Perturbation analysis of second-order cone programming problems

Abstract

We discuss first and second order optimality conditions for nonlinear second-order cone programming problems, and their relation with semidefinite programming problems. For doing this we extend in an abstract setting the notion of optimal partition. Then we state a characterization of strong regularity in terms of second order optimality conditions.

Differentiable Exact Penalty Functions for Nonlinear Second-Order Cone Programs

“…We point out that the above result can be established if we replace the strong secondorder sucient condition by the strict complementarity, together with the second-order sucient condition [7]. In such a case, from (5.2), (5.5) and Corollary 5.5, ∂ B ∇w c (x * )…”

“…Thus, (3.2) is actually a linear least squares problem. Let us present now some properties associated to this estimate, under the following assumption [7].…”

An SQP-type algorithm for nonlinear second-order cone programs

“…Next we consider the relation between the regularity of a solution and the second-order conditions for NSOCP (1). We recall the notion of nondegeneracy in second-order cone programming [4].…”

“…It is showed in [4] that when a local optimal solution x * of NSOCP (1) is nondegenerate, (x * , ζ * , η * ) is a regular solution of the generalized equation representing the KKT conditions (2) of NSOCP (1) if and only if (x * , ζ * , η * ) satisfies the following second-order condition:…”

“…However, the study of nonlinear second-order cone programming (NSOCP) and nonlinear semidefinite programming (NSDP), which are natural extensions of LSOCP and LSDP, respectively, are much more recent and still in its preliminary phase. Optimality conditions for NSOCP and NSDP are studied in [4,5]. An interior point method has been proposed for NSOCP in [17].…”