SeDuMi is an add-on for MATLAB, that lets you solve optimization problems with linear, quadratic and semidefiniteness constraints. It is possible to have complex valued data and variables in SeDuMi. Moreover, large scale optimization problems are solved efficiently, by exploiting sparsity. This paper describes how to work with this toolbox.

1 Introduction Consider the linear programming (LP) problem in the standard form: (LP) minimize cT x

We present polynomial-time interior-point algorithms for solving the Fisher and Arrow-Debreu competitive market equilibrium problems with linear utilities and n players. Both of them have the arithmetic operation complexity bound of O(n 4 log(1/ɛ)) for computing an ɛ-equilibrium solution. If the problem data are rational numbers and their bit-length is L, then the bound to generate an exact solution is O(n 4 L) which is in line with the best complexity bound for linear programming of the same dimension and size. This is a significant improvement over the previously best bound O(n 8 log(1/ɛ)) for approximating the two problems using other methods. The key ingredient to derive these results is to show that these problems admit convex optimization formulations, efficient barrier functions and fast rounding techniques. We also present a continuous path leading to the set of the Arrow-Debreu equilibrium, similar to the central path developed for linear programming interior-point methods. This path is derived from the weighted logarithmic utility and barrier functions and the Brouwer fixed-point theorem. The defining equations are bilinear and possess some primal-dual structure for the application of the Newton-based path-following method.

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We consider an infeasible-interior-point algorithm, endowed with a finite termination scheme, applied to random linear programs generated according to a model of Todd. Such problems have degenerate optimal solutions, and possess no feasible starting point. We use no information regarding an optimal solution in the initialization of the algorithm. Our main result is that the expected number of iterations before termination with an exact optimal solution is O(n ln(n)). Keywords: Linear Programming, Average-Case Behavior, Infeasible-Interior-Point Algorithm. Running Title: Probabilistic Analysis of an LP Algorithm 1 Dept. of Management Sciences, University of Iowa. Supported by an Interdisciplinary Research Grant from the Center for Advanced Studies, University of Iowa. 2 Dept. of Mathematics, Valdosta State University. Supported by an Interdisciplinary Research Grant from the Center for Advanced Studies, University of Iowa. 3 Dept. of Mathematics, University of Iowa. Supported by ...

this paper wehave selected the primal-dual logarithmic barrier algorithm to present our ideas, because it and its modified versions are considered, in general, to be the most efficient in practice. The computational results presented in this paper were obtained using implementations of this algorithm. It is to be noted, however, that this choice has notational consequences only. Practically,anyinterior point method, even nonlinear ones can be discussed in a similar linear algebra framework. Let us consider the linear programming problem

We describe Fortran subroutines for network flow optimization using an interior point network flow algorithm, that, together with a Fortran language driver, make up PDNET. The algorithm is described in detail and its implementation is outlined. Usage of the package is described and some computational experiments are reported. Source code for the software can be downloaded at

This paper studies a primal-dual interior/exterior-point path-following approach for linearprogramming that is motivated on using an iterative solver rather than a direct solver for the search direction. We begin with the usual perturbed primal-dual optimality equations Fu(x, y, z) = 0. Under nondegeneracy assumptions, this nonlinear system is well-posed,i.e. it has a nonsingular Jacobian at optimality and is not necessarily ill-conditioned as the iterates approach optimality. We use a simple preprocessing step to eliminate boththe primal and dual feasibility equations. This results in a single bilinear equation that maintains the well-posedness property. We then apply both a direct solution techniqueas well as a preconditioned conjugate gradient method (PCG), within an inexact Newton framework, directly on the linearized equations. This is done without forming the usualnormal equations, NEQ, or augmented system. Sparsity is maintained. The work of aniteration for the PCG approach consists almost entirely in the (approximate) solution of this well-posed linearized system. Therefore, improvements depend on efficient preconditioning.

We study numerical stability for interior-point methods applied to Linear Programming, LP, and Semidefinite Programming, SDP. We analyze the di#culties inherent in current methods and present robust algorithms.

Interior-point methods in mathematical programming have been the largest and most dramatic area of research in optimization since the development of the simplex method... Interior-point methods have permanently changed the landscape of mathematical programming theory, practice and computation... (Freund & Mizuno 1996). Major impacts on The linear programming problem (LP) The quadratic programming problem (QP) The linear complementarity problem (LCP) The semi-definite programming problem (SDP) Some classes of convex programming problems UMBC ◭ � ◮ Interior Point Methods – p.2/30The linear programming problem min x c T x s.t. Ax = b, x ≥ 0. Dantzig (1947-1951): the simplex method – good practical performance – exponential worst case complexity (Klee and Minty (1972)) UMBC ◭ � ◮ Interior Point Methods – p.3/30The linear programming problem min x c T x s.t. Ax = b, x ≥ 0. Dantzig (1947-1951): the simplex method – good practical performance – exponential worst case complexity (Klee and Minty (1972)) Question: Is (LP) solvable in polynomial time? (in terms of: L = bitlength of data, and n = dim(x)) UMBC ◭ � ◮ Interior Point Methods – p.3/30The linear programming problem min x c T x s.t. Ax = b, x ≥ 0. Dantzig (1947-1951): the simplex method – good practical performance – exponential worst case complexity (Klee and Minty (1972)) Question: Is (LP) solvable in polynomial time? (in terms of: L = bitlength of data, and n = dim(x))