Algorithms for optimizing a function over a cone

This paper is about a special type of Convex Programming Problem for which the set of constraints is a closed and unbounded cone generated by a compact convex set : cone[W] = conv { µw: µεR+, wεW}. Allright shows that, for the case where the objective function v is a norm function, an equivalent problem, with the same solution, can be derived wherein minimization of v is carried over a new, compact convex and bounded, set S, actually a suitably truncated version of cone[W]. A generalization of Allwright's results to the case where v is a general quadratic is presented and a convergence rate is derived which depends on the ratio between the smallest and the largest eigenvalue of the second derivative matrix. For the cases where the objective function v is a general convex function, whose Hessian is upper and lower bounded, it is shown that a similar equivalent problem can also be formulated. An algorithm to solve the equivalent problem is stated and a convergence rate depending on both lower and upper bounds is derived