Giovanni Giallombardo

Minimizing Nonconvex Nonsmooth Functions via Cutting Planes and Proximity Control

We describe an extension of the classical cutting plane algorithm to tackle the unconstrained minimization of a nonconvex, not necessarily differentiable function of several variables. The method is based on the construction of both a lower and an upper polyhedral approximation to the objective function and is related to the use of the concept of proximal trajectory. Convergence to a stationary point is proved for weakly semismooth functions.

A DC piecewise affine model and a bundling technique in nonconvex nonsmooth minimization

We introduce an algorithm to minimize a function of several variables with no convexity nor smoothness assumptions. The main peculiarity of our approach is the use of an objective function model which is the difference of two piecewise affine convex functions. Bundling and trust region concepts are embedded into the algorithm. Convergence of the algorithm to a stationary point is proved and some numerical results are reported.

An Incremental Method for Solving Convex Finite Min-Max Problems

We introduce a new approach to minimizing a function defined as the pointwise maximum over finitely many convex real functions (next referred to as the component functions), with the aim of working on the basis of incomplete knowledge of the objective function. A descent algorithm is proposed, which need not require at the current point the evaluation of the actual value of the objective function, namely, of all the component functions, thus extending to min-max problems the philosophy of the incremental approaches, widely adopted in the nonlinear least squares literature. Given the nonsmooth nature of the problem, we resort to the well-established machinery of bundle methods. We provide global convergence analysis of our method, and in addition, we study a subgradient aggregation scheme aimed at simplifying the problem of finding a tentative step. This paper is completed by the numerical results obtained on a set of standard test problems.

On solving the Lagrangian dual of integer programs via an incremental approach

Abstract The Lagrangian dual of an integer program can be formulated as a min-max problem where the objective function is convex, piecewise affine and, hence, nonsmooth. It is usually tackled by means of subgradient algorithms, or multiplier adjustment techniques, or even more sophisticated nonsmooth optimization methods such as bundle-type algorithms. Recently a new approach to solving unconstrained convex finite min-max problems has been proposed, which has the nice property of working almost independently of the exact evaluation of the objective function at every iterate-point. In the paper we adapt the method, which is of the descent type, to the solution of the Lagrangian dual. Since the Lagrangian relaxation need not be solved exactly, the approach appears suitable whenever the Lagrangian dual must be solved many times (e.g., to improve the bound at each node of a branch-and-bound tree), and effective heuristic algorithms at low computational cost are available for solving the Lagrangian relaxation. We present an application to the Generalized Assignment Problem (GAP) and discuss the results of our numerical experimentation on a set of standard test problems.

Minimizing nonsmooth DC functions via successive DC piecewise-affine approximations

We introduce a proximal bundle method for the numerical minimization of a nonsmooth difference-of-convex (DC) function. Exploiting some classic ideas coming from cutting-plane approaches for the convex case, we iteratively build two separate piecewise-affine approximations of the component functions, grouping the corresponding information in two separate bundles. In the bundle of the first component, only information related to points close to the current iterate are maintained, while the second bundle only refers to a global model of the corresponding component function. We combine the two convex piecewise-affine approximations, and generate a DC piecewise-affine model, which can also be seen as the pointwise maximum of several concave piecewise-affine functions. Such a nonconvex model is locally approximated by means of an auxiliary quadratic program, whose solution is used to certify approximate criticality or to generate a descent search-direction, along with a predicted reduction, that is next explored in a line-search setting. To improve the approximation properties at points that are far from the current iterate a supplementary quadratic program is also introduced to generate an alternative more promising search-direction. We discuss the main convergence issues of the line-search based proximal bundle method, and provide computational results on a set of academic benchmark test problems.

Numerical infinitesimals in a variable metric method for convex nonsmooth optimization

Abstract The objective of the paper is to evaluate the impact of the infinity computing paradigm on practical solution of nonsmooth unconstrained optimization problems, where the objective function is assumed to be convex and not necessarily differentiable. For such family of problems, the occurrence of discontinuities in the derivatives may result in failures of the algorithms suited for smooth problems. We focus on a family of nonsmooth optimization methods based on a variable metric approach, and we use the infinity computing techniques for numerically dealing with some quantities which can assume values arbitrarily small or large, as a consequence of nonsmoothness. In particular we consider the case, treated in the literature, where the metric is defined via a diagonal matrix with positive entries. We provide the computational results of our implementation on a set of benchmark test-problems from scientific literature.

Minimizing Piecewise-Concave Functions Over Polyhedra

We introduce an iterative method for solving linearly constrained optimization problems, whose nonsmooth nonconvex objective function is defined as the pointwise maximum of finitely many concave functions. Such problems often arise from reformulations of certain constraint structures (e.g., binary constraints, finite max-min constraints) in diverse areas of optimization. We state a local optimization strategy, which exploits piecewise-concavity of the objective function, giving rise to a linearized model corrected by a proximity term. In addition we introduce an approximate line-search strategy, based on a curvilinear model, which, similarly to bundle methods, can return either a satisfactory descent or a null-step declaration. Termination at a point satisfying an approximate stationarity condition is proved. We embed the local minimization algorithm into a Variable Neighborhood Search scheme and a Coordinate Direction Search heuristic, whose aim is to improve the current estimate of the global minimizer ...

New Formulations for the Conflict Resolution Problem in the Scheduling of Television Commercials

We consider the conflict-resolution problem arising in the allocation of commercial advertisements to television program breaks. Due to the competition-avoidance requirements issued by advertisers, broadcasters aim to allocate any pairs of commercials promoting highly conflicting products to different breaks. Hence, the problem consists of assigning commercials to breaks, subject to time capacity constraints, with the aim of maximizing a total measure of the conflicts among commercials assigned to different breaks. Since the existing reformulation can hardly be solved via exact methods, we introduce three new and efficient (mixed-)integer programming reformulations of the problem. Our computational study is based on two sets of test problems, one from the literature and another that we generate. Numerical results show the excellent performance of the proposed reformulations in terms of solution quality and computation times, when compared against an existing reformulation and an effective heuristic approach. We also provide theoretical evidences to demonstrate why some of our new reformulations should outperform the existing reformulation.

A savings-based model for two-shipper cooperative routing

We introduce a methodology for operational planning of cooperation between two independent shippers who manage their own fleets of vehicles in a given geographic area. We assume that shippers are willing to establish partial cooperation by sharing only a subset of customers. Our approach is based on the iterative attempt of identifying subsets of shareable customers which can be fruitfully exchanged between shippers. We resort to classic concepts of vehicle routing literature such as savings and insertion costs, providing both a heuristic and an exact approach.

Optimal Replenishment Order Placement in a Finite Time Horizon

Import companies operating in the globalized market and in a multi-item context are frequently faced with the need of aggregating their orders to benefit from scale economies associated to the use of containers for freight shipping.We introduce the problem of optimal replenishment order placement, namely the problem of scheduling and aggregating a fixed number of replenishment orders, along a given planning horizon, with the aim of minimizing the total inventory and backorder costs.For this problem, a continuous optimization formulation is provided, which is characterized by a nonconvex piecewise affine objective function. We introduce an ad hoc algorithm, based on the coordinate search approach, for the approximate numerical solution of the continuous model, and present the numerical results obtained on a set of randomly generated instances. In order to evaluate the quality of solutions returned by the algorithm, a discrete formulation of the problem is also provided, which falls into the well-known class of uncapacitated location models of the p-median type, whose exact solution on the same instance set can be obtained by means of an integer programming commercial solver.