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Properties of equation reformulation of the Karush–Kuhn–Tucker condition for nonlinear second order cone optimization problems

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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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Authors and Affiliations

  1. Department of Applied Mathematics, Dalian University of Technology, 116024, Dalian, China

    Yun Wang & Liwei Zhang

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  1. Yun Wang

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  2. Liwei Zhang

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Correspondence toYun Wang.

Additional information

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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