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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 thescope of CVX beyond convex minimization -- to convex-concave saddle pointproblems and variational inequalities with monotone operators. To attain thisgoal, given a family K of cones (e.g., Lorentz, semidefinite{, geometric,etc.), we introduce the notions of K-conic representation of a convex-concavesaddle point problem and of variational inequality with monotone operator. Wedemonstrate that given such a representation of the problem of interest, thelatter can be reduced straightforwardly to a conic problem on a cone from K andthus can be solved by (any) solver capable to handle conic problems on conesfrom K (e.g., Mosek or SDPT3 in the case of semidefinite cones). We also showthat K-representations of convex-concave functions and monotone vector fieldsadmit a fully algorithmic calculus which helps to recognize the cases when asaddle point problem or variational inequality can be converted into a conicproblem on a cone from K and to carry out such conversion.

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