N. Theodorakopoulos- Nonlinear physics (solitons, chaos, discrete breathers)
Nonlinear Discrete OptimizationThe many aspects of nonlinear discrete optimization To begin with...
Transcript of Nonlinear Discrete OptimizationThe many aspects of nonlinear discrete optimization To begin with...
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Nonlinear Discrete Optimization
Robert Weismantel
Otto-von-Guericke-Universitat Magdeburg
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 1 / 18
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The many aspects of nonlinear discrete optimization
To begin with
max/min f (x1, . . . , xn)
subject to (x1, . . . , xn) ∈ P ∩ Zn.
Parametric non-linear optimization
A borderline case from the point of viewof computational complexity
max f (Wx)
s.t. x ∈ P ∩ Zn
with W ∈ Zm×n with fixed m. “Mapshigh-dimensional problems to fixeddimension.” [Onn, Rothblum ’05]
Convex maximization Polynomial optimization Convex minimization
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 2 / 18
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The many aspects of nonlinear discrete optimization
To begin with
max/min f (x1, . . . , xn)
subject to (x1, . . . , xn) ∈ P ∩ Zn.
Parametric non-linear optimization
A borderline case from the point of viewof computational complexity
max f (Wx)
s.t. x ∈ P ∩ Zn
with W ∈ Zm×n with fixed m. “Mapshigh-dimensional problems to fixeddimension.” [Onn, Rothblum ’05]
Convex maximization Polynomial optimization Convex minimization
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 2 / 18
![Page 4: Nonlinear Discrete OptimizationThe many aspects of nonlinear discrete optimization To begin with max/min f(x 1,...,x n) subject to (x 1,...,x n) ∈ P ∩ Z n. Parametric non-linear](https://reader034.fdocuments.us/reader034/viewer/2022042219/5ec53f38320cc925d949c7ae/html5/thumbnails/4.jpg)
The many aspects of nonlinear discrete optimization
To begin with
max/min f (x1, . . . , xn)
subject to (x1, . . . , xn) ∈ P ∩ Zn.
Parametric non-linear optimization
A borderline case from the point of viewof computational complexity
max f (Wx)
s.t. x ∈ P ∩ Zn
with W ∈ Zm×n with fixed m. “Mapshigh-dimensional problems to fixeddimension.” [Onn, Rothblum ’05]
Convex maximization Polynomial optimization Convex minimization
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 2 / 18
![Page 5: Nonlinear Discrete OptimizationThe many aspects of nonlinear discrete optimization To begin with max/min f(x 1,...,x n) subject to (x 1,...,x n) ∈ P ∩ Z n. Parametric non-linear](https://reader034.fdocuments.us/reader034/viewer/2022042219/5ec53f38320cc925d949c7ae/html5/thumbnails/5.jpg)
The many aspects of nonlinear discrete optimization
To begin with
max/min f (x1, . . . , xn)
subject to (x1, . . . , xn) ∈ P ∩ Zn.
Parametric non-linear optimization
A borderline case from the point of viewof computational complexity
max f (Wx)
s.t. x ∈ P ∩ Zn
with W ∈ Zm×n with fixed m. “Mapshigh-dimensional problems to fixeddimension.” [Onn, Rothblum ’05]
Convex maximization Polynomial optimization Convex minimization
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 2 / 18
![Page 6: Nonlinear Discrete OptimizationThe many aspects of nonlinear discrete optimization To begin with max/min f(x 1,...,x n) subject to (x 1,...,x n) ∈ P ∩ Z n. Parametric non-linear](https://reader034.fdocuments.us/reader034/viewer/2022042219/5ec53f38320cc925d949c7ae/html5/thumbnails/6.jpg)
The many aspects of nonlinear discrete optimization
To begin with
max/min f (x1, . . . , xn)
subject to (x1, . . . , xn) ∈ P ∩ Zn.
Parametric non-linear optimization
A borderline case from the point of viewof computational complexity
max f (Wx)
s.t. x ∈ P ∩ Zn
with W ∈ Zm×n with fixed m. “Mapshigh-dimensional problems to fixeddimension.” [Onn, Rothblum ’05]
Convex maximization Polynomial optimization Convex minimization
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 2 / 18
![Page 7: Nonlinear Discrete OptimizationThe many aspects of nonlinear discrete optimization To begin with max/min f(x 1,...,x n) subject to (x 1,...,x n) ∈ P ∩ Z n. Parametric non-linear](https://reader034.fdocuments.us/reader034/viewer/2022042219/5ec53f38320cc925d949c7ae/html5/thumbnails/7.jpg)
To get started: Convex integer maximization
Problem type
max f (x1, . . . , xd)
subject to (x1, . . . , xd) ∈ P ∩ Zd ,
where
P is a polytope with P ⊆ B(box),
f : Zd → R is convex
1 2 3 4
f (x) = x2
0
Observation
Let
G(f ) = conv˘
(y , f (y)) : y ∈ B ∩ Zd ¯.f is convex over B⇐⇒ every (x , π) ∈ G(f ) satisfies f (x) ≤ π.
Reduction Scheme
maxx∈P∩Zd
f (x) = minx∈P∩Zd
f (x)≤π
π = minx∈P∩Zd
(x,π)∈G(f )
π
G(f ) is a polyhedron describing thenonlinearities!
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 3 / 18
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To get started: Convex integer maximization
Problem type
max f (x1, . . . , xd)
subject to (x1, . . . , xd) ∈ P ∩ Zd ,
where
P is a polytope with P ⊆ B(box),
f : Zd → R is convex
1 2 3 4
f (x) = x2
0
Observation
Let
G(f ) = conv˘
(y , f (y)) : y ∈ B ∩ Zd ¯.f is convex over B⇐⇒ every (x , π) ∈ G(f ) satisfies f (x) ≤ π.
Reduction Scheme
maxx∈P∩Zd
f (x) = minx∈P∩Zd
f (x)≤π
π = minx∈P∩Zd
(x,π)∈G(f )
π
G(f ) is a polyhedron describing thenonlinearities!
