Interior methods for optimization were widely used in the 1960s, primarily in the form of barrier methods. However, they were not seriously applied to linear programming because of the dominance of the simplex method. Barrier methods fell from favour during the 1970s for a variety of reasons, including their apparent inefficiency compared with the best available alternatives. In 1984, Karmarkar's announcement of a fast polynomial-time interior method for linear programming caused tremendous excitement in the field of optimization. A formal connection can be shown between his method and classical barrier methods, which have consequently undergone a renaissance in interest and popularity. Most papers published since 1984 have concentrated on issues of computational complexity in interior methods for linear programming. During the same period, implementations of interior methods have displayed great efficiency in solving many large linear programs of ever-increasing size. Interior methods...

In this paper we describe an interior point mathematical programming approach to inductive inference. We list several versions of this problem and study in detail the formulation based on hidden Boolean logic. We consider the problem of identifying a hidden Boolean function F : f0; 1g n ! f0; 1g using outputs obtained by applying a limited number of random inputs to the hidden function. Given this input-output sample, we give a method to synthesize a Boolean function that describes the sample. We pose the Boolean Function Synthesis Problem as a particular type of Satisfiability Problem. The Satisfiability Problem is translated into an integer programming feasibility problem, that is solved with an interior point algorithm for integer programming. A similar integer programming implementation has been used in a previous study to solve randomly generated instances of the Satisfiability Problem. In this paper we introduce a new variant of this algorithm, where the Riemannian metric used...

We consider the Riemannian geometry defined on a convex set by the Hessian of a selfconcordant barrier function, and its associated geodesic curves. These provide guidance for the construction of efficient interior-point methods for optimizing a linear function over the intersection of the set with an affine manifold. We show that algorithms that follow the primal-dual central path are in some sense close to optimal. The same is true for methods that follow the shifted primal-dual central path among certain infeasible-interior-point methods. We also compute the geodesics in several simple sets. ∗ Copyright (C) by Springer-Verlag. Foundations of Computational Mathewmatics 2 (2002), 333–361.

We consider the problem of optimizing a quadratic function subject to integer constraints. This problem is NP-hard in the general case. We present a new polynomial time algorithm for computing bounds on the solutions to such optimization problems. We transform the problem into a problem for minimizing the trace of a matrix subject to positive definiteness condition. We then propose an interior-point method to solve this problem. We show that the algorithm takes no more than O(nL) iterations (where L is the the number of bits required to represent the input). The algorithm does two matrix inversions in each iteration . Keywords: Bounds, complexity, quadratic optimization, interior point methods. 1 Outline The second section of the paper shall introduce the problem of computing upper bounds on a quadratic optimization problem. We shall also motivate an interior point approach to solving the problem. The third section gives an interior point method for solving the problem. The algorith...

Abstract. We prove a linear bound on the average total curvature of the central path of linear programming theory in terms on the number of variables. 1 Introduction. In this paper we study the curvature of the central path of linear programming theory. We establish that for a linear programming problem defined on a compact polytope contained in R n, the total curvature of the central path is less than or

This study examines two different barrier functions and their use in both path-following and potential-reduction interior-point algorithms for solving a linear program of the form: minimize c T x subject to Ax = b and ` x u, where components of ` and u can be nonfinite, so the variables x j can have 0\Gamma; 1\Gamma;or 2-sided bounds, j = 1; :::; n: The barrier functions that we study include an extension of the standard logarithmic barrier function and an extension of a barrier function introduced by Nesterov. In the case when both ` and u have all of their components finite, these barrier functions are \Psi(x) = X j f\Gamma ln(u j \Gamma x j ) \Gamma ln(x j \Gamma ` j )g and \Psi(x) = X j f\Gamma ln(minfu j \Gamma x j ; x j \Gamma ` j g) + minfu j \Gamma x j ; x j \Gamma ` j g=((u j \Gamma ` j )=2)g: Each of these barrier functions gives rise to suitable primal and dual metrics that are used to develop both path-following and potential-reduction interior-point algorithms ...