Sample Average Approximation Method for Chance Constrained ...

frontier of portfolios can be constructed in an objective function value versus confidencelevel plot, that is, for every confidence level α we associate the optimal value of problem(14). The efficient frontier dates back to Markowitz [15].If r follows a multivariate lognormal distribution, then no closed form solution isavailable. The sample average approximation (SAA) of problem (14) can be written asmaxx∈X rT x, s.t. ˆp N (x) ≤ γ, (15)where∑ Nˆp N (x) := N −1 1l (0,∞) (v − ri T x)i=1and γ ∈ [0, 1). The reason we use γ instead of α is to suggest that for a fixed α, adifferent choice of the parameter γ in (15) might be suitable. For instance, if γ = 0,then the SAA problem (15) becomes the linear programmaxx∈X rT x, s.t. ri T x ≥ v, i = 1, . . .,N. (16)A recent paper by Campi and Garatti [17], building on the work of Calafiore and Campi[18], provides an expression for the probability of an optimal solution ˆx N of the SAA17