Levitin–Polyak Type Well-Posedness in Constrained Optimization

Abstract

AbstractWell-posedness of unconstrained and constrained scalar optimization problems was first introduced and studied in Tykhonov [30] and Levitin and Polyak [21], respectively. Since then, various notions of well-posedness, 28, 32, 33, 34]). Recent studies on well-posedness of optimization problems have been extended to vector optimization problems (see, e.g., [3, 7, 12, 13, 22, 24]). The study of LP well-posedness for convex scalar optimization problems with functional constraints originates from [ 19]. In Sect. 10.2 of this chapter, we will introduce three types of (generalized) LP well-posedness for convex scalar optimization problems with functional constraints. Characterizations and criteria for the three types of (generalized) LP well-posedness will be derived. Relations among these three types of (generalized) LP well-posednesses will be established. We will also present convergence results for a class of penalty methods and a class of augmented Lagrangian methods under the assumption of one of the three types of LP well-posedness. In Sect. 10.3, we will introduce several types of (generalized) LP well-posedness for vector optimization problems with functional constraints. Criteria and characterizations for these types of well-posednesses will be given. Relations among these types of well-posedness will be presented. We will also carry out convergence analysis for a class of penalty methods under the assumption of a type of generalized LP well-posedness. KeywordsConvex Scalar Optimization ProblemsAugmented Lagrangian MethodConstrained Vector Optimization ProblemPenalty Type MethodsLevitin Polyak Well-posednessThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.