Subdifferentials of value functions and optimality conditions for DC and bilevel infinite and semi-infinite programs

• Published in: Mathematical programming Vol. 123; no. 1; pp. 101 - 138
• Main Authors: Dinh, N., Mordukhovich, B., Nghia, T. T. A.
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.05.2010; Springer; Springer Nature B.V
• Subjects: Applied sciences; Banach spaces; Calculus of variations and optimal control; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Control theory; Convex analysis; Exact sciences and technology; Full Length Paper; Mathematical analysis; Mathematical and Computational Physics; Mathematical functions; Mathematical Methods in Physics; Mathematical programming; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Operational research and scientific management; Operational research. Management science; Optimization; Parameter optimization; Partial differential equations; Sciences and techniques of general use; Studies; Theoretical; Optimality conditions; Marginal and value functions; Semi-infinite and infinite programming; 90C30; Convex inequality constraints; 49J53; 49J52; Bilevel programming; Generalized differentiation; Variational analysis and parametric optimization; Well-posedness and sensitivity; Lipschitz condition; Parametric analysis; Subdifferential; Parametric programming; Semi infinite programming; Non linear programming; Convex programming; Optimization; Nonsmooth analysis; Inequality constraint; Variational calculus; Lipschitz function; Convex function; Infinite dimension; Mathematical programming; Existence theorem; Well posed problem; Continuous function; Program specification; Optimality condition; Optimality criterion
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-009-0323-4

The paper concerns the study of new classes of parametric optimization problems of the so-called infinite programming that are generally defined on infinite-dimensional spaces of decision variables and contain, among other constraints, infinitely many inequality constraints. These problems reduce to semi-infinite programs in the case of finite-dimensional spaces of decision variables. We focus on DC infinite programs with objectives given as the difference of convex functions subject to convex inequality constraints. The main results establish efficient upper estimates of certain subdifferentials of (intrinsically nonsmooth) value functions in DC infinite programs based on advanced tools of variational analysis and generalized differentiation. The value/marginal functions and their subdifferential estimates play a crucial role in many aspects of parametric optimization including well-posedness and sensitivity . In this paper we apply the obtained subdifferential estimates to establishing verifiable conditions for the local Lipschitz continuity of the value functions and deriving necessary optimality conditions in parametric DC infinite programs and their remarkable specifications. Finally, we employ the value function approach and the established subdifferential estimates to the study of bilevel finite and infinite programs with convex data on both lower and upper level of hierarchical optimization. The results obtained in the paper are new not only for the classes of infinite programs under consideration but also for their semi-infinite counterparts.