Multiple objective function optimizationR.T. Marker, J.S. Arora, Survey of multi-objective optimization methods for engineering

Structural and Multidisciplinary OptimizationVolume 26,Number 6, April 2004 , pp. 369-395(27)

Assume all f,g,h are differentiableMultiple Objective Functions

Feasible design space - satisfies all constraintsPreliminariesFeasible criterion space - objective function values of feasible design space regionPreferences - users opinion about points in criterion spaceScalarization methods v. vector methods

rugged fitness landscapesensitivity issuehttp://www.calresco.org/lucas/pmo.htm

Strange Attractorsnon-linear cross-couplingM( t+1 ) = a * M(t) + b * I ( t ) + c * T ( t )I ( t+1 ) = d * I ( t ) + e * T ( t ) + f * M( t )T ( t+1 ) = g * T (t) + h * M( t ) + j * I ( t )economic resourcesmoneyideastimehttp://www.calresco.org/lucas/pmo.htm

a priori articulation of preferencesa posteriori articulation of preferencesprogressive articulation of preferencesgenetic algorithmsOrganization

compromise solutionutopia (ideal) pointpoint that optimizes all objective functionsoften doesnt existone or more objective functions not optimalclose as possible to utopia pointF0

x1 is superior to x2 iffx1 dominates x2x1 > x2

Pareto optimal solutionif there does not exist another feasible design objective vector such that all objective functions are better than or equal to and at least one objective function is betteri.e., there is no x such that x > xi.e., it is not dominated by any other point

Weakly Pareto Optimalno other point with better object valuesProperly Pareto Optimal

Pareto optimal setSet of all Pareto optimal pointspossibly infinite setVarious ApproachesIdentify Pareto optimal setIdentify some subset of optimal setseek a single final point

Solving multiple objective optimization provides: Necessary condition for Pareto optimalityand / orSufficient condition for Pareto optimality

Common function transformation methodsto remove dimensions or balance magnitude differences

Methods with a priori articulation of preferencesAllow user to specify preferences for, or relative importance of, objective functions

Weighted Sum MethodSufficient for Pareto optimalityno guarantee of final result acceptableimpossible to find points in non-convex sectionsnot even distribution

Weighted global criterion method

Lexicographic Methodobjective functions arranged in order of importancesolve following optimization problems one at a time

Goal Programming Method

Goal Attainment Methodcomputationally faster than typical goal programming methods

Physcial ProgrammingClass function for each metricmonotonically increasing, monotonically decresing, or unimodal functionspecify numeric ranges for degrees of preferencedesirable, tolerable, undesirable, etc.

Methods for a posteriori articualtion of preferencegenerate first, choose later approachesgenerate representative Pareto optimal setuser selects from palette of solutions

Physical Programmingsystematically vary parameterstraverses criterion space

Normal boundary intersection method

Normal constraint methoddetermine utopia pointnormalize objective functionsindividual minimization of objective functionsform vertices of utopia hyperplane

Methods no articulation of preferencesGlobal criterion methodswith wi = 1.0similar to a priori techniques with no weights

Min max methodprovides weakly Pareto optimal pointtreat as single objective function

Objective sum methodTo avoid additional constraints and discontinuities

Nast arbitration and objective product methodMaximizewhere si >= Fi(x)

Raos methodnormalize so Finorm is between zero and oneand Finorm=1 is worst possible

Genetic Algorithmsno derivative information neededglobal optimizatione.g., generate sub-populations by optimizing one objective function

directions in shaded area reduce both objective functions

plot one function in terms of 2 variablessensitive to initial conditionsdifficult to find representative spread of good solutions4 classifications of algorithmsa priori - combine individual functions into one