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 first result is used to construct confidence intervals on the optimality gap for any candidate solution (x) over cap to SP, e.g., (x) over cap=x(n)*. A sampling procedure based on common random numbers ensures nonnegative gap estimates and provides significant variance reduction over naive sampling on four test problems. (C) 1999 Elsevier Science B.V. All rights reserved.