Minimization of convex functions on the convex hull of a point set

A basic algorithm for the minimization of a differentiable convex function (in particular, a strictly convex quadratic function) defined on the convex hull of m points in R n is outlined. Each iteration of the algorithm is implemented in barycentric coordinates, the number of which is equal to m. Th...

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Bibliographic Details
Published inMathematical methods of operations research (Heidelberg, Germany) Vol. 62; no. 2; pp. 167 - 185
Main Authors Botkin, Nikolai D., Stoer, Josef
Format Journal Article
LanguageEnglish
Published Heidelberg Springer Nature B.V 01.11.2005
Subjects
Algorithms
Convex analysis
Euclidean space
Mathematical analysis
Mathematical functions
Operations research
Theory
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Summary:A basic algorithm for the minimization of a differentiable convex function (in particular, a strictly convex quadratic function) defined on the convex hull of m points in R n is outlined. Each iteration of the algorithm is implemented in barycentric coordinates, the number of which is equal to m. The method is based on a new procedure for finding the projection of the gradient of the objective function onto a simplicial cone in R m , which is the tangent cone at the current point to the simplex defined by the usual constraints on barycentric coordinates. It is shown that this projection can be computed in O(m log m) operations. For strictly convex quadratic functions, the basic method can be refined to a noniterative method terminating with the optimal solution. [PUBLICATION ABSTRACT]
ISSN:1432-2994
1432-5217
DOI:10.1007/s00186-005-0018-4