A framework for discrete-time dynamic programming with multiple objectives.

Abstract

The investigation reported in this dissertation attempts to determine the feasibility of using a distance-based approach like compromise programming for discrete-time dynamic programming problems with multiple objectives. In compromise programming, a function measuring the distance from a generally infeasible ideal solution to the feasible set of the problem is the single objective acting as a surrogate for the set of multiple objectives. Since, in general, there is no single best solution to a multiple objective problem, a framework to generate a family of compromise solutions interactively on a computer is proposed. Various quantities relevant to dynamic compromise programming are defined in precise terms. Dynamic compromise programming problems are computationally difficult to solve because in order to make the distance function decomposable over stages, dimensionality of the state-space must be increased by the number of objectives. To generate compromise solutions, quasi-Newton differential dynamic programming (QDDP), a recently developed variable-metric method for discrete-time optimal control, was employed. QDDP is attractive because no second order or Hessian information is required as input. Instead, Hessian matrices are approximated by first order or gradient information. Since very little is known about its numerical properties, computational experiments were conducted on QDDP. A new strategy for updating Hessian matrix approximations was computationally tested. A constrained QDDP algorithm is proposed, computationally tested, and applied to solve a multiobjective dynamic programming problem with inequality constraints at each stage. The algorithm has the potential for application to the more general discrete-time optimal control problem with stage constraints. The framework for generating compromise solutions interactively was implemented for prototype problems. Because decision maker interaction is crucial in a multiple objective situation, special attention was paid towards developing a man-machine interface using on-screen windows. All implementation and computational testing were done on a UNIX based personal computer.