Sampling-Based Approximation Schemes for Capacitated Stochastic Inventory Control Models

We provide a refined convergence analysis for the SAA (sample average approximation) method applied to stochastic optimization problems with either single or mixed CVaR (conditional value-at-risk) measures. Under certain regularity conditions, it is shown that any accumulation point of the weak GKKT (generalized Karush-Kuhn-Tucker) points produced by the SAA method is almost surely a weak stationary point of the original CVaR or mixed CVaR optimization problems. In addition, it is shown that, as the sample size increases, the difference of the optimal values between the SAA problems and the original problem tends to zero with probability approaching one exponentially fast.

This paper aims to optimise healthcare resource pre-positioning, patient scheduling, and patient transferring under uncertain demands and stochastic resource consumption. We propose a two-stage stochastic programming model that formulates the patient scheduling problem as a Markov decision process. To address complexities and uncertainties, we use Artificial Intelligence (AI) techniques to improve both model formulation and algorithmic performance. To tackle the limited data challenge, we introduce a Wasserstein distance-based ambiguity set and propose a two-stage distributionally robust optimisation (DRO) model, which derives a deterministic equivalent using the Lagrangian dual of non-anticipativity constraints. The solution process is accelerated with a scenario decomposition approach and the K-means clustering method. Both theoretical and numerical results demonstrate the consistency of the two-stage DRO model with the sample average approximation (SAA) method. The potential of AI to improve the model's performance is evident through the significant reduction in computation time achieved with the K-means clustering approach without compromising solution quality. Compared to the SAA method, the DRO model exhibits a considerable reduction in the waiting penalty cost for out-of-sample cases, ranging from 43.06 % to 81.23 % . Numerical results show that the proposed algorithm outperforms SAA methods and several benchmark policies in terms of computational efficiency and solutions.

A note on the sample average approximation method for stochastic mathematical programs with complementarity constraints

The aim of this paper is to investigate the convergence properties for Mordukhovich’s coderivative of the solution map of the sample average approximation (SAA) problem for a parametric stochastic variational inequality with equality and inequality constraints. The notion of integrated deviation is introduced to characterize the outer limit of a sequence of sets. It is demonstrated that, under suitable conditions, both the cosmic deviation and the integrated deviation between the coderivative of the solution mapping to SAA problem and that of the solution mapping to the parametric stochastic variational inequality converge almost surely to zero as the sample size tends to infinity. Moreover, the exponential convergence rate of coderivatives of the solution maps to the SAA parametric stochastic variational inequality is established. The results are used to develop sufficient conditions for the consistency of the Lipschitz-like property of the solution map of SAA problem and the consistency of stationary points of the SAA estimator for a stochastic bilevel program.

To reflect uncertain data in practical problems, stochastic versions of the mathematical program with complementarity constraints (MPCC) have drawn much attention in the recent literature. Our concern is the detailed analysis of convergence properties of a regularization sample average approximation (SAA) method for solving a stochastic mathematical program with complementarity constraints (SMPCC). The analysis of this regularization method is carried out in three steps: First, the almost sure convergence of optimal solutions of the regularized SAA problem to that of the true problem is established by the notion of epiconvergence in variational analysis. Second, under MPCC-MFCQ, which is weaker than MPCC-LICQ, we show that any accumulation point of Karash-Kuhn-Tucker points of the regularized SAA problem is almost surely a kind of stationary point of SMPCC as the sample size tends to infinity. Finally, some numerical results are reported to show the efficiency of the method proposed.

In this paper, we investigate scenario generation methods to establish lower bounds on the optimal objective value for stochastic scheduling problems that contain random parameters with continuous distributions. In contrast to the Sample Average Approximation (SAA) approach, which yields probabilistic bound values, we use an alternative bounding method that relies on the ideas of discrete bounding and recursive stratified sampling. Theoretical support is provided for deriving exact lower bounds for both expectation and conditional value-at-risk objectives. We illustrate the use of our method on the single machine total weighted tardiness problem. The results of our numerical investigation demonstrate good properties of our bounding method, compared with the SAA method and an earlier discrete bounding method.

This paper applies sample average approximation (SAA) method based on ‐space decomposition theory to solve stochastic convex minimax problems. Under some moderate conditions, the SAA solution converges to its true counterpart with probability approaching one and convergence is exponentially fast with the increase of sample size. Based on the ‐theory, a superlinear convergent ‐algorithm frame is designed to solve the SAA problem.

