New Perspective on Slack Variables Applications to Singular Optimization Problems

Abstract

This paper is devoted to a new approach for solving nonlinear programming (NLP) problems for which the Kuhn-Tucker optimality conditions system of equations is singular. It happens when the strict complementarity condition (SCC), a constrained qualification (CQ), and a second-order sufficient condition (SOSC) for optimality is not necessarily satisfied at a solution. Our approach is based on the construction of p-regularity and on reformulating the inequality constraints as equality. Namely, by introducing the slack variables, we get the equality constrained problem, for which the Lagrange optimality system is singular at the solution of the NLP problem in the case of the violation of the CQs, SCC and/or SOSC. To overcome the difficulty of singularity, we propose the p-factor method for solving the Lagrange system. The method has a superlinear rate of convergence under a mild assumption. We show that our assumption is always satisfied under a standard second-order sufficient optimality condition.