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 3 / 18
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To get started: Convex integer maximization
Problem type
max f (x1, . . . , xd)
subject to (x1, . . . , xd) ∈ P ∩ Zd ,
where
P is a polytope with P ⊆ B(box),
f : Zd → R is convex
1 2 3 4
f (x) = x2
0
Observation
Let
G(f ) = conv˘
(y , f (y)) : y ∈ B ∩ Zd ¯.f is convex over B⇐⇒ every (x , π) ∈ G(f ) satisfies f (x) ≤ π.
Reduction Scheme
maxx∈P∩Zd
f (x) = minx∈P∩Zd
f (x)≤π
π = minx∈P∩Zd
(x,π)∈G(f )
π
G(f ) is a polyhedron describing thenonlinearities!
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 3 / 18
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Grey areas of computational complexity
Variables and constraintsObjectivefunction Fixed dimension Combinatorial Arbitrary
Convex max Cook, Hartmann, Kan-
nan, McDiarmid ’89
Reduction techniqueto ILP: Michaels ’07
Convex min Khachiyan, Porkolab ’00 Outer approxima-tion: Grossmann ’86;
Fletcher, Leyffer ’94
Parametric
Polynomial SOS programming[Shor ’87; Parrilo ’03;
Laurent ’01], Posi-tivstellensatz [Putinar
’93]
Arbitrary
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 4 / 18
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Parametric nonlinear optimization
The setting
max f (Wx)
s.t. x ∈ P ∩ Zn
with W ∈ Zm×n with fixed m.
Would like to work with
max f (w)
s.t. w ∈ WP ∩ Zm;
but need to have control over the “holes”,i.e., a w ∈ Zm such that Wx = w ,x ∈ P ∩ Zn has no solution
Observation
No control over the holes if W is arbitraryand encoded in binary
(Reduction to Subset Sum)
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 5 / 18
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Parametric nonlinear optimization
The setting
max f (Wx)
s.t. x ∈ P ∩ Zn
with W ∈ Zm×n with fixed m.
Would like to work with
max f (w)
s.t. w ∈ WP ∩ Zm;
but need to have control over the “holes”,i.e., a w ∈ Zm such that Wx = w ,x ∈ P ∩ Zn has no solution
Observation
No control over the holes if W is arbitraryand encoded in binary
(Reduction to Subset Sum)
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 5 / 18
![Page 13: Nonlinear Discrete OptimizationThe many aspects of nonlinear discrete optimization To begin with max/min f(x 1,...,x n) subject to (x 1,...,x n) ∈ P ∩ Z n. Parametric non-linear](https://reader034.fdocuments.us/reader034/viewer/2022042219/5ec53f38320cc925d949c7ae/html5/thumbnails/13.jpg)
Parametric nonlinear optimization
The setting
max f (Wx)
s.t. x ∈ P ∩ Zn
with W ∈ Zm×n with fixed m.
Would like to work with
max f (w)
s.t. w ∈ WP ∩ Zm;
but need to have control over the “holes”,i.e., a w ∈ Zm such that Wx = w ,x ∈ P ∩ Zn has no solution
Observation
No control over the holes if W is arbitraryand encoded in binary
(Reduction to Subset Sum)
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 5 / 18
![Page 14: Nonlinear Discrete OptimizationThe many aspects of nonlinear discrete optimization To begin with max/min f(x 1,...,x n) subject to (x 1,...,x n) ∈ P ∩ Z n. Parametric non-linear](https://reader034.fdocuments.us/reader034/viewer/2022042219/5ec53f38320cc925d949c7ae/html5/thumbnails/14.jpg)
Parametric nonlinear optimization
The setting
max f (Wx)
s.t. x ∈ P ∩ Zn
with W ∈ Zm×n with fixed m.
Would like to work with
max f (w)
s.t. w ∈ WP ∩ Zm;
but need to have control over the “holes”,i.e., a w ∈ Zm such that Wx = w ,x ∈ P ∩ Zn has no solution
Observation
No control over the holes if W is arbitraryand encoded in binary
(Reduction to Subset Sum)
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 5 / 18
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An algorithmic framework for optimization over independence systems
Independencesystem
S ⊆ 2N closedunder takingsubsets
endowed with anoracle foroptimizing linearfunctions over S .
Problem
Let m = 1,w ∈ {a1, . . . , ap}n.
Consider
max f (w>x)
s.t. x ∈ S
Algorithmic Framework
Step 1: Define appropriate upper bounds λi for i = 1, . . . , p.Step 2: For all S1 ⊆ N1, . . . , Sp ⊆ Np such that |Si | ≤ λi :
2.1 Determine
max w>x s.t. x ∈ Sxj = 1 for all j ∈ Si and i = 1, . . . , p.
Let x(S1, . . . , Sp) denote the optimal solution.
2.2 Determine a sequence of at most n vectors
x0 = x(S1, . . . , Sp) ≥ x1 ≥ . . . ≥ x t ∈ Zn+
and select the one, x ′(S1, . . . , Sp) say, attaining
max{f (wx i ) | i = 0, . . . , t}.
Step 3: Return x ′ attaining
max f`wx ′(S1, . . . , Sp)
´s.t. S1 ⊆ N1, . . . , Sp ⊆ Np, |Si | ≤ λi .
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 6 / 18
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Some results along these lines
Let S be a matroid
m = 1, p = 2, a1 = 0, a2 = 1
max f (w>χI )
s.t. I ∈ Sreduces to
max w>χI
s.t. I ∈ S
Theorem 1 (Lee, Onn, W. ’07)
For every fixed m and p, there is an algorithm that, given a1, . . . , ap ∈ Z,W ∈ {a1, . . . , ap}m×n, and a function f : Rn → R, finds a matroid base B minimizingf (WχB) in time polynomial in n and 〈a1, . . . , ap〉.
