Based originally on work of McCormick, a number of recent global optimization algorithms have relied on replacing an original nonconvex nonlinear program by convex or linear relaxations. Such linear relaxations can be generated automatically through an automatic differentiation process. This process decomposes the objective and constraints (if any) into convex and nonconvex unary and binary ope...

This paper proposes a geometrical analysis of the polyhedral feasible domains for the predictive control laws under constraints. The state vector is interpreted as a vector of parameters for the optimization problem to be solved at each sampling instant and its influence can be fully described by the use of parameterized polyhedra and their dual constraints/generators representation. The constr...

A wide variety of machine learning algorithms such as support vector machine (SVM), minimax probability machine (MPM), and Fisher discriminant analysis (FDA), exist for binary classification. The purpose of this paper is to provide a unified classification model that includes the above models through a robust optimization approach. This unified model has several benefits. One is that the extens...

Consider the problem of minimizing the expected value of a (possibly nonconvex) cost function parameterized by a random (vector) variable, when the expectation cannot be computed accurately (e.g., because the statistics of the random variables are unknown and/or the computational complexity is prohibitive). Classical sample stochastic gradient methods for solving this problem may empirically su...

Proximity control is a well-known mechanism in bundle method for nonsmooth optimization. Here we show that it can be used to optimize a large class of nonconvex and nonsmooth functions with additional structure. This includes for instance nonconvex maximum eigenvalue functions, and also infinite suprema of such functions.

The problem of reconstructing a sparse signal vector from magnitude-only measurements (a.k.a., compressive phase retrieval), emerges naturally in diverse applications, but it is NP-hard in general. Building on recent advances in nonconvex optimization, this paper puts forth a new algorithm that is termed compressive reweighted amplitude flow and abbreviated as CRAF, for compressive phase retrie...