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Abstract
We give an equation reformulation of the Karush–Kuhn–Tucker (KKT) condition for the second order cone optimization problem. The equation is strongly semismooth and its Clarke subdifferential at the KKT point is proved to be nonsingular under the constraint nondegeneracy condition and a strong second order sufficient optimality condition. This property is used in an implicit function theorem of semismooth functions to analyze the convergence properties of a local sequential quadratic programming type (for short, SQP-type) method by Kato and Fukushima (Optim Lett 1:129–144, 2007). Moreover, we prove that, a local solutionx* to the second order cone optimization problem is a strict minimizer of the Han penalty merit function when the constraint nondegeneracy condition and the strong second order optimality condition are satisfied atx*.
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References
Alizadeh F, Goldfarb D (2003) Second-order cone programming. Math Program 95: 3–51
Boggs PT, Tolle JW (1995) Sequential quadratic programming. Acta Numer 4: 1–51
Bonnans JF, Gilbert JC, Lemaréchal C, Sagastizábal CA (2003) Numerical optimization. Springer, New York
Bonnans JF, Ramírez CH (2005) Perturbation analysis of second-order-cone programming problems. Math Program 104: 205–227
Bonnans JF, Shapiro A (2000) Perturbation analysis of optimization problems. Springer, New York
Chan ZX, Sun D (2008) Constraint nondegeneracy, strong regularity and nonsingularity in semidefinite programming. SIAM J Optim 19: 370–396
Chen XD, Sun D, Sun J (2003) Complementarity functions and numerical experiments for second-order cone complementarity problems. Comput Optim Appl 25: 39–56
Clarke FH (1976) On the inverse function theorem. Pac J Math 64: 97–102
Clarke FH (1983) Optimization and nonsmooth analysis. Wiley, New York
Kato H, Fukushima M (2007) An SQP-type algorithm for nonlinear second-order cone programs. Optim Lett 1: 129–144
Kummer B (1991) Lipschitzian inverse functions, directional derivatives, and applications inC1,1-optimization. J Optim Theory Appl 70: 559–580
Liu Y, Zhang L (2007a) On the convergence of the augmented Lagrangian method for nonlinear optimization problems over second-order cones. J Optim Theory Appl (to appear)
Liu Y, Zhang L (2007b) Convergence analysis of the augmented Lagrangian method for nonlinear second-order cone optimization problems. Nonlinear Anal 67: 1359–1373
Meng F, Sun D, Zhao G (2005) Semismoothness of solutions to generalized equations and the Moreau-Yosida regularization. Math Program 104: 561–581
Mifflin R (1977) Semismooth and semiconvex functions in constrained optimization. SIAM J Control Optim 15: 957–972
Pang JS, Sun D, Sun J (2003) Semismooth homeomorphisms and strong stability of semidefinite and Lorentz cone complementarity problems. Math Oper Res 28: 39–63
Qi L, Sun J (1993) A nonsmooth version of Newton’s method. Math Program 58: 353–367
Robinson SM (1976) First order conditions for general nonlinear optimization. SIAM J Appl Math 30: 597–607
Robinson SM (1983) Generalized equations. In: Bachem A et al (eds) Mathematical Programming: The State of the Art. Springer, Berlin, pp 346–367
Rockafellar RT (1970) Convex analysis. Princeton University Press, Princeton
Sun D (2001) A further result on an implicit function theorem for locally Lipschitz functions. Oper Res Lett 28: 193–198
Sun D (2006) The strong second order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications. Math Oper Res 31: 761–776
Zarantonello EH (1971) Projections on convex sets in Hilbert space and spectral theory I and II. In: Zarantonello EH (eds.) Contributions to Nonlinear Functional Analysis. Academic Press, New York, pp 237–424
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Authors and Affiliations
Department of Applied Mathematics, Dalian University of Technology, 116024, Dalian, China
Yun Wang & Liwei Zhang
- Yun Wang
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Correspondence toYun Wang.
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The research is supported by the National Natural Science Foundation of China under project No. 10771026 and by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry of China.
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Wang, Y., Zhang, L. Properties of equation reformulation of the Karush–Kuhn–Tucker condition for nonlinear second order cone optimization problems.Math Meth Oper Res70, 195–218 (2009). https://doi.org/10.1007/s00186-008-0241-x
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