We analyze two popular semidefinite programming relaxations for quadratically constrained quadratic programs with matrix variables. These relaxations are based on vector lifting and on matrix lifting; they are of different size and expense. We prove, under mild assumptions, that these two relaxations provide equivalent bounds. Thus, our results provide a theoretical guideline for how to choose ...

Burer has shown that completely positive relaxations of nonconvex quadratic programs with nonnegative and binary variables are exact when the binary variables satisfy a so-called key assumption. Here we show that introducing binary variables to obtain an equivalent problem that satisfies the key assumption will not improve the semidefinite relaxation, and only marginally improve the doubly nonn...

We compare algorithms for global optimization of polynomial functions in many variables. It is demonstrated that existing algebraic methods (Gröbner bases, resultants, homotopy methods) are dramatically outperformed by a relaxation technique, due to N.Z. Shor and the first author, which involves sums of squares and semidefinite programming. This opens up the possibility of using semidefinite pr...

In this paper we summarize recent results on finding tight semidefinite programming relaxations for the Max-Cut problem and hence tight upper bounds on its optimal value. Our results hold for every instance of Max-Cut and in particular we make no assumptions on the edge weights. We present two strengthenings of the well-known semidefinite programming relaxation of Max-Cut studied by Goemans and...

We produce approximation bounds on a semidefinite programming relaxation for sparse principal component analysis. These bounds control approximation ratios for tractable statistics in hypothesis testing problems where data points are sampled from Gaussian models with a single sparse leading component. We study approximation bounds for a semidefinite relaxation of the sparse eigenvalue problem, ...

This paper studies the relationship between the optimal value of a homogeneous quadratic optimization problem and that of its Semidefinite Programming (SDP) relaxation. We consider two quadratic optimization models: (1) min{x∗Cx | x∗Akx ≥ 1, k = 0, 1, ...,m, x ∈ Fn} and (2) max{x∗Cx | x∗Akx ≤ 1, k = 0, 1, ..., m, x ∈ Fn}, where F is either the real field R or the complex field C, and Ak, C are ...

We prove a quantitative bi-Lipschitz nonembedding theorem for the Heisenberg group with its Carnot-Carathéodory metric and apply it to give a lower bound on the integrality gap of the Goemans-Linial semidefinite relaxation of the Sparsest Cut problem.

In the Goemans-Williamson semidefinite relaxation of MAX-CUT, the gradient of the dual barrier objective function has a term of the form diag(Z), where Z is the slack matrix. The purpose of this note is to show that this term can be computed in time and space proportional to the time and space for computing a sparse Cholesky factor of Z using an algorithm due to Erisman and Tinney. The algorith...

Semidefinite programming relaxations of combinatorial problems date back to the work of Lovász [17] from 1979, who proposed a semidefinite programming relaxation for the maximum stable set problem which now is known as the Lovász theta number. More recently, Goemans and Williamson [9] showed how to use semidefinite programming to provide an approximation algorithm for the maximum-cut problem; t...