Finding a maximal element of a convex set through its characteristic cone: An application to finding a strictly complementary solution

Naive constant rank-type constraint qualifications for multifold second-order cone programming and semidefinite programming

The constant rank constraint qualification, introduced by Janin in 1984 for nonlinear programming, has been extensively used for sensitivity analysis, global convergence of first- and second-order algorithms, and for computing the derivative of the value function. In this paper we discuss naive extensions of constant rank-type constraint qualifications to second-order cone programming and semidefinite programming, which are based on the Approximate-Karush-Kuhn-Tucker necessary optimality condition and on the application of the reduction approach. Our definitions are strictly weaker than Robins…

Interior-point algorithms for convex optimization based on primal-dual metrics

We propose and analyse primal-dual interior-point algorithms for convex optimization problems in conic form. The families of algorithms we analyse are so-called short-step algorithms and they match the current best iteration complexity bounds for primal-dual symmetric interior-point algorithm of Nesterov and Todd, for symmetric cone programming problems with given self-scaled barriers. Our results apply to any self-concordant barrier for any convex cone. We also prove that certain specializations of our algorithms to hyperbolic cone programming problems (which lie strictly between symmetric co…

Geometric characterizations of the strict Hadamard differentiability of sets

Let $S$ be a closed subset of a Banach space $X$. Assuming that $S$ is epi-Lipschitzian at $\bar{x}$ in the boundary $ \bd S$ of $S$, we show that $S$ is strictly Hadamard differentiable at $\bar{x}$ IFF the Clarke tangent cone $T(S, \bar{x})$ to $S$ at $\bar{x}$ contains a closed hyperplane IFF the Clarke tangent cone $T(\bd S, \bar{x})$ to $\bd S$ at $\bar{x}$ is a closed hyperplane. Moreover when $X$ is of finite dimension, $Y$ is a Banach space and $g: X \mapsto Y$ is a locally Lipschitz mapping around $\bar{x}$, we show that $g$ is strictly Hadamard differentiable at $\bar{x}$ IFF $T(\mat…

随机共正矩阵以正概率完全正

A random copositive matrix is completely positive with positive probability

本文证明随维度增大，随机共正矩阵属于完全正矩阵锥的概率具有正下界，并推广到夹在两锥间的各类矩阵锥。

非线性二阶锥规划问题的二阶序列最优性条件

A second-order sequential optimality condition for nonlinear second-order cone programming problems

本文为非线性二阶锥规划问题显式定义了AKKT2和CAKKT2二阶序列最优性条件，证明其在无约束规范下于局部最优点成立，并给出两种全局收敛算法。

$K$-洛伦兹多项式、半正锥与锥稳定EVI系统

$K-$Lorentzian Polynomials, Semipositive Cones, and Cone-Stable EVI Systems

本文提出$K$-洛伦兹多项式和半正锥理论，用于分析锥约束动态系统的稳定性与负依赖性，提供新的李雅普诺夫稳定性准则。

平方和基追寻与线性及二次锥规划

Sum of Squares Basis Pursuit with Linear and Second Order Cone Programming

本文提出通过迭代求解线性或二次锥规划来近似半正定或平方和规划的最优值，利用DSOS和SDSOS多项式改进界，并应用于独立集和划分问题。