Sample Average Approximation Method for Chance Constrained ...

problem (5), with γ = 0, to be infeasible for the true problem (4). That is, under theassumptions that the set X and functions f(·) and G(·, ξ), ξ ∈ Ξ, are convex and thatw.p.1 the SAA problem attains unique optimal solution, we have that for N ≥ n,prob {p(ˆx N ) > α} ≤ B(n − 1; α, N), (17)and the above bound is tight. Thus, for a confidence parameter β ∈ (0, 1) and a samplesize N ∗ such thatB(n − 1; α, N ∗ ) ≤ β, (18)the optimal solution of problem (16) is feasible for the corresponding true problem (14)with probability at least 1 − β.For γ > 0, problem (15) can be written as the mixed-integer linear programmaxx,zr T x,(19a)s.t. r T i x + vz i ≥ v, (19b)N∑z i ≤ Nγ,i=1x ∈ X, z ∈ {0, 1} N ,(19c)(19d)18