A Modified L-Shaped Method

Solution of Stochastic Quadratic Programming with Imperfect Probability Distribution Using Nelder-Mead Simplex Method

“…This method is essentially an external approximation algorithm that can effectively solve the large-scale problems that occur after the stochastic programming is transformed into deterministic mathematical programming. Abaffy and Allevi present a modified version of the L-shaped method in [4], used to solve two-stage stochastic linear programs with fixed recourse.…”

Augmented Lagrangian method within L-shaped method for stochastic linear programs

“…The classical two-stage stochastic linear program was introduced by Dantzig in 1955 [6]. After that, many scholars such as Van Slyke and Wets [19], Higle and Sen [10], Dantzig and Wolfe [7], Prekopa [18], Birge [4] and Abaffy and Allevi [1] presented different methods for solving this problem.…”

“…In this paper, matrices A and W are assumed to have full row rank and m n and m 2 n 2 . Assuming that the uncertainties are represented by a finite scenario set, the stochastic program (1) and (2) can be reformulated [12] as the following deterministic equivalent program:…”

“…x is a deterministic variable and an optimum x is optimal for all scenarios ω ∈ while y(ω) is a random variable and an optimum y(ω) is optimal for a single scenario ω. The decomposition methods exploit the nature of random variables and the structure of problem (1). Decomposition algorithms such as L-shaped [19] are based on Benders decomposition [3] whose original goal was to solve mixed integer programming problems.…”

“…The use of the L-shaped method to solve problem (1) was proposed originally by Van Slyke and Wets in [12]. After that, many scholars such as Abaffy and Allevi in [1] and Nielsen and Zenios in [17] modified and extended the L-shaped method by introducing new algorithms to generate optimality and feasibility cuts although the method in [12] is still the most common method, and new methods are based on.…”