Optimization with Semidefinite, Quadratic and Linear Constraints

We consider optimization problems where variables have either linear, or convex quadratic or semidefinite constraints. First, we define and characterize primal and dual nondegeneracy and strict complementarity conditions. Next, we develop primal-dual interior point methods for such problems and show that in the absence of degeneracy these algorithms are numerically stable. Finally we describe an implementation of our method and present numerical experiments with both degenerate and nondegenerate problems

Associative Algebras, Symmetric Cones and Polynomial Time Interior Point Algorithms

We present a generalization of Monteiro's polynomiality proof of a large class of primal-dual interior point algorithms for semidefinite programs. We show that this proof, essentially verbatim, applies to all optimization problems over almost all symmetric cones, that is those cones that can be derived from classes of normed associative algebras and certain Jordan algebras obtained from them. As a consequence, we show that Monteiro's proof can be extended to convex quadratically constrained quadratic optimization problems. We also extend the notion of Zhang family of algorithms, and show that it can be applied to all symmetric cones, in particular the quadratic cone