Cuts and semidefinite liftings for the complex cut polytope

\(\) We consider the complex cut polytope: the convex hull of Hermitian rank 1 matrices \(xx^{\mathrm{H}}\), where the elements of \(x \in \mathbb{C}^n\) are \(m\)th unit roots. These polytopes have applications in \({\text{MAX-3-CUT}}\), digital communication technology, angular synchronization and more generally, complex quadratic programming. For \({m=2}\), the complex cut polytope corresponds to the well-known cut … Read more

Higher-Order Newton Methods with Polynomial Work per Iteration

\(\) We present generalizations of Newton’s method that incorporate derivatives of an arbitrary order \(d\) but maintain a polynomial dependence on dimension in their cost per iteration. At each step, our \(d^{\text{th}}\)-order method uses semidefinite programming to construct and minimize a sum of squares-convex approximation to the \(d^{\text{th}}\)-order Taylor expansion of the function we wish … Read more

A more efficient reformulation of complex SDP as real SDP

This note proposes a novel reformulation of complex semidefinite programs (SDPs) as real SDPs. As an application, we present an economical reformulation of complex SDP relaxations of complex polynomial optimization problems as real SDPs and derive some further reductions by exploiting structure of the complex SDP relaxations. Various numerical examples demonstrate that our new reformulation … Read more

A Moment-SOS Hierarchy for Robust Polynomial Matrix Inequality Optimization with SOS-Convexity

We study a class of polynomial optimization problems with a robust polynomial matrix inequality constraint for which the uncertainty set is defined also by a polynomial matrix inequality (including robust polynomial semidefinite programs as a special case). Under certain SOS-convexity assumptions, we construct a hierarchy of moment-SOS relaxations for this problem to obtain convergent upper … Read more

Optimized Dimensionality Reduction for Moment-based Distributionally Robust Optimization

Moment-based distributionally robust optimization (DRO) provides an optimization framework to integrate statistical information with traditional optimization approaches. Under this framework, one assumes that the underlying joint distribution of random parameters runs in a distributional ambiguity set constructed by moment information and makes decisions against the worst-case distribution within the set. Although most moment-based DRO problems … 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

Semidefinite approximations for bicliques and biindependent pairs

\(\) We investigate some graph parameters dealing with biindependent pairs $(A,B)$ in a bipartite graph $G=(V_1\cup V_2,E)$, i.e., pairs $(A,B)$ where $A\subseteq V_1$, $B\subseteq V_2$ and $A\cup B$ is independent. These parameters also allow to study bicliques in general graphs. When maximizing the cardinality $|A\cup B|$ one finds the stability number $\alpha(G)$, well-known to be … 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

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