To address difficult optimization problems, convex relaxations based on semidefinite programming are now common place in many fields. Although solvable in polynomial time, large semidefinite programs tend to be computationally challenging. Over a decade ago, exploiting the fact that in many applications of interest the desired solutions are low rank, Burer and Monteiro proposed a heuristic to s...

A semidefinite program (SDP) is an optimization problem over n × n symmetric matrices where a linear function of the entries is to be minimized subject to linear equality constraints, and the condition that the unknown matrix is positive semidefinite. Standard techniques for solving SDP’s require O(n) operations per iteration. We introduce subspace algorithms that greatly reduce the cost os sol...

While semidefinite relaxations are known to deliver good approximations for combinatorial optimization problems like graph bisection, their practical scope is mostly associated with small dense instances. For large sparse instances, cutting plane techniques are considered the method of choice. These are also applicable for semidefinite relaxations via the spectral bundle method, which allows to...

In this article we present the moment based approach for computing all real solutions of a given system of polynomial equations. This approach builds upon a lifting method for constructing semidefinite relaxations of several nonconvex optimization problems, using sums of squares of polynomials and the dual theory of moments. A crucial ingredient is a semidefinite characterization of the real ra...

Considering that preprocessing is an important phase in linear programming, it should be systematically more incorporated in semidefinite programming solvers. The conversion method proposed by the authors (SIAM Journal on Optimization, vol. 11, pp. 647–674, 2000, and Mathematical Programming, Series B, vol. 95, pp. 303–327, 2003) is a preprocessing of sparse semidefinite programs based on matri...

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 semidefinit...

This manuscript develops a new framework to analyze and design iterative optimization algorithms built on the notion of Integral Quadratic Constraints (IQC) from robust control theory. IQCs provide sufficient conditions for the stability of complicated interconnected systems, and these conditions can be checked by semidefinite programming. We discuss how to adapt IQC theory to study optimizatio...

Interior point methods have proven very successful at solving linear programming problems. When an explicit linear programming formulation is either not available or is too large to employ directly, a column generation approach can be used. Examples of column generation approaches include cutting plane methods for integer programming and decomposition methods for many classes of optimization pr...

We discuss the effectiveness of linear and semidefinite relaxations in approximating the optimum for combinatorial optimization problems. Various hierarchies of these relaxations, such as the ones defined by Lovász and Schrijver [47], Sherali and Adams [55] and Lasserre [42] generate increasingly strong linear and semidefinite programming relaxations starting from a basic one. We survey some po...

In recent years semidefinite programming has become awidely used tool for designing more efficient algorithms for approximating hard combinatorial optimization problems and, more generally, polynomial optimization problems, which deal with optimizing a polynomial objective function over a basic closed semi-algebraic set. The underlying paradigm is that while testing nonnegativity of a polynomia...