Optimality Conditions for Problems over Symmetric Cones and a Simple Augmented Lagrangian Method

Abstract

In this work, we are interested in nonlinear symmetric cone problems (NSCPs), which contain as special cases nonlinear semidefinite programming, nonlinear second-order cone programming, and the classical nonlinear programming problems. We explore the possibility of reformulating NSCPs as common nonlinear programs (NLPs), with the aid of squared slack variables. Through this connection, we show how to obtain second-order optimality conditions for NSCPs in an easy manner, thus bypassing a number of difficulties associated to the usual variational analytical approach. We then discuss several aspects of this connection. In particular, we show a “sharp” criterion for membership in a symmetric cone that also encodes rank information. Also, we discuss the possibility of importing convergence results from nonlinear programming to NSCPs, which we illustrate by discussing a simple augmented Lagrangian method for nonlinear symmetric cones. We show that, employing the slack variable approach, we can use the results for NLPs to prove convergence results, thus extending a special case (i.e., the case with strict complementarity) of an earlier result by Sun et al. [Sun D, Sun J, Zhang L (2008) The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming. Math. Programming 114(2):349–391] for nonlinear semidefinite programs.