# Hidden convexity in some nonconvex quadratically constrained quadratic programming

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Aharon Ben-Tal, Feb 10, 2015 Available from:- [Show abstract] [Hide abstract]

**ABSTRACT:**Variational inequality and complementarity problems have found utility in modeling a range of optimization and equilibrium problems. Yet, while there has been tremendous growth in addressing uncertainty in optimization, relatively less progress has been seen in the context of variational inequality problems, exceptions being efforts to solve variational inequality problems with expectation-valued maps as well as suitably defined expected residual minimization (ERM) problems. Both avenues necessitate distributional information associated with the uncertainty and neither approach is explicitly designed to provide robust solutions. Motivated by this gap, we define a robust solution to a complementarity problem as one that minimizes the worst-case of the gap function over a prescribed uncertainty set and requires feasibility for every point in the uncertainty set. In what we believe is amongst the first efforts to comprehensively address such problems in a distribution-free environment, we present an avenue for obtaining robust solutions to uncertain monotone and a subclass of non-monotone linear complementarity problems via a low-dimensional convex program. In more general non-monotone regimes, we prove that robust counterparts are low-dimensional nonconvex quadratically constrained quadratic programs. By customizing an existing scheme to these problems, robust solutions to uncertain non-monotone LCPs can be provided. Importantly, these results can be extended to account for uncertainty in the underlying sets by generalizing the results to uncertain affine variational inequality problems defined over uncertain polyhedral sets as well as to hierarchical regimes captured by mathematical programs with uncertain complementarity constraints. Preliminary numerics on uncertain linear complementarity problems and a traffic equilibrium case study suggest that such avenues hold promise. -
##### Article: Global solutions to spherically constrained quadratic minimization via canonical duality theory

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**ABSTRACT:**This paper presents a detailed study on global optimal solutions to a nonconvex quadratic minimization problem with a spherical constraint, which is well-known as a trust region subproblem and has been studied extensively for decades. The main challenge is solving the ’hard case’, i.e. the problem has multiple solutions on the boundary of the sphere. By canonical duality-triality theory, this challenging problem can be reformulated as a one-dimensional canonical dual problem, without any duality gaps. Results show that this problem is in the hard case if and only if certain conditions are satisfied by both the direction and norm of coefficient of the linear item in the objective function. A perturbation method and associated algorithms are proposed to solve these hard case problems. Theoretical results and methods are verified by numerical examples.Mathematics and Mechanics of Solids 01/2015; DOI:10.1177/1081286515577122 · 0.86 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper we discuss problems with quadratic objective function, one or two quadratic constraints, and, possibly, some additional linear constraints. In particular, we consider cases where the Hessian of the quadratic functions are simultaneously diagonalizable, so that the objective and constraint functions can all be converted into separable functions. We give conditions under which a simple convex relaxation of these problems returns their optimal values.Operations Research Letters 12/2014; 43(2). DOI:10.1016/j.orl.2014.12.002 · 0.62 Impact Factor