Complete Recourse Research Articles

Comparisons of Two-Stage Models for Flood Mitigation of Electrical Substations

We compare stochastic programming and robust optimization decision models for informing the deployment of ad hoc flood mitigation measures to protect electrical substations prior to an imminent and uncertain hurricane. In our models, the first stage captures the deployment of a fixed quantity of flood mitigation resources, and the second stage captures the operation of a potentially degraded power grid with the primary goal of minimizing load shed. To model grid operation, we introduce adaptations of the direct current (DC) and linear programming alternating current (LPAC) power flow approximation models that feature relatively complete recourse by way of an indicator variable. We apply our models to a pair of geographically realistic flooding case studies, one based on Hurricane Harvey and the other on Tropical Storm Imelda. We investigate the effect of the mitigation budget, the choice of power flow model, and the uncertainty perspective on the optimal mitigation strategy. Our results indicate the mitigation budget and uncertainty perspective are impactful, whereas choosing between the DC and LPAC power flow models is of little to no consequence. To validate our models, we assess the performance of the mitigation solutions they prescribe in an alternating current (AC) power flow model. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Funding: This work was supported by the Energy Institute, The University of Texas at Austin. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0125 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0125 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Duality and sensitivity analysis of multistage linear stochastic programs

In this paper we investigate the dual of a Multistage Stochastic Linear Program (MSLP). By writing Dynamic Programming equations for the dual, we can employ an SDDP type method, called Dual SDDP, which solves these Dynamic Programming equations. allows us to compute a sequence of nonincreasing deterministic upper bounds for the optimal value of the problem. Since the Relatively Complete Recourse (RCR) condition may fail to hold for the dual (even for simple problems), we design two variants of Dual SDDP, namely Dual SDDP with penalizations and Dual SDDP with feasibility cuts, that converge to the optimal value of the dual (and therefore primal when there is no duality gap) problem under mild assumptions. We also show that optimal dual solutions can be obtained using dual information from Primal SDDP (applied to the original primal MSLP) subproblems.As a byproduct of the developed methodology we study sensitivity of the optimal value of the problem as a function of the involved parameters. For the sensitivity analysis we provide formulas for the derivatives of the value function with respect to the parameters and illustrate their application on an inventory problem. Since these formulas involve optimal dual solutions, we need an algorithm that computes such solutions to use them, i.e., we need to solve the dual problem.Finally, as a proof of concept of the tools developed, we present the results of numerical experiments computing the sensitivity of the optimal value of an inventory problem as a function of parameters of the demand process and compare Primal and Dual SDDP on the inventory and a hydro-thermal planning problems.

Stochastic dual dynamic programming for multistage stochastic mixed-integer nonlinear optimization

In this paper, we study multistage stochastic mixed-integer nonlinear programs (MS-MINLP). This general class of problems encompasses, as important special cases, multistage stochastic convex optimization with non-Lipschitzian value functions and multistage stochastic mixed-integer linear optimization. We develop stochastic dual dynamic programming (SDDP) type algorithms with nested decomposition, deterministic sampling, and stochastic sampling. The key ingredient is a new type of cuts based on generalized conjugacy. Several interesting classes of MS-MINLP are identified, where the new algorithms are guaranteed to obtain the global optimum without the assumption of complete recourse. This significantly generalizes the classic SDDP algorithms. We also characterize the iteration complexity of the proposed algorithms. In particular, for a (T+1)-stage stochastic MINLP satisfying L-exact Lipschitz regularization with d-dimensional state spaces, to obtain an varepsilon -optimal root node solution, we prove that the number of iterations of the proposed deterministic sampling algorithm is upper bounded by {mathcal {O}}((frac{2LT}{varepsilon })^d), and is lower bounded by {mathcal {O}}((frac{LT}{4varepsilon })^d) for the general case or by {mathcal {O}}((frac{LT}{8varepsilon })^{d/2-1}) for the convex case. This shows that the obtained complexity bounds are rather sharp. It also reveals that the iteration complexity depends polynomially on the number of stages. We further show that the iteration complexity depends linearly on T, if all the state spaces are finite sets, or if we seek a (Tvarepsilon )-optimal solution when the state spaces are infinite sets, i.e. allowing the optimality gap to scale with T. To the best of our knowledge, this is the first work that reports global optimization algorithms as well as iteration complexity results for solving such a large class of multistage stochastic programs. The iteration complexity study resolves a conjecture by the late Prof. Shabbir Ahmed in the general setting of multistage stochastic mixed-integer optimization.

