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:

Abstract. Mixed integer optimization is a powerful tool for modeling many optimization problems arising from real-world applications. Finding a first feasible solution represents the first step for several mixed integer programming (MIP) solvers. The feasibility pump is a heuristic for finding feasible solutions to mixed integer linear programming (MILP) problems which is effective even when dealing with hard MIP instances. In this work, we start by interpreting the feasibility pump as a Frank-Wolfe method applied to a nonsmooth concave merit function. Then we define a general class of functions that can be included in the feasibility pump scheme for measuring solution integrality, and we identify some merit functions belonging to this class. We further extend our approach by dynamically combining two different merit functions. Finally, we define a new version of the feasibility pump algorithm, which includes the original version of the feasibility pump as a special case, and we present computational results on binary MILP problems showing the effectiveness of our approach. In the literature, several heuristics methods for finding a first feasible solution for an MIP problem have been proposed (see, e.g., [4,5,6,10,21,22,23,24,25,26,29]). Recently, Fischetti, Glover, and Lodi [18] proposed a new heuristic, the well-known feasibility pump (FP), which turned out to be very useful in finding a first feasible solution even when dealing with hard MIP instances. The FP heuristic is implemented in various MIP solvers such as BONMIN [11].The basic idea of the FP is that of generating two sequences of points {x k } and {x k } such thatx k is LP-feasible, but may not be integer feasible, andx k is integer, but not necessarily LP-feasible. To be more specific, the algorithm starts with a solution of the LP relaxationx 0 and setsx 0 equal to the rounding ofx 0 . Then at each iterationx k+1 is chosen as the nearest LP-feasible point in 1 -norm tox k , and x k+1 is obtained as the rounding ofx k+1 . The aim of the algorithm is to reduce at each iteration the distance between the points of the two sequences, until the two points are the same and an integer feasible solution is found. Unfortunately, it can happen that the distance betweenx k+1 andx k is greater than zero andx k+1 =x k , and the strategy can stall. In order to overcome this drawback, random perturbations

In this paper, we describe a two-stage method for solving optimization problems with bound constraints. It combines the active-set estimate described in [15] with a modification of the non-monotone line search framework recently proposed in [14]. In the first stage, the algorithm exploits a property of the active-set estimate that ensures a significant reduction in the objective function when setting to the bounds all those variables estimated active. In the second stage, a truncated-Newton strategy is used in the subspace of the variables estimated non-active. In order to properly combine the two phases, a proximity check is included in the scheme. This new tool, together with the other theoretical features of the two stages, enables us to prove global convergence. Furthermore, under additional standard assumptions, we can show that the algorithm converges at a superlinear rate. Promising experimental results demonstrate the effectiveness of the proposed method.

We propose a branch-and-bound algorithm for minimizing a not necessarily convex quadratic function over integer variables. The algorithm is based on lower bounds computed as continuous minima of the objective function over appropriate ellipsoids. In the nonconvex case, we use ellipsoids enclosing the feasible region of the problem. In spite of the nonconvexity, these minima can be computed quickly; the corresponding optimization problems are equivalent to trust-region subproblems. We present several ideas that allow to accelerate the solution of the continuous relaxation within a branch-and-bound scheme and examine the performance of the overall algorithm by computational experiments. Good computational performance is shown especially for ternary instances.

Multiobjective mixed integer convex optimization refers to mathematical programming problems where more than one convex objective function needs to be optimized simultaneously and some of the variables are constrained to take integer values. We present a branch-and-bound method based on the use of properly defined lower bounds. We do not simply rely on convex relaxations, but we built linear outer approximations of the image set in an adaptive way. We are able to guarantee correctness in terms of detecting both the efficient and the nondominated set of multiobjective mixed integer convex problems according to a prescribed precision. As far as we know, the procedure we present is the first deterministic algorithm devised to handle this class of problems. Our numerical experiments show results on biobjective and triobjective mixed integer convex instances.