For nonconvex optimization problems, possibly having mixed-integer variables, a convergent primal-dual solution algorithm is proposed. The approach applies a proximal bundle method to certain augmented Lagrangian dual that arises in the context of the so-called generalized augmented Lagrangians. We recast these Lagrangians into the framework of a classical Lagrangian, by means of a special reformulation … Read more
We study minimization of a structured objective function, being the sum of a smooth function and a composition of a weakly convex function with a linear operator. Applications include image reconstruction problems with regularizers that introduce less bias than the standard convex regularizers. We develop a variable smoothing algorithm, based on the Moreau envelope with … Read more
First-order operator splitting methods are ubiquitous among many fields through science and engineering, such as inverse problems, signal/image processing, statistics, data science and machine learning, to name a few. In this paper, we study a geometric property of first-order methods when applying to solve non-smooth optimization problems. With the tool of “partial smoothness”, we design … Read more
In this paper, we bring forward a completely perturbed nonconvex Schatten $p$-minimization to address a model of completely perturbed low-rank matrix recovery. The paper that based on the restricted isometry property generalizes the investigation to a complete perturbation model thinking over not only noise but also perturbation, gives the restricted isometry property condition that guarantees … Read more
This paper discusses an outer-approximation guided optimization method for constrained neural network inverse problems with rectified linear units. The constrained neural network inverse problems refer to an optimization problem to find the best set of input values of a given trained neural network in order to produce a predefined desired output in presence of constraints … Read more
This paper deals with a general framework for inexact forward–backward algorithms aimed at minimizing the sum of an analytic function and a lower semicontinuous, subanalytic, convex term. Such framework relies on an implementable inexactness condition for the computation of the proximal operator, and a linesearch procedure which is possibly performed whenever a variable metric is … Read more
In this paper we consider stochastic weakly convex composite problems, however without the existence of a stochastic subgradient oracle. We present a derivative free algorithm that uses a two point approximation for computing a gradient estimate of the smoothed function. We prove convergence at a similar rate as state of the art methods, however with … Read more
This work considers nonsmooth and nonconvex optimization problems whose objective and constraint functions are defined by difference-of-convex (DC) functions. We consider an infeasible bundle method based on the so-called improvement functions to compute critical points for problems of this class. Our algorithm neither employs penalization techniques nor solves subproblems with linearized constraints. The approach, which … Read more
This paper continues the study of general nonlinear transversality properties of collections of sets and focuses on primal space necessary (in some cases also sufficient) characterizations of the properties. We formulate geometric, metric and slope characterizations, particularly in the convex setting. The Holder case is given a special attention. Quantitative relations between the nonlinear transversality … Read more
Convex optimization problems, whose solutions live in very high dimensional spaces, have become ubiquitous. To solve them, proximal splitting algorithms are particularly adequate: they consist of simple operations, by handling the terms in the objective function separately. We present several existing proximal splitting algorithms and we derive new ones, within a unified framework, which consists … Read more