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.