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We describe a primal-dual interior point algorithm for linear programming problems which requires a total of O(~fnL) number of iterations, where L is the input size. Each iteration updates a penalty parameter and finds the Newton direction associated with the Karush-Kuhn-Tucker system of equations which characterizes a solution of the logarithmic barrier… (More)

This paper describes the implementation of power series dual affine scaling variants of Karmarkar's algorithm for linear programming. Based on a continuous version of Karmarkar's algorithm, two variants resulting from first and second order approximations of the continuous trajectory are implemented and tested. Linear programs are expressed in an inequality… (More)

We describe an algorithm for linear and convex quadratic programming problems that uses power series approximation of the weighted barrier path that passes through the current iterate in order to find the next iterate. If r > 1 is the order of approximation used, we show that our algorithm has time complexity O(n t(+l/r)L(l+l/r)) iterations and O(n3 + n2r)… (More)

We consider the continuous trajectories of the vector field induced by the primal affine scaling algorithm as applied to linear programming problems in standard form. By characterizing these trajectories as solutions of certain parametrized logarithmic barrier families of problems, we show that these trajectories tend to an optimal solution which in general… (More)

The linear complementarity problem, LCP (q, M), is defined as follows. For given M ∈ R m×m , q ∈ R m , find z such that q + M z ≥ 0, z ≥ 0, z ⊺ (q + M z) = 0, or certify that there is no such z. It is well known that the problem of finding a Nash equilibrium for a bimatrix game (2-NASH) can be formulated as a linear complementarity problem (LCP). In… (More)

We describe a primal-dual interior point algorithm for convex quadratic programming problems which requires a total of O(~/nL) number of iterations, where L is the input size. Each iteration updates a penalty parameter and finds an approximate Newton direction associated with the Karush-Kuhn-Tucker system of equations which characterizes a solution of the… (More)

It has been a challenge for mathematicians to confirm theoretically the extremely good performance of simplex-type algorithms for linear programming. In this paper the average number of steps performed by a simplex algorithm, the so-called <italic>self-dual method</italic>, is analyzed. The algorithm is not started at the traditional point (1, … ,… (More)

A model of two-settlement electricity markets is introduced, which accounts for flow congestion, demand uncertainty, system contingencies, and market power. We formulate the subgame perfect Nash equilibrium for this model as an equilibrium problem with equilibrium constraints (EPEC), in which each firm solves a mathematical program with equilibrium… (More)