Nonconvex optimization problems are frequently encountered in much of statistics, business, science and engineering, but they are not yet widely recognized as a technology. A reason for this relatively low degree of popularity is the lack of a well developed system of theory and algorithms to support the applications, as is the case for its convex counterpart. This paper aims to take one step i...

We consider nonconvex constrained optimization problems and propose a new approach to the convergence analysis based on penalty functions. make use of classical functions in an unconventional way, that only enter theoretical while algorithm itself is free. Based this idea, we are able establish several results, including first general for diminishing stepsize methods nonconvex, optimization, sh...

We describe a version of randomization technique within a general framework of Euclidean Jordan algebras. It is shown how to use this technique to evaluate the quality of symmetric relaxations for several nonconvex optimization problems.

We provide theoretical analysis of the statistical and computational properties of penalized M-estimators that can be formulated as the solution to a possibly nonconvex optimization problem. Many important estimators fall in this category, including least squares regression with nonconvex regularization, generalized linear models with nonconvex regularization and sparse elliptical random design...

We report on the theory and implementation of a global optimization solver for general constrained nonlinear programming problems based on Variable Neighbourhood Search, and we give comparative computational results on several instances of continuous nonconvex problems. Compared to an efficient multi-start global optimization solver, the VNS solver proposed appears to be significantly faster.

This paper studies a fundamental bicriteria optimization problem for variable selection in statistical learning; the two criteria are a loss/residual function and a model control (also called regularization, penalty). The former function measures the fitness of the learning model to data and the latter function is employed as a control of the complexity of the model. We focus on the case where ...

A heuristic method for optimizing a solar power tower system is proposed, in which both heliostat field (heliostat locations and number) and the tower (tower height and receiver size) are simultaneously considered. Maximizing the thermal energy collected per unit cost leads to a difficult optimization problem due to its characteristics: it has a nonconvex black-box objective function with compu...

Solving statistical learning problems often involves nonconvex optimization. Despite the empirical success of nonconvex statistical optimization methods, their global dynamics, especially convergence to the desirable local minima, remain less well understood in theory. In this paper, we propose a new analytic paradigm based on diffusion processes to characterize the global dynamics of nonconvex...

The problem of cluster analysis is formulated as a problem of nonsmooth, nonconvex optimization. An algorithm for solving the latter optimization problem is developed which allows one to significantly reduce the computational efforts. This algorithm is based on the so-called discrete gradient method. Results of numerical experiments are presented which demonstrate the effectiveness of the propo...

In this paper bearing the same title as our earlier survey-paper [11] we pursue the goal of characterizing the global solutions of an optimization problem, i.e. getting at necessary and sufficient conditions for a feasible point to be a global minimizer (or maximizer) of the objective function. We emphasize nonconvex optimization problems presenting some specific structures like ‘convexanticonv...