Yuan’s theorem of the alternative is an important theoretical tool in optimization, which provides a checkable certificate for the infeasibility of a strict inequality system involving two homogeneous quadratic functions. In this paper, we provide a tractable extension of Yuan’s theorem of the alternative to the symmetric tensor setting. As an application, we establish that the optimal value of...

This paper studies a scalar minimization problem with an integral functional of the gradient under affine boundary conditions. A new approach is proposed using a minimal and a maximal solution to the convexified problem. We prove a density result: any relaxed solution continuously depending on boundary data may be approximated uniformly by solutions of the nonconvex problem keeping continuity r...

A Mathematical Program with Linear Complementarity Constraints (MPLCC) is an optimization problem where a continuously differentiable function is minimized on a set defined by linear constraints and complementarity conditions on pairs of complementary variables. This problem finds many applications in several areas of science, engineering and economics and is also an important tool for the solu...

We consider the nonconvex problem (RQ) of minimizing the ratio of two nonconvex quadratic functions over a possibly degenerate ellipsoid. This formulation is motivated by the so-called Regularized Total Least Squares problem (RTLS), which is a special case of the problem’s class we study. We prove that under a certain mild assumption on the problem’s data, problem (RQ) admits an exact semidefin...

A nonconvex generalized semi-infinite programming problem is considered involving parametric max-functions in both, the objective and the constraints. For a fixed vector of parameters, the values of these parametric max-functions are given as optimal values of convex quadratic programming problems. Assuming that for each parameter the parametric quadratic problems satisfy the strong duality rel...

The problem of optimizing a real valued function over the efficient set of a multiple objective linear program has some applications in multiple objective decision making. The main difficulty of this problem arises from the fact that its feasible domain, in general, is nonconvex and not given explicitly. In this paper we formulate this problem as a linearly constrained convex-concave program wh...

This paper presents a set of complete solutions and optimality conditions for a nonconvex quadratic-exponential optimization problem. By using the canonical duality theory developed by the first author, the nonconvex primal problem in n-dimensional space can be converted into an one-dimensional canonical dual problem with zero duality gap, which can be solved easily to obtain all dual solutions...

Earlier research has shown that the problem of optimal weighted median filtering with structural constraints can be formulated as a nonconvex nonlinear programming problem in general. However, its high computational complexity and poor performance due to its nonconvex nature prohibit it from practical applications. In this paper, we shall show that the design problem can be formulated as a conv...

The problem of minimizing sum-of-nonconvex functions (i.e., convex functions that are average of non-convex ones) is becoming increasingly important in machine learning, and is the core machinery for PCA, SVD, regularized Newton’s method, accelerated non-convex optimization, and more. We show how to provably obtain an accelerated stochastic algorithm for minimizing sumof-nonconvex functions, by...

Although many methods exist for intensity modulated radiotherapy (IMRT) fluence map optimization for crisp data, based on clinical practice, some of the involved parameters are fuzzy. In this paper, the best fluence maps for an IMRT procedure were identifed as a solution of an optimization problem with a quadratic objective function, where the prescribed target dose vector was fuzzy. First, a d...