Optimization on nonlinear surfaces

Hung-Chieh Chang, Purdue University

Abstract

In this work a new method for maintaining feasibility in optimization problems on nonlinear surfaces is presented. Especially when the dimension of the feasible region is lower than that of the ambient space, most algorithms expend a significant fraction of the total computation maintaining feasibility, i.e., confining the trajectories of the algorithms to the feasible region. The method presented dynamically re-coordinatizes the surface as the algorithm progresses. In each step, a coordinate system most appropriate for the local geometry of the surface is chosen, rendering feasibility concerns computationally trivial. The method utilizes a complexification of the (real) feasible region, which then allows the method to draw upon results from differential geometry, several complex variables, and systems of ordinary differential equations. Along with the presentation of the new algorithm, its mathematical basis and a proof of its convergence, an important class of surfaces—Lie manifolds—is given special attention because it provides a natural framework for describing the invariant transformation groups in applications of pattern recognition, and gives a class of examples in which the method yields dramatic reduction in computational complexity. For the general nonlinear problem, several numerical test cases are included to demonstrated the accuracy of the algorithm.

Degree

Ph.D.

Advisors

Prabhu, Purdue University.

Subject Area

Operations research|Industrial engineering

COinS