Mathematical Methods of Operations Research

Recht, P.

doi: 10.1007/BF01415887pmid: N/A

In this paper we present a concept of the construction of generalized gradients by considering a development of directional derivatives into spherical harmonics. This leads to a derivation system as a system of generalized partial derivatives. Necessary conditions for local extrema for a broad class of not necessarily differentiable function can be given and a characterization of points of differentiability can be proved by using generalized gradients.

Bücker, M.

doi: 10.1007/BF01415888pmid: N/A

This paper deals with single machine scheduling problems with stochastic precedence relations (so calledGERT networks). Until now most investigations on such problems, dealt with algorithms running in polynomial time. On the other hand, for scheduling problems with deterministic precedence relations exist a lot of results about time complexity. Therefore, the object of this paper is to consider time complexity of scheduling problems with stochastic precedence constraints and to describe the boundary between theNP-hard problems and those which can be solved in polynomial time.

Wagner, Frank

doi: 10.1007/BF01415889pmid: N/A

Sheu, R.; Fang, S.

doi: 10.1007/BF01415890pmid: N/A

In this paper, we study the search directions of three important interior-point algorithms, namely, the primal-affine scaling method (with logarithmic barrier function), the dual-affine scaling method (with logarithmic barrier function), and the primal-dual interior point method. From an algebraic point of view, we show that the search directions of these three algorithms are merely Newton directions along three different “paths” that lead to a solution of the Karush-Kuhn-Tucker conditions of a given linear programming problem. From a geometric point of view, we show that these directions can be obtained by solving certain well-defined subproblems. Both views provide a general platform for studying the existing interior-point methods and deriving new interior-point algorithms. We illustrate the derivation of new interior-point algorithms by replacing the logarithmic barrier function with an entropic barrier function. The results have been generalized and discussed.

Formann, Michael

doi: 10.1007/BF01415891pmid: N/A

Marti, K.

doi: 10.1007/BF01415892pmid: N/A

In engineering and economics often a certain vectorx of inputs or decisions must be chosen, subject to some constraints, such that the expected costs (or loss) arising from the deviation between the outputA(ω) x of a stochastic linear systemx→A(ω)x and a desired stochastic target vectorb(ω) are minimal. Hence, one has the following stochastic linear optimization problem minimizeF(x)=Eu(A(ω)x b(ω)) s.t.xεD, (1) whereu is a convex loss function on ℝ m , (A(ω), b(ω)) is a random (m,n + 1)-matrix, “E” denotes the expectation operator andD is a convex subset of ℝ n . Concrete problems of this type are e.g. stochastic linear programs with recourse, error minimization and optimal design problems, acid rain abatement methods, problems in scenario analysis and non-least square regression analysis.

Holsapple, C.; Park, S.; Whinston, A.

doi: 10.1007/BF01415893pmid: N/A

The quest for OR interface development tools that are both flexible (in terms of interfaces produced) and high-level (in terms of developer specification accepted) is based on work in the DSS and UIMS fields. The DSS literature offers a foundation for understanding possible relationships between OR procedures, the data they can use, and the problem statements that cause their executions. The UIMS literature offers mechanisms for interface production. Of these, structure editing has been singled out here as a candidate for advancing OR research into interface construction. The first implementation of a structure editor interface for OR problem statements is described. The result is a direct manipulation user interface that presents LP problem statement images resembling traditional forms, but which allows variants of a single form to dynamically unfold in response to a user's expansion actions.