A general formula for the algebraic degree in semidefinite programming

The Algebraic Degree of Semidefinite Programming

Given a generic semidefinite program, specified by matrices with rational entries, each coordinate of its optimal solution is an algebraic number. We study the degree of the minimal polynomials of these algebraic numbers. Geometrically, this degree counts the critical points attained by a linear functional on a fixed rank locus in a linear space of symmetric matrices. We determine this degree using methods from complex algebraic geometry, such as projective duality, determinantal varieties, and their Chern classes.

Gramian Tensor Decomposition via Semidefinite Programming

In this paper we examine a symmetric tensor decomposition problem, the Gramian decomposition, posed as a rank minimization problem. We study the relaxation of the problem and consider cases when the relaxed solution is a solution to the original problem. In some instances of tensor rank and order, we prove generically that the solution to the relaxation will be optimal in the original. In other cases, we present interesting examples and approaches that demonstrate the intricacy of this problem.

A formula for the algebraic degree in semidefinite programming

In this paper we use the Bott residue formula in equivariant cohomology to show a formula for the algebraic degree in semidefinite programming.

半定规划中代数次数的一个刻画

A characterization of the algebraic degree in semidefinite programming

本文通过双对称多项式单项式系数刻画半定规划的代数次数，并利用对称多项式理论简化了相关重要结果的证明。

Sums of Squares and Sparse Semidefinite Programming

We consider two seemingly unrelated questions: the relationship between nonnegative polynomials and sums of squares on real varieties, and sparse semidefinite programming. This connection is natural when a real variety $X$ is defined by a quadratic square-free monomial ideal. In this case nonnegative polynomials and sums of squares on $X$ are also natural objects in positive semidefinite matrix completion. Nonnegative quadratic forms over $X$ naturally correspond to partially specified matrices where all of the fully specified square blocks are PSD, and sums of squares quadratic forms naturall…

多项式不等式判定中的半定规划方法的致命退化

Fatal Degeneracy in the Semidefinite Programming Approach to the Decision of Polynomial Inequalities

本文揭示了将多项式不等式不可满足性判定问题转化为半定规划求解时，通常会导致数值方法难以处理的退化问题。