On extremal behaviors of Murty's least index method

  title={On extremal behaviors of Murty's least index method},
  author={Komei Fukuda and Makoto Namiki},
  journal={Mathematical Programming},
In this small note, we observe some extremal behaviors of Murty's least index method for solving linear complementarity problems. In particular, we show that the expected number of steps for solving Murty's exponential example with a random permutation of variable indices is exactly equal ton, wheren is the size of the input square matrix. 

Pivoting in Linear Complementarity: Two Polynomial-Time Cases

This work solves an open problem of Morris by showing that Murty's least-index pivot rule leads to a quadratic number of iterations on Morris’s highly cyclic P-LCP examples, and shows that on K-matrix LCP instances, all pivot rules require only a linear number of iteration.

The existence of a strongly polynomial time simplex method

It is well known how to clarify whether there is a polynomial time simplex algorithm is the most challenging open problem in optimization and discrete geometry. Under the assumption of

Randomized pivot algorithms for P-matrix linear complementarity problems

  • W. Morris
  • Mathematics, Computer Science
    Math. Program.
  • 2002
It is shown that orientations of the n-cube that come from simple principal pivot algorithms for the linear complementarity problem with a P-matrix properly generalize those that are obtained from linear objective functions on polytopes combinatorially equivalent to the cube.

The existence of a strongly polynomial time simplex algorithm for linear programs

It is shown that there is a simplex algorithm whose number of pivoting steps does not exceed the number of variables of a LP problem.

The existence of a strongly polynomial time simplex algorithm for linear programming problems

A positive answer to the question whether there is a strongly polynomial time simplex algorithm for linear programming (LP) problems is given by using the parameter analysis technique.

On Classes of Unique-Sink Orientations Arising from Pivoting in Linear Complementarity

  • L. Klaus
  • Mathematics, Computer Science
  • 2009
This thesis determines bounds on sizes of USO classes arising from pivoting in P-LCPs and concludes that P-USOs contain much more combinatorial structure, which is needed to prove polynomial runtime of an existing pivot rule or to devise superior rules for simple principal pivoting algorithms.

Criss-cross methods: A fresh view on pivot algorithms

A recent result on the existence of a short admissible pivot path to an optimal basis is given, indicating shortest pivot paths from any basis might be indeed short for criss-cross type algorithms.

The Complexity of the P-Matrix Linear Complementarity Problem

The linear complementarity problem (LCP) is a useful framework for linear and convex programming and has also many direct applications, e.g. in control theory, nance, algorithms and game theory. In

Articles and Scheduling for Student Seminar in Combinatorics: Linear Complementarity

Date Article Presenter(s) September 15 overview, initial planning Komei Fukuda September 22 fixing teams and planning Komei Fukuda September 29 Preparation (no seminar) October 6 QP duality [5] team



A short proof of finiteness of Murty's principal pivoting algorithm

We give a short proof of the finiteness of Murty's principal pivoting algorithm for solving the linear complementarity problemy = Mz + q, y T z = 0,y ≥ 0,z ≥ 0 withP-matrixM.

An exponential example for Terlaky's pivoting rule for the criss-cross simplex method

It is shown that the required number of iterations may be exponential in the number of variables and constraints of the problem.

Some generalizations of the criss-cross method for the linear complementarity problem of oriented matroids

Here some generalizations of Terlaky's finite criss-cross method are presented for oriented matroid quadratic programming and two special cases (oriented matroid linear programming and the definite case) are discussed.


A combinatorial abstraction of the linear complementarity theory in the setting of oriented matroids was first considered by M.J. Todd. In this paper, we take a fresh look at this abstraction, and

Computational complexity of complementary pivot methods

The class of orthants in R n contains 2n orthants. A straight line L in R n is said to cut across an orthant C of R n if L has a nonempty intersection with the interior of C. It is well known that