airliner
Antony Jameson, Juan Alonso Aerospace Computing Lab
MDO Multidisciplinary Design Optimization
ASO Aerodynamic Shape Optimization
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 81/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Blended wing airliner
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 82/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Blended wing airliner
Ilan Kroo, Michael Holden, Aero/Astro Dept, Stanford 1999
Compute control for stable flight of 17ft-span flying modelCollocation model:
minimize wing weight (or move CG as far aft as possible)subject to flutter constraints
9000 nonlinear equations9000 state variables, 7 design variables
MINOS 5.5:
26 major iterations4000 minor iterations (first 3000 = LP)1500 constraint + Jacobian evaluations
= 60000 function evaluations3 days on SGI Octane
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 83/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
News Flash, 3 March 2007
Mike Ross Naval Postgraduate School, Monterey
DIDO: A package for solving optimal control problemsImplemented in MatlabCalls TOMLAB/SNOPT for the optimization
GMT 062:19:26The International Space Station was successfully maneuvered
using DIDO/TOMLAB/SNOPT
Found zero propellant solutions (globally optimal)Saved NASA $1M fuel cost
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 84/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
America’s Cup Yacht RaceAC 95: NPSOL was used in different ways by both AC95 finalists
For Team Dennis Conner, NPSOL was used with Boeing’s TRANAIR CFD system foroptimal hull design of Young America
For Team NZ, Andy Philpott (University of Auckland) used NPSOL to maximize thevelocity around the course of 11 potential hull designs. One design appeared significantlyfaster and was chosen to become Black Magic
AC 2013 (San Francisco Bay): NZ Herald blog
(Score 8–5) Come on whale, where are you?
(Score 8–6) Dratted whales, we can’t trust them anymore
I haven’t had so much fun since the cat died
. . .
(Score 8–9) In the immortal words, ’bugga’
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 85/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
America’s Cup Yacht RaceAC 95: NPSOL was used in different ways by both AC95 finalists
For Team Dennis Conner, NPSOL was used with Boeing’s TRANAIR CFD system foroptimal hull design of Young America
For Team NZ, Andy Philpott (University of Auckland) used NPSOL to maximize thevelocity around the course of 11 potential hull designs. One design appeared significantlyfaster and was chosen to become Black Magic
AC 2013 (San Francisco Bay): NZ Herald blog
(Score 8–5) Come on whale, where are you?
(Score 8–6) Dratted whales, we can’t trust them anymore
I haven’t had so much fun since the cat died
. . .
(Score 8–9) In the immortal words, ’bugga’
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 85/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
America’s Cup Yacht RaceAC 95: NPSOL was used in different ways by both AC95 finalists
For Team Dennis Conner, NPSOL was used with Boeing’s TRANAIR CFD system foroptimal hull design of Young America
For Team NZ, Andy Philpott (University of Auckland) used NPSOL to maximize thevelocity around the course of 11 potential hull designs. One design appeared significantlyfaster and was chosen to become Black Magic
AC 2013 (San Francisco Bay): NZ Herald blog
(Score 8–5) Come on whale, where are you?
(Score 8–6) Dratted whales, we can’t trust them anymore
I haven’t had so much fun since the cat died
. . .
(Score 8–9) In the immortal words, ’bugga’
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 85/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
America’s Cup Yacht RaceAC 95: NPSOL was used in different ways by both AC95 finalists
For Team Dennis Conner, NPSOL was used with Boeing’s TRANAIR CFD system foroptimal hull design of Young America
For Team NZ, Andy Philpott (University of Auckland) used NPSOL to maximize thevelocity around the course of 11 potential hull designs. One design appeared significantlyfaster and was chosen to become Black Magic
AC 2013 (San Francisco Bay): NZ Herald blog
(Score 8–5) Come on whale, where are you?
(Score 8–6) Dratted whales, we can’t trust them anymore
I haven’t had so much fun since the cat died
. . .
(Score 8–9) In the immortal words, ’bugga’
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 85/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
America’s Cup Yacht RaceAC 95: NPSOL was used in different ways by both AC95 finalists
For Team Dennis Conner, NPSOL was used with Boeing’s TRANAIR CFD system foroptimal hull design of Young America
For Team NZ, Andy Philpott (University of Auckland) used NPSOL to maximize thevelocity around the course of 11 potential hull designs. One design appeared significantlyfaster and was chosen to become Black Magic
AC 2013 (San Francisco Bay): NZ Herald blog
(Score 8–5) Come on whale, where are you?
