Joaquim Gromicho

Restricted dynamic programming

Most successful solution methods for solving large vehicle routing and scheduling problems are based on local search. These approaches are designed and optimized for specific types of vehicle routing problems (VRPs). VRPs appearing in practice typically accommodate restrictions that are not accommodated in classical VRP models, such as time-dependent travel times and driving hours regulations. We present a new construction framework for solving VRPs that can handle a wide range of different types of VRPs. In addition, this framework accommodates various restrictions that are not considered in classical vehicle routing models, but that regularly appear in practice. Within this framework, restricted dynamic programming is applied to the VRP through the giant-tour representation. This algorithm is a construction heuristic which for many types of restrictions and objective functions leads to an optimal algorithm when applied in an unrestricted way. We demonstrate the flexibility of the framework for various restrictions appearing in practice. The computational experiments demonstrate that the framework competes with state of the art local search methods when more realistic constraints are considered than in classical VRPs. Therefore, this new framework for solving VRPs is a promising approach for practical applications.

The Role of Robust Optimization in Single-Leg Airline Revenue Management

In this paper, we introduce robust versions of the classical static and dynamic single-leg seat allocation models. These robust models take into account the inaccurate estimates of the underlying probability distributions. As observed by simulation experiments, it turns out that for these robust versions the variability compared to their classical counterparts is considerably reduced with a negligible decrease in average revenue.

A Primal-Dual Decomposition Algorithm for Multistage Stochastic Convex Programming

This paper presents a new and high performance solution method for multistage stochastic convex programming. Stochastic programming is a quantitative tool developed in the field of optimization to cope with the problem of decision-making under uncertainty. Among others, stochastic programming has found many applications in finance, such as asset-liability and bond-portfolio management. However, many stochastic programming applications still remain computationally intractable because of their overwhelming dimensionality. In this paper we propose a new decomposition algorithm for multistage stochastic programming with a convex objective and stochastic recourse matrices, based on the path-following interior point method combined with the homogeneous self-dual embedding technique. Our preliminary numerical experiments show that this approach is very promising in many ways for solving generic multistage stochastic programming, including its superiority in terms of numerical efficiency, as well as the flexibility in testing and analyzing the model.

Solving the job-shop scheduling problem optimally by dynamic programming

Scheduling problems received substantial attention during the last decennia. The job-shop problem is a very important scheduling problem, which is NP-hard in the strong sense and with well-known benchmark instances of relatively small size which attest the practical difficulty in solving it. The literature on the job-shop scheduling problem includes several approximation and exact algorithms. So far, no algorithm is known which solves the job-shop scheduling problem optimally with a lower complexity than the exhaustive enumeration of all feasible solutions. We propose such an algorithm, based on the well-known Bellman equation designed by Held and Karp to find optimal sequences and which offers the best complexity to solve the Traveling Salesman Problem known to this date. For the TSP this means O ( n 2 2 n ) which is exponentially better than O ( n ! ) required to evaluate all feasible solutions. We reach similar results by first recovering the principle of optimality, which is not obtained for free in the case of the job-shop scheduling problem, and by performing a complexity analysis of the resulting algorithm. Our analysis is conservative but nevertheless exponentially better than brute force. We also show very promising results obtained from our implementation of this algorithm, which seem to indicate two things: firstly that there is room for improvement in the complexity analysis (we observe the generation of a number of solutions per state for the benchmark instances considered which is orders of magnitude lower than the bound we could devise) and secondly that the potential practical implications of this approach are at least as exciting as the theoretical ones, since we manage to solve some celebrated benchmark instances in processing times ranging from seconds to minutes.

A deep cut ellipsoid algorithm for convex programming: theory and applications

This paper proposes a deep cut version of the ellipsoid algorithm for solving a general class of continuous convex programming problems. In each step the algorithm does not require more computational effort to construct these deep cuts than its corresponding central cut version. Rules that prevent some of the numerical instabilities and theoretical drawbacks usually associated with the algorithm are also provided. Moreover, for a large class of convex programs a simple proof of its rate of convergence is given and the relation with previously known results is discussed. Finally some computational results of the deep and central cut version of the algorithm applied to a min—max stochastic queue location problem are reported.

Exact Solution of Multiple Traveling Salesman Problem

This paper presents a method developed for the multiple traveling salesman problem (m-TSP), which is a generalization of the well known TSP [6]. In the m-TSP, there are m salesmen who are required to visit n customers in such a way that all customers are visited exactly once by exactly one of the salesmen. Hence, each salesman leaves from and returns to the same point, the depot, and each one of them completes a tour visiting a subset of the customers.

General models in min-max continuous location: theory and solution techniques

In this paper, a class of min-max continuous location problems is discussed. After giving a complete characterization of th stationary points, we propose a simple central and deep-cut ellipsoid algorithm to solve these problems for the quasiconvex case. Moreover, an elementary convergence proof of this algorithm and some computational results are presented.

General models in min-max planar location: checking optimality conditions

AbstractThis paper studies the problem of deciding whether the present iteration point of some algorithm applied to a planar singlefacility min-max location problem, with distances measured by either anlp-norm or a polyhedral gauge, is optimal or not. It turns out that this problem is equivalent to the decision problem of whether 0 belongs to the convex hull of either a finite number of points in the plane or a finite number of differentlq-circles

Duality theory for convex/quasiconvex functions and its application to optimization

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