The minimax theorem for a convex-concave bifunction is a fundamental theorem in optimization and convex analysis, and has a lot of applications in economics. In the last two decades, a nonconvex extension of this minimax theorem has been well studied under various generalized convexity assumptions. In this note, by exploiting the hidden convexity (joint range convexity) of separable homogeneous...

This paper illustrates the fundamental connection between nonconvex quadratic optimization and copositive optimization—a connection that allows the reformulation of nonconvex quadratic problems as convex ones in a unified way. We focus on examples having just a few variables or a few constraints for which the quadratic problem can be formulated as a copositive-style problem, which itself can be...

In binary hashing, one wants to learn a function that maps a high-dimensional feature vector to a vector of binary codes, for application to fast image retrieval. This typically results in a difficult optimization problem, nonconvex and nonsmooth, because of the discrete variables involved. Much work has simply relaxed the problem during training, solving a continuous optimization, and truncati...

Nowadays specialists on optimization observe the persistent demands from the world of applications to create an effective apparatus for finding just a global solution to nonconvex problems in which there may exist local solutions located very far from a global one even up to the values of goal function. As well-known, the conspicuous limitation of convex optimization methods applied to nonconve...

A rigorous foundation is presented for the decomposition method in nonconvex global optimization, including parametric optimization, partly convex, partly monotonic, and monotonic/linear optimization. Incidentally, some errors in the recent literature on this subject are pointed out and fixed.

We describe a general scheme for solving nonconvex optimization problems, where in each iteration the nonconvex feasible set is approximated by an inner convex approximation. The latter is defined using an upper bound on the nonconvex constraint functions. Under appropriate conditions on this upper bounding convex function, a monotone convergence to a KKT point is established. The scheme is app...

(ABSTRACT) Despite recent advances in convex optimization techniques, the areas of discrete and continuous nonconvex optimization remain formidable, particularly when globally optimal solutions are desired. Most solution techniques, such as branch-and-bound, are enumerative in nature, and the rate of their convergence is strongly dependent on the accuracy of the bounds provided, and therefore, ...