(Score 8–6) Dratted whales, we can’t trust them anymore
I haven’t had so much fun since the cat died
. . .
(Score 8–9) In the immortal words, ’bugga’
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 85/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
America’s Cup Yacht RaceAC 95: NPSOL was used in different ways by both AC95 finalists
For Team Dennis Conner, NPSOL was used with Boeing’s TRANAIR CFD system foroptimal hull design of Young America
For Team NZ, Andy Philpott (University of Auckland) used NPSOL to maximize thevelocity around the course of 11 potential hull designs. One design appeared significantlyfaster and was chosen to become Black Magic
AC 2013 (San Francisco Bay): NZ Herald blog
(Score 8–5) Come on whale, where are you?
(Score 8–6) Dratted whales, we can’t trust them anymore
I haven’t had so much fun since the cat died
. . .
(Score 8–9) In the immortal words, ’bugga’
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 85/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
America’s Cup Yacht RaceAC 95: NPSOL was used in different ways by both AC95 finalists
For Team Dennis Conner, NPSOL was used with Boeing’s TRANAIR CFD system foroptimal hull design of Young America
For Team NZ, Andy Philpott (University of Auckland) used NPSOL to maximize thevelocity around the course of 11 potential hull designs. One design appeared significantlyfaster and was chosen to become Black Magic
AC 2013 (San Francisco Bay): NZ Herald blog
(Score 8–5) Come on whale, where are you?
(Score 8–6) Dratted whales, we can’t trust them anymore
I haven’t had so much fun since the cat died
. . .
(Score 8–9) In the immortal words, ’bugga’
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 85/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Optimization in NZ
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 86/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
In New Zealand, the radio/TV guide is called The Listener.Every week a Life in New Zealand column publishes clippings describing local events.The first sender receives a $5 Lotto Lucky Dip. The following clippings illustrate somecharacteristics of optimization problems in the real(?) world.
Robust solutionsRECOVERY CARE gives you financial protection from specified sudden illness.You get cash if you live . . . and cash if you don’t.
No objective functionPeople have been marrying and bringing up children for centuries now. Nothinghas ever come of it. (Evening Post, 1977)
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 87/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
In New Zealand, the radio/TV guide is called The Listener.Every week a Life in New Zealand column publishes clippings describing local events.The first sender receives a $5 Lotto Lucky Dip. The following clippings illustrate somecharacteristics of optimization problems in the real(?) world.
Robust solutionsRECOVERY CARE gives you financial protection from specified sudden illness.You get cash if you live . . . and cash if you don’t.
No objective functionPeople have been marrying and bringing up children for centuries now. Nothinghas ever come of it. (Evening Post, 1977)
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 87/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Multiple objectives“I had the choice of running over my team-mate or going onto the grass, so I ranover my team-mate then ran onto the grass”, Rymer recalled later.
Obvious objectiveHe said the fee was increased from $5 to $20 because some people hadcomplained it was not worth writing a cheque for $5.
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 88/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Equilibrium condition“The pedestrian count was not considered high enough to justify an overbridge”,Helen Ritchie said. “And if there continues to be people knocked down on thecrossing, the number of pedestrians will dwindle.”
ConstraintsENTERTAINERS, DANCE BAND, etc. Vocalist wanted for New Wave rock band,must be able to sing.
DRIVING INSTRUCTOR Part-time position. No experience necessary.
HOUSE FOR REMOVAL in excellent order, $800. Do not disturb tenant.
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 89/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Exactly one feasible solutionMATTHEWS RESTAURANT, open 365 nights. Including Mondays.
Buying your own business might mean working 24 hours a day. But at least whenyou’re self-employed you can decide which 24.
Peters: Oh, it’s not that I don’t want to be helpful. But in this case the answer isthat I don’t want to be helpful. (Listener, 1990)
Sergeant J Johnston said when Hall was stopped by a police patrol the defendantdenied being the driver, but after it was pointed out he was the only person in thecar he admitted to being the driver.
His companion was in fact a transvestite, X, known variously as X or X.
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 90/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Bound your variablesBy the way, have you ever seen a bird transported without the use of a cage? Ifyou don’t use a cage it will fly away and maybe the same could happen to yourcat. Mark my words, we have seen it happen.
Redundant constraintsIf you are decorating before the baby is born, keep in mind that you may have aboy or a girl.
EAR PIERCING while you wait
CONCURRENT TERM FOR BIGAMY (NZ Herald, 1990)
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 91/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Infeasible constraintsI chose to cook myself to be quite sure what was going into the meals.
