We discuss formulations of integer programs with a huge number of variables and their solution by column generation methods, i.e., implicit pricing of nonbasic variables to generate new columns or to prove LP optimality at a node of the branchand-bound tree. We present classes of models for which this approach decomposes the problem, provides tighter LP relaxations, and eliminates symmetry. Wethen discuss computational issues and implementation of column generation, branch-andbound algorithms, including special branching rules and e cient ways to solve the LP relaxation. We also discuss the relationship with Lagrangian duality. 1

We present a survey of nondifferentiable optimization problems and methods with special focus on the analytic center cutting plane method. We propose a self-contained convergence analysis, that uses the formalism of the theory of self-concordant functions, but for the main results, we give direct proofs based on the properties of the logarithmic function. We also provide an in depth analysis of two extensions that are very relevant to practical problems: the case of multiple cuts and the case of deep cuts. We further examine extensions to problems including feasible sets partially described by an explicit barrier function, and to the case of nonlinear cuts. Finally, we review several implementation issues and discuss some applications.

The paper deals with nonlinear multicommodity flow problems with convex costs. A decomposition method is proposed to solve them. The approach applies a potential reduction algorithm to solve the master problem approximately and a column generation technique to define a sequence of primal linear programming problems. Each subproblem consists of finding a minimum cost flow between an origin and a destination node in an uncapacited network. It is thus formulated as a shortest path problem and solved with the Dijkstra's d-heap algorithm. An implementation is described that that takes full advantage of the supersparsity of the network in the linear algebra operations. Computational results show the efficiency of this approach on well-known nondifferentiable problems and also large scale randomly generated problems (up to 1000 arcs and 5000 commodities). This research has been supported by the Fonds National de la Recherche Scientifique Suisse, grant #12 \Gamma 34002:92, NSERC-Canada and ...

The formulation of a wide variety of image recovery problems leads to the minimization of a convex objective over a convex set representing the constraints derived from a priori knowledge and consistency with the observed signals. In recent years, nondifferentiable objectives have become popular due in part to their ability to capture certain features such as sharp edges. They also arise naturally in minimax inconsistent set theoretic recovery problems. At the same time, the issue of developing reliable numerical algorithms to solve such convex programs in the context of image recovery applications has received little attention. In this paper, we address this issue and propose an adaptive level set method for nondifferentiable constrained image recovery. The asymptotic properties of the method are analyzed and its implementation is discussed. Numerical experiments illustrate applications to total variation and minimax set theoretic image restoration and denoising problems.

We consider a problem in which a given good has to be delivered from some origins (say production plants), to some destinations (say nodes at which the transportation mode is changed, or simply customers), during a workday, by means of a given fleet of trucks, at minimum cost. For the purpose of solution the problem is split into two levels, where, at the first level, the decision concerns the planning of trips in order to deliver goods, while, at the second level, the vehicles needed to operate the trips have to be scheduled. The solution approach presented here is based on Lagrangean Decomposition and makes use of a new algorithm for the approximate solution of the Lagrangean Dual. Computational results from a set of real life problems are presented. Keywords: Lagrangean Relaxation; Optimisation; Transportation; Vehicle Scheduling Revised January 1996 1 This research has been supported by "Progetto Finalizzato Trasporti 2" of the Italian National Research Council (C.N.R.) and by Fon...

Optimal control of the 1-D Riemann problem of Euler equations whose solution is characterized by discontinuities is carried out by both nonsmooth and smooth op- timization methods. By matching a desired flow to the numerical model for a given time window we effectively change the location of discontinuities. The control pa- rameters are chosen to be the initial values for pressure and density fields. Existence of solutions for the optimal control problem is proven. A high resolution model and a model with artificial viscosity, implementing two different numerical methods, are used to solve the forward model. The cost functional is the weighted difference be- tween the numerical values and the observations for pressure, density and velocity. The observations are constructed from the analytical solution. We consider either distributed observations in time or observations calculated at the end of the assimi- lation window. We consider two different time horizons and two sets of observations. The gradient (respectively a subgradient) of the cost functional, obtained from the adjoint of the discrete forward model, are employed for the smooth minimization (respectively for the nonsmooth minimization) algorithm. Discontinuity detection improves the performance of the minimizer for the model with artificial viscosity by selecting the points where the shock occurs (and these points are then removed from Preprint submitted to Elsevier Science 26 March 2002 the cost functional and its gradient). The numerical flow obtained with the optimal initial conditions obtained from the nonsmooth minimization matches very well the observations. The algorithm for smooth minimization converges for the shorter time horizon but fails to perform satisfactorily for the longer time horizon.

Abstract. The GlobSol software package combines various ideas from interval analysis, automatic differentiation, and constraint propagation to provide verified solutions to unconstrained and constrained global optimization problems. After briefly reviewing some of these techniques and GlobSol’s development history, we provide the first overall description of GlobSol’s algorithm. Giving advice on use, we point out strengths and weaknesses in GlobSol’s approaches. Through examples, we show how to configure and use GlobSol.

We present the main freight transportation planning and management issues, briefly review the associated literature, describe a number of major developments, and identify trends and challenges. In order to keep the length of the paper within reasonable limits, we focus on long-haul, intercity, freight transportation. Optimization-based operations research methodologies are privileged. The paper starts with an overview of freight transportation systems and planning issues and continues with models which attempt to analyze multimodal, multicommodity transportation systems at the regional, national or global level. We then review location and network design formulations which are often associated with the long-term evolution of transportation systems and also appear prominently when service design issues are considered as described later on. Operational models and methods, particularly those aimed at the allocation and repositioning of resources such as empty vehicles, are then described....

It is with the aim of solving scheduling problems with irregular cost functions that this paper focuses on the continuous assignment problem. It consists in partitioning a region of R into subregions of prescribed volumes so that the total cost is minimized.

Abstract—The classification, monitoring, and compression of electrocardiogram (ECG) signals recorded of a single patient over a relatively long period of time is considered. The particular application we have in mind is high-resolution ECG analysis, such as late potential analysis, morphology changes in QRS during arrythmias,-wave alternants, or the study of drug effects on ventricular activation. We propose to apply a modification of a classical method of cluster analysis or vector quantization. The novelty of our approach is that we use a new distortion measure to quantify the distance of two ECG cycles, and the class-distortion measure is defined using a min-max criterion. The new class-distortion-measure is much more sensitive to outliers than the usual distortion measures using average-distance. The price of this practical advantage is that computational complexity is significantly increased. The resulting nonsmooth optimization problem is solved by an adapted version of the simultaneous perturbation stochastic approximation (SPSA) method of [1]. The main idea is to generate a smooth approximation by a randomization procedure. The viability of the method is demonstrated on both simulated and real data. An experimental comparison with the widely used correlation method is given on real data. Index Terms—Classification, ECG, data-compression, min-max problems, monitoring, nonsmooth optimization, randomization,