Distributionally robust optimization for the berth allocation problem under uncertainty

Berth allocation problems are amongst the most important problems occurring in port terminals, and they are greatly affected by several unpredictable events. As a result, the study of these problems under uncertainty has been a target of more and more researchers. Following this research line, we consider the berth allocation problem under uncertain handling times. A distributionally robust two-stage model is presented to minimize the worst-case of the expected sum of delays with respect to a set of possible probability distributions of the handling times. The solutions of the proposed model are obtained by an exact decomposition algorithm for which several improvements are discussed. An adaptation of the proposed algorithm for the case where the assumption of relatively complete recourse fails is also presented. Extensive computational tests are reported to evaluate the effectiveness of the proposed approach and to compare the solutions obtained with those resulting from the stochastic and robust approaches.

General Feasibility Bounds for Sample Average Approximation via Vapnik--Chervonenkis Dimension

We investigate the feasibility of sample average approximation (SAA) for general stochastic optimization problems, including two-stage stochastic programming without the relatively complete recourse assumption. Instead of analyzing problems with specific structures, we utilize results from the Vapnik-Chervonenkis (VC) dimension and Probably Approximately Correct learning to provide a general framework that offers explicit feasibility bounds for SAA solutions under minimal structural or distributional assumption. We show that, as long as the hypothesis class formed by the feasbible region has a finite VC dimension, the infeasibility of SAA solutions decreases exponentially with computable rates and explicitly identifiable accompanying constants. We demonstrate how our bounds apply more generally and competitively compared to existing results.

A Lagrangian dual method for two-stage robust optimization with binary uncertainties

This paper presents a new exact method to calculate worst-case parameter realizations in two-stage robust optimization problems with categorical or binary-valued uncertain data. Traditional exact algorithms for these problems, notably Benders decomposition and column-and-constraint generation, compute worst-case parameter realizations by solving mixed-integer bilinear optimization subproblems. However, their numerical solution can be computationally expensive not only due to their resulting large size after reformulating the bilinear terms, but also because decision-independent bounds on their variables are typically unknown. We propose an alternative Lagrangian dual method that circumvents these difficulties and is readily integrated in either algorithm. We specialize the method to problems where the binary parameters switch on or off constraints as these are commonly encountered in applications, and discuss extensions to problems that lack relatively complete recourse and to those with integer recourse. Numerical experiments provide evidence of significant computational improvements over existing methods.

Robust Satisficing

We present a general framework for robust satisficing that favors solutions for which a risk-aware objective function would best attain an acceptable target even when the actual probability distribution deviates from the empirical distribution. The satisficing decision maker specifies an acceptable target, or loss of optimality compared with the empirical optimization model, as a trade-off for the model’s ability to withstand greater uncertainty. We axiomatize the decision criterion associated with robust satisficing, termed as the fragility measure, and present its representation theorem. Focusing on Wasserstein distance measure, we present tractable robust satisficing models for risk-based linear optimization, combinatorial optimization, and linear optimization problems with recourse. Serendipitously, the insights to the approximation of the linear optimization problems with recourse also provide a recipe for approximating solutions for hard stochastic optimization problems without relatively complete recourse. We perform numerical studies on a portfolio optimization problem and a network lot-sizing problem. We show that the solutions to the robust satisficing models are more effective in improving the out-of-sample performance evaluated on a variety of metrics, hence alleviating the optimizer’s curse. Funding: D. Z. Long is supported by the Hong Kong Research Grants Council [Grant 14207819]. M. Sim and M. Zhou are supported by the Ministry of Education, Singapore, under its 2019 Academic Research Fund Tier 3 [Grant MOE-2019-T3-1-010]. Supplemental Material: The online appendices are available at https://doi.org/10.1287/opre.2021.2238 .

