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Self-concordant functions for optimization on smooth manifolds

Citation

Jiang, D and Moore, JB and Ji, H, Self-concordant functions for optimization on smooth manifolds, Journal of Global Optimization, 38, (3) pp. 437-457. ISSN 1573-2916 (2007) [Refereed Article]

DOI: doi:10.1007/s10898-006-9095-z

Abstract

This paper discusses self-concordant functions on smooth manifolds. In Euclidean space, such functions are utilized extensively as barrier functions in interior-point methods for polynomial time optimization algorithms. Here, the self-concordant function is carefully defined on a differential manifold in such a way that the properties of self-concordant functions in Euclidean space are preserved. A Newton decrement is defined and analyzed for this class of functions. Based on this, a damped Newton algorithm is proposed for the optimization of self-concordant functions. Under reasonable technical assumptions such as geodesic completeness of the manifold, this algorithm is guaranteed to fall in any given small neighborhood of the optimal solution in a finite number of steps. The existence and uniqueness of the optimal solution is also proved in this paper. Hence, the optimal solution is a global one. Furthermore, it ensures a quadratic convergence within a neighborhood of the minimal point. This neighborhood can be specified in terms of the Newton decrement. The computational complexity bound of the proposed approach is also given explicitly. This complexity bound is shown to be of the order O(-ln(ϵ)), where ϵ is the desired precision. Some interesting optimization problems are given to illustrate the proposed concept and algorithm.

Item Details

Item Type:Refereed Article
Research Division:Mathematical Sciences
Research Group:Numerical and Computational Mathematics
Research Field:Optimisation
Objective Division:Energy
Objective Group:Energy Storage, Distribution and Supply
Objective Field:Energy Systems Analysis
Author:Jiang, D (Dr Danchi Jiang)
ID Code:40686
Year Published:2007 (online first 2006)
Web of Science® Times Cited:1
Deposited By:Engineering
Deposited On:2006-08-01
Last Modified:2017-04-20
Downloads:0

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