Iterated matroid intersection, Edmonds ’79
Theorem 2 (Lee, Onn, W. ’07)
Let a1 < . . . < ap ∈ Z+ be a sequence of divisible integers. Lettingq = max{ ai+1
ai| i = 1, . . . , p − 1}, the algorithm solves the univariate optimization
problem over an independence system by O(npq) calls of a linear optimization oracle.
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 7 / 18
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Some results along these lines
Let S be a matroid
m = 1, p = 2, a1 = 0, a2 = 1
max f (w>χI )
s.t. I ∈ Sreduces to
max w>χI
s.t. I ∈ S
Theorem 1 (Lee, Onn, W. ’07)
For every fixed m and p, there is an algorithm that, given a1, . . . , ap ∈ Z,W ∈ {a1, . . . , ap}m×n, and a function f : Rn → R, finds a matroid base B minimizingf (WχB) in time polynomial in n and 〈a1, . . . , ap〉.
Iterated matroid intersection, Edmonds ’79
Theorem 2 (Lee, Onn, W. ’07)
Let a1 < . . . < ap ∈ Z+ be a sequence of divisible integers. Lettingq = max{ ai+1
ai| i = 1, . . . , p − 1}, the algorithm solves the univariate optimization
problem over an independence system by O(npq) calls of a linear optimization oracle.
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 7 / 18
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Some results along these lines
Let S be a matroid
m = 1, p = 2, a1 = 0, a2 = 1
max f (w>χI )
s.t. I ∈ Sreduces to
max w>χI
s.t. I ∈ S
Theorem 1 (Lee, Onn, W. ’07)
For every fixed m and p, there is an algorithm that, given a1, . . . , ap ∈ Z,W ∈ {a1, . . . , ap}m×n, and a function f : Rn → R, finds a matroid base B minimizingf (WχB) in time polynomial in n and 〈a1, . . . , ap〉.
Iterated matroid intersection, Edmonds ’79
Theorem 2 (Lee, Onn, W. ’07)
Let a1 < . . . < ap ∈ Z+ be a sequence of divisible integers. Lettingq = max{ ai+1
ai| i = 1, . . . , p − 1}, the algorithm solves the univariate optimization
problem over an independence system by O(npq) calls of a linear optimization oracle.
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 7 / 18
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A combinatorial example: Integer convex minimization
Convex N-fold integer minimization
min
(NX
i=1
f (i)“x (i)”
:NX
i=1
x (i) = b(0), Ax (i) = b(i), 0 ≤ x (i) ≤ u(i), x (i) ∈ Zn, i = 1, . . . , N
),
where f (i)(x (i)) :=Ps
j=1 f(i)j
“cᵀj x (i)
”with convex functions f
(i)j : R → R
Ingredient 1: Superadditivity of objective function
Let g : R → R be a convex function and let z , y1, . . . , yr
be real numbers.
If y1, . . . , yr have same signs:
g(z)− g
z +
rXj=1
yj
!≤
rXj=1
[g(z)− g (z + yj)] .
(Murota, Saito, W. ’04)
0BBBBB@In In · · · InA 0 · · · 00 A · · · 0
. . . 00 0 · · · A
1CCCCCA
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 8 / 18
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A combinatorial example: Integer convex minimization
Convex N-fold integer minimization
min
(NX
i=1
f (i)“x (i)”
:NX
i=1
x (i) = b(0), Ax (i) = b(i), 0 ≤ x (i) ≤ u(i), x (i) ∈ Zn, i = 1, . . . , N
),
where f (i)(x (i)) :=Ps
j=1 f(i)j
“cᵀj x (i)
”with convex functions f
(i)j : R → R
Ingredient 1: Superadditivity of objective function
Let g : R → R be a convex function and let z , y1, . . . , yr
be real numbers.
If y1, . . . , yr have same signs:
g(z)− g
z +
rXj=1
yj
!≤
rXj=1
[g(z)− g (z + yj)] .
(Murota, Saito, W. ’04)
0BBBBB@In In · · · InA 0 · · · 00 A · · · 0
. . . 00 0 · · · A
1CCCCCA
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 8 / 18
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Ingredient 2: Graver bases (Graver, 1975)
Hilbert bases and Graver bases
Hilbert basis HB(C) of a rational polyhedral cone C ⊆ Rn
is a set of integer vectors of C that generate all integervectors of C by non-negative integer linear combinations
Graver basis G(A) of a matrix A ∈ Zd×n is the unionSj HB(ker(A) ∩ Oj) over all 2n orthants of Rn.
Integer linear programs
Graver basis G(A) provides anoptimality certificate for linearinteger programs (Graver ’75)
max c>x
s.t. Ax = b
x ∈ Zn+
Convex integer minimization problems
Graver basis G`
A In 0C 0 −Is
´provides an optimality
certificate for convex integer minimizationproblems
min
sXi=1
gi (cᵀi z) : Az ≤ b, z ∈ Zn
+
ffwith convex functions gi : R → R (Murota,Saito, W. ’04).
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 9 / 18
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Ingredient 2: Graver bases (Graver, 1975)
Hilbert bases and Graver bases
Hilbert basis HB(C) of a rational polyhedral cone C ⊆ Rn
is a set of integer vectors of C that generate all integervectors of C by non-negative integer linear combinations
Graver basis G(A) of a matrix A ∈ Zd×n is the unionSj HB(ker(A) ∩ Oj) over all 2n orthants of Rn.
Integer linear programs
Graver basis G(A) provides anoptimality certificate for linearinteger programs (Graver ’75)
max c>x
s.t. Ax = b
x ∈ Zn+
Convex integer minimization problems
Graver basis G`
A In 0C 0 −Is
´provides an optimality
certificate for convex integer minimizationproblems
min
sXi=1
gi (cᵀi z) : Az ≤ b, z ∈ Zn
+
ffwith convex functions gi : R → R (Murota,Saito, W. ’04).
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 9 / 18
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Ingredient 3: Efficient augmentation(cf. Ahuja, Magnanti, Orlin ’93; Schulz, W. ’99)
Theorem (Santos, Sturmfels ’03)
The Graver basis G of
A(N) :=
0BBBBB@In In · · · InA 0 · · · 00 A · · · 0
. . . 00 0 · · · A
1CCCCCAincreases only polynomially in N.
