锥系统的病态性结构化距离

Sparse Conic Reformulation of Structured QCQPs based on Copositive Optimization with Applications in Stochastic Optimization

In an effort to develop an alternative approach to traditional sparse reformulations, we will provide a new type of convex reformulation of a large class of stochastic quadratically constrained quadratic optimization problems that is similar to Burer's reformulation, but lifts the variables into a comparatively lower dimensional space. The reformulation rests on a generalization of the set-completely positive matrix cone. This cone can then be approximated via inner and outer approximations in order to obtain upper and lower bounds, which potentially close the optimality gap, and hence can giv…

关于具有单调算子的凸-凹鞍点问题和变分不等式的良好结构

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

本文提出K-conic表示法，将凸-凹鞍点问题和变分不等式转化为锥问题，从而扩展了CVX的应用范围。

一种求解锥束子问题的预处理迭代内点方法

A Preconditioned Iterative Interior Point Approach to the Conic Bundle Subproblem

针对大规模半定规划锥束子问题求解中Schur补分解的瓶颈，提出两种利用低秩特性的预处理迭代内点法，推导了条件数界，实验显示确定性预处理器效率优异，适用于精度适中或系统规模较大的场景。

Linear conic optimization for nonlinear optimal control

Infinite-dimensional linear conic formulations are described for nonlinear optimal control problems. The primal linear problem consists of finding occupation measures supported on optimal relaxed controlled trajectories, whereas the dual linear problem consists of finding the largest lower bound on the value function of the optimal control problem. Various approximation results relating the original optimal control problem and its linear conic formulations are developed. As illustrated by a couple of simple examples, these results are relevant in the context of finite-dimensional semidefinite …

Linear conic optimization for inverse optimal control

We address the inverse problem of Lagrangian identification based on trajecto-ries in the context of nonlinear optimal control. We propose a general formulation of the inverse problem based on occupation measures and complementarity in linear programming. The use of occupation measures in this context offers several advan-tages from the theoretical, numerical and statistical points of view. We propose an approximation procedure for which strong theoretical guarantees are available. Finally, the relevance of the method is illustrated on academic examples.

The LP-Newton Method and Conic Optimization

We propose that the LP-Newton method can be used to solve conic LPs over a conic box, whenever linear optimization over an otherwise unconstrained conic box is easy. In particular, if $\leq_\mathcal{K}$ is the partial order induced by a proper convex cone $\mathcal{K}$, then optimizing a linear function over the intersection of $[{l},{u}]_\mathcal{K}=\{{l}\leq_\mathcal{K} {x}\leq_\mathcal{K}{u}\}$ and an affine subspace can be done with this method whenever optimizing a linear function over $[{l},{u}]_{\mathcal{K}}$ is efficient. This generalizes the result for the case of $\mathcal{K}=\mathbb…