In this paper, we propose new linesearch-based methods for nonsmooth constrained optimization problems when first-order information on the problem functions is not available. In the first part, we describe a general framework for bound-constrained problems and analyze its convergence toward stationary points, using the Clarke-Jahn directional derivative. In the second part, we consider inequality constrained optimization problems where both objective function and constraints can possibly be nonsmooth. In this case, we first split the constraints into two subsets: difficult general nonlinear constraints and simple bound constraints on the variables. Then, we use an exact penalty function to tackle the difficult constraints and we prove that the original problem can be reformulated as the bound-constrained minimization of the proposed exact penalty function. Finally, we use the framework developed for the bound-constrained case to solve the penalized problem. Moreover, we prove that every accumulation point, under standard assumptions on the search directions, of the generated sequence of iterates is a stationary point of the original constrained problem. In the last part of the paper, we report extended numerical results on both bound-constrained and nonlinearly constrained problems, showing that our approach is promising when compared to some state-of-the-art codes from the literature.

Zero-norm, Concave optimization, Frank-Wolfe algorithm, Feature selection,

In this work, we study exact continuous reformulations of nonlinear integer programming problems. To this aim, we preliminarily state conditions to guarantee the equivalence between pairs of general nonlinear problems. Then, we prove that optimal solutions of a nonlinear integer programming problem can be obtained by using various exact penalty formulations of the original problem in a continuous space.

In the field of global optimization many efforts have been devoted to solve unconstrained global optimization problems. The aim of this paper is to show that unconstrained global optimization methods can be used also for solving constrained optimization problems, by resorting to an exact penalty approach. In particular, we make use of a non-differentiable exact penalty function . We show that, under weak assumptions, there exists a threshold value of the penalty parameter such that, for any , any global minimizer of P (q) is a global solution of the related constrained problem and conversely. On these bases, we describe an algorithm that, by combining an unconstrained global minimization technique for minimizing P (q) for given values of the penalty parameter and an automatic updating of that occurs only a finite number of times, produces a sequence {x (k) } such that any limit point of the sequence is a global solution of the related constrained problem. In the algorithm any efficient unconstrained global minimization technique can be used. In particular, we adopt an improved version of the DIRECT algorithm. Some numerical experimentation confirms the effectiveness of the approach

This paper proposes the design optimization procedure of three-phase interior permanent magnet (IPM) synchronous motors with minimum weight, maximum power output, and suitability for wide constant-power region operation. The particular rotor geometry of the IPM synchronous motor and the presence of several variables and constraints make the design problem very complicated. The authors propose to combine an accurate finite-element analysis with a multiobjective optimization procedure using a new algorithm belonging to the class of controlled random search algorithms. The optimization procedure has been employed to design two IPM motors for industrial application and a city electrical scooter. A prototype has been realized and tested. The comparison between the predicted and measured performances shows the reliability of the simulation results and the effectiveness, versatility, and robustness of the proposed procedure

In this work, we consider multiobjective optimization problems with both bound constraints on the variables and general nonlinear constraints, where objective and constraint function values can only be obtained by querying a black box. We define a linesearch-based solution method, and we show that it converges to a set of Pareto stationary points. To this aim, we carry out a theoretical analysis of the problem by only assuming Lipschitz continuity of the functions; more specifically, we give new optimality conditions that take explicitly into account the bound constraints, and prove that the original problem is equivalent to a bound constrained problem obtained by penalizing the nonlinear constraints with an exact merit function. Finally, we present the results of some numerical experiments on bound constrained and nonlinearly constrained problems, showing that our approach is promising when compared to a state-of-the-art method from the literature.

The paper is concerned with black-box nonlinear constrained multi-objective optimization problems. Our interest is the definition of a multi-objective deterministic partition-based algorithm. The main target of the proposed algorithm is the solution of a real ship hull optimization problem. To this purpose and in pursuit of an efficient method, we develop an hybrid algorithm by coupling a multi-objective DIRECT-type algorithm with an efficient derivative-free local algorithm. The results obtained on a set of "hard" nonlinear constrained multi-objective test problems show viability of the proposed approach. Results on a hull-form optimization of a high-speed catamaran (sailing in head waves in the North Pacific Ocean) are also presented. In order to consider a real ocean environment, stochastic sea state and speed are taken into account. The problem is formulated as a multi-objective optimization aimed at (i) the reduction of the expected value of the mean total resistance in irregular head waves, at variable speed and (ii) the increase of the ship operability, with respect to a set of motion-related constraints. We show that the hybrid method performs well also on this industrial problem.

Abstract. The problem of finding sparse solutions to underdetermined systems of linear equations arises in several applications (e.g. signal and image processing, compressive sensing, statistical inference). A standard tool for dealing with sparse recovery is the ℓ 1 -regularized least-squares approach that has been recently attracting the attention of many researchers.In this paper, we describe an active set estimate (i.e. an estimate of the indices of the zero variables in the optimal solution) for the considered problem that tries to quickly identify as many active variables as possible at a given point, while guaranteeing that some approximate optimality conditions are satisfied. A relevant feature of the estimate is that it gives a significant reduction of the objective function when setting to zero all those variables estimated active. This enables to easily embed it into a given globally converging algorithmic framework.In particular, we include our estimate into a block coordinate descent algorithm for ℓ 1 -regularized least squares, analyze the convergence properties of this new active set method, and prove that its basic version converges with linear rate.Finally, we report some numerical results showing the effectiveness of the approach.Key words. ℓ 1 -regularized least squares, active set, sparse optimization AMS subject classifications. 65K05, 90C25, 90C061. Introduction. The problem of finding sparse solutions to large underdetermined linear systems of equations has received a lot of attention in the last decades. This is due to the fact that several real-world applications can be formulated as linear inverse problems. A standard approach is the so called ℓ 2 -ℓ 1 unconstrained optimization problem: