Interior point and outer approximation methods for conic optimization

Abstract

Any convex optimization problem may be represented as a conic problem that minimizes a linear function over the intersection of an affine subspace with a convex cone. An advantage of representing convex problems in conic form is that, under certain regularity conditions, a conic problem has a simple and easily checkable certificate of optimality, primal infeasibility, or dual infeasibility. As a natural generalization of linear programming duality, conic duality allows us to design powerful algorithms for continuous and mixed-integer convex optimization.

The main goal of this thesis is to improve the generality and practical performance of (i) interior point methods for continuous conic problems and (ii) outer approximation methods for mixed-integer conic problems. We implement our algorithms in extensible open source solvers accessible through the convenient modeling language JuMP. From around 50 applied examples, we formulate continuous and mixed-integer problems over two dozen different convex cone types, many of which are new. Our extensive computational experiments with these examples explore which algorithmic features and what types of equivalent conic formulations lead to the best performance.