Integer line search algorithm
To solve min{f (z0 + αt) : α ∈ Z+} forconvex f , use binary search with acomparison oracle for f → polynomiallymany bisection steps
Greedy augmentation algorithm
1 Find feasible solution z0.2 While z0 is not optimal, find t ∈ G
and α ∈ Z+ such thatz0 + αt is a feasible and
f (z0 + αt) is minimal.
3 Return z0 as optimal solution.
Theorem (Hemmecke, Onn, W. ’07)
For fixed A, C , convex N-fold integer minimization is polynomial-time solvable.
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 10 / 18
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Ingredient 3: Efficient augmentation(cf. Ahuja, Magnanti, Orlin ’93; Schulz, W. ’99)
Theorem (Santos, Sturmfels ’03)
The Graver basis G of
A(N) :=
0BBBBB@In In · · · InA 0 · · · 00 A · · · 0
. . . 00 0 · · · A
1CCCCCAincreases only polynomially in N.
Integer line search algorithm
To solve min{f (z0 + αt) : α ∈ Z+} forconvex f , use binary search with acomparison oracle for f → polynomiallymany bisection steps
Greedy augmentation algorithm
1 Find feasible solution z0.2 While z0 is not optimal, find t ∈ G
and α ∈ Z+ such thatz0 + αt is a feasible and
f (z0 + αt) is minimal.
3 Return z0 as optimal solution.
Theorem (Hemmecke, Onn, W. ’07)
For fixed A, C , convex N-fold integer minimization is polynomial-time solvable.
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General polynomial optimization
Problem type
max f (x1, . . . , xn)
subject to (x1, . . . , xn) ∈ P ∩ Zn,
where
P is a polytope,
f is a polynomial functionnon-negative over P ∩ Zn,
the dimension n is fixed.
Ingredient 1: Generating functions
gP(z1, . . . , zn) =X
α∈P∩Zn
zα
0 1 2 3 4
gP(z) = z0 + z1 + z2 + z3 + z4
=1− z5
1− zfor z 6= 1
Theorem (Barvinok, 1994)
Let the dimension n be fixed. There is a polynomial-time algorithm for computing gP(z)of P ∩ Zn in the form of a rational function. In particular, N = |P ∩ Zn| = gP(1) can becomputed in polynomial time.
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General polynomial optimization
Problem type
max f (x1, . . . , xn)
subject to (x1, . . . , xn) ∈ P ∩ Zn,
where
P is a polytope,
f is a polynomial functionnon-negative over P ∩ Zn,
the dimension n is fixed.
Ingredient 1: Generating functions
gP(z1, . . . , zn) =X
α∈P∩Zn
zα
0 1 2 3 4
gP(z) = z0 + z1 + z2 + z3 + z4
=1− z5
1− zfor z 6= 1
Theorem (Barvinok, 1994)
Let the dimension n be fixed. There is a polynomial-time algorithm for computing gP(z)of P ∩ Zn in the form of a rational function. In particular, N = |P ∩ Zn| = gP(1) can becomputed in polynomial time.
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 11 / 18
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General polynomial optimization
Problem type
max f (x1, . . . , xn)
subject to (x1, . . . , xn) ∈ P ∩ Zn,
where
P is a polytope,
f is a polynomial functionnon-negative over P ∩ Zn,
the dimension n is fixed.
Ingredient 1: Generating functions
gP(z1, . . . , zn) =X
α∈P∩Zn
zα
0 1 2 3 4
gP(z) = z0 + z1 + z2 + z3 + z4
=1− z5
1− zfor z 6= 1
Theorem (Barvinok, 1994)
Let the dimension n be fixed. There is a polynomial-time algorithm for computing gP(z)of P ∩ Zn in the form of a rational function. In particular, N = |P ∩ Zn| = gP(1) can becomputed in polynomial time.
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 11 / 18
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Ingredient 2: Approximation properties of `p norms
Approximation properties of `p norms
Let
P ∩ Zn = {x1, . . . , xN} and f =
0B@ f (x1)...
f (xN)
1CA ∈ RN .
Then:Lk := N−1/k ‖f ‖k ≤ ‖f ‖∞ ≤ ‖f ‖k =: Uk
Convergence
Uk − Lk ≤“
k√
N − 1”
f (xmax) ≤ ε f (xmax) for k ≥ (1 + 1/ε) log N.
k = 1
k = 2
Complexity
encoding length and degree of f k are boundedpolynomially in the input size and 1
ε
How to evaluate?
‖f ‖k =
„ Xx∈P∩Zn
f k(x)
«1/k
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Ingredient 2: Approximation properties of `p norms
Approximation properties of `p norms
Let
P ∩ Zn = {x1, . . . , xN} and f =
0B@ f (x1)...
f (xN)
1CA ∈ RN .
Then:Lk := N−1/k ‖f ‖k ≤ ‖f ‖∞ ≤ ‖f ‖k =: Uk
Convergence
Uk − Lk ≤“
k√
N − 1”
f (xmax) ≤ ε f (xmax) for k ≥ (1 + 1/ε) log N.
k = 1
k = 2
Complexity
encoding length and degree of f k are boundedpolynomially in the input size and 1
ε
How to evaluate?
‖f ‖k =
„ Xx∈P∩Zn
f k(x)
«1/k
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Ingredient 2: Approximation properties of `p norms
Approximation properties of `p norms
Let
P ∩ Zn = {x1, . . . , xN} and f =
0B@ f (x1)...
f (xN)
1CA ∈ RN .
Then:Lk := N−1/k ‖f ‖k ≤ ‖f ‖∞ ≤ ‖f ‖k =: Uk
Convergence
Uk − Lk ≤“
k√
N − 1”
f (xmax) ≤ ε f (xmax) for k ≥ (1 + 1/ε) log N.
k = 1
k = 2
Complexity
encoding length and degree of f k are boundedpolynomially in the input size and 1
ε
How to evaluate?
