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