| The development of computational methods in optimization has focused mostly on algorithms for solving problems of the canonical form{dollar}{dollar}minlimitssbsp{lcub}xinIRsp{lcub}n{rcub}{rcub} {lcub}fsp0(x)vert f(x)le0, g(x)=0{rcub},eqno(1){dollar}{dollar}where the functions{dollar}{dollar}fsp0 :IRsp{lcub}n{rcub}toIR,f :IRsp{lcub}n{rcub}toIRsp{lcub}q{rcub}, {lcub}rm and{rcub} g :IRsp{lcub}n{rcub}toIRsp{lcub}p{rcub}{dollar}{dollar}are smooth.; Unfortunately, many complex engineering-design optimization problems cannot be stated in the form (1), and hence there is a dearth of algorithms for their solution. In this dissertation we develop algorithms for three such classes of problems.; In Chapter 2, we deal with a class of optimal Euler-Bemoulli beam design problems with continuum constraints. We replace the original, infinite dimensional problem by an infinite family of finite dimensional, approximating problems, and show that any accumulation point of the sequence of the stationary points of the family of approximating problems is a stationary point of the original, infinite-dimensional problem. The approximating problems are tractable by standard nonlinear programming algorithms.; In Chapter 3, we present a new mathematical formulation for a class of structural optimization problems with reliability constraints, and show that in this formulation these problems can be solved using outer approximations algorithms that use adaptive continuum discretization techniques.; In Chapter 4 we present a simple transcription of problems with maximum constraints into a nonlinear programming problem with smooth constraints. Equivalence between global and local minimizers of the problem with maximum constraints and those of the problem obtained through the proposed transcription is proved. This transcription facilitates the solution of path planning in the presence of obstacles, and other problems with "exclusion" constraints.; Finally, in Chapter 5, two versions of a new one-dimensional, globally convergent optimization algorithm that achieves quadratic convergence by using cubic interpolation to construct approximations of the second derivative are presented. The algorithm does not require an initial determination of a bounded interval (satisfying certain bracketing conditions) containing a local minimizer.; The robustness and efficiency of all the algorithms and techniques presented in this dissertation are illustrated by numerical results. |
