Optimality conditions for problems over symmetric cones and a simple augmented Lagrangian method

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 … Read more

A note on the squared slack variables technique for nonlinear optimization

In constrained nonlinear optimization, the squared slack variables can be used to transform a problem with inequality constraints into a problem containing only equality constraints. This reformulation is usually not considered in the modern literature, mainly because of possible numerical instabilities. However, this argument only concerns the development of algorithms, and nothing stops us in … Read more

The use of squared slack variables in nonlinear second-order cone programming

In traditional nonlinear programming, the technique of converting a problem with inequality constraints into a problem containing only equality constraints, by the addition of squared slack variables, is well-known. Unfortunately, it is considered to be an avoided technique in the optimization community, since the advantages usually do not compensate for the disadvantages, like the increase … Read more

Optimality conditions for nonlinear semidefinite programming via squared slack variables

In this work, we derive second-order optimality conditions for nonlinear semidefinite programming (NSDP) problems, by reformulating it as an ordinary nonlinear programming problem using squared slack variables. We first consider the correspondence between Karush-Kuhn-Tucker points and regularity conditions for the general NSDP and its reformulation via slack variables. Then, we obtain a pair of “no-gap” … Read more

Differentiable exact penalty functions for nonlinear second-order cone programs

We propose a method to solve nonlinear second-order cone programs (SOCPs), based on a continuously differentiable exact penalty function. The construction of the penalty function is given by incorporating a multipliers estimate in the augmented Lagrangian for SOCPs. Under the nondegeneracy assumption and the strong second-order sufficient condition, we show that a generalized Newton method … Read more