The goal of this paper is to formulate and solve free material optimization problems with constraints on the smallest eigenfrequency of the optimal structure. A natural formulation of this problem as linear semidefinite program turns out to be numerically intractable. As an alternative, we propose a new approach, which is based on a nonlinear semidefinite low-rank approximation of the semidefin...

7 Constraint generation and quadratic inequalities 29 7.1 Example: the stable set polytope again . . . . . . . . . . . . . . . . . . . . . . . . . . 29 7.2 Strong insolvability of quadratic equations . . . . . . . . . . . . . . . . . . . . . . . . 30 7.3 Inference rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 7.4 Algorithmic aspects of inference rules . ...

Recently, there have been several newdevelopments in discrepancy theory based on connections to semidefinite programming. This connection has been useful in several ways. It gives efficient polynomial time algorithms for several problems for which only non-constructive results were previously known. It also leads to several new structural results in discrepancy itself, such as tightness of the ...

‎It is a well-known fact that finding a minimum dominating set and consequently the domination number of a general graph is an NP-complete problem‎. ‎In this paper‎, ‎we first model it as a nonlinear binary optimization problem and then extract two closely related semidefinite relaxations‎. ‎For each of these relaxations‎, ‎different rounding algorithm is exp...

This paper deals with the computation of exact solutions of a classical NP-hard problem in combinatorial optimization, the k-cluster problem. This problem consists in finding a heaviest subgraph with k nodes in an edge weighted graph. We present a branch-and-bound algorithm that applies a novel bounding procedure, based on recent semidefinite programming techniques. We use new semidefinite boun...

In this paper, we try to solve the semidefinite program with box constraints. Since the traditional projection method for constrained optimization with box constraints is not suitable to the semidefinite constraints, we present a new algorithm based on the feasible directionmethod. In the paper, we discuss two cases: the objective function in semidefinite programming is linear and nonlinear, re...

The purpose of this paper is to solve the 0-1 k-item quadratic knapsack problem (kQKP ), a problem of maximizing a quadratic function subject to two linear constraints. We propose an exact method based on semidefinite optimization. The semidefinite relaxation used in our approach includes simple rank one constraints, which can be handled efficiently by interior point methods. Furthermore, we st...

4 Obtaining semidefinite programs 20 4.1 Unit distance graphs and orthogonal representations . . . . . . . . . . . . . . . . . . 20 4.2 Discrete linear and quadratic programs . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.3 Spectra of graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.4 Engineering applications . . . . . . . . . . . . . . . . . . ....