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Critical multipliers in variational systems via second-order generalized differentiation. (English) Zbl 1407.90314
The following equation system is studied: \[ \Psi(x,v) = 0, \;\;v \in \partial\theta(\Phi), \] where \(\Phi: \mathbb R^n \longrightarrow \mathbb R^m\), \(\Psi : \mathbb R^n \times \mathbb R^m \longrightarrow \mathbb R^l\), \(\theta: \mathbb R^m \longrightarrow \overline{\mathbb R}\) is an extended real-valued function, \(\partial\) denotes an appropriate subdifferential. The authors use mainly a convex function \(\theta\) so that \(\partial\) is the classical subdifferential.
The authors introduce the notion of critical and non-critical mulltipliers for the variational system. A complete general characterization of the critical and non-critical multipliers is obtained. Especially comprehensive results are presented for the case that \(\theta\) belongs to the class of convex piecewise linear functions.

MSC:
90C31 Sensitivity, stability, parametric optimization
49J52 Nonsmooth analysis
49J53 Set-valued and variational analysis
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References:
[1] Bonnans, JF, Local analysis of Newton-type methods for variational inequalities and nonlinear programming, Appl. Math. Optim., 29, 161-186, (1994) · Zbl 0809.90115
[2] Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000) · Zbl 0966.49001
[3] Chieu, NH; Hien, LV, Computation of graphical derivative for a class of normal cone mappings under a very weak condition, SIAM J. Optim., 27, 190-204, (2017) · Zbl 1357.49072
[4] Ding, C; Sun, D; Zhang, L, Characterization of the robust isolated calmness for a class of conic programming problems, SIAM J. Optim., 27, 67-90, (2017) · Zbl 1357.49100
[5] Dontchev, AL; Rockafellar, RT; Fiacco, AV (ed.), Characterizations of Lipschitzian stability in nonlinear programming, 65-82, (1997), New York · Zbl 0891.90146
[6] Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings: A View from Variational Analysis, 2nd edn. Springer, New York (2014) · Zbl 1337.26003
[7] Facchinei, F., Pang, J.-S.: Finite-Dimesional Variational Inequalities and Complementarity Problems. Springer, New York (2003) · Zbl 1062.90001
[8] Fischer, A, Local behavior of an iterative framework for generalized equations with nonisolated solutions, Math. Program., 94, 91-124, (2002) · Zbl 1023.90067
[9] Fusek, P, Isolated zeros of Lipschitzian metrically regular \({\mathbb{R}}^n\)-functions, Optimization, 49, 425-446, (2001) · Zbl 0987.49009
[10] Gfrerer, H, First-order and second-order characterizations of metric subregularity and calmness of constraint mappings, SIAM J. Optim., 21, 1439-1474, (2011) · Zbl 1254.90246
[11] Gfrerer, H; Mordukhovich, BS, Complete characterizations of tilt stability in nonlinear programming under weakest qualification conditions, SIAM J. Optim., 25, 2081-2119, (2015) · Zbl 1357.49074
[12] Gfrerer, H; Mordukhovich, BS, Robinson stability of parametric constraint systems via variational analysis, SIAM J. Optim., 27, 438-465, (2017) · Zbl 1368.49014
[13] Gfrerer, H; Outrata, JV, On computation of generalized derivatives of the normal cone mapping and their applications, Math. Oper. Res., 41, 1535-1556, (2016) · Zbl 1352.49018
[14] Henrion, R; Outrata, JV, Calmness of constraint systems with applications, Math. Program., 104, 437-464, (2005) · Zbl 1093.90058
[15] Ioffe, AD; Outrata, JV, On metric and calmness qualification conditions in subdifferential calculus, Set-Valued Anal., 16, 199-227, (2008) · Zbl 1156.49013
[16] Izmailov, AF, On the analytical and numerical stability of critical Lagrange multipliers, Comput. Math. Math. Phys., 45, 930-946, (2005)
[17] Izmailov, A.F.: Tilt and full stability in constrained optimization and the existence of critical Lagrange multipliers, unpublished manuscript, (2015) · Zbl 1304.49047
[18] Izmailov, AF; Solodov, MV, Stabilized SQP revisited, Math. Program., 133, 93-120, (2012) · Zbl 1245.90145
[19] Izmailov, A.F., Solodov, M.V.: Newton-Type Methods for Optimization and Variational Problems. Springer, New York (2014) · Zbl 1304.49001
[20] Izmailov, AF; Solodov, MV, Critical Lagrange multipliers: what we currently know about them, how they spoil our lives, and what we can do about it, TOP, 23, 1-26, (2015) · Zbl 1317.90279
