Sufficient Optimality Conditions in Bilevel Programming

This paper is concerned with the derivation of first- and second-order sufficient optimality conditions for optimistic bilevel optimization problems involving smooth functions. First-order sufficient optimality conditions are obtained by estimating the tangent cone to the feasible set of the bilevel program in terms of initial problem data. This is done by exploiting several different reformulations of the hierarchical model as a single-level problem. To obtain second-order sufficient optimality conditions, we exploit the so-called value function reformulation of the bilevel optimization problem, which is then tackled with the aid of second-order directional derivatives. The resulting conditions can be stated in terms of initial problem data in several interesting situations comprising the settings where the lower level is linear or possesses strongly stable solutions.

[1]  Samir Adly,et al.  New necessary and sufficient optimality conditions for strong bilevel programming problems , 2018, J. Glob. Optim..

[2]  Stephan Dempe,et al.  Directional derivatives of the solution of a parametric nonlinear program , 1995, Math. Program..

[3]  Stephan Dempe,et al.  Is bilevel programming a special case of a mathematical program with complementarity constraints? , 2012, Math. Program..

[4]  S. M. Robinson.  Some continuity properties of polyhedral multifunctions , 1981 .

[5]  Christian Kanzow,et al.  Theorie und Numerik restringierter Optimierungsaufgaben , 2002 .

[6]  A. Ben-Tal.  Second-order and related extremality conditions in nonlinear programming , 1980 .

[7]  Gerd Wachsmuth,et al.  On LICQ and the uniqueness of Lagrange multipliers , 2013, Oper. Res. Lett..

[8]  N. Gadhi,et al.  Sufficient optimality conditions and duality results for a bilevel multiobjective optimization problem via a Ψ reformulation , 2020, Optimization.

[9]  M. Studniarski.  Necessary and sufficient conditions for isolated local minima of nonsmooth functions , 1986 .

[10]  B. Mordukhovich.  Variational Analysis and Applications , 2018 .

[11]  R. Janin.  Directional derivative of the marginal function in nonlinear programming , 1984 .

[12]  Stefan Scholtes,et al.  Mathematical Programs with Complementarity Constraints: Stationarity, Optimality, and Sensitivity , 2000, Math. Oper. Res..

[13]  Stephan Dempe,et al.  Foundations of Bilevel Programming , 2002 .

[14]  J. Penot.  Second-Order Conditions for Optimization Problems with Constraints , 1999 .

[15]  Boris S. Mordukhovich,et al.  Second-Order Variational Analysis of Parametric Constraint and Variational Systems , 2019, SIAM J. Optim..

[16]  René Henrion,et al.  Calmness of constraint systems with applications , 2005, Math. Program..

[17]  Anthony V. Fiacco,et al.  Introduction to Sensitivity and Stability Analysis in Nonlinear Programming , 2012 .

[18]  Bergakademie Freiberg,et al.  BILEVEL PROGRAMMING: REFORMULATIONS, REGULARITY, AND STATIONARITY , 2012 .

[19]  Alexander Shapiro,et al.  Second Order Optimality Conditions Based on Parabolic Second Order Tangent Sets , 1999, SIAM J. Optim..

[20]  R. Henrion,et al.  On calmness conditions in convex bilevel programming , 2011 .

[21]  Helmut Gfrerer,et al.  Lipschitz and Hölder stability of optimization problems and generalized equations , 2016, Math. Program..

[22]  Olvi L. Mangasarian,et al.  Exact penalty functions in nonlinear programming , 1979, Math. Program..

[23]  Jane J. Ye,et al.  Nondifferentiable Multiplier Rules for Optimization and Bilevel Optimization Problems , 2004, SIAM J. Optim..

[24]  Nataliya I. Kalashnykova,et al.  Bilevel Programming Problems: Theory, Algorithms and Applications to Energy Networks , 2015 .

[25]  A. Shapiro.  Sensitivity analysis of nonlinear programs and differentiability properties of metric projections , 1988 .

[26]  Helmut Gfrerer,et al.  On Computation of Generalized Derivatives of the Normal-Cone Mapping and Their Applications , 2016, Math. Oper. Res..

[27]  Jonathan F. Bard,et al.  Practical Bilevel Optimization , 1998 .

[28]  S. M. Robinson.  Generalized equations and their solutions, part II: Applications to nonlinear programming , 1982 .

[29]  Milan Vlach,et al.  SECOND ORDER TANGENT SETS AND OPTIMALITY CONDITIONS , 1997 .

[30]  Boris S. Mordukhovich,et al.  Parabolic regularity in geometric variational analysis , 2020, Transactions of the American Mathematical Society.

[31]  Jonathan F. Bard,et al.  Practical Bilevel Optimization: Algorithms and Applications , 1998 .

[32]  J. Bard,et al.  Nondifferentiable and Two-Level Mathematical Programming , 1996 .

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

[34]  A. Shapiro.  Perturbation theory of nonlinear programs when the set of optimal solutions is not a singleton , 1988 .

[35]  Jane J. Ye,et al.  Optimality conditions for bilevel programming problems , 1995 .

[36]  Stephan Dempe,et al.  The bilevel programming problem: reformulations, constraint qualifications and optimality conditions , 2013, Math. Program..

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

[38]  G. McCormick.  Second Order Conditions for Constrained Minima , 1967 .

[39]  A. Ben-Tal,et al.  Necessary and sufficient optimality conditions for a class of nonsmooth minimization problems , 1982, Math. Program..

[40]  Christian Kanzow,et al.  Abadie-Type Constraint Qualification for Mathematical Programs with Equilibrium Constraints , 2005 .

[41]  Mokhtar S. Bazaraa,et al.  Nonlinear Programming: Theory and Algorithms , 1993 .

[42]  Jerzy Kyparisis,et al.  On uniqueness of Kuhn-Tucker multipliers in nonlinear programming , 1985, Math. Program..

[43]  Nataliya I. Kalashnykova,et al.  Optimality conditions for bilevel programming problems , 2006 .

[44]  Stephan Dempe,et al.  Second order optimality conditions for bilevel set optimization problems , 2010, J. Glob. Optim..