A stable multiple exchange algorithm for linear sip

Abstract

The subject of linear optimization belongs to those branches in mathematics which are widely applied in economics and natural sciences. Especially a “good” solving of linear optimization problems on a computer is of importance today and users are looking for fast and stable methods. So it gives much reason for further research in this subject.

There are different methods to compute an optimal solution of a linear optimization problem, of which the exchange algorithms (simplex method by DANTZIG[51]) and the descent algorithms (gradient methods) are the most important ones. And it seems today that the exchange methods have outrun the descent methods cosidering computing times and exactness of the solutions.

The fundamental exchange algorithm for solving linear SIP is the simplex method. And the multiple exchange algorithm is a generalization of the simplex procedure, allowing more than one vector per iteration step to enter the basis.

The crucial point for the execution of exchange algorithms for linear SIP is the solving of linear equations. Solution methods should be numerical stable: A use of a very time and storage sparing method has at least no sense, if this method can even in stable problems deliver solutions, which are of no use because of heavy rounding errors.

We note that all other exchange methods for linear SIP, considering their main idea, can be subordinated to the simplex algorithm. It can even be shown that with few exceptions these methods deliver the same iteration solutions as the simplex method or the multiple exchange procedure. By specialisation on special problem classes the execution of the simplex or multiple exchange iterations may be simplified.

So it seems convenient to give a brief summary of the simplex procedure, whereby we will follow essentially the recently published book by GLASHOFF/GUSTAFSON[78]. The multiple exchange algorithm later will be treated total analogously.