Post on 25-Feb-2019
Game Theory with gradient based
optimization algorithm
L. Parussini*, V. Pediroda+
University of Trieste, Mechanical Engineering Dep., Via Valerio 10, 34100 Trieste, Italy* E-mail: lparussini@units.it+ E-mail: pediroda@units.it
OUTLINE
• INTRODUCTION
• THEORYNASH THEORYSIMPLE GRADIENT METHODDACE MODEL
• TEST
• APPLICATION
• CONCLUSION
INTRODUCTION
REDUCTION OF NUMBER OF DESIGNS TO COMPUTE FOR OBTAINING GOOD RESULTS
BY OPTIMIZATION
INTEGRATION OF CAD, CAE, CFD, etc. TECHNIQUES WITH NUMERIC OPTIMIZATION METHODS WITH
ADVANTAGE FOR INDUSTRY
NASH THEORY
GRADIENT METHOD
multi objective optimization
mono objectiveoptimization
RESPONSE SURFACES
INTRODUCTION
GAME THEORY
COMPETITION INDIVIDUALS
DESIGN
PROCESS
DESIGN
TEAMS
NASH THEORY
NASH THEORYConsider two-player Nash game
A search space for first player B search space for second player
AxByx ∈),( ** is NASH EQUILIBRIUM if:
),(inf),( *** yxfyxf AAxA ∈=
),(inf),( *** yxfyxf BByB ∈=
where Af gain for first player
Bf gain for second player
ℜ⇒AxByxf A :),(
ℜ⇒AxByxf B :),(are the objective functions to minimize
GRADIENT METHOD
)( )()()()1( KKKK XSXX α+=+
iterative method :
• actual variables
• estimated variables for next step
• step length
• searching function of right direction
)( KX)1( +KX
)(Kα)( XS
GRADIENT METHODSIMPLE GRADIENT METHOD
where α is costant and gradient is the searching function
)( )()()()1( KKKK XfXX ∇−=+ α
computing of partial derivatives by
central finite difference method
on DACE model
COMPUTATIONAL SAVING
DACE MODEL
simplifing hypoteses
modelling phenomena bySRF Spatial Random Fields
∑=
+=k
hhh xxfxY
1
)()()( εβ
linear estimators
BEST LINEAR UNBIASED PREDICTOR
MEAN SQUARED ERROR OF THE PREDICTOR
)ˆ1y(R'rˆ*)x(y 1 µ−+µ= −
−+−σ= −
−−
1R'1)rR11(
rR'r1*)x(s 1
21122
CORRELATION FUNCTION between errors:
[ ] [ ]))(),((exp))(()),(( jxixdjxixCorr −=εε
∑=
−=k
h
phhh
hjxixjxixd1
)()())(),(( θ 0>hθ [ ]2,1∈hp
LIKELYHOOD:
2
1
2/122/
)1()'1(exp
)R(det)2(
12/
σ
µµ
σπ
−− − yRynn θ,pMAX
ADAPTIVE DACE MODEL
ERROR INDEX
standard deviation extrapolated function
ADAPTIVE RESPONSE SURFACES
If f is to maximize:
maxminmax
max
σσ
⋅−
−=
ffff
IE
NASH THEORY
GRADIENT METHOD
multi objective optimization
mono objectiveoptimization
RESPONSE SURFACES
ALGORITHMResuming..
