## Solving low-rank semidefinite programs via manifold optimization

We propose a manifold optimization approach to solve linear semidefinite programs (SDP) with low-rank solutions. This approach incorporates the augmented Lagrangian method and the Burer-Monteiro factorization, and features the adaptive strategies for updating the factorization size and the penalty parameter. We prove that the present algorithm can solve SDPs to global optimality, despite of the … Read more

## Stable Set Polytopes with High Lift-and-Project Ranks for the Lovász-Schrijver SDP Operator

 We study the lift-and-project rank of the stable set polytopes of graphs with respect to the Lovász-Schrijver SDP operator $$\text{LS}_+$$, with a particular focus on a search for relatively small graphs with high $$\text{LS}_+$$-rank (the least number of iterations of the $$\text{LS}_+$$ operator on the fractional stable set polytope to compute … Read more

## Equivalent Sufficient Conditions for Global Optimality of Quadratically Constrained Quadratic Program

 We study the equivalence of several well-known sufficient optimality conditions for a general quadratically constrained quadratic program (QCQP). The conditions are classified in two categories. The first one is for determining an optimal solution and the second one is for finding an optimal value. The first category of conditions includes the existence of a … Read more

## Semidefinite approximations for bicliques and biindependent pairs

 We investigate some graph parameters asking to maximize the size of biindependent pairs (A,B) in a bipartite graph G = (V1 \cup V2;E), where A\subseteq V1, B \subseteq V2 and A \cup B is independent. These parameters also allow to study bicliques in general graphs (via bipartite double graphs). When the size is the … Read more

## On solving the MAX-SAT using sum of squares

We consider semidefinite programming (SDP) approaches for solving the maximum satisfiabilityproblem (MAX-SAT) and the weighted partial MAX-SAT. It is widely known that SDP is well-suitedto approximate the (MAX-)2-SAT. Our work shows the potential of SDP also for other satisfiabilityproblems, by being competitive with some of the best solvers in the yearly MAX-SAT competition.Our solver combines … Read more

## Polynomial argmin for recovery and approximation of multivariate discontinuous functions

We propose to approximate a (possibly discontinuous) multivariate function f(x) on a compact set by the partial minimizer arg min_y p(x,y) of an appropriate polynomial p whose construction can be cast in a univariate sum of squares (SOS) framework, resulting in a highly structured convex semidefinite program. In a number of non-trivial cases (e.g. when … Read more

## Singularity degree of non-facially exposed faces

In this paper, we study the facial structure of the linear image of a cone. We define the singularity degree of a face of a cone to be the minimum number of steps it takes to expose it using exposing vectors from the dual cone. We show that the singularity degree of the linear image … Read more

## Handling Symmetries in Mixed-Integer Semidefinite Programs

Symmetry handling is a key technique for reducing the running time of branch-and-bound methods for solving mixed-integer linear programs. In this paper, we generalize the notion of (permutation) symmetries to mixed-integer semidefinite programs (MISDPs). We first discuss how symmetries of MISDPs can be automatically detected by finding automorphisms of a suitably colored auxiliary graph. Then … Read more

## A Note on Semidefinite Representable Reformulations for Two Variants of the Trust-Region Subproblem

Motivated by encouraging numerical results in the literature, in this note we consider two specific variants of the trust-region subproblem and provide exact semidefinite representable reformulations. The first is over the intersection of two balls; the second is over the intersection of a ball and a special second-order conic representable set. Different from the technique … Read more

## Superlinear convergence of an infeasible interior point algorithm on the homogeneous feasibility model of a semi-definite program

In the literature, superlinear convergence of implementable polynomial-time interior point algorithms to solve semi-definite programs (SDPs) can only be shown by (i) assuming that the given SDP is nondegenerate and modifying these algorithms, or (ii) considering special classes of SDPs, such as the class of linear semi-definite feasibility problems, when a suitable initial iterate is … Read more