Mathematics Research Developments: Robust and Constrained Optimization: Methods and Applications (PDF)

In recent years, the volume of available data has grown exponentially and paved the way for new models in decision-making, particularly decision making under uncertainty. Thus, the opening chapter of Robust and Constrained Optimization: Methods and Applications introduces different robust models induced by three well-known data-driven uncertainty sets: distributional, clustering-oriented, and cutting hyperplanes uncertainty sets. Following this, the authors describe a model of an uncertain vector optimization problem and define robust solutions. Scalarization and vectorization techniques are proposed as efficient ways to compute robust solutions. In one study, a rain-fall optimization algorithm has been applied as a new naturally-inspired algorithm based on the behavior of raindrops. This algorithm has been developed with the goal of finding a simpler and more effective search algorithm to optimize multi-dimensional numerical test functions. The process considers the numerical differential of the cost function rather than the mathematical computation of the gradient. The authors examine the preconditioned iterative solution of a particular type of linear systems, mainly involving matrices of a two-by-two block form with square matrix blocks. Such systems arise in the finite element solution of optimal control problems for partial differential equations in various applications. Finally, it is shown how various metaheuristic algorithms (including memetic, interval, and random search optimization methods) can be applied to solve different types of optimal control problems(e.g., satellite stabilization, solar sail control, interception problems). Hybrid global optimization methods, which combine strategies from several different metaheuristic random search algorithms, aresuggested in an attempt to improve accuracy of the obtained solution.