A Differential Inclusion for Solving Nonsmooth Convex Optimization Problems in Hilbert Space
by
Wei Bian
Dept. of Math, Harbin Institute of Technology, Harbin, China
Coauthors: Xiaoping Xue

This paper is concerned with the asymptotic analysis of the ajectories of a differential inclusion proposed to solve a class of constrained convex optimization problems in Hilbert space. Compared with the existing systems for solving convex optimization problems in Hilbert space, this system has a wider domain for implementation, because the objective and constrained functions needn't smooth and the initial point needn't be chosen in the feasible region. Moreover, as far as the authors know, there is not a paper to investigate the constrained convex optimization problems in Hilbert space, where the constrains are defined by the convex functions. This system is described by a differential inclusion, which is rigorous analyzed by means of theoretical tools from set-valued analysis and nonsmooth analysis. Under a suitable assumption on the objection function and the feasible region, it is proved that for a given nonsmooth convex optimization problem in Hilbert space and a sufficiently large penalty parameter, the unique trajectory of the differential inclusion can reach the feasible region in finite time and stays there thereafter. Moreover, we can prove that the trajectory of the system constructed by a differential inclusion converges to the set consisted by the quilibrium points of the differential inclusion. In particular, we also prove that the trajectory of the differential inclusion converges weakly to a constrained minimizer whenever it exists.

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Date received: February 5, 2008