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Graphical derivatives and stability analysis for parameterized equilibria with conic constraints. (English) Zbl 1330.49013
Summary: The paper concerns parameterized equilibria governed by generalized equations whose multivalued parts are modeled via regular normals to nonconvex conic constraints. Our main goal is to derive a precise pointwise second-order formula for calculating the graphical derivative of the solution maps to such generalized equations that involves Lagrange multipliers of the corresponding KKT systems and critical cone directions. Then, we apply the obtained formula to characterizing a Lipschitzian stability notion for the solution maps that is known as isolated calmness.

49J53 Set-valued and variational analysis
49J52 Nonsmooth analysis
49K40 Sensitivity, stability, well-posedness
90C31 Sensitivity, stability, parametric optimization
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[1] Bonnans, JF; Cominetti, R; Shapiro, A, Sensitivity analysis of optimization problems under second order regular constraints, Math. Oper. Res., 23, 806-831, (1998) · Zbl 0977.90053
[2] Bonnans, JF; Ramírez, HC, Perturbation analysis of second-order cone programming problems, Math. Program., 104, 205-227, (2005) · Zbl 1124.90039
[3] Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000) · Zbl 0966.49001
[4] Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. Springer, Dordrecht (2009) · Zbl 1178.26001
[5] Drusvyatskiy, D., Mordukhovich, B.S., Nghia, T.T.A.: Second-order growth, tilt stability, and metric regularity of the subdifferential. J. Convex Anal. 21(4) (2014) · Zbl 1311.49035
[6] Henrion, R; Kruger, A; Outrata, JV, Some remarks on stability on generalized equations, J. Optim. Theory Appl., 159, 681-697, (2013) · Zbl 1297.90151
[7] King, AJ; Rockafellar, RT, Sensitivity analysis for nonsmooth generalized equations, Math. Program., 55, 193-212, (1992) · Zbl 0766.90075
[8] Klatte, D., Kummer, B.: Nonsmooth Equations in Optimization. Kluwer, Dordrecht (2002) · Zbl 1173.49300
[9] Kummer, B.: Newton’s method based on generalized derivatives for nonsmooth functions: Convergence analysis. In: Oettli, W., Pallaschke, D. (eds.) Advances in Optimization, Lecture Notes in Economics and Mathematical Systems, vol. 382, pp 171-194. Springer, Berlin (1992) · Zbl 0768.49012
[10] Levy, AB, Implicit multifunction theorems for the sensitivity analysis of variational conditions, Math. Program., 74, 333-350, (1996) · Zbl 0864.49003
[11] Levy, AB; Poliquin, RA; Rockafellar, RT, Stability of locally optimal solutions, SIAM J. Optim., 10, 580-604, (2000) · Zbl 0965.49018
[12] Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory. Springer, Berlin (2006)
[13] Mordukhovich, BS; Nghia, TTA, Full Lipschitzian and Hölderian stability in optimization with applications to mathematical programming and optimal control, SIAM J. Optim., 24, 1344-1381, (2014) · Zbl 1304.49047
[14] Mordukhovich, B.S., Nghia, T.T.A., Rockafellar, R.T.: Full stability in finite-dimensional optimization. Math. Oper. Res. to appear. doi:10.1287/moor.2014.0669 (2014) · Zbl 1308.90126
[15] Mordukhovich, B.S., Outrata, J.V., Ramírez, C., H.: Second-order variational analysis in conic programming with applications to optimality and stability. SIAM J. Optim. 25(1), 76-101 (2015) · Zbl 1356.49021
[16] Mordukhovich, B.S., Outrata, J.V., Sarabi, M.E.: Full stability of locally optimal solutions in second-order cone programs. SIAM J. Optim. 24(4), 1581-613 (2014) · Zbl 1311.49062
[17] Mordukhovich, B.S., Rockafellar, R.T., Sarabi, M.E.: Characterizations of full stability in constrained optimization. SIAM J. Optim. 23, 1810-1849 (2013) · Zbl 1284.49032
[18] Mordukhovich, B.S., Sarabi, M.E.: Variational analysis and full stability of optimal solutions to constrained and minimax problem. Nonlinear Anal. doi:10.1016/jna.2014.10.013 · Zbl 1326.90083
[19] Outrata, JV; Ramírez, HC, On the Aubin property of perturbed second-order cone programs, SIAM J. Optim., 21, 798-823, (2011) · Zbl 1247.90256
[20] Poliquin, RA; Rockafellar, RT, Tilt stability of a local minimum, SIAM J. Optim., 8, 287-299, (1998) · Zbl 0918.49016
[21] Poliquin, RA; Rockafellar, RT; Thibault, L, Local differentiability of distance functions, Trans. Amer. Math. Soc., 352, 5231-5249, (2000) · Zbl 0960.49018
[22] Robinson, SM, Generalized equations and theire solutions, part I: basic theory, Math. Program. Study, 10, 128-141, (1979) · Zbl 0404.90093
[23] Robinson, SM, Strongly regular generalized equations, Math. Oper. Res., 5, 43-62, (1980) · Zbl 0437.90094
[24] Robinson, SM, An implicit-function theorem for a class of nonsmooth functions, Math. Oper. Res., 16, 292-309, (1991) · Zbl 0746.46039
[25] Rockafellar, R.T., Wets, R. J-B.: Variational Analysis. Springer, Berlin (1998) · Zbl 0888.49001
[26] Shapiro, A. Differentiability properties of metric projections onto convex sets, Optimization-online. http://www.optimization-online.org/DB_HTML/2013/11/4119.html · Zbl 1342.90192
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