Janez Povh

Copositive and semidefinite relaxations of the quadratic assignment problem

Semidefinite relaxations of the quadratic assignment problem (QAP) have recently turned out to provide good approximations to the optimal value of QAP. We take a systematic look at various conic relaxations of QAP. We first show that QAP can equivalently be formulated as a linear program over the cone of completely positive matrices. Since it is hard to optimize over this cone, we also look at tractable approximations and compare with several relaxations from the literature. We show that several of the well-studied models are in fact equivalent. It is still a challenging task to solve the strongest of these models to reasonable accuracy on instances of moderate size.

Regularization Methods for Semidefinite Programming

We introduce a new class of algorithms for solving linear semidefinite programming (SDP) problems. Our approach is based on classical tools from convex optimization such as quadratic regularization and augmented Lagrangian techniques. We study the theoretical properties and we show that practical implementations behave very well on some instances of SDP having a large number of constraints. We also show that the “boundary point method” from Povh, Rendl, and Wiegele [Computing, 78 (2006), pp. 277-286] is an instance of this class.

A Boundary Point Method to Solve Semidefinite Programs

We investigate the augmented Lagrangian penalty function approach to solve semidefinite programs. It turns out that this method generates iterates which lie on the boundary of the cone of semidefinite matrices which are driven to the affine subspace described by the linear equations defining the semidefinite program. We provide some computational experience with this method and show in particular, that it allows to compute the theta number of a graph to reasonably high accuracy for instances which are beyond reach by other methods.

A Copositive Programming Approach to Graph Partitioning

We consider 3-partitioning the vertices of a graph into sets

NCSOStools: a computer algebra system for symbolic and numerical computation with noncommutative polynomials

S_1

The tracial moment problem and trace-optimization of polynomials

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A note on the nonexistence of sum of squares certificates for the Bessis–Moussa–Villani conjecture

S_2

Constrained trace-optimization of polynomials in freely noncommuting variables

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Constrained Polynomial Optimization Problems with Noncommuting Variables

S_3

Optimization of Polynomials in Non-Commuting Variables

of specified cardinalities, such that the total weight of all edges joining