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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| Published in | Mathematical methods of operations research (Heidelberg, Germany) Vol. 62; no. 2; pp. 167 - 185 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Heidelberg
Springer Nature B.V
01.11.2005
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| Subjects | |
| Online Access | Get full text |
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| Abstract | 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] |
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| AbstractList | 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] |
| Author | Botkin, Nikolai D. Stoer, Josef |
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| Cites_doi | 10.1007/BF02592073 10.1007/BF00938486 10.1137/0109044 10.1137/0320018 10.1137/0108011 |
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| DOI | 10.1007/s00186-005-0018-4 |
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| References | JB Rosen (18_CR4) 1960; 8 C Michelot (18_CR3) 1986; 50 PH Calamai (18_CR2) 1987; 39 JB Rosen (18_CR5) 1961; 9 DP Bertsekas (18_CR1) 1982; 20 |
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| SubjectTerms | Algorithms Convex analysis Euclidean space Mathematical analysis Mathematical functions Operations research Theory |
| Title | Minimization of convex functions on the convex hull of a point set |
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