A line search algorithm for minimizing nonconvex and/or nonsmooth objective functions is presented. The algorithm is a hybrid between a standard Broyden–Fletcher–Goldfarb–Shanno (BFGS) and an adaptive gradient sampling (GS) method. The BFGS strategy is employed because it typically yields fast convergence to the vicinity of a stationary point, and together with the adaptive GS strategy … Read more
In this paper, we propose an inexact proximal multiplier method using proximal distances for solving convex minimization problems with a separable structure. The proposed method unified the work of Chen and Teboulle (PCPM method), Kyono and Fukushima (NPCPMM) and Auslender and Teboulle (EPDM) and extends the convergence properties for a class of phi-divergence distances. We … Read more
We consider the superiorization methodology, which can be thought of as lying between feasibility-seeking and constrained minimization. It is not quite trying to solve the full fledged constrained minimization problem; rather, the task is to find a feasible point which is superior (with respect to the objective function value) to one returned by a feasibility-seeking … Read more
The Mesh Adaptive Direct Search (Mads) algorithm is designed for blackbox optimization problems subject to general inequality constraints. Currently, Mads does not support equalities, neither in theory nor in practice. The present work proposes extensions to treat problems with linear equalities whose expression is known. The main idea consists in reformulating the optimization problem into … Read more
We consider symmetrized KKT systems arising in the solution of convex quadratic programming problems in standard form by Interior Point methods. Their coefficient matrices usually have 3×3 block structure and, under suitable conditions on both the quadratic programming problem and the solution, they are nonsingular in the limit. We present new spectral estimates for these … Read more
In this paper we are interested in the solution of Compressed Sensing (CS) problems where the signals to be recovered are sparse in coherent and redundant dictionaries. CS problems of this type are convex with non-smooth and non-separable regularization term, therefore a specialized solver is required. We propose a primal-dual Newton Conjugate Gradients (pdNCG) method. … Read more
The convex feasibility problem (CFP) is at the core of the modeling of many problems in various areas of science. Subgradient projection methods are important tools for solving the CFP because they enable the use of subgradient calculations instead of orthogonal projections onto the individual sets of the problem. Working in a real Hilbert space, … Read more
This article deals with a study of the MPEC problem based on a reformulation to a DC problem (Difference of Convex functions). This reformulation is obtained by a partial penalization of the constraints. In this article we prove that a classical optimality condition for a DC program, if a constraint qualification is satisfied for MPEC, … Read more
We consider power networks in which it is not possible to satisfy all loads at the demand nodes, due to some attack or disturbance to the network. We formulate a model, based on AC power flow equations, to restore the network to feasibility by shedding load at demand nodes, but doing so in a way … Read more
A lot of real-world optimization models comprise nonconvex and nonlinear as well as nonsmooth functions leading to very hard classes of optimization models. In this article a new interior-point method for the special but practically relevant class of optimization problems with locatable and separable nonsmooth aspects is presented. After motivating and formalizing the problems under … Read more