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Multiple objective function optimization. R.T. Marker, J.S. Arora, “Survey of multi-objective optimization methods for engineering” Structural and Multidisciplinary Optimization Volume 26, Number 6, April 2004 , pp. 369-395(27). Multiple Objective Functions. - PowerPoint PPT Presentation
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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