3 Why Use SDP? 5 3.1 Tractable Relaxations of Max-Cut . . . . . . . . . . . . . . . . . . . . . . . . 5 3.1.1 Simple Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.1.2 Trust Region Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.1.3 Box Constraint Relaxation . . . . . . . . . . . . . . . . . . . . . . . . 6 3.1.4 Eigenvalue Bound . . . . . . . . . . . . ...

Copositive programming is a relatively young field in mathematical optimization. It can be seen as a generalization of semidefinite programming, since it means optimizing over the cone of so called copositive matrices. Like semidefinite programming, it has proved particularly useful in combinatorial and quadratic optimization. The purpose of this survey is to introduce the field to interested r...

Recently, semidefinite programming has been used to bound the price of a single-asset European call option at a fixed time. Given the first n moments, a tight bound can be obtained by solving a single semidefinite programming problem of dimension n + 1. In this paper, we study the multi-asset case, which is generally more practical than the single-asset case. We construct a sequence of semidefi...

Constraint programming uses enumeration and search tree pruning to solve combinatorial optimization problems. In order to speed up this solution process, we investigate the use of semidefinite relaxations within constraint programming. In principle, we use the solution of a semidefinite relaxation to guide the traversal of the search tree, using a limited discrepancy search strategy. Furthermor...

In the last few lectures, we considered constant-factor approximation algorithms that relied on linear programming and greedy algorithms. In this lecture, we will analyze algorithms that use a more powerful mathematical programming technique called semidefinite programming. We will illustrate the power of semidefinite programming by looking at the maximum cut problem. However, SDPs are useful f...

In this paper, we propose a new sequential quadratic semidefinite programming (SQSDP) method for solving degenerate nonlinear programs (NSDPs), in which produce iteration points by sequence of stabilized (QSDP) subproblems, derive from the minimax problem associated with NSDP. Unlike existing SQSDP methods, proposed one allows us to solve those QSDP subproblems inexactly, and each is feasible. ...