This paper answers in affirmative the long-standing question of nonlinear analysis, concerning the existence of a continuous single-valued local selection of the right inverse to a locally Lipschitzian mapping. Moreover, we develop a much more general result, providing conditions for the existence of a continuous single-valued selection not only locally, but rather on any given … Read more
The paper observes a similarity between the stochastic optimal control of discrete dynamical systems and the learning multilayer neural networks. It focuses on contemporary deep networks with nonconvex nonsmooth loss and activation functions. The machine learning problems are treated as nonconvex nonsmooth stochastic optimization problems. As a model of nonsmooth nonconvex dependences, the so-called generalized … Read more
The superiorization methodology is intended to work with input data of constrained minimization problems, i.e., a target function and a constraints set. However, it is based on an antipodal way of thinking to the thinking that leads constrained minimization methods. Instead of adapting unconstrained minimization algorithms to handling constraints, it adapts feasibility-seeking algorithms to reduce … Read more
A convex form of degree larger than one is always nonnegative since it vanishes together with its gradient at the origin. In 2007, Parrilo asked if convex forms are always sums of squares. A few years later, Blekherman answered the question in the negative by showing through volume arguments that for high enough number of … Read more
The paper observes the similarity between the stochastic optimal control of discrete dynamical systems and the training multilayer neural networks. It focuses on contemporary deep networks with nonconvex nonsmooth loss and activation functions. In the paper, the machine learning problems are treated as nonconvex nonsmooth stochastic optimization problems. As a model of nonsmooth nonconvex dependences, … Read more
We develop a general methodology to derive probabilistic guarantees for solutions of robust optimization problems. Our analysis applies broadly to any convex compact uncertainty set and to any constraint affected by uncertainty in a concave manner, under minimal assumptions on the underlying stochastic process. Namely, we assume that the coordinates of the noise vector are … Read more
In this work, nonconvex nonsmooth problems of dynamic optimization, optimal control in discrete time (including feedback control), and machine learning are considered from a common point of view. An analogy is observed between tasks of controlling discrete dynamic systems and training multilayer neural networks with nonsmooth target function and connections. Methods for calculating generalized gradients … Read more
This paper presents an accelerated composite gradient (ACG) variant, referred to as the AC-ACG method, for solving nonconvex smooth composite minimization problems. As opposed to well-known ACG variants that are either based on a known Lipschitz gradient constant or a sequence of maximum observed curvatures, the current one is based on a sequence of average … Read more
The superiorization methodology is intended to work with input data of constrained minimization problems, that is, a target function and a set of constraints. However, it is based on an antipodal way of thinking to what leads to constrained minimization methods. Instead of adapting unconstrained minimization algorithms to handling constraints, it adapts feasibility-seeking algorithms to … Read more
Domain-Driven Solver (DDS) is a MATLAB-based software package for convex optimization problems in Domain-Driven form [11]. The current version of DDS accepts every combination of the following function/set constraints: (1) symmetric cones (LP, SOCP, and SDP); (2) quadratic constraints; (3) direct sums of an arbitrary collection of 2-dimensional convex sets defined as the epigraphs of … Read more