On the Probabilistic Complexity of Finding an Approximate Solution for Linear Programming



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We consider the problem of finding an ε{lunate}-optimal solution of a standard linear program with real data, i.e., of finding a feasible point at which the objective function value differs by at most ε{lunate} from the optimal value. In the worst-case scenario the best complexity result to date guarantees that such a point is obtained in at most O (sqrt(n) | ln ε{lunate} |) steps of an interior-point method. We show that the expected value of the number of steps required to obtain an ε{lunate}-optimal solution for a probabilistic linear programming model is at most O (min { n1.5, m sqrt(n) ln (n) }) + log2 (| ln ε{lunate} |)