Online Surrogate Problem Methodology for Stochastic Discrete Resource Allocation Problems

Abstract

We consider stochastic discrete optimization problems where the decision variables are non-negative integers. We propose and analyze an on-line control scheme which transforms the problem into a "surrogate" continuous optimization problem and proceeds to solve the latter using standard gradient-based approaches while simultaneously updating both actual and surrogate system states. It is shown that the solution of the original problem is recovered as an element of the discrete state neighborhood of the optimal surrogate state. For the special case of separable cost functions, we show that this methodology becomes particularly efficient. Finally, convergence of the proposed algorithm is established under standard technical conditions and numerical results are included in the paper to illustrate the fast convergence properties of this approach.

Resource allocation in satellite networks: certainty equivalent approaches versus sensitivity estimation algorithms

“…As proposed in References [3,4] the discrete functional cost defined over Y d is transformed into a 'surrogate' one that works over Y c : We construct an estimation of the gradient, according to the current measured sample path by Equations (15) and (16), and we apply a sequence of minimization steps until the optimum is reached.…”

“…Even when the setting is deterministic and the expectation is not requested, this class of problems is NP-hard (see e.g. References [3,4] and references therein). In some cases, depending upon the form of the objective function Jðh d Þ (e.g.…”

“…In order to face such heavy drawbacks, following the theoretical framework of Gokbayrak and Cassandras [3,4], we shall formulate a new optimization algorithm, based on a sensitivity estimation procedure. We shall compare it with the optimization technique employed in References [8,20], based on a closed-form expression of the performance index and on the adoption of a dynamic programming algorithm.…”

“…As is shown in Reference [4], the nearest feasible neighbour h d ½k # t t 2 Y d of h c ½ðk À 1Þ # t t can be determined, at step 5, by using an algorithm based on the Simplex method. However, it is possible to apply a simpler OðN þ 1Þ algorithm based on the N þ 1 discrete neighbours of h c ½ðk À 1Þ # t t; not necessarily all feasible, and on the selection of one of them, which satisfies the discrete constraint set Y d (see Reference [3] for further details).…”

Adaptive cross‐layer bandwidth allocation in a rain‐faded satellite environment

“…The comparison with the simple SP method has shown that the cross-layer adaptive policies can yield a significant performance improvement in all quantities of interest (by reducing cell loss and call dropping, and by keeping call blocking probabilities evenly distributed among the traffic stations). Further research could be carried out along the following lines: (i) to extend the policies to stations that experience both up-and down-link fades towards multiple destinations, which create a multi-rate environment, due to the presence of different data redundancies at the same station; (ii) to investigate the effect of the adoption of on-line gradient descent techniques of the type considered in Reference [34], to relax the discrete integer optimization problem to a continuous one, whose solution can be spread over time, instead of being concentrated at the beginning of a fixed time interval (a comparison has been actually already done in Reference [35], in the presence of best-effort packet traffic only); (iii) to compare CP, CS, and hybrid allocation policies, always taking into account the presence of two basic traffic types; (iv) to include models of TCP elastic traffic and their related performance optimization (see, e.g. Reference [36]).…”