Descent Directions and Efficient Solutions in Discretely Distributed Stochastic ProgramsIn engineering and economics a certain vector of inputs or decisions must often be chosen, subject to some constraints, such that the expected costs arising from the deviation between the output of a stochastic linear system and a desired stochastic target vector are minimal. In many cases the loss function u is convex and the occuring random variables have, at least approximately, a joint discrete distribution. Concrete problems of this type are stochastic linear programs with recourse, portfolio optimization problems, error minimization and optimal design problems. In solving stochastic optimization problems of this type by standard optimization software, the main difficulty is that the objective function F and its derivatives are defined by multiple integrals. Hence, one wants to omit, as much as possible, the time-consuming computation of derivatives of F. Using the special structure of the problem, the mathematical foundations and several concrete methods for the computation of feasible descent directions, in a certain part of the feasible domain, are presented first, without any derivatives of the objective function F. It can also be used to support other methods for solving discretely distributed stochastic programs, especially large scale linear programming and stochastic approximation methods. |
Other editions - View all
Descent Directions and Efficient Solutions in Discretely Distributed ... Kurt Marti No preview available - 1988 |
Descent Directions and Efficient Solutions in Discretely Distributed ... Kurt Marti No preview available - 2014 |
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A¹-Ã A³y A³y-b³ A³y=A³x assumptions Consequently constraint convex function Corollary D-stationary point defined Definition 4.1 denotes elements extreme point feasible descent directions feasible direction feasible solution fulfills Furthermore given n-vector Hence holds true implies integer J₁ J₁(x J₂(x JERO jɛR kernA kernA(t Lemma linear equations linear independent linear program loss function minimize Note objective function obtain one-point measure optimal solution point xD programs with recourse Proof Px,D random matrix random variables relations representation resp satisfies solution y,B solving stochastic approximation stochastic dominance stochastic linear program Stochastic Optimization Stochastic Programs strict inequality sign submatrix subset suppose system of linear Theorem 2.2 tuple vector yields z¹+b³-A³y νευ Σ α Σ σ ај