Advances in Optimization With Applications to Biodiversity Conservation

In this dissertation, we explore the intersection of the fields of operations research and biodiversity conservation. The first part of this work involves improving unconstrained optimization methods for nonlinear programming, with special emphasis on methods for solving large-scale machine learning problems. The second part of this work uses optimization models and methods to make conservation decisions that achieve ecological goals. Conjugate gradient minimization methods (CGM) and their accelerated variants are widely used in machine learning applications. We focused on the use of cubic regularization to improve the CGM direction independent of the step length (learning rate) computation. Using Shanno's reformulation of CGM as a memoryless BFGS method, we derive new formulas for the regularized step direction, which can be evaluated without additional computational effort. The new step directions are shown to improve iteration counts and runtimes and reduce the need to restart the CGM. When CGM is applied to machine learning problems, numerical testing shows similar performance to existing algorithms. The resulting method has been implemented in Python, MATLAB, and C for distribution. Then, we examine spatial conservation planning as an application of optimization. To promote biodiversity and combat climate change, decision-makers collaborate to select which parcels of land to protect in order to meet ecological goals with various constraints. We propose an integer programming optimization model that considers the presence of multiple species on these parcels, subject to predator-prey relationships and crowding effects. By simulating these interactions, we find that the optimal reserve differs from reserves designed without these conditions. Finally, we explore a derivative-free optimization framework paired with a nonlinear component, population viability analysis (PVA). Formulated as a mixed-integer nonlinear programming (MINLP) problem, our model allows for linear and nonlinear inputs, continuous and discrete variables, and can be paired with existing ecological software. Connectivity, competition, crowding, and other similar concerns are handled by the PVA software, rather than expressed as constraints of the optimization model. In addition, we present numerical results that serve as a proof of concept, showing our models yield protected areas with similar expected risk to that of preserving every parcel in a habitat, but at a significantly lower cost. Our contributions to the field include improving CGM for unconstrained nonlinear programming problems and machine learning applications, incorporating more advanced species interactions in spatial conservation planning optimization models, and merging state-of-the-art optimization with state-of-the-art ecology models to promote interdisciplinary collaboration.