constraints

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.

A convex optimization problem in conic linear form is an optimization problem of the form CP (d) : maximize c T

In this paper we study practical solution methods for finding the maximum-volume ellipsoid inscribing a given full-dimensional polytope in ! n defined by a finite set of linear inequalities. Our goal is to design a general-purpose algorithmic framework that is reliable and efficient in practice. To evaluate the merit of a practical algorithm, we consider two key factors: the computational cost per iteration and the typical number of iterations required for convergence. In addition, numerical stability is also an important factor. We investigate some new formulations upon which we build primal-dual type, interior-point algorithms, and we provide theoretical justifications for the proposed formulations and algorithmic framework. Extensive numerical experiments have shown that one of the new algorithms should be the method of choice among the tested algorithms.

We present an integrated procedure to build and solve big stochastic programming models. The individual components of the system --the modeling language, the solver and the hardware-- are easily accessible, or a least affordable to a large audience. The procedure is applied to a simple financial model, which can be expanded to arbitrarily large sizes by enlarging the number of scenarios. We generated a model with one million scenarios, whose deterministic equivalent linear program has 1,111,112 constraints and 2,555,556 variables. We have been able to solve it on the cluster of ten PCs in less than 3 hours. Key words. Algebraic modeling language, decomposition methods, distributed systems, large-scale optimization, stochastic programming. 1 Introduction Practical implementations of stochastic programming involve two big challenges. First, we have to build the model: its size is almost invariably large, if not huge, and this task is in itself a challenge for This research was suppor...

Let P = fx j Ax bg, where A is an m \Theta n matrix. We assume that P contains a ball of radius one centered at the origin, and is contained in a ball of radius R centered at the origin. We consider the problem of approximating the maximum volume ellipsoid inscribed in P . Such ellipsoids have a number of interesting applications, including the inscribed ellipsoid method for convex optimization. We reduce the complexity of finding an ellipsoid whose volume is at least a factor e \Gammaffl of the maximum possible to O(m 3:5 ln(mR=ffl)) operations, improving on previous results of Nesterov and Nemirovskii, and Khachiyan and Todd. A further reduction in complexity is obtained by first computing an approximation of the analytic center of P . Keywords: Maximum volume inscribed ellipsoid, inscribed ellisoid method. 1 1

.<F3.866e+05> The feasibility problem for a system of linear inequalities can be converted into an unconstrained optimization problem by using ideas from the ellipsoid method, which can be viewed as a very simple minimization technique for the resulting nonlinear function. This function is related to the volume of an ellipsoid containing all feasible solutions, which is parametrized by certain weights which we choose to minimize the function. The center of the resulting ellipsoid turns out to be a feasible solution to the inequalities. Using more sophisticated nonlinear minimization algorithms, we develop and investigate more e#cient methods, which lead to two kinds of weighted centers for the feasible set. Using these centers, we develop new algorithms for solving linear programming problems.<F3.977e+05> Key words.<F3.866e+05> weighted center, the ellipsoid method, Newton's method, linear programming<F3.977e+05> AMS subject classifications.<F3.866e+05> 65K, 90C<F5.13e+05> 1. Introduct...

The convergence and the complexity of a primal-dual column generation and cutting plane algorithm from approximate analytic centers for solving convex feasibility problems defined by a "deep cut" separation oracle is studied. The primal-dual infeasible Newton method is used to generate a primal-dual updating direction. The number of recentering steps is O(1) for cuts as deep as half way to the deepest cut, where the deepest cut is a cut that is tangent to the primal-dual variant of Dikin's ellipsoid. Keywords: Convex feasibility problem, analytic center, column generation, cutting planes, deep cut. AMS subject classification: 90C25, 90C26, 90C60. 1 This research is supported by the Natural Sciences and Engineering Research Council of Canada, grant number OPG0004152, by the FCAR of Quebec and by an Obermann fellowship at the Center for Advanced Studies at the University of Iowa. 1 Introduction The convex feasibility problem defined by a separation oracle is: find an interior poin...

We consider the problem of approximating fixed points of non-smooth contractive functions with using of the absolute error criterion. In [12]we proved that the upper bound on the number of function evaluations to compute "-approximations is O(n +ln n)) in the worstcase, where 0 !q!1 is the contraction factor and n is the dimension of the problem. This upper bound is achieved by the circumscribed ellipsoid (CE) algorithm combined with a dimensional deflation process. In this paper we present an inscribed ellipsoid (IE) algorithm that enjoys O(n +lnn)) bound. Therefore the IE algorithm has almost the same (modulo multiplicative constant) number of function evaluations as the (nonconstructive) centroid method [11]. We conjecture that this bound is the best possible for mildly contractive functions (q 1) in moderate dimensional case. Affirmative solution of this conjecture would imply that the IE algorithm and the centroid algorithms are almost optimal in the worst case. In particular they are much faster than the simple iteration method, that requires l ln(1=") ln(1=q) m function evaluations to solve the problem.

We study the problem of computing the Minimum Volume Enclosing Ellipsoid (MVEE) containing a given set of ellipsoids S = {E1, E2,..., En} ⊆ Rd. We show how to efficiently compute a small set X ⊆ S of size at most α = |X | = O ( d2) whose minimum volume