We apologize to Wellington listeners who may not be receiving this broadcast.
The model 200 is British all the way from its stylish roofline to its French-madeMichelin tyres. (NZ Car Magazine)
BALD, 36 yr old, handsome male seeking social times and fun with bald 22 yearsand upwards female Napier Courier, 28/2/02
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 92/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
≥ or ≤?BUY NOW! At $29.95 these jeans will not last long!
NOT TOO GOOD TO BE TRUE! We can sell your home for much less thanyou’d expect! (NZ Property Weekly)
The BA 146’s landing at Hamilton airport was barely audible above airportbackground noise, which admittedly included a Boeing 737 idling in theforeground.
Yesterday Mr Palmer said “The Australian reports are not correct that I’ve seen,although I can’t say that I’ve seen them”.
It will be a chance for all women of this parish to get rid of anything that is notworth keeping but is too good to throw away. Don’t forget to bring your husbands.
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 93/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
≥ or ≤?The French were often more blatant and more active, particularly prop X andnumber eight Y, but at least oneAll Black was seen getting his retaliation in first.
WHAT EVERY TEENAGER SHOULD KNOW — PARENTS ONLY
“Love Under 17” Persons under 18 not admitted.
“Keeping young people in the dark would not stop them having sex—in fact itusually had the opposite effect,” she said.
NELSON, approximately 5 minutes from airport. Golf course adjacent. Sleepsseven all in single beds. Ideal for honeymoons. (Air NZ News, 1978)
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 94/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Hard or soft constraintsThe two have run their farm as equal partners for 10 years, with Jan in charge ofgrass management, Lindsay looking after fertilizer, and both working in the milkshed. “We used to have our staff meetings in bed. That got more difficult whenwe employed staff!” (NZ country paper)
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 95/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Elastic constraintsThe Stationary Engine Drivers Union is planning rolling stoppages.
When this happens there are set procedures to be followed and they areestablished procedures, provided they are followed.
APATHY RAMPANT? Not in Albany—the closing of the electoral rolls saw fully103.49 percent of the area’s eligible voters signed up.
Auckland City ratepayers are to be reminded that they can pay their rates afterthey die. (Auckland Herald, 1990)
He was remanded in custody to appear again on Tuesday if he is still in thecountry.
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 96/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Convergence“There is a trend to open libraries when people can use them”, he says.
Mayor for 15 years, Sir Dove-Myer wants a final three yearsat the helm “to restore sanity and stability in the affairs ofthe city”.
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 97/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Applications(Yachting) It is not particularly dangerous, as it only causes vomiting, hot andcold flushes, diarrhoea, muscle cramping, paralysis, and sometimes death . . . (Boating
New Zealand, 1990)
(Ecological models) CAR POLLUTION SOARS IN CHRISTCHURCH—BUTCAUSE REMAINS MYSTERY
Nappies wanted for window cleaning. Must be used.
(Optimal control) Almost half the women seeking fertility investigations at theclinic knew what to do to get pregnant
, but not when to do it.
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 98/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Applications(Yachting) It is not particularly dangerous, as it only causes vomiting, hot andcold flushes, diarrhoea, muscle cramping, paralysis, and sometimes death . . . (Boating
New Zealand, 1990)
(Ecological models) CAR POLLUTION SOARS IN CHRISTCHURCH—BUTCAUSE REMAINS MYSTERY
Nappies wanted for window cleaning. Must be used.
(Optimal control) Almost half the women seeking fertility investigations at theclinic knew what to do to get pregnant, but not when to do it.
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 98/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Binary variables0 or 1 Sometimes neither is optimal
Integer variables0 or 1 or 2 . . .
When Taupo police arrested a Bay of Plenty manfor driving over the limit,they discovered he was a bigamist. Nelson Mail, 5/04
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 99/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Binary variables0 or 1 Sometimes neither is optimal
Integer variables0 or 1 or 2 . . .
When Taupo police arrested a Bay of Plenty manfor driving over the limit,they discovered he was a bigamist. Nelson Mail, 5/04
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 99/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Binary variables0 or 1 Sometimes neither is optimal
Integer variables0 or 1 or 2 . . .
When Taupo police arrested a Bay of Plenty manfor driving over the limit,they discovered he was a bigamist. Nelson Mail, 5/04
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 99/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Always room for improvementThe owner Craig Andrew said the three main qualities for the job were speed,agility and driving skills. “Actually, Merv has none of those, but he’s still the bestdelivery boy we’ve had”, he said.
