DescriptionWe consider optimization problems featuring nonnegativity constraints on functions associated with the decision variables. The first part of the thesis is devoted to the topic of optimization over the cone of positive polynomials. We present a method for constructing non-negative spline approximations to the arrival rate of a non-homogenous Poisson process based on observed arrival data, along with numerical results and a comparison to previous approaches. Our results are obtained by formulating the problem as a
semidefinite program; we explore the theoretical obstacles to a more direct method by proving that only a constant number of linearly independent bilinear optimality conditions exist for cones of positive polynomials, regardless of dimension.
In the second part we look at optimization with second order stochastic dominance constraints. Here the functional inequalities appear naturally, featuring the integral of the distribution function of a random variable defined by the decision variables. We develop new duality results as well as cutting plane methods that
are shown to perform well on a class of portfolio optimization problems. Finally we point out an interesting connection, arising as part of our duality considerations, between the theory of measures with given marginals and network feasibility. This results in a new proof of Strassen's Theorem from its trivial discrete case.