Back to Departmental Colloquium: Spring 2007
Departmental Colloquium
Date: Monday, Jan 15, 2007
Time: 4:15PM
Location: JWB 335
Coralia Cartis
University of Edinburgh
Title |
Adaptive regularization methods for nonlinear optimization |
Abstract |
Nonlinear optimization problems represent the bed-rock of numerous real-life applications, such as data assimilation for weather prediction, radiation therapy treatment planning, optimal design of energy transmission networks, and many more. The solution of these problems usually involves iteratively constructing easier-to-solve local models of the function to be optimized, with the optimizer of the model taken as an estimate of the sought-after solution. Linear or quadratic models are usually employed locally in this context; however, these approximations are often unsatisfactory either because they are unbounded in the presence of nonconvexity and hence cannot be meaningfully optimized, or they are accurate representations of the function only in a small neighbourhood, yielding only small or no iterative improvements. Hence such models require some form of regularization to improve algorithm performance and avoid failure; traditionally, linesearchand trust-region techniques have been employed for this purpose and represent the state-of-the-art. Here, a new class of methods for nonlinear nonconvex unconstrained problems will be presented that approximately globally minimize a quadratic model of the objective regularized by a cubic term, inspired by earlier regularization approaches of Nesterov (2007) and Griewank (1982). An overestimation property of functions with Lipschitz-continuous Hessians underlies and justifies the model construction in the work to be presented. Preliminary numerical experiments show our methods to perform better than a trust-region implementation, while our convergence and complexity results show it to be at least as reliable as the latter approach. Extensions to problems with simple constraints and a simplified application to the subclass of nonlinear least-squares problems will also be presented. This is joint work with Nick Gould (Rutherford Appleton Laboratory, UK), Philippe Toint (University of Namur, Belgium), and partly, also with Stefania Bellavia and Benedetta Morini (University of Florence, Italy). February 5: (Special Colloquium) |
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