‖f ‖k =
„ Xx∈P∩Zn
f k(x)
«1/k
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 12 / 18
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Ingredient 2: Approximation properties of `p norms
Approximation properties of `p norms
Let
P ∩ Zn = {x1, . . . , xN} and f =
0B@ f (x1)...
f (xN)
1CA ∈ RN .
Then:Lk := N−1/k ‖f ‖k ≤ ‖f ‖∞ ≤ ‖f ‖k =: Uk
Convergence
Uk − Lk ≤“
k√
N − 1”
f (xmax) ≤ ε f (xmax) for k ≥ (1 + 1/ε) log N.
k = 1
k = 2
Complexity
encoding length and degree of f k are boundedpolynomially in the input size and 1
ε
How to evaluate?
‖f ‖k =
„ Xx∈P∩Zn
f k(x)
«1/k
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 12 / 18
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Ingredient 3: Differential operators on generating functions
Euler differential operator
gP(z) = z0 + z1 + z2 + z3 + z4
=1
1− z− z5
1− z
Apply differential operator:„z
d
dz
«gP(z) = 1z1 + 2z2 + 3z3 + 4z4
=1
(1− z)2− −4z5 + 5z4
(1− z)2
Apply differential operator again:„z
d
dz
«„z
d
dz
«gP(z) = 1z1 + 4z2 + 9z3 + 16z4
=z + z2
(1− z)3− 25z5 − 39z6 + 16z7
(1− z)3
Theorem (De Loera, Hemmecke, Koppe, W., 2004 / Barvinok, 2004)
Let gP(z) be the Barvinok representation of the generating function of the lattice pointsof P. Let f be a polynomial in Z[x1, . . . , xn] of maximum total degree D.
We can compute, in time polynomial in D and the size of the input data, a Barvinokrational function representation gP,f (z) for the function
Pα∈P∩Zn f (α)zα.
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 13 / 18
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Ingredient 3: Differential operators on generating functions
Euler differential operator
gP(z) = z0 + z1 + z2 + z3 + z4 =1
1− z− z5
1− z
Apply differential operator:„z
d
dz
«gP(z) = 1z1 + 2z2 + 3z3 + 4z4 =
1
(1− z)2− −4z5 + 5z4
(1− z)2
Apply differential operator again:„z
d
dz
«„z
d
dz
«gP(z) = 1z1 + 4z2 + 9z3 + 16z4 =
z + z2
(1− z)3− 25z5 − 39z6 + 16z7
(1− z)3
Theorem (De Loera, Hemmecke, Koppe, W., 2004 / Barvinok, 2004)
Let gP(z) be the Barvinok representation of the generating function of the lattice pointsof P. Let f be a polynomial in Z[x1, . . . , xn] of maximum total degree D.
We can compute, in time polynomial in D and the size of the input data, a Barvinokrational function representation gP,f (z) for the function
Pα∈P∩Zn f (α)zα.
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 13 / 18
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Ingredient 3: Differential operators on generating functions
Euler differential operator
gP(z) = z0 + z1 + z2 + z3 + z4 =1
1− z− z5
1− z
Apply differential operator:„z
d
dz
«gP(z) = 1z1 + 2z2 + 3z3 + 4z4 =
1
(1− z)2− −4z5 + 5z4
(1− z)2
Apply differential operator again:„z
d
dz
«„z
d
dz
«gP(z) = 1z1 + 4z2 + 9z3 + 16z4 =
z + z2
(1− z)3− 25z5 − 39z6 + 16z7
(1− z)3
Theorem (De Loera, Hemmecke, Koppe, W., 2004 / Barvinok, 2004)
Let gP(z) be the Barvinok representation of the generating function of the lattice pointsof P. Let f be a polynomial in Z[x1, . . . , xn] of maximum total degree D.
We can compute, in time polynomial in D and the size of the input data, a Barvinokrational function representation gP,f (z) for the function
Pα∈P∩Zn f (α)zα.
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 13 / 18
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Taking all together
Theorem (De Loera, Hemmecke, Koppe, W., 2006)
Let the dimension n be fixed. There exists a fully polynomial-time approximation schemefor the problem
max f (x1, . . . , xn)
subject to (x1, . . . , xn) ∈ P ∩ Zn,
where
P is a polytope,
f is a polynomial function non-negative over P ∩ Zn.
Fully Polynomial-Time Approximation Scheme (FPTAS)
For every ε > 0, there exists an algorithm Aε with running time polynomial in the inputsize and 1/ε, which computes an approximation xε with
|f (xε)− f (xmax)| ≤ ε f (xmax),
where xmax denotes an optimal solution of the optimization problem.
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 14 / 18
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A view towards the mixed integer world: convex mixed integer programs.
An operation
For a lattice point free polyhedron L = {x ∈ Rn | Πx ≤ π} ⊆ Rn and a closed convex setC , define
R(L) := conv (C \ rint(L)) .
Basic properties of such anoperation:
R(L) ⊆ C .
R(L) ⊇ conv`C ∩ (Zd × Rn−d)
´.
R(L) 6= C iff there exists anextreme point (x , y) of C such thatx is in the interior of L.
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 15 / 18
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A view towards the mixed integer world: convex mixed integer programs.
An operation
For a lattice point free polyhedron L = {x ∈ Rn | Πx ≤ π} ⊆ Rn and a closed convex setC , define
R(L) := conv (C \ rint(L)) .
Basic properties of such anoperation:
R(L) ⊆ C .
R(L) ⊇ conv`C ∩ (Zd × Rn−d)
´.
R(L) 6= C iff there exists anextreme point (x , y) of C such thatx is in the interior of L.
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 15 / 18
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A view towards the mixed integer world: convex mixed integer programs.
An operation
For a lattice point free polyhedron L = {x ∈ Rn | Πx ≤ π} ⊆ Rn and a closed convex setC , define
R(L) := conv (C \ rint(L)) .
Basic properties of such anoperation:
R(L) ⊆ C .
