this paper we examine the underlying ideas of the SQP method and the theory that establishes it as a framework from which effective algorithms can

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.

This paper is concerned with the longstanding problem of optimal unit commitment in an electric power system. We follow the traditional formulation of this problem which gives rise to a large-scale, dynamic, mixed-integer programming problem. We describe a solution methodology based on duality, Lagrangian relaxation and nondifferentiable optimization that has two unique features. First, computational requirements typically grow only linearly witb the number of generating units. Second, the duality gap decreases in relative terms as the number of units increases, and as a result our algorithm tends to actually perform better for problems of large size. This allows for the first time consistently reliable solution of large practical problems involving several hundreds of units within realistic time constraints. Aside from the unit commilment problem. this methodology is applicable to a broad class of large-scale dpamic scheduling and resource allocation problems involving integer variables.

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There are many problems related to the design of networks. Among them, the message routing problem plays a determinant role in the optimization of network performance. Much of the motivation for this work comes from this problem which is shown to belong to the class of nonlinear convex multicommodity flow problems. This paper emphasizes the message routing problem in data networks, but it includes a broader literature overview of convex multicommodity flow problems. We present and discuss the main solution techniques proposed for solving this class of largescale convex optimization problems. We conduct some numerical experiments on the message routing problem with some different techniques. 1 Introduction The literature dealing with multicommodity flow problems is rich since the publication of the works of Ford and Fulkerson's [19] and T.C. Hu [30] in the beginning of the 1960s. These problems usually have a very large number of variables and constraints and arise in a great variety o...

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We describe the analytic center cutting plane method and its relationship to classical methods of nondifferentiable optimization and column generation. Implementation issues are also discussed, and current applications listed. Keywords Projective Algorithm, Analytic Center, Cutting Plane Method. This work has been completed with support from the Fonds National Suisse de la Recherche Scientifique, grant 12-42503.94, from the Natural Sciences and Engineering Research Council of Canada, grant number OPG0004152 and from the FCAR of Quebec. GERAD/Faculty of Management, McGill University, 1001, Sherbrooke West, Montreal, Que., H3A 1G5, Canada. E-mail: ma56@musica.mcgill.ca. LOGILAB/Management Studies, University of Geneva, 102, Bd Carl-Vogt, CH-1211 Gen`eve 4, Switzerland. E-mail: jpvial@hec.unige.ch. 1 Introduction Nondifferentiable convex optimization may be deemed an arcane topic in the field of optimization. Truly enough, many a times problems that are formulated as nondiffere...

In this paper, we present a study of the proximal point algorithm using very general regularizations for minimizing possibly nondierentiable and nonconvex locally Lipschitz functions. We deduce from the proximal point scheme simple and implementable bundle methods for the convex and nonconvex cases. The originality of our bundle method is that the bundle information incorporates the subgradients of both the objective and the regularization function. The resulting method opens up a broad class of regularizations which are not restricted to quadratic, convex or even dierentiable functions. Keywords: mathematical programming, proximal point, bundle methods, nonsmooth regularization This work was partially supported by the Department of Defense Research & Engineering (DDR&E) Multidisciplinary University Research Initiative (MURI) on "Reduced Signature Target Recognition" managed by the Air Force OÆce of Scientic Research (AFOSR) under AFOSR grant AFOSR F49620-96-0028. Chretien and He...

INTRODUCTION, APPLICATIONS AND ALGORITHMS, Nondifferentiable Optimization Introduction. Nondifferentiable, also known as nonsmooth, optimization (NDO) is concerned with problems where the smoothness assumption on the functions involved is relaxed. Nondifferentiability means that the gradient does not exist, implying that the function may have kinks or corner points. Consequently, the function cannot be approximated locally by a tangent hyperplane, or by a quadratic approximation. In NDO, the smoothness assumption is usually replaced by weaker ones, which at least guarantee the existence of directional derivatives. NDO problems arise in a variety of contexts, and methods designed for smooth optimization may fail to solve them. This justifies developing a specialized theory and methods that are the object of this short introduction. In the sequel, we will often refer to convex NDO, a subclass of nondifferentiable optimization,