A modification of karmarkar's linear programming algorithm

Abstract

Abstract. We present a modification of Karmarkar's linear programming algorithm. Our algorithm uses a recentered projected gradient approach thereby obviating a priori knowledge of the optimal objective function value. Assuming primal and dual nondegeneracy, we prove that our algorithm converges. We present computational comparisons between our algorithm and the revised simplex method. For small, dense constraint matrices we saw little difference between the two methods.Key Words. Linear programming, Karmarkar's algorithm, Projected gradient methods, Least squares.1. Introduction. This paper proposes a modification to Karmarkar's original algorithm [6] for solving linear programs. Our algorithm is formulated in the positive orthant instead of the simplex. This makes it easier to conceptualize and leads to computational simplicity. Karmarkar's sliding-objective-function method is replaced by a projected gradient search for the optimum. Empirically, this leads to a decrease in the number of iterations the algorithm requires to solve a problem.In describing our algorithm, we show how to start it, when to stop it, and how to identify infeasibility and unboundedness easily. Assuming primal and dual nondegeneracy, we prove convergence to the optimal solution. (In practice, the algorithm works equally well on problems not satisfying these assumptions.) We also show that duality plays an important role.Finally, we present results comparing the performance of our algorithm with the revised simplex method for problems with fewer than 200 variables and randomly generated, dense constraint matrices. For this class of problems we saw little difference between the two methods. However, our results may not be indicative of the behavior for large sparse problems.

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