R(L) ⊇ conv`C ∩ (Zd × Rn−d)
´.
R(L) 6= C iff there exists anextreme point (x , y) of C such thatx is in the interior of L.
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 15 / 18
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A view towards the mixed integer world: convex mixed integer programs.
An operation
For a lattice point free polyhedron L = {x ∈ Rn | Πx ≤ π} ⊆ Rn and a closed convex setC , define
R(L) := conv (C \ rint(L)) .
Basic properties of such anoperation:
R(L) ⊆ C .
R(L) ⊇ conv`C ∩ (Zd × Rn−d)
´.
R(L) 6= C iff there exists anextreme point (x , y) of C such thatx is in the interior of L.
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 15 / 18
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A view towards the mixed integer world: convex mixed integer programs.
An operation
For a lattice point free polyhedron L = {x ∈ Rn | Πx ≤ π} ⊆ Rn and a closed convex setC , define
R(L) := conv (C \ rint(L)) .
Basic properties of such anoperation:
R(L) ⊆ C .
R(L) ⊇ conv`C ∩ (Zd × Rn−d)
´.
R(L) 6= C iff there exists anextreme point (x , y) of C such thatx is in the interior of L.
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 15 / 18
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A view towards the mixed integer world: convex mixed integer programs.
An operation
For a lattice point free polyhedron L = {x ∈ Rn | Πx ≤ π} ⊆ Rn and a closed convex setC , define
R(L) := conv (C \ rint(L)) .
Basic properties of such anoperation:
R(L) ⊆ C .
R(L) ⊇ conv`C ∩ (Zd × Rn−d)
´.
R(L) 6= C iff there exists anextreme point (x , y) of C such thatx is in the interior of L.
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 15 / 18
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A view towards the mixed integer world: convex mixed integer programs.
An operation
For a lattice point free polyhedron L = {x ∈ Rn | Πx ≤ π} ⊆ Rn and a closed convex setC , define
R(L) := conv (C \ rint(L)) .
Basic properties of such anoperation:
R(L) ⊆ C .
R(L) ⊇ conv`C ∩ (Zd × Rn−d)
´.
R(L) 6= C iff there exists anextreme point (x , y) of C such thatx is in the interior of L.
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 15 / 18
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A view towards the mixed integer world: convex mixed integer programs.
An operation
For a lattice point free polyhedron L = {x ∈ Rn | Πx ≤ π} ⊆ Rn and a closed convex setC , define
R(L) := conv (C \ rint(L)) .
Basic properties of such anoperation:
R(L) ⊆ C .
R(L) ⊇ conv`C ∩ (Zd × Rn−d)
´.
R(L) 6= C iff there exists anextreme point (x , y) of C such thatx is in the interior of L.
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 15 / 18
![Page 44: Nonlinear Discrete OptimizationThe many aspects of nonlinear discrete optimization To begin with max/min f(x 1,...,x n) subject to (x 1,...,x n) ∈ P ∩ Z n. Parametric non-linear](https://reader034.fdocuments.us/reader034/viewer/2022042219/5ec53f38320cc925d949c7ae/html5/thumbnails/44.jpg)
about when convex sets are closed w.r.t. one such operation
A closed convex set C is called ...
LMI-representable if C = {x | A0 + A1x1 + . . . + Anxn � 0} with real symmetricmatrices Ai .
SDP-representable if C is the projection of a LMI-representable set.
For a closed convex set C and
L = {x ∈ Rn | Πx ≤ π} ⊆ Rn full dimensional and lattice point free in its interior, defining
Ck := C ∩ {x | ΠTk·x ≥ πk}, we observe that C \ rint(L) =
[k
Ck .
Theorem: When C is xxx, then R(L) = conv(C \ rint(L)) is xxx.
with xxx = polyhedral
[Balas 1979]
with xxx = SDP-representable
[Helton, Nie 2007]
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 16 / 18
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about when convex sets are closed w.r.t. one such operation
A closed convex set C is called ...
LMI-representable if C = {x | A0 + A1x1 + . . . + Anxn � 0} with real symmetricmatrices Ai .
SDP-representable if C is the projection of a LMI-representable set.
For a closed convex set C and
L = {x ∈ Rn | Πx ≤ π} ⊆ Rn full dimensional and lattice point free in its interior, defining
Ck := C ∩ {x | ΠTk·x ≥ πk}, we observe that C \ rint(L) =
[k
Ck .
Theorem: When C is xxx, then R(L) = conv(C \ rint(L)) is xxx.
with xxx = polyhedral
[Balas 1979]
with xxx = SDP-representable
[Helton, Nie 2007]
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 16 / 18
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about when convex sets are closed w.r.t. one such operation
A closed convex set C is called ...
LMI-representable if C = {x | A0 + A1x1 + . . . + Anxn � 0} with real symmetricmatrices Ai .
SDP-representable if C is the projection of a LMI-representable set.
For a closed convex set C and
L = {x ∈ Rn | Πx ≤ π} ⊆ Rn full dimensional and lattice point free in its interior, defining
Ck := C ∩ {x | ΠTk·x ≥ πk}, we observe that C \ rint(L) =
[k
Ck .
Theorem: When C is xxx, then R(L) = conv(C \ rint(L)) is xxx.
with xxx = polyhedral
[Balas 1979]
with xxx = SDP-representable
[Helton, Nie 2007]
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 16 / 18
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about when convex sets are closed w.r.t. one such operation
A closed convex set C is called ...
LMI-representable if C = {x | A0 + A1x1 + . . . + Anxn � 0} with real symmetricmatrices Ai .
SDP-representable if C is the projection of a LMI-representable set.
For a closed convex set C and
L = {x ∈ Rn | Πx ≤ π} ⊆ Rn full dimensional and lattice point free in its interior, defining
Ck := C ∩ {x | ΠTk·x ≥ πk}, we observe that C \ rint(L) =
[k
Ck .