TEST2
222
11 )()(1)(1 BABATest −+−+=x
22
21 )1()3()(2 +++= ββxTest
∑=
+=2
1
))cos()sin((j
jijjiji baA αα for i=1,2
∑=
+=2
1
))cos()sin((j
jijjiji baB ββ for i=1,2
=
0.25.1
0.15.0a
−−−−
=5.00.15.10.2
b
( )0.2,0.1=α
∑∑=
+==
n
nii
n
ii xx
12/
2/
1,β
where
n
nn-
∈
2/,
2/ππ
x and 16=n
OPTIMIZATION OF Test1
with
TESTMULTI OBJECTIVE
MINIMIZATION
OF Test1 AND Test2
Comparison between Nash equilibrium and Pareto frontier
computed by MOGA (Multi Objective Genetic Algorithm)
12860.00717.982103nash4
322240.01615.13232nash3
246140.02514.73533nash2
356160.04517.87035nash1
n°computations
n°exchangesTest2Test1
init.data set
DACEσ
APPLICATION
DESIGN OF A TRANSONIC AIRFOIL IN STOCHASTIC OPERATIVE
CONDITIONS
ROBUST DESIGN
APPLICATION
OVER-OPTIMIZATIONDETERMINISTIC DESIGN:
COMPARISON BETWEEN DETERMINISTIC DESIGN
AND PROBABILISTIC DESIGN
ROBUST DESIGN
function to optimizewherex deterministic variablesy stochastic variables
n OPTIMIZATION OF EXPECTED VALUE
n MINIMIZATION OF STANDARD DEVIATION
∑=
=n
iifn
xf1
1)(
1
)()( 1
2
−
−=
∑=
n
ffx
n
ii
fσ
ℜ→ℜ +mnxyF :)(
ROBUST DESIGNSTOCHASTIC VARIABLES
n ANGLE OF ATTACKn MACH NUMBER OF FREE STREAM
DETERMINISTIC VARIABLES
OBJECTIVES AND CONSTRAINTS
Cd
dC
σmin
min
≥
≤≥
≤
≤
RAE
ClRAECl
lRAEl
CdRAECd
dRAEd
tt
CC
CC
maxmax
σσ
σσwith
05.073.0 ±=∞M
°° ±= 5.02α
ROBUST DESIGN∑
=
=n
iifn
xf1
1)(
1
)()( 1
2
−
−=
∑=
n
ffx
n
ii
fσ RESPONSE SURFACES
°÷°=
÷==
∞
5.25.1
78.068.0065.1Re
α
ME
COMPUTING OF , , AND
COMPUTATIONALLY and TIME EXPENSIVE
dC CdσlC Clσ
DACE MODEL OF Cd AND Cl
ROBUST DESIGNEXPLORATIVE DESIGN
OPTIMIZED AIRFOIL
RAE2822 Comparison between objectives of optimized airfoil (explorative design) and RAE2822.
dCσ
lCσ
maxt 0.1212940.121223
0.0537600.053835
0.6105510.604603
0.0060870.006128
0.0172970.017351
RAE2822 airfoiloptimized airfoil
dC
lC
• 4 exchanges of variables between players
• 87 fluid dynamics simulations
• Nash frequency equal to 2
ROBUST DESIGNCONSERVATIVE DESIGN
Comparison between objectives of optimized airfoil (conservative design) and RAE2822.
dC
dCσ
lC
lCσ
maxt 0.1212940.121334
0.0537600.053552
0.6105510.608349
0.0060870.005996
0.0172970.017133
RAE2822 airfoiloptimized airfoil • 3 exchanges of variables between players
• 134 fluid dynamics simulations
• Nash frequency equal to 3
ROBUST DESIGN
• Cause of constant step lenght of SGM difficult to find solutions inside constraints.
• Choice of domain space decomposition is very important for the efficiency of optimization as it has been demonstrated in “Application of Game Strategy in Multi-objective Robust Design Optimisation Implementing Self-adaptive Search Space Decomposition by Statistical Analysis”, A.Clarich, V.Pediroda, L.Padovan, C.Poloni, J. Periaux, (2004), European Congress on Computational Methods in Applied Sciences and Engineering ECCOMAS 2004. An adaptive strategy for variables distribution is needed.
REMARKS
CONCLUSION• IMPLEMENTATION OF A NASH THEORY WITH GRADIENT BASED OPTIMIZATION ALGORITHM FOR MULTI OBJECTIVE PROBLEMS
• SGM TO OPTIMIZE EACH ONE OF OBJECTIVES
• ADAPTIVE DACE MODEL, A PARTICULAR EFFICIENT EXTRAPOLATION METHOD, TO CALCULATE THE DERIVATIVES REQUIRED BY SGM
v TESTS ON MATHEMATICAL FUNCTIONS
v APPLICATION TO DESIGN UNDER UNCERTAINTIES OF AN AIRFOIL IN TRANSONIC FIELD
CONCLUSIONFURTHER STEPS
• in mathematical test cases great efficiency : quick and accurate, but limits due to lack of adaptivity to different problems.In particular - constrained problems require more investigation - introduction of a criteria for the re-distribution of variables to players
• in Robust Design application:- explorative design has been really successful: we can think of using
Nash/SGM to investigate an unknown problem, with the advantage to get quickly an optimum point, and next, starting from the Nash equilibrium, to obtain more accurate solutions by Pareto Games