Optimization Methods
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Abstract
In this paper, we study approximate optimality conditions for the Canonical DC (CDC) optimization problemandtheirrelationshipswithstoppingcriteriaforalargeclassofsolutionalgorithmsfortheproblem. Infact,globaloptimalityconditionsforCDCareveryoftenrestatedintermsofanon-convexoptimizationproblem,whichhastobesolvedeachtimetheoptimalityofagivententativesolutionhastobechecked. Sincethisisinprincipleacostlytask,itmakessensetoonlysolvetheproblemapproximately,leadingtoaninexactstoppingcriteriaandthereforetoapproximateoptimalityconditions.Inthisframework,itis importanttostudytherelationshipsbetweentheapproximationinthestoppingcriteriaandthequalityofthesolutionsthatthecorrespondingapproximatedoptimalityconditionsmayeventuallyacceptasoptimal, inordertoensurethatasmalltoleranceinthestoppingcriteriadoesnotleadtoadisproportionallylargeapproximationoftheoptimalvalueoftheCDCproblem.Wedevelopconditionsensuringthatthisisthe case;theseturnouttobecloselyrelatedwiththewell-knownconceptof regularity of a CDC problem, actually coinciding with the latter if the reverse-constraint set is a polyhedron.