Theorem: When C is xxx, then R(L) = conv(C \ rint(L)) is xxx.
with xxx = polyhedral
[Balas 1979]
with xxx = SDP-representable
[Helton, Nie 2007]
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 16 / 18
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about when convex sets are closed w.r.t. one such operation
A closed convex set C is called ...
LMI-representable if C = {x | A0 + A1x1 + . . . + Anxn � 0} with real symmetricmatrices Ai .
SDP-representable if C is the projection of a LMI-representable set.
For a closed convex set C and
L = {x ∈ Rn | Πx ≤ π} ⊆ Rn full dimensional and lattice point free in its interior, defining
Ck := C ∩ {x | ΠTk·x ≥ πk}, we observe that C \ rint(L) =
[k
Ck .
Theorem: When C is xxx, then R(L) = conv(C \ rint(L)) is xxx.
with xxx = polyhedral
[Balas 1979]
with xxx = SDP-representable
[Helton, Nie 2007]
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 16 / 18
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about when convex sets are closed w.r.t. one such operation
A closed convex set C is called ...
LMI-representable if C = {x | A0 + A1x1 + . . . + Anxn � 0} with real symmetricmatrices Ai .
SDP-representable if C is the projection of a LMI-representable set.
For a closed convex set C and
L = {x ∈ Rn | Πx ≤ π} ⊆ Rn full dimensional and lattice point free in its interior, defining
Ck := C ∩ {x | ΠTk·x ≥ πk}, we observe that C \ rint(L) =
[k
Ck .
Theorem: When C is xxx, then R(L) = conv(C \ rint(L)) is xxx.
with xxx = polyhedral
[Balas 1979]
with xxx = SDP-representable
[Helton, Nie 2007]
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 16 / 18
![Page 50: Nonlinear Discrete OptimizationThe many aspects of nonlinear discrete optimization To begin with max/min f(x 1,...,x n) subject to (x 1,...,x n) ∈ P ∩ Z n. Parametric non-linear](https://reader034.fdocuments.us/reader034/viewer/2022042219/5ec53f38320cc925d949c7ae/html5/thumbnails/50.jpg)
about when convex sets are closed w.r.t. one such operation
A closed convex set C is called ...
LMI-representable if C = {x | A0 + A1x1 + . . . + Anxn � 0} with real symmetricmatrices Ai .
SDP-representable if C is the projection of a LMI-representable set.
For a closed convex set C and
L = {x ∈ Rn | Πx ≤ π} ⊆ Rn full dimensional and lattice point free in its interior, defining
Ck := C ∩ {x | ΠTk·x ≥ πk}, we observe that C \ rint(L) =
[k
Ck .
Theorem: When C is xxx, then R(L) = conv(C \ rint(L)) is xxx.
with xxx = polyhedral
[Balas 1979]
with xxx = SDP-representable
[Helton, Nie 2007]
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 16 / 18
![Page 51: Nonlinear Discrete OptimizationThe many aspects of nonlinear discrete optimization To begin with max/min f(x 1,...,x n) subject to (x 1,...,x n) ∈ P ∩ Z n. Parametric non-linear](https://reader034.fdocuments.us/reader034/viewer/2022042219/5ec53f38320cc925d949c7ae/html5/thumbnails/51.jpg)
What happens if one applies this operation over and over again?
The closure of split bodies
For a family F of split bodies, let
Cl(F , C) :=\L∈F
R(L).
Letting C 0(F , C) = C , define for i ≥ 1,
C i (F , C) = Cl(F , C i−1(F , C)).
We search for meta-theorems of Type A ...
Let F be any family of split bodies satisfying meaningful property xxx and let C be anyconvex set satisfying a meaningful property yyy, so that yyy is closed under theR(L)-operation. Then Cl(F , C) satisfies again property yyy.
and also for meta-theorems of Type B ...
Let F be any family of split bodies satisfying xxx. Let C be any convex set satisfying yyythat it is closed under the Cl(F , C) -operation. There exists finite t such thatconv(C ∩ (Zd × Rn−d) = C t(F , C).
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 17 / 18
![Page 52: Nonlinear Discrete OptimizationThe many aspects of nonlinear discrete optimization To begin with max/min f(x 1,...,x n) subject to (x 1,...,x n) ∈ P ∩ Z n. Parametric non-linear](https://reader034.fdocuments.us/reader034/viewer/2022042219/5ec53f38320cc925d949c7ae/html5/thumbnails/52.jpg)
What happens if one applies this operation over and over again?
The closure of split bodies
For a family F of split bodies, let
Cl(F , C) :=\L∈F
R(L).
Letting C 0(F , C) = C , define for i ≥ 1,
C i (F , C) = Cl(F , C i−1(F , C)).
We search for meta-theorems of Type A ...
Let F be any family of split bodies satisfying meaningful property xxx and let C be anyconvex set satisfying a meaningful property yyy, so that yyy is closed under theR(L)-operation. Then Cl(F , C) satisfies again property yyy.
and also for meta-theorems of Type B ...
Let F be any family of split bodies satisfying xxx. Let C be any convex set satisfying yyythat it is closed under the Cl(F , C) -operation. There exists finite t such thatconv(C ∩ (Zd × Rn−d) = C t(F , C).
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 17 / 18
![Page 53: Nonlinear Discrete OptimizationThe many aspects of nonlinear discrete optimization To begin with max/min f(x 1,...,x n) subject to (x 1,...,x n) ∈ P ∩ Z n. Parametric non-linear](https://reader034.fdocuments.us/reader034/viewer/2022042219/5ec53f38320cc925d949c7ae/html5/thumbnails/53.jpg)
What happens if one applies this operation over and over again?
The closure of split bodies
For a family F of split bodies, let
Cl(F , C) :=\L∈F
R(L).
Letting C 0(F , C) = C , define for i ≥ 1,
C i (F , C) = Cl(F , C i−1(F , C)).
We search for meta-theorems of Type A ...
Let F be any family of split bodies satisfying meaningful property xxx and let C be anyconvex set satisfying a meaningful property yyy, so that yyy is closed under theR(L)-operation. Then Cl(F , C) satisfies again property yyy.
and also for meta-theorems of Type B ...
