Quadratically constrained quadratic programs (QQPs) play an important modeling role for many diverse problems. These problems are in general NP hard and numerically intractable. Lagrangian relaxations often provide good approximate solutions to these hard problems. Such relaxations are equivalent to semidefinite programming relaxations. For several special cases of QQP, e.g., convex programs an...

The quadratic assignment problem (QAP ) is arguably one of the hardest of the NP-hard discrete optimization problems. Problems of dimension greater than 20 are considered to be large scale. Current successful solution techniques depend on branch and bound methods. However, it is difficult to get strong and inexpensive bounds. In this paper we introduce a new semidefinite programming (SDP ) rela...

Variable selection is a fundamental task in statistical data analysis. Sparsity-inducing regularization methods are a popular class of methods that simultaneously perform variable selection and model estimation. The central problem is a quadratic optimization problem with an `0-norm penalty. Exactly enforcing the `0-norm penalty is computationally intractable for larger scale problems, so diffe...

We develop a spatial branch-and-cut approach for nonconvex Quadratically Constrained Quadratic Programs with bounded complex variables (CQCQP). Linear valid inequalities are added at each node of the search tree to strengthen semidefinite programming relaxations of CQCQP. These valid inequalities are derived from the convex hull description of a nonconvex set of 2 × 2 positive semidefinite Herm...

This paper is concerned with the study of an arbitrary polynomial optimization via a convex relaxation, namely a semidefinite program (SDP). The existence of a rank-1 matrix solution for the SDP relaxation guarantees the recovery of a global solution of the original problem. The main objective of this work is to show that an arbitrary polynomial optimization has an equivalent formulation in the...

We consider the problem of scheduling unrelated parallel machines so as to minimize the total weighted completion time of jobs. Whereas the best previously known approximation algorithms for this problem are based on LP relaxations, we give a 2 –approximation algorithm that relies on a convex quadratic programming relaxation. For the special case of two machines we present a further improvement...

In this paper we study Support Vector Machine(SVM) classifiers in the face of uncertain knowledge sets and show how data uncertainty in knowledge sets can be treated in SVM classification by employing robust optimization. We present knowledge-based SVM classifiers with uncertain knowledge sets using convex quadratic optimization duality. We show that the knowledge-based SVM, where prior knowled...

In this paper, we consider the optimal design of photonic crystal band structures for twodimensional square lattices. The mathematical formulation of the band gap optimization problem leads to an infinite-dimensional Hermitian eigenvalue optimization problem parametrized by the dielectric material and the wave vector. To make the problem tractable, the original eigenvalue problem is discretized...

Format (5.1.1) covers all uncertain optimization problems considered so far; moreover, in these latter problems the objective f and the right hand side F of the constraints always were bi-affine in x, ζ, (that is, affine in x when ζ is fixed, and affine in ζ, x being fixed), and K was a “simple” convex cone (a direct product of nonnegative rays/Lorentz cones/Semidefinite cones, depending on whe...