Balder Von Hohenbalken

Simplicial decomposition in nonlinear programming algorithms

Simplicial decomposition is a special version of the Dantzig—Wolfe decomposition principle, based on Carathéodorys theorem. The associated class of algorithms has the following features and advantages: The master and the subprogram are constructed without dual variables; the methods remain therefore well-defined for non-concave objective functions, and pseudo-concavity suffices for convergence to global maxima. The subprogram produces affinely independent sets of feasible generator points defining simplices, which the master program keeps minimal by dropping redundant generator points and finding maximizers in the relative interiors of the resulting subsimplices. The use of parallel subspaces allows the direct application of any unrestricted optimization method in the master program; thus the best unconstrained procedure for any type of objective function can be used to find constrained maximizers for it.The paper presents the theory for this class of algorithms, the APL-code of a “demonstration” method and some computational experience with Colvilles test problems.

A finite algorithm to maximize certain pseudoconcave functions on polytopes

AbstractThis paper develops and proves an algorithm that finds the exact maximum of certain nonlinear functions on polytopes by performing a finite number of logical and arithmetic operations. Permissible objective functions need to be pseudoconcave and allow the closed-form solution of sets of equations

Least distance methods for the scheme of polytopes

\partial f(Dy + \hat x^k )/\partial y = 0

Manhattan versus Euclid: Market areas computed and compared

, which are first order conditions associated with the unconstrained, but affinely transformed objective function. Examples are pseudoconcave quadratics and especially the homogeneous functioncx +m(xVx)1/2,m < 0, V positive definite, for which sofar no finite algorithm existed.In distinction to most available methods, this algorithm uses the internal representation [6]|of the feasible set to selectively decompose it into simplices of varying dimensions; linear programming and a gradient criterion are used to select a sequence of these simplices, which contain a corresponding sequence of strictly increasing, relative and relatively interior maxima, the greatest of which is shown to be the global maximum on the feasible set. To find the interior maxima on these simplices in a finite way, calculus maximizations on the affine hulls of subsets of their vertices are necessary; thus the above requirement that

Least distance methods for the frame of homogeneous equation systems

\partial f(Dy + \hat x^k )/\partial y = 0

An econometric-spatial analysis of the growth and decline of shopping centers

be explicitly solvable.The paper presents a flow structure of the algorithm, its supporting theory, its decision-theoretic use, and an example, computed by an APL-version of the method.