The objective of the proposed dissertation is to improve the performance of interior point methods in specialized settings related to the optimal power flow (OPF) problem of power engineering and privacy-preserving computation. The OPF is a constrained optimization problem of critical importance for efficient operation of power systems. However, the formulation of the OPF problem involves sensitive data from generators, and also from end users in systems implementing “smart grid” technology. As the OPF problem is traditionally solved by system operators, they have access to this data, which is a privacy concern. Consequently, recent research has focused not only on improving solution methods for the OPF, but also on addressing these privacy concerns. Currently, the most popular techniques for constrained optimization (like OPF and its related problems) are interior point methods. However, to this day there is not an efficient, privacy-preserving implementation of these methods. To this end, new specialized interior point methods aimed at tackling some of the aforementioned issues are presented in this dissertation. First, we consider the problem of driving the state of a power system to a desired state (usually an OPF solution). This is equivalent to a discretized shortest path problem constrained by the feasible region of a non-linear OPF. We propose an interior point method that exhibits block tri-diagonal structure, and we show how this structure can be used to solve this problem efficiently. Next, we study the theoretical properties of an interior point method designed to have reduced computation time in privacy-preserving settings. Lastly, we develop an interior point method for linear programs that uses a novel type of constraint barrier, where the cost of matrix factorizations is reduced in proportion to the amount of constraints away from the current iterate.