Always room for improvementThe dean Jim Plummer said the three main qualities for the job were speed, agilityand drinking skills. “Actually, Gene has none of those, but he’s still the bestseminar organizer and party host we’ve had”, he said.
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 100/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Always room for improvementThe owner Craig Andrew said the three main qualities for the job were speed,agility and driving skills. “Actually, Merv has none of those, but he’s still the bestdelivery boy we’ve had”, he said.
Always room for improvementThe dean Jim Plummer said the three main qualities for the job were speed, agilityand drinking skills. “Actually, Gene has none of those, but he’s still the bestseminar organizer and party host we’ve had”, he said.
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 100/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Always room for improvementThe owner Craig Andrew said the three main qualities for the job were speed,agility and driving skills. “Actually, Merv has none of those, but he’s still the bestdelivery boy we’ve had”, he said.
Always room for improvementThe dean Jim Plummer said the three main qualities for the job were speed, agilityand drinking skills. “Actually, Gene has none of those, but he’s still the bestseminar organizer and party host we’ve had”, he said.
SCCM
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 100/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Always room for improvementThe owner Craig Andrew said the three main qualities for the job were speed,agility and driving skills. “Actually, Merv has none of those, but he’s still the bestdelivery boy we’ve had”, he said.
Always room for improvementThe dean Jim Plummer said the three main qualities for the job were speed, agilityand drinking skills. “Actually, Gene has none of those, but he’s still the bestseminar organizer and party host we’ve had”, he said.
SCCMEI
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 100/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Always room for improvementThe owner Craig Andrew said the three main qualities for the job were speed,agility and driving skills. “Actually, Merv has none of those, but he’s still the bestdelivery boy we’ve had”, he said.
Always room for improvementThe dean Jim Plummer said the three main qualities for the job were speed, agilityand drinking skills. “Actually, Gene has none of those, but he’s still the bestseminar organizer and party host we’ve had”, he said.
SCCMEI
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 100/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Always room for improvementThe owner Craig Andrew said the three main qualities for the job were speed,agility and driving skills. “Actually, Merv has none of those, but he’s still the bestdelivery boy we’ve had”, he said.
Always room for improvementThe dean Jim Plummer said the three main qualities for the job were speed, agilityand drinking skills. “Actually, Gene has none of those, but he’s still the bestseminar organizer and party host we’ve had”, he said.
SCCMEI
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 100/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Always room for improvementThe owner Craig Andrew said the three main qualities for the job were speed,agility and driving skills. “Actually, Merv has none of those, but he’s still the bestdelivery boy we’ve had”, he said.
Always room for improvementThe dean Jim Plummer said the three main qualities for the job were speed, agilityand drinking skills. “Actually, Gene has none of those, but he’s still the bestseminar organizer and party host we’ve had”, he said.
SCCMEI
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 100/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Final thoughtsWe learn from history
that we don’t learn from history– G. W. F. Hegel
If you’re not fired with enthusiasm,you’ll be fired with enthusiasm– Vince Lombardi
Special thanks!
Jacek GondzioRachael TappendenEPSRC, Maxwell Institute
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 101/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Final thoughtsWe learn from historythat we don’t learn from history– G. W. F. Hegel
If you’re not fired with enthusiasm,you’ll be fired with enthusiasm– Vince Lombardi
Special thanks!
Jacek GondzioRachael TappendenEPSRC, Maxwell Institute
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 101/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Final thoughtsWe learn from historythat we don’t learn from history– G. W. F. Hegel
If you’re not fired with enthusiasm,
you’ll be fired with enthusiasm– Vince Lombardi
Special thanks!
Jacek GondzioRachael TappendenEPSRC, Maxwell Institute
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 101/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Final thoughtsWe learn from historythat we don’t learn from history– G. W. F. Hegel
If you’re not fired with enthusiasm,you’ll be fired with enthusiasm– Vince Lombardi
Special thanks!
Jacek GondzioRachael TappendenEPSRC, Maxwell Institute
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 101/101
SOL Solvers (Ax ≈ b) Applications (Ax ≈ b) rank(S) Solvers (optimization) Applications (optimization) Aerospace AC NZ
Final thoughtsWe learn from historythat we don’t learn from history– G. W. F. Hegel
If you’re not fired with enthusiasm,you’ll be fired with enthusiasm– Vince Lombardi
Special thanks!
Jacek GondzioRachael TappendenEPSRC, Maxwell Institute
Michael Saunders: Optimization software at SOL CLAODE, ICMS, Edinburgh Oct 31–Nov 1, 2013 101/101