ABSTRACT

For those acquainted with CVX (aka disciplined convex programming) of Grant and Boyd (Matlab software for disciplined convex programming, version 2.2, CVX Research, Inc., 2020. http://cvxr.com/cvx/doc/), 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 $\mathcal{K}$ of cones (e.g. Lorentz, semidefinite, geometric, etc.), we introduce the notions of $\mathcal{K}$-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 $\mathcal{K}$ and thus can be solved by (any) solver capable to handle conic problems on cones from $\mathcal{K}$ (e.g. Mosek or SDPT3 in the case of semidefinite cones). We also show that $\mathcal{K}$-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 $\mathcal{K}$ and to carry out such conversion.

摘要

对于那些熟悉Grant 和 Boyd 的CVX（又名纪律性凸编程）（用于纪律性凸编程的 Matlab 软件，版本 2.2，CVX Research, Inc.，2020 年。http://cvxr.com/cvx/doc/）的人，动机这项工作的目的是希望将CVX的范围扩展到凸最小化之外 - 凸凹鞍点问题和单调算子的变分不等式。为了实现这个目标，给定一个家庭$\mathcal{钾}$ 锥体（例如洛伦兹、半定、几何等），我们引入了 $\mathcal{钾}$-凸凹鞍点问题和单调算子变分不等式的圆锥表示。我们证明，给定感兴趣的问题的这种表示，后者可以直接简化为圆锥上的圆锥问题，从$\mathcal{钾}$ 因此可以通过（任何）能够处理圆锥上的圆锥问题的求解器来解决 $\mathcal{钾}$（例如，在半定锥体的情况下，Mosek或SDPT3）。我们还表明$\mathcal{钾}$-凸凹函数和单调向量场的表示承认完全算法演算，这有助于识别鞍点问题或变分不等式可以从以下角度转换为圆锥上的圆锥问题的情况 $\mathcal{钾}$ 并进行这种转换。