Authors: Rommel G Regis
Publish Date: 2016/07/18
Volume: 170, Issue: 3, Pages: 932-959
Abstract
Stochastic search methods for global optimization and multiobjective optimization are widely used in practice especially on problems with blackbox objective and constraint functions Although there are many theoretical results on the convergence of stochastic search methods relatively few deal with blackbox constraints and multiple blackbox objectives and previous convergence analyses require feasible iterates Moreover some of the convergence conditions are difficult to verify for practical stochastic algorithms and some of the theoretical results only apply to specific algorithms First this article presents some technical conditions that guarantee the convergence of a general class of adaptive stochastic algorithms for constrained blackbox global optimization that do not require iterates to be always feasible and applies them to practical algorithms including an evolutionary algorithm The conditions are only required for a subsequence of the iterations and provide a recipe for making any algorithm converge to the global minimum in a probabilistic sense Second it uses the results for constrained optimization to derive convergence results for stochastic search methods for constrained multiobjective optimizationClearly Pmathcal Icup mathcal H le Pmathcal I+Pmathcal H=0 and so Pmathcal Icup mathcal H=0 Next fix omega in Omega setminus mathcal Icup mathcal H Since omega in Omega setminus mathcal I exists komega such that X n komega omega in mathcal D and so X nomega in mathcal D for all n ge n komega Hence fX nomega n ge n komega is monotonically nonincreasing Moreover fX nomega ge f for all n ge n komega Hence lim n rightarrow infty fX nomega exists Also since omega in Omega setminus mathcal H it follows that lim i rightarrow infty fX n kiomega =f Hence lim n rightarrow infty fX nomega =f This shows that fX n longrightarrow f as square Fix epsilon 0 and let widetildef = inf fx x in mathcal D Vert xxVert ge epsilon Since fX n longrightarrow fx as it follows that exists mathcal Nsubseteq Omega with Pmathcal N=0 such that fX nomega longrightarrow fx for all omega in Omega setminus mathcal N As in the proof of Proposition 21 define mathcal I= omega in Omega X n komega not in mathcal D text for text all k = bigcap k=1infty X n k not in mathcal D It was shown thatPmathcal I=0Now for any n ge max Nomega komega we have X nomega in mathcal D and fX nomega f Note that we must have Vert X nomega xVert epsilon Otherwise if Vert X nomega xVert ge epsilon then fX nomega ge inf fx x in mathcal D Vert xxVert ge epsilon = widetildef which is a contradiction This shows that X nomega ~longrightarrow ~x for each omega in Omega setminus mathcal Icup mathcal N Thus X n longrightarrow x as square Since S ne emptyset the given assumption implies that text int S ne emptyset otherwise text cl S=emptyset Next if text bd S=emptyset then the above statement is vacuously true so assume that text bd S ne emptyset and let x in text bd S Then x in text cl S Since text cl text int S=text cl S it follows that x in text cl text int S Since x not in text int S it follows that x is a limit point of text int S Thus every neighborhood of x contains an interior point of S
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