Shinji Mizuno

A primal-dual interior point algorithm for linear programming

This chapter presents an algorithm that works simultaneously on primal and dual linear programming problems and generates a sequence of pairs of their interior feasible solutions. Along the sequence generated, the duality gap converges to zero at least linearly with a global convergence ratio (1 — η/n); each iteration reduces the duality gap by at least η/n. Here n denotes the size of the problems and η a positive number depending on initial interior feasible solutions of the problems. The algorithm is based on an application of the classical logarithmic barrier function method to primal and dual linear programs, which has recently been proposed and studied by Megiddo.

On Adaptive-Step Primal-Dual Interior-Point Algorithms for Linear Programming

We describe several adaptive-step primal-dual interior point algorithms for linear programming. All have polynomial time complexity while some allow very long steps in favorable circumstances. We provide heuristic reasoning for expecting that the algorithms will perform much better in practice than guaranteed by the worst-case estimates, based on an analysis using a nonrigorous probabilistic assumption.

A polynomial-time algorithm for a class of linear complementary problems

Given ann × n matrixM and ann-dimensional vectorq, the problem of findingn-dimensional vectorsx andy satisfyingy = Mx + q, x ≥ 0,y ≥ 0,xiyi = 0 (i = 1, 2,⋯,n) is known as a linear complementarity problem. Under the assumption thatM is positive semidefinite, this paper presents an algorithm that solves the problem in O(n3L) arithmetic operations by tracing the path of centers,{(x, y) ∈ S: xiyi =μ (i = 1, 2,⋯,n) for some μ > 0} of the feasible regionS = {(x, y) ≥ 0:y = Mx + q}, whereL denotes the size of the input data of the problem.

A primal-dual infeasible-interior-point algorithm for linear programming

As in many primal—dual interior-point algorithms, a primal—dual infeasible-interior-point algorithm chooses a new point along the Newton direction towards a point on the central trajectory, but it does not confine the iterates within the feasible region. This paper proposes a step length rule with which the algorithm takes large distinct step lengths in the primal and dual spaces and enjoys the global convergence.

An O n L iteration potential reduction algorithm for linear complementary problems

AbstractThis paper proposes an interior point algorithm for a positive semi-definite linear complementarity problem: find an (x, y)∈ℝ2n such thaty=Mx+q, (x,y)⩾0 andxTy=0. The algorithm reduces the potential function

Infeasible-Interior-Point Primal-Dual Potential-Reduction Algorithms for Linear Programming

f(x,y) = (n + \sqrt n )\log x^T y - \sum\limits_{i = 1}^n {\log x_i y_i }

A new polynomial time method for a linear complementarity problem

by at least 0.2 in each iteration requiring O(n3) arithmetic operations. If it starts from an interior feasible solution with the potential function value bounded by