Abstract
We shall extend Iri's multiplicative penalty function method for linear programming [4] so that it can handle the problem of unknown optimum value of the objective function, without solving both primal and dual problems simultaneously, and generate convergent dual solutions. By making use of these dual variables, lower bounds of the optimum objective function value are updated efficiently, which makes the total number of iterations required in the extended algorithm small. In doing so, a new duality on the multiplicative penalty function is discussed. A sufficient condition for a constraint to be inactive at all optimum solutions is given, which can be checked in the extended algorithm. Several computational techniques for enhancing the efficiency of the algorithm are also discussed. Some connection of the proposed algorithm with Sonnevend's and Renegar's methods [ 10, 11] is touched upon. Furthermore a method of estimating the optimum objective function value is given. Preliminary computational results on the random linear programming problem are finally shown.
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