A survey on second-order conditions for nonlinear symmetric cone programming via squared slack variables (New Trends of Numerical Optimization in Advanced Information-Oriented Society)

Abstract

The nonlinear symmetric cone programming (NSCP) problems contain as special cases the nonlinear semidefinite programming, the nonlinear second-order cone programming and the nonlinear programming problems. In this survey, we first explain the basics about Euclidean Jordan algebras and symmetric cones. Then, we observe that NSCP problems can be reformulated as nonlinear programming problems with the use of squared slack variables. Using such reformulations, we show how to obtain second-order optimality conditions for NSCP problems. In particular, under the strict complementarity condition, we lead to a description of the so-called sigma-term.