In this paper, we study the problem of minimizing the ratio of two quadratic functions subject to a quadratic constraint. First we introduce a parametric equivalent of the problem. Then a bisection and a generalized Newton-based method algorithms are presented to solve it. In order to solve the quadratically constrained quadratic minimization problem within both algorithms, a semidefinite optim...

Using recent progress on moment problems, and their connections with semidefinite optimization, we present in this paper a new methodology based on semidefinite optimization, to obtain a hierarchy of upper and lower bounds on linear functionals defined on solutions of linear partial differential equations. We apply the proposed method to examples of PDEs in one and two dimensions, with very enc...

Multi-way partitioning of an undirected weighted graph where pairwise similarities are assigned as edge weights, provides an important tool for data clustering, but is an NP-hard problem. Spectral relaxation is a popular way of relaxation, leading to spectral clustering where the clustering is performed by the eigen-decomposition of the (normalized) graph Laplacian. On the other hand, semidefin...

Consider the N -person non-cooperative game in which each player’s cost function and the opponents’ strategies are uncertain. For such an incomplete information game, the new solution concept called a robust Nash equilibrium has attracted much attention over the past several years. The robust Nash equilibrium results from each player’s decision-making based on the robust optimization policy. In...

We study how to solve semidefinite programming (SDP) relaxations for large scale polynomial optimization. When interior-point methods are used, typically only small or moderately large problems could be solved. This paper studies regularization methods for solving polynomial optimization problems. We describe these methods for semidefinite optimization with block structures, and then apply them...

This survey paper is intended for the graduate students and researchers who are interested in Operations Research, have solid understanding of linear optimization but are not familiar with Semidefinite Programming (SDP). Here, I provide a very gentle introduction to SDP, some entry points for further look into the SDP literature, and brief introductions to some selected well-known applications ...

The Quadratic Convex Reformulation (QCR) method is used to solve quadratic unconstrained binary optimization problems. In this method, the semidefinite relaxation is used to reformulate it to a convex binary quadratic program which is solved using mixed integer quadratic programming solvers. We extend this method to random quadratic unconstrained binary optimization problems. We develop a Penal...

Spectrahedra are sets defined by linear matrix inequalities. Projections of spectrahedra are called semidefinite representable sets. Both kinds of sets are of practical use in polynomial optimization, since they occur as feasible sets in semidefinite programming. There are several recent results on the question which sets are semidefinite representable. So far, all results focus on the case of ...

In previous lectures, we discussed a few fairly direct connections between quantum information theoretic notions and semidefinite programs. For instance, the semidefinite program associated with an optimization over measurements is quite simple and direct, and if you are familiar with the Choi representation of channels the same can be said about our semidefinite program for optimizing over cha...

In this paper we extend the results obtained for a class of finite kernel functions by Y.Q. Bai M. El Ghami and C.Roos published in SIAM Journal of Optimization, 13(3):766–782, 2003 [3] for linear optimization to semidefinite optimization. We show that the iteration bound for primal dual methods is O( √ n log n log n ), for large-update methods andO( √ n log n ), for small-update methods. The i...