1 Abbreviations and Notations 3

Semidefinite programming (SDP) bounds for the quadratic assignment problem (QAP) were introduced in: [Q. Zhao, S.E. Karisch, F. Rendl, and H. Wolkowicz. Semidefinite Programming Relaxations for the Quadratic Assignment Problem. Journal of Combinatorial Optimization, 2, 71–109, 1998.] Empirically, these bounds are often quite good in practice, but computationally demanding, even for relatively s...

In this paper, we consider semidefinite programs where the data is only known to belong to some uncertainty set U . Following recent work by the authors, we develop the notion of robust solution to such problems, which are required to satisfy the (uncertain) constraints whatever the value of the data in U . Even when the decision variable is fixed, checking robust feasibility is in general NP-h...

We consider a new semidefinite programming (SDP) relaxation of the symmetric traveling salesman problem (TSP), that may be obtained via an SDP relaxation of the more general quadratic assignment problem (QAP). We show that the new relaxation dominates the one in the paper: [D. Cvetković, M. Cangalović and V. Kovačević-Vujčić. Semidefinite Programming Methods for the Symmetric Traveling Salesman...

We consider optimization problems with polynomial inequality constraints in non-commuting variables. These non-commuting variables are viewed as bounded operators on a Hilbert space whose dimension is not fixed and the associated polynomial inequalities as semidefinite positivity constraints. Such problems arise naturally in quantum theory and quantum information science. To solve them, we intr...

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 ...

This paper proposes a new semidefinite programming relaxation for the satisfiability problem. This relaxation is an extension of previous relaxations arising from the paradigm of partial semidefinite liftings for 0/1 optimization problems. The construction of the relaxation depends on a choice of permutations of the clauses, and different choices may lead to different relaxations. We then consi...

2. Max-Cut Revisited As in last week’s lecture, we approximate solutions to Max-Cut using Goemans’s and Williamson’s αGW = 0.878-approximation. Specifically, we seek max ∑ (i,j)∈E 1 4 ∥∥vi − vj∥∥2 subject to the constraint that ∀i, ‖vi‖ = 1. We can visualize this by drawing the vectors restricted to a unit circle, as seen in the figure to the left. There is an appealing geometric intuition here...