Let F be any family of split bodies satisfying xxx. Let C be any convex set satisfying yyythat it is closed under the Cl(F , C) -operation. There exists finite t such thatconv(C ∩ (Zd × Rn−d) = C t(F , C).
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 17 / 18
![Page 54: Nonlinear Discrete OptimizationThe many aspects of nonlinear discrete optimization To begin with max/min f(x 1,...,x n) subject to (x 1,...,x n) ∈ P ∩ Z n. Parametric non-linear](https://reader034.fdocuments.us/reader034/viewer/2022042219/5ec53f38320cc925d949c7ae/html5/thumbnails/54.jpg)
Meaningful properties: C being polyhedral sets and F ?
The max facet width
For a split body L = {x ∈ Rn | Πx ≤ π}, let
w(k) = πk −min{ΠTk·x : x ∈ L}.
The number maxk{w(k)} is a measure of complexity of L.
Theorem [Andersen, Louveaux, W 08].Let F be the family of all split bodies of maxfacet width at most ω∗. For every polyhedronP, Cl(F , P) is a polyhedron.The special case of ω∗ = 1 was shown in[Cook, Kannan, Schrijver 90].
and for a result of type B ...
Theorem [Balas et al. 91]. For the family F of all splits, there exists a finite t such that
conv(P ∩ ({0, 1}d × Rn−d) = C t(F , P).
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 18 / 18
![Page 55: Nonlinear Discrete OptimizationThe many aspects of nonlinear discrete optimization To begin with max/min f(x 1,...,x n) subject to (x 1,...,x n) ∈ P ∩ Z n. Parametric non-linear](https://reader034.fdocuments.us/reader034/viewer/2022042219/5ec53f38320cc925d949c7ae/html5/thumbnails/55.jpg)
Meaningful properties: C being polyhedral sets and F ?
The max facet width
For a split body L = {x ∈ Rn | Πx ≤ π}, let
w(k) = πk −min{ΠTk·x : x ∈ L}.
The number maxk{w(k)} is a measure of complexity of L.
Theorem [Andersen, Louveaux, W 08].Let F be the family of all split bodies of maxfacet width at most ω∗. For every polyhedronP, Cl(F , P) is a polyhedron.The special case of ω∗ = 1 was shown in[Cook, Kannan, Schrijver 90].
and for a result of type B ...
Theorem [Balas et al. 91]. For the family F of all splits, there exists a finite t such that
conv(P ∩ ({0, 1}d × Rn−d) = C t(F , P).
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 18 / 18
![Page 56: Nonlinear Discrete OptimizationThe many aspects of nonlinear discrete optimization To begin with max/min f(x 1,...,x n) subject to (x 1,...,x n) ∈ P ∩ Z n. Parametric non-linear](https://reader034.fdocuments.us/reader034/viewer/2022042219/5ec53f38320cc925d949c7ae/html5/thumbnails/56.jpg)
Meaningful properties: C being polyhedral sets and F ?
The max facet width
For a split body L = {x ∈ Rn | Πx ≤ π}, let
w(k) = πk −min{ΠTk·x : x ∈ L}.
The number maxk{w(k)} is a measure of complexity of L.
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Theorem [Andersen, Louveaux, W 08].Let F be the family of all split bodies of maxfacet width at most ω∗. For every polyhedronP, Cl(F , P) is a polyhedron.The special case of ω∗ = 1 was shown in[Cook, Kannan, Schrijver 90].
and for a result of type B ...
Theorem [Balas et al. 91]. For the family F of all splits, there exists a finite t such that
conv(P ∩ ({0, 1}d × Rn−d) = C t(F , P).
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 18 / 18
![Page 57: Nonlinear Discrete OptimizationThe many aspects of nonlinear discrete optimization To begin with max/min f(x 1,...,x n) subject to (x 1,...,x n) ∈ P ∩ Z n. Parametric non-linear](https://reader034.fdocuments.us/reader034/viewer/2022042219/5ec53f38320cc925d949c7ae/html5/thumbnails/57.jpg)
Meaningful properties: C being polyhedral sets and F ?
The max facet width
For a split body L = {x ∈ Rn | Πx ≤ π}, let
w(k) = πk −min{ΠTk·x : x ∈ L}.
The number maxk{w(k)} is a measure of complexity of L.
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Theorem [Andersen, Louveaux, W 08].Let F be the family of all split bodies of maxfacet width at most ω∗. For every polyhedronP, Cl(F , P) is a polyhedron.The special case of ω∗ = 1 was shown in[Cook, Kannan, Schrijver 90].
and for a result of type B ...
Theorem [Balas et al. 91]. For the family F of all splits, there exists a finite t such that
conv(P ∩ ({0, 1}d × Rn−d) = C t(F , P).
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 18 / 18
![Page 58: Nonlinear Discrete OptimizationThe many aspects of nonlinear discrete optimization To begin with max/min f(x 1,...,x n) subject to (x 1,...,x n) ∈ P ∩ Z n. Parametric non-linear](https://reader034.fdocuments.us/reader034/viewer/2022042219/5ec53f38320cc925d949c7ae/html5/thumbnails/58.jpg)
Meaningful properties: C being polyhedral sets and F ?
The max facet width
For a split body L = {x ∈ Rn | Πx ≤ π}, let
w(k) = πk −min{ΠTk·x : x ∈ L}.
The number maxk{w(k)} is a measure of complexity of L.
����
����
����
����
����
����
����
Theorem [Andersen, Louveaux, W 08].Let F be the family of all split bodies of maxfacet width at most ω∗. For every polyhedronP, Cl(F , P) is a polyhedron.The special case of ω∗ = 1 was shown in[Cook, Kannan, Schrijver 90].
and for a result of type B ...
Theorem [Balas et al. 91]. For the family F of all splits, there exists a finite t such that
conv(P ∩ ({0, 1}d × Rn−d) = C t(F , P).
Robert Weismantel (Magdeburg) Nonlinear Discrete Optimization 18 / 18