Hessian Riemannian gradient flows in convex programming *
| Autor: | Felipe Alvarez, Jérôme Bolte, Olivier Brahic |
|---|---|
| Přispěvatelé: | Departamento de Ingeniería Matemática [Santiago] (DIM), Universidad de Chile = University of Chile [Santiago] (UCHILE)-Centre National de la Recherche Scientifique (CNRS), Groupe de recherche en économie mathématique et quantitative (GREMAQ), Centre National de la Recherche Scientifique (CNRS)-École des hautes études en sciences sociales (EHESS)-Institut National de la Recherche Agronomique (INRA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées, Departamento de Ingenierı́a Matemática, Universidad de Chile [Santiago], GREMAQ, Centre National de la Recherche Scientifique (CNRS)-École des hautes études en sciences sociales (EHESS)-Institut National de la Recherche Agronomique (INRA)-Université Toulouse 1 Capitole (UT1)-Centre National de la Recherche Scientifique (CNRS)-École des hautes études en sciences sociales (EHESS)-Institut National de la Recherche Agronomique (INRA)-Université Toulouse 1 Capitole (UT1) |
| Jazyk: | angličtina |
| Rok vydání: | 2004 |
| Předmět: |
Hessian matrix
0209 industrial biotechnology Control and Optimization Geodesic Hessian Riemannian metric 0211 other engineering and technologies 02 engineering and technology 34D05 Legendre type convex function 90C25 Hamiltonian system convex and linear programming symbols.namesake 020901 industrial engineering & automation Differential inclusion Gradient flow Liapounov functional FOS: Mathematics Applied mathematics quasi-convex mini-mization Mathematics - Optimization and Control Mathematics ex-istence 021103 operations research Applied Mathematics 34A12 Bregman distance Legendre transform coordinates global convergence Lagrange and Hamilton equations AMS classification: 34G20 Rate of convergence Optimization and Control (math.OC) Convex optimization Variational inequality symbols [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] Convex function |
| Zdroj: | SIAM Journal on Control and Optimization SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2004, 43 (2), pp.477-501. ⟨10.1137/S0363012902419977⟩ SIAM JOURNAL ON CONTROL AND OPTIMIZATION Artículos CONICYT CONICYT Chile instacron:CONICYT ResearcherID |
| ISSN: | 0363-0129 1095-7138 |
| DOI: | 10.1137/S0363012902419977⟩ |
| Popis: | International audience; Motivated by a constrained minimization problem, it is studied the gradient flows with respect to Hessian Riemannian metrics induced by convex functions of Legendre type. The first result characterizes Hessian Riemannian structures on convex sets as those metrics that have a specific integration property with respect to variational inequalities, giving a new motivation for the introduction of Bregman-type distances. Then, the general evolution problem is introduced and a differential inclusion reformulation is given. A general existence result is proved and global convergence is established under quasi-convexity conditions, with interesting refinements in the case of convex minimization. Some explicit examples of these gradient flows are discussed. Dual trajectories are identified and sufficient conditions for dual convergence are examined for a convex program with positivity and equality constraints. Some convergence rate results are established. In the case of a linear objective function, several optimality characterizations of the orbits are given: optimal path of viscosity methods, continuous-time model of Bregman-type proximal algorithms, geodesics for some adequate metrics and projections of ˙ q-trajectories of some Lagrange equations and completely integrable Hamiltonian systems. |
| Databáze: | OpenAIRE |
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