Exact optimal designs for estimating the minimum of a quadratic response surface via stochastic programming.

Abstract

This dissertation introduces a new class of stochastic programming algorithms that find solutions for an optimal design problem under small sample conditions. Our stochastic programming algorithm proceeds in two stages: first, the design criterion is approximated using a finite sample from a simulation, and second, the approximating function is minimized using standard optimization techniques. The need for simulation based approximation arises in the context of optimal design because decision theoretic design criteria often involve analytically intractable high dimensional integrals. This dissertation establishes the theoretical validity of the above approach and develops a computer code for implementing it. We apply our methodology to the problem of finding an optimal design for estimating the point which minimizes a convex quadratic response surface. Chapter I provides an introduction and motivation. Chapter II develops the theory underlying our approach in terms of the above specific problem and in general terms. Chapter III uses the theory of sensitivity analysis for parametric mathematical programming to derive gradients and subgradients of our design criteria. Chapter IV adapts the theory of epiconvergence for random lower semicontinuous functions to the problem of establishing almost sure convergence results under weak conditions; special attention is given to complications that result from multimodality. Chapter V applies the theory of constrained M-estimation to establish asymptotic distribution theory for the algorithm: particular attention is paid to problems which arise when the approximate form of the design criterion is not differentiable. Chapter VI treats the theory of accuracy assessment for global optimization via the multistart method: we propose a new class of estimators for this problem and give a numerical example involving our optimal design code. Chapter VII reports two computational experiments that were carried out to investigate the properties of our optimal design code. Finally, Chapter VIII provides a conclusion and gives directions for future research.