This thesis addresses the t-avoiding-set problem on the complex n-dimensional unit sphere, which asks for the maximal surface measure of a set where no pair of points has an inner product equal to t. By first interpreting the t-avoiding-set problem as an independence-number problem, we use a formulation for the Lovasz theta number to find upper bounds for the maximum measure. In order to adapt the formulation for use in standard optimization solvers, we construct real-valued disk polynomials and use them as a basis for solutions. We further improve the upper bounds by extending the formulation for the Lovasz theta number with a set of constraints derived using the Boolean Quadric Polytope (BQP). In this thesis, we find an optimal construction for eiϕ-avoiding sets and analyze the behavior of the upper bounds for t-avoiding sets. Given that the 0-avoiding set problem corresponds to Witsenhausen’s problem on the complex sphere, we investigate this problem and its upper bounds in depth.

Packages to encode Machine Learned models into optimization problems is an underdeveloped area, despite the advantages is could provide. The main draw of implementing Machine Learned models into optimization models, is that it allows the optimizer to better account for the human experience.
Maragno D., Wiberg H. et al. constructed an implementation of the encoding with their package OptiCL. In order to verify their implementation and provide principles for (re)designing packages with similar functions, an amount of components of OptiCL were replicated within this paper. The requirements for
the program were first constructed before detailing the implementation process. After the program was implemented, both OptiCL and the found program were tested in order to compare performances. Using the results and an investigation of the two implementations, a framework for encoding similar packages
was provided using the insights gained. Using mathematical formulations supplied by Maragno D., Wiberg H. et al., design principles outlined in this report and research into the encoding of other Machine Learned models, other developers could construct robust packages that allow for easy integration of
valuable information gained from Machine Learning into optimization problems. This in turn allows for frequently used optimization models to account for more human understanding.