A natural way to handle optimization problem with data affected by stochastic uncertainty is to pass to a chance constrained version of the problem, where candidate solutions should satisfy the randomly perturbed constraints with probability at least 1− . While being attractive from modeling viewpoint, chance constrained problems “as they are” are, in general, computationally intractable. In th...

Robust optimization is a common framework in optimization under uncertainty when the problem parameters are not known, but it is rather known that the parameters belong to some given uncertainty set. In the robust optimization framework the problem solved is a min-max problem where a solution is judged according to its performance on the worst possible realization of the parameters. In many cas...

The underestimation of data points by a convex quadratic function is a useful tool for approximating the location of the global minima of potential energy functions that arise in protein-ligand docking problems. Determining the parameters that define the underestimator can be formulated as a convex quadratically constrained quadratic program and solved efficiently using algorithms for semidefin...

We consider semidefinite programming relaxations of the quadratic assignment problem, and show how to exploit group symmetry in the problem data. Thus we are able to compute the best known lower bounds for several instances of quadratic assignment problems from the problem library: [R.E. Burkard, S.E. Karisch, F. Rendl. QAPLIB — a quadratic assignment problem library. Journal on Global Optimiza...

Linear projection equations arise in many optimization problems and have important applications in science and engineering. In this paper, we present a recurrent neural network for solving linear projection equations in real time. The proposed neural network has two layers and is amenable to parallel implementation with simple hardware. In the theoretical aspect, we prove that the proposed neur...

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...

Semidefinite optimization problems are an expressive family of convex optimization problems that can be solved efficiently. We develop semidefinite optimization-based formulations and approximations for a number of families of optimization problems, including problems arising in spacecraft attitude estimation and in learning tree-structured statistical models. We construct explicit exact reform...

Linear projection equations arise in many optimization problems and have important applications in science and engineering. In this paper, we present a recurrent neural network for solving linear projection equations in real time. The proposed neural network has two layers and is amenable to parallel implementation with simple hardware. In the theoretical aspect, we prove that the proposed neur...

In this paper, we present a novel semidefinite programming approach for multiple-instance learning. We first formulate the multipleinstance learning as a combinatorial maximummargin optimization problem with additional instance selection constraints within the framework of support vector machines. Although solving this primal problem requires non-convex programming, we nevertheless can then der...

Optimization problems arise in widely varying contexts. The general optimization problem is to find the minimum value of a certain function, the objective, on a certain set, defined by constraints. To make such problems amenable to analysis, further restrictions must be imposed on the kinds of objectives and constraints that may arise. A priori, it might seem useful to require them to be polyno...