On sample average approximation for two-stage stochastic programs without relatively complete recourse

We investigate sample average approximation (SAA) for two-stage stochastic programs without relatively complete recourse, i.e., for problems in which there are first-stage feasible solutions that are not guaranteed to have a feasible recourse action. As a feasibility measure of the SAA solution, we consider the “recourse likelihood”, which is the probability that the solution has a feasible recourse action. For $$\epsilon \in (0,1)$$ , we demonstrate that the probability that a SAA solution has recourse likelihood below $$1-\epsilon $$ converges to zero exponentially fast with the sample size. Next, we analyze the rate of convergence of optimal solutions of the SAA to optimal solutions of the true problem for problems with a finite feasible region, such as bounded integer programming problems. For problems with non-finite feasible region, we propose modified “padded” SAA problems and demonstrate in two cases that such problems can yield, with high confidence, solutions that are certain to have a feasible recourse decision. Finally, we conduct a numerical study on a two-stage resource planning problem that illustrates the results, and also suggests there may be room for improvement in some of the theoretical analysis.

Technical Note—Two-Stage Sample Robust Optimization

In “Two-Stage Sample Robust Optimization,” Bertsimas, Shtern, and Sturt investigate a simple approximation scheme, based on overlapping linear decision rules, for solving data-driven two-stage distributionally robust optimization problems with the type-infinity Wasserstein ambiguity set. Their main result establishes that this approximation scheme is asymptotically optimal for two-stage stochastic linear optimization problems; that is, under mild assumptions, the optimal cost and optimal first-stage decisions obtained by approximating the robust optimization problem converge to those of the underlying stochastic problem as the number of data points grows to infinity. These guarantees notably apply to two-stage stochastic problems that do not have relatively complete recourse, which arise frequently in applications. In this context, the authors show through numerical experiments that the approximation scheme is practically tractable and produces decisions that significantly outperform those obtained from state-of-the-art data-driven alternatives.

Stochastic batch dispersion model to optimize traceability and enhance transparency using Blockchain

The absence of food traceability has led to severe problems like a product recall, consumer dissatisfaction, and contamination insecurities in the past. We consider a five-level supply chain for sausage where at each level, the output product is manufactured by combining/mixing correct proportions of the raw materials from the previous stages. The demand for the final product is uncertain. Using stochastic models, we improve the supply chain's traceability and optimize dispersion among the sausage material batches. We also derive theoretical results in terms of proving a relatively complete recourse structure in the proposed model. Furthermore, to provide insights regarding the supply chain network's transparency, we integrate the Blockchain framework in our model data storage. Using a simple case study, where an attacker tries to alter or delete data inside the Blockchain, we quantitatively measure this immutability of this decentralized data. The case study reveals that decentralized databases could make data accessible to peers (retailers, suppliers, manufacturers) while mitigating data tampering by Blockchain technology. This ensures transparency among the peers and the privacy and security of the data present in the decentralized database of different supply chain networks.

Robust Conic Satisficing

Inspired by the principle of satisficing (Simon 1955), Long et al. (2021) propose an alternative framework for optimization under uncertainty, which we term as a robust satisficing model. Instead of sizing the uncertainty set in robust optimization, the robust satisficing model is specified by a target objective with the aim of delivering the solution that is least impacted by uncertainty in achieving the target. At the heart of this framework, we minimize the level of constraint violation under all possible realizations within the support set. Our framework is based on a constraint function that evaluates to the optimal objective value of a standard conic optimization problem, which can be used to model a wide range of constraint functions that are convex in the decision variables but can be either convex or concave in the uncertain parameters. We derive an exact semidefinite optimization formulation when the constraint is biconvex quadratic with quadratic penalty and the support set is ellipsoidal. We also show the equivalence between the more general robust satisficing problems and the classical adaptive robust linear optimization models with conic uncertainty sets, where the latter can be solved approximately using affine recourse adaptation. More importantly, under complete recourse, and reasonably chosen polyhedral support set and penalty function, we show that the exact reformulation and safe approximations do not lead to infeasible problems if the chosen target is above the optimum objective obtained when the nominal optimization problem is minimized. Finally, we extend our framework to the data-driven setting and showcase the modelling and the computational benefits of the robust satisficing framework over robust optimization with three numerical examples: portfolio selection, log-sum-exp optimization and adaptive lot-sizing problem.

