Deterministic Solutions for a Class of Chance-Constrained Programming Problems

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Abstract
It is desired to select a deterministic decision vector x to maximize a function Z(x) subject to one or more independent probability constraints of the form Pr[y(x) ≦ 0] ≧ α, where y(x) is a random vector for each given vector x, and α is a given probability. For this purpose we develop a certainty equivalent model without random variables for our chance-constrained model such that feasible and optimal solutions of a chance-constrained problem and of its associated certainty equivalent problem coincide. Problems with any number of single chance-constraints of the form Pr[y(x) ≦ 0] ≧ α, i ϵ I, where the yı(x) are independent single random variables for given vector x, can be solved directly after converting them to the certainty equivalent form. Problems with joint chance-constraints of the form Pr[y(x) ≦ 0] ≧ α, where y(x) is a random vector for given vector x, require selection from a set of certainty equivalent values in order to optimize the objective function. For the case of linear chance-constrained problems with joint constraints, and where only the b bector contains random variables, we develop a method for transforming any linear programming problem of the form max cx, Ax = b, x ≧ 0 into a cost unitized linear programming problem of the form max 1y, AT−1y = b, y ≧ 0. We then determine how to solve the chance-constrained problem max cx, Pr[A(x) ≦ b] ≧ α, where b is a random vector.

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