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From: paver@ut-ngp.UUCP (Bob Paver)
Newsgroups: net.math,net.math.stat
Subject: Re: New Linear Programming Alogrithm
Message-ID: <1114@ut-ngp.UUCP>
Date: Mon, 10-Dec-84 10:43:20 EST
Article-I.D.: ut-ngp.1114
Posted: Mon Dec 10 10:43:20 1984
Date-Received: Thu, 13-Dec-84 01:39:54 EST
References: <311@sdchema.UUCP>
Organization: U.Texas Computation Center, Austin, Texas
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Xref: watmath net.math:1641 net.math.stat:72


Someone had mentioned this new algorithm to me.  Here's what
I found out.

From "OR/MS Today" (A joint publication of ORSA and TIMS),
Vol 11, No. 5.

Karmarkar Develops Super Fast LP Solution;  Session on Method
Nov. 27 at Dallas Meeting

	Narendra Karmarkar, a young mathematician at AT&T Bell
Laboratories, has developed a new method for solving linear programming
problems that promises to be much faster than the current simplex method
devixed by George Dantzig forty years ago.
	In one test, where Karmarkar ran his program in FORTRAN (a higher
level language and therefore slower) against the fastest simplex program
available (in IBM assembly language) on a 5,000 variable LP problem,
Karmarkar's program was 50 times faster.
	As LP problems become larger, Karmarkar's method becomes even 
more superior.  This is because the simplex method is an exponential time
algorithm, with the solution time increasing as an exponential function
of the size of the problem.
	Karmarkar's method is a polynomial time algorithm where the running
time is a polynomial function of the number variables in the problem.
	The difference between the two types of algorithms can be
dramatic.  For example, a polynomial time algorithm with CHI**3 (chi cubed)
steps can take 1/5 of a second of computer time when CHI is 100.  An
exponential time algorithm with 3**CHI (3 to the CHI power) steps might
require billions of centuries of computer time when CHI is 100.
	In the simplex approach, the computer searches from vertex to
vertex of a multidimensional polytope to find the best one.  In
Karmarkar's approach, the computer goes into the interior and creates
a sequence of spheres inside the polytope that converge on the desired 
vertex.
	Karmarkar's method works in a strange mathematical space,
somewhat akin to the hyperbolic geometry of relativity theory, in
which the polytope is transformed to a regular geometric figure that is 
the multidimensional analog of an equilateral triangle.
	The method has the promise of opening up whole new areas of
inquiry and solving problems which previously were too large to handle.
	A special seminar on the method will be given by Karmarkar
at the ORSA/TIMS meeting in Dallas on Tuesday, November 27, 1984 at
8 a.m.  The announcement of Karmarkar's discovery came too recently
to be included in the Meeting Bulletin.