Abstract We give asymptotically converging semidefinite programming hierarchies of outer bounds on bilinear programs the form $${\mathrm {Tr}}\big [H(D\otimes E)\big ]$$ Tr [ H ( D ? E )</m...

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...

(ABSTRACT) Despite recent advances in convex optimization techniques, the areas of discrete and continuous nonconvex optimization remain formidable, particularly when globally optimal solutions are desired. Most solution techniques, such as branch-and-bound, are enumerative in nature, and the rate of their convergence is strongly dependent on the accuracy of the bounds provided, and therefore, ...

We introduce basic ideas of a nonsmooth Newton’s method and its application in solving semidefinite optimization (SDO) problems. In particular, the method can be used to solve both linear and nonlinear semidefinite complementarity problems. We also survey recent theoretical results in matrix functions and stability of SDO that are stemed from the research on the matrix form of the nonsmooth New...

These notes give an introduction to duality theory in the context of linear and positive semidefinite programming. These notes are based on material from Convex Analysis and Nonlinear Optimization by Borwein and Lewis and Numerical Optimization by Nocedal and Wright. Two examples are given to show how duality can be used. The first optimization application is to find the matrix in an affine fam...

In this paper we describe an algorithm to approximately solve a class of semidefinite programs called covering semidefinite programs. This class includes many semidefinite programs that arise in the context of developing algorithms for important optimization problems such as Undirected SPARSEST CUT, wireless multicasting, and pattern classification. We give algorithms for covering SDPs whose de...