We give an overview of cone optimization software, with special attention to the differences between the existing packages. We assume the reader is familiar with the theory and algorithms of cone optimization, thus technical details are kept at a minimum. We also outline current research trends and area of potential improvement. 1 Problem description Conic optimization solvers target problems o...

Designing phononic crystals by creating frequency bandgaps is of particular interest in elastic and acoustic metamaterial engineering. Mathematically, the problem of optimizing the frequency bandgaps is often nonconvex, as it requires the maximization of the higher indexed eigenfrequency and the minimization of the lower indexed eigenfrequency. A novel algorithm [1] has been previously develope...

Representation of a given nonnegative multivariate polynomial in terms of a sum of squares of polynomials has become an essential subject in recent developments of sums of squares optimization and SDP (semidefinite programming) relaxation of polynomial optimization problems. We disscuss effective methods to obtain a simpler representation of a “sparse” polynomial as a sum of squares of sparse p...

Several problems in optimization and control involve a matrix of decision variables to be subject to a rank constraint. Although semidefinite programming is used as a generalpurpose tool to provide strong relaxations of such problems, finding feasible solutions mostly relies on algorithmic techniques specific to the problem at hand. We present models for expressing rank constraints using mathem...

Matrix variables are ubiquitous in modern optimization, in part because variational properties of useful matrix functions often expedite standard optimization algorithms. Convexity is one important such property: permutation-invariant convex functions of the eigenvalues of a symmetric matrix are convex, leading to the wide applicability of semidefinite programming algorithms. We prove the analo...

Lecture notes for the tutorial at the workshop HPOPT 2008 — 10th International Workshop on High Performance Optimization Techniques (Algebraic Structure in Semidefinite Programming), June 11th to 13th, 2008, Tilburg University, The Netherlands.

There are similarities between semidefinite programming and linear programming in theory and practice, e.g., duality theory (see Bellman and Fan [5]), the role of complementary slackness, and efficient solution techniques using interior-point methods (see Nesterov and Nemirovski [11], Wright, [20]). Semidefinite programming has been developed both theoretically and practically for the past few ...

We address the multi-period portfolio optimization problem with the constant rebalanc-ing strategy. This problem is formulated as a polynomial optimization problem (POP) by using a mean-variance criterion. In order to solve the POPs of high degree, we develop a cutting-plane algorithm based on semidefinite programming. Our algorithm can solve problems that can not be handled by any of known pol...

The verification of matrix copositivity is a well known computationally hard problem, with many applications in continuous and combinatorial optimization. In this paper, we present a hierarchy of semidefinite programming based sufficient conditions for a real matrix to be copositive. These conditions are obtained through the use of a sum of squares decomposition for multivariable forms. As can ...

We describe a conic interior point decomposition approach for solving a large scale semidefinite program (SDP) whose primal feasible set is bounded. The idea is to solve such an SDP using existing primal-dual interior point methods, in an iterative fashion between amaster problem and a subproblem. In our case, the master problem is a mixed conic problem over linear and smaller sized semidefinit...