Monte Carlo bounding techniques for determining solution quality in stochastic programs

A stochastic program SP with solution value z∗ can be approximately solved by sampling n realizations of the program’s stochastic parameters, and by solving the resulting “approximating problem” for (x∗ n ; z ∗ n ). We show that, in expectation, z ∗ n is a lower bound on z∗ and that this bound monotonically improves as n increases. The rst result is used to construct con dence intervals on the optimality gap for any candidate solution x̂ to SP, e.g., x̂ = x∗ n . A sampling procedure based on common random numbers ensures nonnegative gap estimates and provides signi cant variance reduction over naive sampling on four test problems. c © 1999 Elsevier Science B.V. All rights reserved.