The purpose of this semester project is to investigate the Spectral Bundle Method, which is a specialized subgradient method particularly suited for solving large scale semidefinite programs that can be cast as eigenvalue optimization problems of the form min y∈R aλmax(C − m ∑ i=1 Aiyi) + b T y, where C and Ai are given real symmetric matrices, b ∈ R allows to specify a linear cost term, and a ...

We show that the Goemans-Linial semidefinite relaxation of the Sparsest Cut problem with general demands has integrality gap (log n)Ω(1). This is achieved by exhibiting n-point metric spaces of negative type whose L1 distortion is (log n)Ω(1). Our result is based on quantitative bounds on the rate of degeneration of Lipschitz maps from the Heisenberg group to L1 when restricted to cosets of the...

We present semidefinite relaxations of nonconvex, box-constrained quadratic programming, which incorporate the firstand second-order necessary optimality conditions, and establish theoretical relationships between the new relaxations and a basic semidefinite relaxation due to Shor. We compare these relaxations in the context of branch-and-bound to determine a global optimal solution, where it i...

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...

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...