Multi-objective portfolio optimization problems with complex objectives and expensive function evaluations can be solved through the sample efficient solution method of Bayesian optimization. In realistic settings, inclusion of many assets coupled with a limited evaluation budget negatively impacts the standard ARD Gaussian process surrogate model, as in high-dimensional environments this kernel no longer permits meaningful inference without a vast number of observations. The ARD kernel overestimates problem complexity, as financial assets display significant structure or dependence, for instance by belonging to the same asset class. This work presents a methodology to learn and exploit these asset similarities through a Mahalanobis kernel. Within this kernel, we treat the partitioning structure as a latent variable, and learn this structure through a coordinate-wise greedy optimization approach based on k-fold cross-validation using previously observed data points. We examine our proposed methodology based on ability to retrieve synthetic similarities, surrogate predictive and uncertainty quantification performance, along with attained hypervolume during Bayesian optimization. Our results show we successfully find similarities both in the synthetic and authentic settings, leading to improved predictive performance of Gaussian processes equipped with the learned Mahalanobis kernel. Furthermore, we showcase the importance of correct uncertainty quantification for Bayesian optimization, and achieve improved results for models averaged over multiple structures - thus taking model uncertainty into account - compared to the use of a single locally optimal partitioning. For the moderate dimensionalities considered within this work, the ARD kernel achieves competitive hypervolume results. However, this work indicates the exploitation of similarity structures within portfolio optimization is a promising avenue, and we provide multiple ways forward to capture the full potential the main ideas presented have to offer.

Upper bounds for the kissing number can be written as a semidefinite program (SDP) through the Delsarte-Goethals-Seidel method for spherical codes. This thesis solves the resulting SDP with a cutting plane approach, in which a sequence of linear programs (LPs) is solved with the addition of linear constraints every round. We study the computational efficiency of dense and sparser cuts. Sparse cuts are obtained through a relation to the $k$-Sparse Principal Component Analysis problem. For the modest polynomial degrees considered, the dense and sparse methods show similar performance. Upper bounds are obtained through calculations in standard and where necessary quadruple precision. Lastly, it is shown that under a linear cutting plane approach the SDP is solved quicker if not every subsequent LP is solved till optimality.