This thesis explores four methods for convex optimization. The first two are an interior point method and a simulated annealing algorithm that share a theoretical foundation. This connection is due to the interior point method’s use of the so-called entropic barrier, whose derivatives can be approximated through sampling. Here, the sampling will be carried out with a technique known as hit-and-run. By carefully analyzing the properties of hit-and-run sampling, it is shown that both the interior point method and the simulated annealing algorithm can solve a convex optimization problem in the membership oracle setting. The number of oracle calls made by these methods is bounded by a polynomial in the input size. The third method is an analytic center cutting plane method that shows promising performance for copositive optimization. It outperforms the first two methods by a significant margin on the problem of separating a matrix from the completely positive cone. The final method is based on Mosek’s algorithm for nonsymmetric conic optimization. With their scaling matrix, search direction, and neighborhood, we define a method that converges to a near-optimal solution in polynomial time
