Global optimization of nonconvex generalized polynomial design models.

Global optimization is a branch of mathematical programming with fewer computational techniques and less theoretical foundation than standard nonlinear programming (local optimization). In engineering optimization, there is a frequent need for solving a highly competitive design problem beyond a local optimum. Since many fundamental questions have yet to be answered, algorithms developed for global optimization problems have been heavily based on a stochastic rather than a deterministic approach. On the other hand, most deterministic algorithms are restricted to certain classes of problems. In this dissertation, a deterministic approach is investigated for a special class of problems called generalized polynomial problems which occur often in engineering applications. Our approach is to investigate the underlying structure of a generalized polynomial problem so that certain model transformations can convert such a problem into one that is in the form of a Difference-of-two Convex functions (DC), solvable with existing techniques. Furthermore, we take advantage of the special structure of a positive generalized polynomial function to develop an improved algorithm based on certain convex outer approximations. The abstraction of our approach for solving a global optimization problem is presented as a set of conditions on an algorithm. An algorithm satisfying those conditions is able to locate a feasible solution if it exists or to prove a problem is infeasible if it does not. Such an algorithm is called a Global Feasible Search Algorithm. Two algorithms, LINCUT and CONCUT, were developed and implemented based on Global Feasible Search approach with linear and convex outer approximations respectively. Computational results for several test problems are included. Four engineering applications including air tank, flywheel, speed reducer and corrugated bulkhead designs, are solved to demonstrate the applicability of such algorithms to practical engineering problems.Ph.D.Applied SciencesMathematicsMechanical engineeringOperations researchPure SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/128824/2/9208601.pd