Biconvex sets and optimization with biconvex functions: a survey and extensions

Parameterizations of Infinite Biconvex Sets in Affine Root Systems

We investigate detailed relationships between the set W of all infinite “reduced” sequences of elements of a canonical generating set of an arbitrary affine Weyl group and the set B of all infinite sets having a certain “biconvex” property in a positive root system ∆+ of the corresponding untwisted affine Lie algebra g. The study is related to the classification of convex orders on ∆+, which ar...

Jointly Constrained Biconvex Programming

This paper presents a branch-and-bound algorithm for minimizing the sum of a convex function in x, a convex function in y and a bilinear term in x andy over a closed set. Such an objective function is called biconvex with biconcave functions similarly defined. The feasible region of this model permits joint constraints in x and y to be expressed. The bilinear programming problem becomes a speci...

Parametrizations of Infinite Biconvex Sets in Affine Root Systems

We investigate in detail relationships between the set B of all infinite “biconvex” sets in the positive root system ∆+ of an arbitrary untwisted affine Lie algebra g and the set W of all infinite “reduced word” of the Weyl group of g. The study is applied to the classification of “convex orders” on ∆+ (cf. [Ito]), which are indispensable to construct “convex bases” of PoincaréBirkhoff-Witt typ...

Biconvex programming approach to optimization over the weakly efficient set of a multiple objective affine fractional problem

We formulate the problem of optimizing a convex function over the weakly e$cient set of a multicriteria a$ne fractional program as a special biconvex problem. We propose a decomposition algorithm for solving the latter problem. The proposed algorithm is a branch-and-bound procedure taking into account the a$ne fractionality of the criterion functions. c © 2001 Published by Elsevier Science B.V.

Experiments with Biconvex Aerof 6 ils in Low - Density , and Double - Wedge Supersonic Flow