Online Primal-Dual Algorithms For Stochastic Resource Allocation Problems

Abstract

This paper studies the online stochastic resource allocation problem (RAP) with chance constraints and conditional expectation constraints. The online RAP is an integer linear programming problem where resource consumption coefficients are revealed column by column along with the corresponding revenue coefficients. When a column is revealed, the corresponding decision variables are determined instantaneously without future information. In online applications, the resource consumption coefficients are often obtained by prediction. An application for such scenario rises from the online order fulfilment task. When the timeliness constraints are considered, the coefficients are generated by the prediction for the transportation time from origin to destination. To model their uncertainties, we take the chance constraints and conditional expectation constraints into the consideration. Assuming that the uncertain variables have known Gaussian distributions, the stochastic RAP can be transformed into a deterministic but nonlinear problem with integer second-order cone constraints. Next, we linearize this nonlinear problem and theoretically analyze the performance of vanilla online primal-dual algorithm for solving the linearized stochastic RAP. Under mild technical assumptions, the optimality gap and constraint violation are both on the order of √ n. Then, to further improve the performance of the algorithm, several modified online primal-dual algorithms with heuristic corrections are proposed. Finally, extensive numerical experiments demonstrate the applicability and effectiveness of our methods.

An Online Algorithm for Chance Constrained Resource Allocation

“…Modified OPD Algorithm for CCP Input: d = b/n Output: x = (x1, ..., xn) 1 Initialize: p1 = 0, d1 = d 2 for t = 1, ..., n do 3Compute βt via equation (7)Set vt = max l=1,...,k (c ⊤ t − p ⊤ t Ât(βt))elPick an index lt randomly froml : vt = (c ⊤ t − p ⊤ t Ât(βt))el pt+1 = max pt + 1 √ n Ât(βt)xt − dt , 0stochastic subgradient descent method where (d − Ãtxt) is the subgradient corresponding to pt.The following Theorem 1 states that Algorithm 1 achieves O(√ n) regret and constraint violation compared to the optimal solution of the ISOCP problem (3). The detailed proof of Theorem 1 as well as Proposition 1 is presented in the full-length version[29]. Assume coefficient sets {ctj , ātj, Ktj}s are bounded and sampled i.i.d.…”