Reformulations for nonlinear programming, second-order cone programming and semidefinite programming

Abstract

This work consists in developing methods to solve three well-known problems in Optimization: nonlinear programming, second-order cone programming and semidefinite programming. For each one of these problems, we consider its necessary optimality conditions and reformulate it as a system of semismooth equations. The reformulation is based on a Fischer-Burmeister-type function or a differentiable exact penalty function. Then, the generalized Newton method can be applied to this system of equations, and a merit function must be constructed in order to globalize the method. From the theoretical point of view, the method should converge globally with superlinear (or quadratic) convergence rate, and without requiring strong assumptions like the strict complementarity. In practice, under numerical experiments, the method should also be sufficiently robust and efficient. (AU)