Giovanna Miglionico

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.

Strategic and operational decisions in restaurant revenue management

The paper addresses restaurant revenue management from both a strategic and an operational point of view. Strategic decisions in restaurants are mainly related to defining the most profitable combination of tables that will constitute the restaurant. We propose new formulations of the so-called “Tables Mix Problem” by taking into account several features of the real setting. We compare the proposed models in a computational study showing that restaurants, with the capacity of managing tables as renewable resources and of combining different-sized tables, can improve expected revenue performances. Operational decisions are mainly concerned with the more profitable assignment of tables to customers. Indeed, the “Parties Mix Problem” consists of deciding on accepting or denying a booking request from different groups of customers, with the aim of maximizing the total expected revenue. A dynamic formulation of the “Parties Mix Problem” is presented together with a linear programming approximation, whose solutions can be used to define capacity control policies based on booking limits and bid prices. Computational results compare the proposed policies and show that they lead to higher revenues than the traditional strategies used to support decision makers.

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.

A bundle modification strategy for convex minimization

We present a new bundle algorithm for minimizing convex not necessarily smooth functions. The novelty of our approach is based on a bundle modification strategy that we apply whenever the stability center is updated and which is aimed at substituting the points of the bundle by new points characterized by possibly better values of the objective function. Convergence of the algorithm is proved and numerical results are presented.

A Revenue Management Approach to Address a Truck Rental Problem

This paper describes an application of revenue management techniques and policies in the field of logistics and distribution. In particular, the problem of transportation operators, who offer products for hire, is considered. A product is a truck of a given capacity, which can be rented for one or several time periods, throughout a multi-period planning horizon. The logistic operator can satisfy the demand of a given product with trucks with a capacity greater than that initially required, that is an ‘upgrade’ can take place. In this context, the logistic operator has to decide whether to accept or reject a request and which type of truck should be used to address it. For this purpose, a dynamic programming (DP) formulation of the problem under consideration is devised. The ‘course of dimensionality’ leads to the necessity of introducing different mathematical programming models to represent the problem. The mathematical models we presented are an extension of the well-known approximations for the DP of traditional network capacity management analysis. Based on these models and exploiting revenue management concepts, primal and dual acceptance policies are developed and compared in a computational study.

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.

Location and reorganization problems: The Calabrian health care system case

In the last few years, the Italian healthcare system has been coping with radical changes, aimed at guaranteeing more efficiency while containing costs. Starting from the actual service network organization, we discuss the problem faced by the Italian authorities of reorganizing the healthcare service network and we propose some optimization models to support the decision-making process. In the first part of the work, we compare the existing health care service network of the northern area of Calabria (Italy) with the configurations determined by solving well-known facility location models. In the second part, taking into account the healthcare reorganization plans imposed by local governments, we consider the problem of reorganizing the public health care service network of the northern area of Calabria. Indeed, we propose two ad-hoc optimization models that consider national and regional guidelines and constraints. The behavior of the proposed models, in terms of solution quality, is evaluated on the basis of an extensive computational study on real data.

Managing TV commercials inventory in the Italian advertising market

This paper studies the problem of a TV broadcaster, managing its programs schedule. A TV network is considered that has to decide on accepting and scheduling requests from its advertisers. A request is formulated both in terms of people to be reached and of budget available for the advertisement campaign. Moreover, several formulations of the problem are presented and defined in such a way as to handle different aspects of the real process: compatibility among advertisements that are scheduled in the same break; compatibility between advertisements and breaks; price discrimination due to the different positions of the advertisements within a break. Since the introduced models are computationally intractable, several heuristics are presented whose performance are evaluated in an extensive computational study based on test problems defined by considering the peculiarities of the Italian advertisements market.

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.