Sample average approximation (SAA) method has recently been applied to solve stochastic programs with second order stochastic dominance (SSD) constraints. In particular, Hu et al. (Math Program 133:171–201, 2012) presented a detailed convergence analysis of \(\epsilon \)-optimal values and \(\epsilon \)-optimal solutions of sample average approximated stochastic programs with polyhedral SSD constraints. In this paper, we complement the existing research by presenting convergence analysis of stationary points when SAA is applied to a class of stochastic minimization problems with SSD constraints. Specifically, under some moderate conditions we prove that optimal solutions and stationary points obtained from solving sample average approximated problems converge with probability one to their true counterparts. Moreover, by exploiting some recent results on large deviation of random functions and sensitivity analysis of generalized equations, we derive exponential rate of convergence of stationary points.

In this paper, we combine two approaches to handling uncertainty: we use techniques for finding optimal solutions in the expected sense to solve combinatorial optimization problems in an online setting.

A scenario-based planning for the pickup and delivery problem with time windows, scheduled lines and stochastic demands

Various stochastic programming problems can be formulated as problems of optimization of an expected value function. Quite often the corresponding expectation function cannot be computed exactly and should be approximated, say by Monte Carlo sampling methods. In fact, in many practical applications, Monte Carlo simulation is the only reasonable way of estimating the expectation function. We discuss converges properties of the sample average approximation (SAA) approach to stochastic programming. We argue that the SAA method is easily implementable and can be surprisingly efficient for some classes of stochastic programming problems.

Sample average approximation method is one of the well-behaved methods in the stochastic optimization.This paper presents a sample average approximation method based on a D-gap function for stochastic variational inequality problems.An unconstrained optimization reformulation is proposed for the expected-value formulation of stochastic variational inequality problems based on the D-gap function. An implementable sample average approximation method for the reformulation is established andit is proven that the optimal values and the optimal solutions of the approximation problems converge to their true counterpart with probability one as the sample size increases under some moderate assumptions.Finally, the preliminary numerical results for some test examples are reported, which show that the proposed method is promising.

Nowadays, due to the increasing complexity of business environment, especially demand uncertainty, supply chain managers need to establish more-effective sourcing and distribution strategies to ensure high customer service and low stock costs. To overcome this challenge multi-echelon network structures and alternative distribution strategies such as lateral transshipments and multiple sourcing should be considered in inventory optimisation models. In this article, we propose a scenario-based modelling approach to solve a two-stage multi-echelon inventory optimisation problem with a non-stationary demand. The model is based on a distribution requirements planning (DRP) approach and minimises the expected total cost that is composed of the fixed allocation, inventory holding, procurement, transportation, and back-ordering costs. Alternative inventory optimisation models, including the lateral transshipment strategy and multiple sourcing, are thus built, and the corresponding stochastic programmes are solved using the sample average approximation method. Through a numerical investigation conducted with several generated instances and an empirical investigation based on the case of a major French retailer’s distribution network, we show the substantial benefit of lateral transshipments and multiple sourcing in reducing the expected total costs of the distribution network.

In this paper, we develop a stochastic programming model for economic dispatch of a power system with operational reliability and risk control constraints. By defining a severity-index function, we propose to use conditional value-at-risk (CVaR) for measuring the reliability and risk control of the system. The economic dispatch is subsequently formulated as a stochastic program with CVaR constraint. To solve the stochastic optimization model, we propose a penalized sample average approximation (SAA) scheme which incorporates specific features of smoothing technique and level function method. Under some moderate conditions, we demonstrate that with probability approaching to 1 at an exponential rate with the increase of sample size, the optimal solution of the smoothing SAA problem converges to its true counterpart. Numerical tests have been carried out for a standard IEEE-30 DC power system.

The drone swarm that performs missions in an autonomous and intelligent overall collaborative manner is an important operational factor in the complex flight space. However, the complex and changing battlefield environment greatly affects the mission execution and survivability of drone swarm. In this paper, we formulated the problem of formation reconfiguration to avoid collision with obstacles with minimum cost in a complex flight space as a two-stage stochastic programming with considering uncertain moving obstacles, where we first calculate the optimal formation parameters of the drone swarm that can avoid certain fixed obstacles with minimum reconfiguration cost in the first-stage. For the uncertain moving obstacles, we calculate the extra cost for avoiding in the second-stage after the distribution of moving obstacles is known, and add this part of the cost to the first-stage to ensure the overall reconfiguration cost is minimized. Moreover, the sample average approximation (SAA) method is exploited to solve this stochastic programming problem. Extensive experimental results from the OMNeT ++ simulation environment indicate both the feasibility and effectiveness of our proposed two-stage stochastic programming approach and the better performs, compared with the existing approach.