[21] King, A; Rockafellar, RT, Sensitivity analysis for nonsmooth generalized equations, Math. Oper. Res., 55, 341-364, (1992) · Zbl 0766.90075
[22] Klatte, D., Kummer, B.: Nonsmooth Equations in Optimization: Regularity, Calculus, Methods and Applications. Kluwer, Dordrecht (2002) · Zbl 1173.49300
[23] Klatte, D; Kummer, B, Aubin property and uniqueness of solutions in cone constrained optimization, Math. Methods Oper. Res., 77, 291-304, (2013) · Zbl 1269.49039
[24] Levy, AB, Implicit multifunction theorems for the sensitivity analysis of variational conditions, Math. Program., 74, 333-350, (1996) · Zbl 0864.49003
[25] Levy, AB; Poliquin, RA; Rockafellar, RT, Stability of locally optimal solutions, SIAM J. Optim., 10, 580-604, (2000) · Zbl 0965.49018
[26] Mordukhovich, BS, Complete characterizations of openness, metric regularity, and Lipschitzian properties of multifunctions, Trans. Am. Math. Soc., 340, 1-35, (1993) · Zbl 0791.49018
[27] Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory; II: Applications. Springer, Berlin (2006) · Zbl 1100.49002
[28] Mordukhovich, BS, Comments on: critical Lagrange multipliers: what we currently know about them, how they spoil our lives, and what we can do about it, TOP, 23, 35-42, (2015) · Zbl 1329.90141
[29] Mordukhovich, BS; Nghia, TTA, Full lipschitzian and Holderian stability in optimization with applications to mathematical programming and optimal control, SIAM J. Optim., 24, 1344-1381, (2014) · Zbl 1304.49047
[30] Mordukhovich, BS; Nghia, TTA, Second-order characterizations of tilt stability with applications to nonlinear programming, Math. Program., 149, 83-104, (2015) · Zbl 1312.49016
[31] Mordukhovich, BS; Nghia, TTA, Local monotonicity and full stability for parametric variational systems, SIAM J. Optim., 26, 1032-1059, (2016) · Zbl 1337.49030
[32] Mordukhovich, BS; Nghia, TTA; Rockafellar, RT, Full stability in finite-dimensional optimization, Math. Oper. Res., 40, 226-252, (2015) · Zbl 1308.90126
[33] Mordukhovich, BS; Outrata, JV; Ramírez, CH, Second-order variational analysis in conic programming with applications to optimality and stability, SIAM J. Optim., 25, 76-101, (2015) · Zbl 1356.49021
[34] Mordukhovich, BS; Outrata, JV; Ramírez, CH, Graphical derivatives and stability analysis for parameterized equilibria with conic constraints, Set-Valued Var. Anal., 23, 687-704, (2015) · Zbl 1330.49013
[35] Mordukhovich, BS; Rockafellar, RT; Sarabi, ME, Characterizations of full stability in constrained optimization, SIAM J. Optim., 23, 1810-1849, (2013) · Zbl 1284.49032
[36] Mordukhovich, BS; Sarabi, ME, Variational analysis and full stability of optimal solutions to constrained and minimax problems, Nonlinear Anal., 121, 36-53, (2015) · Zbl 1326.90083
[37] Mordukhovich, BS; Sarabi, ME, Generalized differentiation of piecewise linear functions in second-order variational analysis, Nonlinear Anal., 132, 240-273, (2016) · Zbl 1329.49024
[38] Mordukhovich, BS; Sarabi, ME, Second-order analysis of piecewise linear functions with applications to optimization and stability, J. Optim. Theory Appl., 171, 504-526, (2016) · Zbl 1355.49011
[39] Mordukhovich, BS; Sarabi, ME, Stability analysis for composite optimization problems and parametric variational systems, to appear in, J. Optim. Theory Appl., 172, 554-577, (2017) · Zbl 1431.49016
[40] Pang, J-S, Convergence of splitting and Newton methods for complementarity problems: an application of some sensitivity results, Math. Program., 58, 149-160, (1993) · Zbl 0784.90089
[41] Poliquin, RA; Rockafellar, RT, Tilt stability of a local minimum, SIAM J. Optim., 8, 287-299, (1998) · Zbl 0918.49016
[42] Robinson, SM, Generalized equations and their solutions, part I: basic theory, Math. Program. Stud., 10, 128-141, (1979) · Zbl 0404.90093
[43] Robinson, SM, Some continuity properties of polyhedral multifunctions, Math. Program. Stud., 14, 206-214, (1981) · Zbl 0449.90090
[44] Rockafellar, RT, First- and second-order epi-differentiability in nonlinear programming, Trans. Am. Math. Soc., 307, 75-108, (1988) · Zbl 0655.49010
[45] Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998) · Zbl 0888.49001
[46] Rockafellar, RT; Zagrodny, D, A derivative-coderivative inclusion in second-order nonsmooth analysis, Set-Valued Anal., 5, 1-17, (1997) · Zbl 0897.49014
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