Nonsingularity Conditions for FB System of Reformulating Nonlinear Second-Order Cone Programming

Shaohua Pan,Shujun Bi,Jein-Shan Chen

doi:10.1155/2013/602735

Abstract

This paper is a counterpart of Bi et al., 2011. For a locally optimal solution to the nonlinear second-order cone programming (SOCP), specifically, under Robinson’s constraint qualification, we establish the equivalence among the following three conditions: the nonsingularity of Clarke’s Jacobian of Fischer-Burmeister (FB) nonsmooth system for the Karush-Kuhn-Tucker conditions, the strong second-order sufficient condition and constraint nondegeneracy, and the strong regularity of the Karush-Kuhn-Tucker point.

Highlights

IntroductionWith the help of [7, Theorem 30] and [8, Lemma 11], Wang and Zhang [9] gave a characterization for the strong regularity of the KKT point of the second-order cone programming (SOCP) (1) via the nonsingularity study of Clarke’s Jacobian of the natural residual (NR) nonsmooth system
The nonlinear second-order cone programming (SOCP) problem can be stated as min f (ζ) ζ∈Rn s.t. h (ζ) = 0, (1)g (ζ) ∈ K, where f : Rn → R, h : Rn → Rm, and g : Rn → Rn are given twice continuously differentiable functions, and K is the Cartesian product of some second-order cones, that is,K := Kn1 × Kn2 × ⋅ ⋅ ⋅ × Knr, (2)with n1 + (SOC) in ⋅⋅⋅ Rnj+ nr = n and defined byKnj being the second-order coneKnj := {(xj1, xj2) ∈ R × Rnj−1 | xj1 ≥ 󵄩󵄩󵄩󵄩󵄩xj2󵄩󵄩󵄩󵄩󵄩} . (3)By introducing a slack variable to the second constraint, the SOCP (1) is equivalent to min f (ζ) ζ,x∈Rn s.t. h (ζ) = 0
With the help of [7, Theorem 30] and [8, Lemma 11], Wang and Zhang [9] gave a characterization for the strong regularity of the KKT point of the SOCP (1) via the nonsingularity study of Clarke’s Jacobian of the natural residual (NR) nonsmooth system

Summary

Introduction

With the help of [7, Theorem 30] and [8, Lemma 11], Wang and Zhang [9] gave a characterization for the strong regularity of the KKT point of the SOCP (1) via the nonsingularity study of Clarke’s Jacobian of the NR nonsmooth system They showed that the strong regularity of the KKT point, the nonsingularity of Clarke’s Jacobian of ENR at the KKT point, and the strong second-order sufficient condition and constraint nondegeneracy [7] are all equivalent. For a locally optimal solution to the nonlinear SOCP (4), under Robinson’s constraint qualification, we show that the strong second-order sufficient condition and constraint nondegeneracy introduced in [7], the nonsingularity of Clarke’s Jacobian of EFB at the KKT point, and the strong regularity of the KKT point are equivalent to each other. Jωf(ω) and J2ωωf(ω) denote the derivative and the second-order derivative, respectively, of a twice differentiable function f with respect to the variable ω

Preliminary Results

Directional Derivative and B-Subdifferential

Nonsingularity Conditions

Conclusions