Robust Stability and Optimality Conditions for Parametric Infinite and Semi-Infinite Programs

This paper primarily concerns the study of parametric problems of infinite and semi-infinite programming, where functional constraints are given by systems of infinitely many linear inequalities indexed by an arbitrary set T, where decision variables run over Banach (infinite programming) or finite-dimensional (semi-infinite case) spaces, and where objectives are generally described by nonsmooth and nonconvex cost functions. The parameter space of admissible perturbations in such problems is formed by all bounded functions on T equipped with the standard supremum norm. Unless the index set T is finite, this space is intrinsically infinite-dimensional (nonreflexive and nonseparable) of the l=-type. By using advanced tools of variational analysis and generalized differentiation and largely exploiting underlying specific features of linear infinite constraints, we establish complete characterizations of robust Lipschitzian stability (with computing the exact bound of Lipschitzian moduli) for parametric maps of feasible solutions governed by linear infinite inequality systems and then derive verifiable necessary optimality conditions for the infinite and semi-infinite programs under consideration expressed in terms of their initial data. A crucial part of our analysis addresses the precise computation of coderivatives and their norms for infinite systems of parametric linear inequalities in general Banach spaces of decision variables. The results obtained are new in both frameworks of infinite and semi-infinite programming.

[1]  Bethany L. Nicholson,et al.  Mathematical Programs with Equilibrium Constraints , 2021, Pyomo — Optimization Modeling in Python.

[2]  Marco A. López,et al.  Metric regularity of semi-infinite constraint systems , 2005, Math. Program..

[3]  Xiaoqi Yang,et al.  Lagrange Multipliers in Nonsmooth Semi-Infinite Optimization Problems , 2007, Math. Oper. Res..

[4]  E. Beckenbach.  CONVEX FUNCTIONS , 2007 .

[5]  A. Ioffe.  Approximate subdifferentials and applications 3: the metric theory , 1989 .

[6]  B. Mordukhovich.  Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions , 1993 .

[7]  J. Borwein,et al.  Techniques of variational analysis , 2005 .

[8]  C. Zălinescu.  Convex analysis in general vector spaces , 2002 .

[9]  Hubertus Th. Jongen,et al.  One-Parametric Semi-Infinite Optimization: On the Stability of the Feasible Set , 1994, SIAM J. Optim..

[10]  D. Varberg.  Convex Functions , 1973 .

[11]  Gerald Beer,et al.  Topologies on Closed and Closed Convex Sets , 1993 .

[12]  N. Dinhy,et al.  From linear to convex systems: Consistency, Farkas Lemma and applications , 2005 .

[13]  Gautam Appa,et al.  Linear Programming in Infinite-Dimensional Spaces , 1989 .

[14]  Jiří V. Outrata,et al.  Mathematical Programs with Equilibrium Constraints: Theory and Numerical Methods , 2006 .

[15]  Boris S. Mordukhovich,et al.  Necessary Conditions in Nonsmooth Minimization via Lower and Upper Subgradients , 2004 .

[16]  Bruno Brosowski,et al.  Parametric semi-infinite optimization , 1982 .

[17]  Marco A. López,et al.  LIPSCHITZ MODULUS IN CONVEX SEMI-INFINITE OPTIMIZATION VIA D.C. FUNCTIONS ∗ , 2009 .

[18]  M. A. López-Cerdá,et al.  Linear Semi-Infinite Optimization , 1998 .

[19]  René Henrion,et al.  Regularity and Stability in Nonlinear Semi-Infinite Optimization , 1998 .

[20]  Marco A. López,et al.  On the Stability of the Feasible Set in Optimization Problems , 2010, SIAM J. Optim..

[21]  B. Mordukhovich.  Variational Analysis and Generalized Differentiation II: Applications , 2006 .

[22]  Александр Давидович Иоффе,et al.  Метрическая регулярность и субдифференциальное исчисление@@@Metric regularity and subdifferential calculus , 2000 .

[23]  D. Klatte.  Nonsmooth equations in optimization , 2002 .

[24]  Marco A. López,et al.  Metric Regularity in Convex Semi-Infinite Optimization under Canonical Perturbations , 2007, SIAM J. Optim..

[25]  Diethard Klatte,et al.  Nonsmooth Equations in Optimization: "Regularity, Calculus, Methods And Applications" , 2006 .

[26]  S. M. Robinson.  Generalized equations and their solutions, Part I: Basic theory , 1979 .

[27]  Boris S. Mordukhovich,et al.  Coderivatives of set-valued mappings: Calculus and applications , 1997 .

[28]  J. Schwartz,et al.  Linear Operators. Part I: General Theory. , 1960 .

[29]  Marco A. López,et al.  Stability and Well-Posedness in Linear Semi-Infinite Programming , 1999, SIAM J. Optim..

[30]  A. Ioffe.  Approximate subdifferentials and applications II , 1986 .

[31]  F. Giannessi.  Variational Analysis and Generalized Differentiation , 2006 .

[32]  María J. Cánovas,et al.  Regularity modulus of arbitrarily perturbed linear inequality systems , 2008 .

[33]  Oliver Stein,et al.  Bi-Level Strategies in Semi-Infinite Programming , 2003 .

[34]  M. Fabian,et al.  Functional Analysis and Infinite-Dimensional Geometry , 2001 .

[35]  B. Mordukhovich.  Maximum principle in the problem of time optimal response with nonsmooth constraints PMM vol. 40, n≗ 6, 1976, pp. 1014-1023 , 1976 .

[36]  Boris S. Mordukhovich,et al.  Subdifferentials of value functions and optimality conditions for DC and bilevel infinite and semi-infinite programs , 2010, Math. Program..