A two-stage stochastic mixed-integer program for reliable supply chain network design under uncertain disruptions and demand

Having a reliable supply chain network is one of the most effective ways to plan and deal with facility failures and demand uncertainty due to very limited options available in a facility location problem. This research addresses issues bordering on capacitated supply chain problems, specifically on how reliable supply chain networks can be designed in the face of random facility disruptions and uncertain demand with a combination of hardening selected facilities and product reassignment. The proposed multi-period capacitated facility location and allocation problem is modeled as a two-stage stochastic mixed-integer formulation that minimizes the total establishing and transportation cost. The L-shaped method of stochastic linear programming is applied by integrating with two types of optimality and feasibility cuts for solving the stochastic model. This research aims to improve the proposed algorithm in two different ways: replacing the single-cut approach with a multi-cut and showing relatively complete recourse in the stochastic model by reformulating the original model. Finally, to illustrate the applicability of the model and computational effectiveness of the improved algorithm, a case study with different scenarios is presented and the results are discussed. Our computational results show that the proposed solution algorithm is not only capable of solving large-scale problems, but also avoids long run times. It is also demonstrated that substantial improvements in reliability of the system can often be possible with minimal increases in facility cost.

Benders Cut Classification via Support Vector Machines for Solving Two-Stage Stochastic Programs

We consider Benders decomposition for solving two-stage stochastic programs with complete recourse based on finite samples of the uncertain parameters. We define the Benders cuts binding at the final optimal solution or the ones significantly improving bounds over iterations as valuable cuts. We propose a learning-enhanced Benders decomposition (LearnBD) algorithm, which adds a cut classification step in each iteration to selectively generate cuts that are more likely to be valuable cuts. The LearnBD algorithm includes two phases: (i) sampling cuts and collecting information from training problems and (ii) solving testing problems with a support vector machine (SVM) cut classifier. We run the LearnBD algorithm on instances of capacitated facility location and multicommodity network design under uncertain demand. Our results show that SVM cut classifier works effectively for identifying valuable cuts, and the LearnBD algorithm reduces the total solving time of all instances for different problems with various sizes and complexities.

On Feasibility of Sample Average Approximation Solutions

When there are infinitely many scenarios, the current studies of two-stage stochastic programming problems rely on the relatively complete recourse assumption. However, such an assumption can be un...

The Dao of Robustness

We present a general framework for data-driven optimization called robustness optimization that favors solutions for which a risk-aware objective function would best attain an acceptable target even when the actual probability distribution deviates from the empirical distribution. Unlike robust optimization approaches, the decision maker does not have to size the ambiguity set, but specifies an acceptable target, or loss of optimality compared to the empirical optimization model, as a trade off for the model’s ability to withstand greater uncertainty. We axiomatize the decision criterion associated with robustness optimization, termed as the fragility measure and present its representation theorem. Focusing on Wasserstein distance measure with l1-norm, we present tractable robustness optimization models for risk-based linear optimization, combinatorial optimization, and linear optimization problems with recourse. Serendipitously, the insights to the approximation also provide a recipe for approximating solutions for hard stochastic optimization prob- lems without relatively complete recourse. We illustrate in a portfolio optimization problem and a network lot-sizing problem on how we can set targets in the robustness optimization model, which can be more intuitive and effective than specifying the hyper-parameter used in a robust optimization model. The numerical studies show that the solutions to the robustness optimization models are more effective in alleviating the Optimizer’s Curse (Smith and Winkler 2006) by improving the out-of-sample performance evaluated on a variety of metrics.

