## Optimality conditions for nonlinear second-order cone programming and symmetric cone programming

Nonlinear symmetric cone programming (NSCP) generalizes important optimization problems such as nonlinear programming, nonlinear semidefinite programming and nonlinear second-order cone programming (NSOCP). In this work, we present two new optimality conditions for NSCP without constraint qualifications, which implies the Karush-Kuhn-Tucker conditions under a condition weaker than Robinson’s constraint qualification. In addition, we show the relationship … Read more

## A relaxed-certificate facial reduction algorithm based on subspace intersection

A “facial reduction”-like regularization algorithm is established for conic optimization problems by relaxing requirements on the reduction certificates. It requires only a linear number of reduction certificates from a constant time-solvable auxiliary problem, but is challenged by representational issues of the exposed reductions. A condition for representability is presented, analyzed for Cartesian product cones, and … Read more

## Exploiting Second-Order Cone Structure for Global Optimization

Identifying and exploiting classes of nonconvex constraints whose feasible region is convex after branching can reduce the time to compute global solutions for nonlinear optimization problems. We develop techniques for identifying quadratic and nonlinear constraints whose feasible region can be represented as the union of a finite number of second-order cones, and we provide necessary … Read more

## A primal-dual second order cone approximations algorithm for symmetric cone programming

This paper presents the new concept of second-order cone approximations for convex conic programming. Given any open convex cone $K$, a logarithmically homogeneous self-concordant barrier for $K$ and any positive real number $r \le 1$, we associate, with each direction $x \in K$, a second-order cone $\hat K_r(x)$ containing $K$. We show that $K$ is … Read more