On Second-Order Subdifferentials and Their Applications

We study second-order subdifferentials of nonsmooth functions that are particularly important for applications to sensitivity analysis in optimization and related problems. First we develop various calculus rules for these subdifferentials in rather general settings. Then we obtain exact formulas for computing the second-order subdifferentials for a class of separable piecewise smooth functions. Functions of this class arise, in particular, in equilibrium models related to some practical problems of continuum mechanics. Finally we provide applications of the obtained results to Lipschitzian stability of parametric variational and hemivariational inequalities and efficiently express the derived conditions in terms of the initial data for selected problems of continuum mechanics.

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

[2]  B. Mordukhovich.  Lipschitzian stability of constraint systems and generalized equations , 1994 .

[3]  Ruoxin Zhang,et al.  Problems of Hierarchical Optimization in Finite Dimensions , 1994, SIAM J. Optim..

[4]  R. Tyrrell Rockafellar,et al.  Characterizations of Strong Regularity for Variational Inequalities over Polyhedral Convex Sets , 1996, SIAM J. Optim..

[5]  J. Treiman,et al.  Lagrange Multipliers for Nonconvex Generalized Gradients with Equality, Inequality, and Set Constraints , 1999 .

[6]  Adam B. Levy,et al.  Stability of Locally Optimal Solutions , 1999, SIAM J. Optim..

[7]  B. Mordukhovich.  Stability theory for parametric generalized equations and variational inequalities via nonsmooth analysis , 1994 .

[8]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[9]  R. Rockafellar.  The theory of subgradients and its applications to problems of optimization : convex and nonconvex functions , 1981 .

[10]  B. Mordukhovich.  Sensitivity analysis in nonsmooth optimization , 1992 .

[11]  B. Mordukhovich.  Generalized Differential Calculus for Nonsmooth and Set-Valued Mappings , 1994 .

[12]  R. Tyrrell Rockafellar,et al.  Tilt Stability of a Local Minimum , 1998, SIAM J. Optim..

[13]  I. Hlavácek,et al.  Mathematical Theory of Elastic and Elasto Plastic Bodies: An Introduction , 1981 .

[14]  Jiří V. Outrata.  On Constraint Qualifications for Mathematical Programs with Mixed Complementarity Constraints , 2001 .

[15]  U. Mosco.  Implicit variational problems and quasi variational inequalities , 1976 .

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

[17]  J. Frédéric Bonnans,et al.  Perturbation Analysis of Optimization Problems , 2000, Springer Series in Operations Research.

[18]  J. V. Outrata,et al.  Optimality conditions for a class of mathematical programs with equilibrium constraints: strongly regular case , 1999, Kybernetika.

[19]  Jean-Pierre Aubin,et al.  Lipschitz Behavior of Solutions to Convex Minimization Problems , 1984, Math. Oper. Res..

[20]  J. J. Ye,et al.  Necessary Optimality Conditions for Optimization Problems with Variational Inequality Constraints , 1997, Math. Oper. Res..

[21]  Jirí V. Outrata,et al.  A Generalized Mathematical Program with Equilibrium Constraints , 2000, SIAM J. Control. Optim..

[22]  Jane J. Ye,et al.  Optimality Conditions for Optimization Problems with Complementarity Constraints , 1999, SIAM J. Optim..