On Well-Structured Convex-Concave Saddle Point Problems and Variational Inequalities with Monotone Operators

Abstract

For those acquainted with CVX (aka disciplined convex programming) of M. Grant and S. Boyd, the motivation of this work is the desire to extend the scope of CVX beyond convex minimization -- to convex-concave saddle point problems and variational inequalities with monotone operators. To attain this goal, given a family K of cones (e.g., Lorentz, semidefinite{, geometric, etc.), we introduce the notions of K-conic representation of a convex-concave saddle point problem and of variational inequality with monotone operator. We demonstrate that given such a representation of the problem of interest, the latter can be reduced straightforwardly to a conic problem on a cone from K and thus can be solved by (any) solver capable to handle conic problems on cones from K (e.g., Mosek or SDPT3 in the case of semidefinite cones). We also show that K-representations of convex-concave functions and monotone vector fields admit a fully algorithmic calculus which helps to recognize the cases when a saddle point problem or variational inequality can be converted into a conic problem on a cone from K and to carry out such conversion.