The alternating direction method of multipliers (ADMM) has been popular for solving many signal processing problems, convex or nonconvex. In this paper, we study an asynchronous implementation of the ADMM for solving a nonconvex nonsmooth optimization problem, whose objective is the sum of a number of component functions. The proposed algorithm allows the problem to be solved in a distributed, ...

The two-dimensional stock cutting problem is well known and often studied. A genetic algorithm approach to the problem is developed that is capable of handling some of the more intractable forms of the problem: nonconvex parts; non-convex sheets; multiple irregularly shaped sheets; and so on. An integrated system is developed that incorporates a machine vision module for acquiring the images of...

We study the error landscape of deep linear and nonlinear neural networks with the squared error loss. Minimizing the loss of a deep linear neural network is a nonconvex problem, and despite recent progress, our understanding of this loss surface is still incomplete. For deep linear networks, we present necessary and sufficient conditions for a critical point of the risk function to be a global...

In this paper, we consider the div-curl problem posed on nonconvex polyhedral domains. We propose a least-squares method based on discontinuous elements with normal and tangential continuity across interior faces, as well as boundary conditions, weakly enforced through a properly designed least-squares functional. Discontinuous elements make it possible to take advantage of regularity of given ...

We consider a proximal algorithm with quasi distance applied to nonconvex and nonsmooth functions involving analytic properties for an unconstrained minimization problem. We show the behavioral importance of this proximal point model for habit’s formation in Decision and Making Sciences.

We give an efficient, quick algorithm for the minimization of a quadratic function over Stiefel manifold. We reduce the original (nonconvex) problem, to an SDP, by computing a convex hull of the certain set of matrices.

For a class of nonconvex nonsmooth functions, we consider the problem of computing an approximate critical point, in the case when only inexact information about the function and subgradient values is available. We assume that the errors in function and subgradient evaluations are merely bounded, and in principle need not vanish in the limit. We examine the redistributed proximal bundle approac...

A novel class of variational models with nonconvex `q-normtype regularizations (0 < q < 1) is considered, which typically outperforms popular models with convex regularizations in restoring sparse images. Due to the fact that the objective function is nonconvex and nonLipschitz, such models are very challenging from an analytical as well as numerical point of view. In this work a smoothing desc...

We provide a unifying geometric framework for the analysis of general classes of duality schemes and penalty methods for nonconvex constrained optimization problems. We present a separation result for nonconvex sets via general concave surfaces. We use this separation result to provide necessary and sufficient conditions for establishing strong duality between geometric primal and dual problems...

We consider a general multiobjective optimization problem with five basic optimality principles: efficiency, weak and proper Pareto optimality, strong efficiency and lexicographic optimality. We generalize the concept of tradeoff directions defining them as some optimal surface of appropriate cones. In convex optimization, the contingent cone can be used for all optimality principles except lex...