A simple randomised algorithm for convex optimisation: Application to two-stage stochastic programming

We consider maximising a concave function over a convex set by a simple randomised algorithm. The strength of the algorithm is that it requires only approximate function evaluations for the concave function and a weak membership oracle for the convex set. Under smoothness conditions on the function and the feasible set, we show that our algorithm computes a near-optimal point in a number of operations which is bounded by a polynomial function of all relevant input parameters and the reciprocal of the desired precision, with high probability. As an application to which the features of our algorithm are particularly useful we study two-stage stochastic programming problems. These problems have the property that evaluation of the objective function is #P-hard under appropriate assumptions on the models. Therefore, as a tool within our randomised algorithm, we devise a fully polynomial randomised approximation scheme for these function evaluations, under appropriate assumptions on the models. Moreover, we deal with smoothing the feasible set, which in two-stage stochastic programming is a polyhedron

Convergence Analysis of Some Methods for Minimizing a Nonsmooth Convex Function

In this paper, we analyze a class of methods for minimizing a proper lower semicontinuous extended-valued convex function . Instead of the original objective function f , we employ a convex approximation f k + 1 at the k th iteration. Some global convergence rate estimates are obtained. We illustrate our approach by proposing (i) a new family of proximal point algorithms which possesses the global convergence rate estimate even it the iteration points are calculated approximately, where are the proximal parameters, and (ii) a variant proximal bundle method. Applications to stochastic programs are discussed.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45249/1/10957_2004_Article_417694.pd

Stochastic programming and the option of doing it differently

Option theory and stochastic programming are tightly linked. Most options can be analyzed in both frameworks, and the two approaches support each other in many slightly more complex situations. But this similarity hides some central differences in perspective. This short note tries to focus on one of these, namely the fact that option theory can be applied only to options already identified, while stochastic programming is able to help us find options in contexts where it is not at all clear what they are, and where finding might be more important than valuing

Adjustable Robust Optimisation approach to optimise discounts for multi-period supply chain coordination under demand uncertainty

In this research, a problem of supply chain coordination with discounts under demand uncertainty is studied. To solve the problem, an Affinely Adjustable Robust Optimisation model is developed. At the time when decisions about order periods, ordering quantities and discounts to offer are made, only a forecasted value of demand is available to a decision-maker. The proposed model produces a discount schedule, which is robust against the demand uncertainty. The model is also able to utilise the information about the realised demand from the previous periods in order to make decisions for future stages in an adjustable way. We consider both box and budget uncertainty sets. Computational results show the necessity of accounting for uncertainty, as the total costs of the nominal solution increase significantly even when only a small percentage of uncertainty is in place. It is testified that the affinely adjustable model produces solutions, which perform significantly better than the nominal solutions, not only on average, but also in the worst case. The trade-off between reduction of the conservatism of the model and the uncertainty protection is investigated as well