Continuous and discrete optimization techniques for some problems in industrial engineering and materials design

This work comprises several projects that involve optimization of physical systems. By a physical system we understand an object or a process that is governed by physical, mechanical, chemical, biological, etc., laws. Such objects and the related optimization problems are relatively rarely considered in operations research literature, where the traditional subjects of optimization methods are represented by schedules, assignments and allocations, sequences, and queues. The corresponding operations research and management sciences models result in optimization problems of relatively simple structure (for example, linear or quadratic optimization models), but whose difficulty comes from very large number (from hundreds to millions) of optimization variables and constraints. In contrast, in many optimization problems that arise in mechanical engineering, chemical engineering, biomedical engineering, the number of variables or constraints in relatively small (typically, in the range of dozens), but the objective function and constraints have very complex, nonlinear and nonconvex analytical form. In many problems, the analytical expressions for objective function and constraints may not be available, or are obtained as solutions of governing equations (e.g., PDE-onstrained optimization problems), or as results of external simulation runs (black-box optimization). In this dissertation we consider problems of classification of biomedical data, construction of optimal bounds on elastic tensor of composite materials, multiobjective (multi-property) optimization via connection to stochastic orderings, and black-box combinatorial optimization of crystal structures of organic molecules. iii PUBLIC ABSTRACT This work comprises four projects that involve optimization of physical systems, which are governed by physical, mechanical, chemical, biological, etc., laws. The first project is focused on efficient solving of special data classification problems, and the developed methodology was applied to several biomedical data sets in order to improve prediction whether a patient or test subject has a certain type of disease (e.g., diabetes). The second project was concerned with determining and optimizing the ranges of material properties of composite materials. Composite materials typically consist of two or more constituents, which are combined in such a way so as to produce a material whose properties are superior to the properties of the individual constituent materials. The proposed approach was illustrated on nano-composites, or composite materials containing carbon fiber nanotube inclusions. Third part of this research is focused on multiobjective optimization, that can be used, for example, in market portfolio management. The final part discusses the possibility of using heuristic algorithms for crystal structure determination from X-ray diffraction data. iv TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii LIST OF ALGORITHMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x CHAPTER