Formulation of Two-Stage Stochastic Programming with Fixed Recourse

Stochastic Programming is an asset for the next world researchers due to its uncertainty calculations, which has been skipped in deterministic world experiments as it includes complicated calculations. Two-stage stochastic programming concerns two time period decisions based on some random parameters obtained from past experience or some sort of survey. The objective function for formulating two-stage stochastic programming with fixed recourse includes two parts: first-stage forecast and second-stage fixed decisions based on the experiment results. The constraints are similar to the normal optimization techniques rather some adjustments of requirements and technology assets. The fixed recourse decisions are sort of decisions from the deterministic world. Formulation techniques of two-stage stochastic programming with fixed recourse may be used for further complications arises in stochastic programming like complete recourse problems, multi-stage problems, etc. And that’s why Two-stage stochastic programming with fixed recourse is called the primary model for stochastic programming.

Designing networks with resiliency to edge failures using two-stage robust optimization

We study the design of resilient single-commodity flow networks that can remain robust against multiple concurrent edge failures. We model these failures as binary random variables, allowing us to formally formulate the network design problem as a two-stage robust optimization problem. With an objective of minimizing the overall cost of building and operating the network, the capacities of the edges are decided in the first stage, while the optimal flows are determined in the second stage once the uncertainty has been realized. We first examine the standard affine decision rules approach and show that it is not a viable approach when two or more edges are allowed to fail at the same time. We then propose a column and constraint generation algorithm that we tailor to this application. Since the problem does not satisfy the relatively complete recourse assumption, we employ an oracle with two subproblems: one to determine edge failure scenarios that render the required demand satisfaction infeasible, and if no such scenario exists, a second one to determine the flow rerouting plan of highest cost. Our column and constraint generation algorithm is applied to networks adapted from the Survivable Network Design Library. For each instance, we determine sequences of fully adaptive, robust optimal solutions for various levels of resiliency, identifying also the maximum number of concurrent edge failures that can be sustained by these networks. Finally, we demonstrate how our algorithm can be applied to a defender versus attacker context, via the use of a decision-dependent uncertainty set.

Robust Dual Dynamic Programming

In the paper “Robust Dual Dynamic Programming,” Angelos Georghiou, Angelos Tsoukalas, and Wolfram Wiesemann propose a novel solution scheme for addressing planning problems with long horizons. Such problems can be formulated as multistage robust optimization problems. The proposed method takes advantage of the decomposable nature of these problems by bounding the costs arising in the future stages through lower and upper cost-to-go functions. The proposed scheme does not require a relatively complete recourse, and it offers deterministic upper and lower bounds throughout the execution of the algorithm. The promising performance of the algorithm is shown in a stylized inventory-management problem in which the proposed algorithm achieved the optimal solution in problem instances with 100 time stages in a few minutes.

Convergence Analysis of Sample Average Approximation of Two-Stage Stochastic Generalized Equations

A solution of two-stage stochastic generalized equations is a pair: a first stage solution which is independent of realization of the random data and a second stage solution which is a function of random variables.This paper studies convergence of the sample average approximation of two-stage stochastic nonlinear generalized equations. In particular an exponential rate of the convergence is shown by using the perturbed partial linearization of functions. Moreover, sufficient conditions for the existence, uniqueness, continuity and regularity of solutions of two-stage stochastic generalized equations are presented under an assumption of monotonicity of the involved functions. These theoretical results are given without assuming relatively complete recourse, and are illustrated by two-stage stochastic non-cooperative games of two players.

Strong convexity in risk-averse stochastic programs with complete recourse

We give sufficient conditions for the expected excess and the mean-upper-semideviation of recourse functions to be strongly convex. This is done in the setting of two-stage stochastic programs with complete linear recourse and random right-hand side. This work extends results on strong convexity of risk-neutral models.