An upper bound for SLP using first and total second moments (Q1178443)

To avoid evaluating multiple integrals in the solution process of stochastic programming problems lower and upper bounds for those integrals can be used. In the present article the problem of finding an upper bound for the expected recourse \(\int_{R^ n}\psi(\xi)P(d\xi)\) over the set of distributions which preserve the first and the total second moment is considered, i.e., a moment problem \[ \sup_ P\left\{\int_{R^ n}\psi(\xi)P(d\xi)\mid\int_{R^ n}\xi P(d\xi)=\mu, \int_{R ^ n}\|\xi\|^ 2 P(d\xi)=\rho\right\}\tag{1} \] is required to be solved. It is shown that under certain conditions imposed on the dual, problem (1) can be solved as a nonsmooth optimization problem.