量子鸽巢原理与通信复杂性的两种半定松弛

Lower bounds on the size of semidefinite programming relaxations

We introduce a method for proving lower bounds on the efficacy of semidefinite programming (SDP) relaxations for combinatorial problems. In particular, we show that the cut, TSP, and stable set polytopes on $n$-vertex graphs are not the linear image of the feasible region of any SDP (i.e., any spectrahedron) of dimension less than $2^{n^c}$, for some constant $c > 0$. This result yields the first super-polynomial lower bounds on the semidefinite extension complexity of any explicit family of polytopes. Our results follow from a general technique for proving lower bounds on the positive semidef…

Efficient Rank Minimization to Tighten Semidefinite Programming for Unconstrained Binary Quadratic Optimization

We propose a method for low-rank semidefinite programming in application to the semidefinite relaxation of unconstrained binary quadratic problems. The method improves an existing solution of the semidefinite programming relaxation to achieve a lower rank solution. This procedure is computationally efficient as it does not require projecting on the cone of positive-semidefinite matrices. Its performance in terms of objective improvement and rank reduction is tested over multiple graphs of large-scale Gset graph collection and over binary optimization problems from the Biq Mac collection.

Semidefinite Relaxations of Products of Nonnegative Forms on the Sphere

We study the problem of maximizing the geometric mean of $d$ low-degree non-negative forms on the real or complex sphere in $n$ variables. We show that this highly non-convex problem is NP-hard even when the forms are quadratic and is equivalent to optimizing a homogeneous polynomial of degree $O(d)$ on the sphere. The standard Sum-of-Squares based convex relaxation for this polynomial optimization problem requires solving a semidefinite program (SDP) of size $n^{O(d)}$, with multiplicative approximation guarantees of $Ω(\frac{1}{n})$. We exploit the compact representation of this polynomial t…

A sub-constant improvement in approximating the positive semidefinite Grothendieck problem

Semidefinite relaxations are a powerful tool for approximately solving combinatorial optimization problems such as MAX-CUT and the Grothendieck problem. By exploiting a bounded rank property of extreme points in the semidefinite cone, we make a sub-constant improvement in the approximation ratio of one such problem. Precisely, we describe a polynomial-time algorithm for the positive semidefinite Grothendieck problem -- based on rounding from the standard relaxation -- that achieves a ratio of $2/π+ Θ(1/{\sqrt n})$, whereas the previous best is $2/π+ Θ(1/n)$. We further show a corresponding int…

半定规划与线性方程组对同态问题

Semidefinite programming and linear equations vs. homomorphism problems

本文提出结合半定规划与线性丢番图方程的同态问题松弛方法，并基于结合方案谱理论证明其无法解决近似图同态及近似图着色问题。

多项式Optimization的代数松弛和严格性结果及李雅普诺夫分析

Algebraic Relaxations and Hardness Results in Polynomial Optimization and Lyapunov Analysis

本论文解决了优化与控制中有关计算复杂性和基于代数、半定规划的松弛问题。