Mathematical Sciences: Quasi-Second-Order Methods for Nonsmooth Optimization

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9402018 Mifflin The proposed research concerns developing a way to combine polyhedral and quadratic approximation to produce a rapidly convergent algorithm for minimizing a nonsmooth function of several variables. The idea is to combine bundle and proximal point methods to determine a space decomposition such that on one subspace the cutting-plane aspect of bundling works well and on the other subspace a quasi-Newton approximation of a certain reduced Hessian can be employed. This research can be applied in many practical decision-making situations. These include decomposition of large or complicated models such as those occurring in resource allocation, manufacturing, logistics and civil infrastructure design. For example, a water pressure control problem of finding values for flow rates and pump pressure gaps to minimize pumping cost subject to maintaining water pressures in allowable ranges at all points in a pipe network can be solved via decomposition. If the flow rates are fixed then the subproblem of finding optimal values for the pressure gaps is an easy-to-solve linear minimization problem. With this separation of variables approach the outer problem of finding optimal values for the flow variables is a nonsmooth problem whose solution can be found efficiently via the techniques of this proposal.