Many combinatorial optimization problems have relaxations that are semidefinite programming problems. In principle, the combinatorial optimization problem can then be solved by using a branch-and-cut procedure, where the problems to be solved at the nodes of the tree are semidefinite programs. It is desirable that the solution to one node of the tree should be exploited at the child node in ord...

This paper discusses nonlinear optimization techniques in robust control synthesis, with special emphasis on design problems which may be cast as minimizing a linear objective function under linear matrix inequality (LMI) constraints in tandem with nonlinear matrix equality constraints. The latter type of constraints renders the design numerically and algorithmically difficult. We solve the opt...

This report reviews some approximation algorithms for combinatorial optimization problems, based on a semidefinite relaxation followed by randomized rounding.

In recent years semidefinite optimization has become a tool of major importance in various optimization and machine learning problems. In many of these problems the amount of data in practice is so large that there is a constant need for faster algorithms. In this work we present the first sublinear time approximation algorithm for semidefinite programs which we believe may be useful for such p...

We present two algorithms for large-scale noisy low-rank Euclidean distance matrix completion problems, based on semidefinite optimization. Our first method works by relating cliques in the graph of the known distances to faces of the positive semidefinite cone, yielding a combinatorial procedure that is provably robust and partly parallelizable. Our second algorithm is a first order method for...

In this paper, we present a novel semidefinite programming approach for multiple-instance learning. We first formulate the multipleinstance learning as a combinatorial maximummargin optimization problem with additional instance selection constraints within the framework of support vector machines. Although solving this primal problem requires non-convex programming, we nevertheless can then der...

In this paper, we propose a coordinate descent approach to low-rank structured semidefinite programming. The approach, which we call the Mixing method, is extremely simple to implement, has no free parameters, and typically attains an order of magnitude or better improvement in optimization performance over the current state of the art. We show that for certain problems, the method is strictly ...

We focus on a family of quantum coin-flipping protocols based on quantum bit-commitment. We discuss how the semidefinite programming formulations of cheating strategies can be reduced to optimizing a linear combination of fidelity functions over a polytope. These turn out to be much simpler semidefinite programs which can be modelled using second-order cone programming problems. We then use the...

During the last two decades, major developments in convex optimization were focusing on conic programming, primarily, on linear, conic quadratic and semidefinite optimization. Conic programming allows to reveal rich structure which usually is possessed by a convex program and to exploit this structure in order to process the program efficiently. In the paper, we overview the major components of...

Our goal will be to optimize over all behaviors for Bob, maximizing the probability with which he causes Alice to output the measurement outcome a. (Later we will consider a variant of this scenario where Bob also produces a measurement outcome, but for now our focus will be on the case where only Alice produces a measurement outcome.) In the remainder of this lecture we will discuss how such a...