Robust Discrete Optimization Research Articles

Benchmarking problems for robust discrete optimization

Robust discrete optimization is a highly active field of research where a plenitude of combinations between decision criteria, uncertainty sets and underlying nominal problems are considered. Usually, a robust problem becomes harder to solve than its nominal counterpart, even if it remains in the same complexity class. For this reason, specialized solution algorithms have been developed. To further drive the development of stronger solution algorithms and to facilitate the comparison between methods, a set of benchmark instances is necessary but so far missing. In this paper we propose a further step towards this goal by proposing several instance generation procedures for combinations of min–max, min–max regret, two-stage and recoverable robustness with interval, discrete, budgeted or ellipsoidal uncertainty sets. Besides sampling methods that go beyond the simple uniform sampling method that is the de-facto standard to produce instances, also optimization models to construct hard instances are considered. Using a selection problem for the nominal ground problem, we are able to generate instances that are several orders of magnitudes harder to solve than uniformly sampled instances when solving them with a general mixed-integer programming solver. All instances and generator codes are made available online.

Robust Operating Room Scheduling Model with Violation Probability Consideration under Uncertain Surgery Duration.

This paper proposes an operating room (OR) scheduling model to assign a group of next-day patients to ORs while adhering to OR availability, priorities, and OR overtime constraints. Existing studies usually consider OR scheduling problems by ignoring the influence of uncertainties in surgery durations on the OR assignment. In this paper, we address this issue by formulating accurate patient waiting times as the cumulative sum of uncertain surgery durations from the robust discrete approach point of view. Specifically, by considering the patients’ uncertain surgery duration, we formulate the robust OR scheduling model to minimize the sum of the fixed OR opening cost, the patient waiting penalty cost, and the OR overtime cost. Then, we adopt the box uncertainty set to specify the uncertain surgery duration, and a robustness coefficient is introduced to control the robustness of the model. This resulting robust model is essentially intractable in its original form because there are uncertain variables in both the objective function and constraint. To make this model solvable, we then transform it into a Mixed Integer Linear Programming (MILP) model by employing the robust discrete optimization theory and the strong dual theory. Moreover, to evaluate the reliability of the robust OR scheduling model under different robustness coefficients, we theoretically analyze the constraint violation probability associated with overtime constraints. Finally, an in-depth numerical analysis is conducted to verify the proposed model’s effectiveness and to evaluate the robustness coefficient’s impact on the model performance. Our analytical results indicate the following: (1) With the robustness coefficient, we obtain the tradeoff relationship between the total management cost and the constraint violation probability, i.e., a smaller robustness coefficient yields remarkably lower total management cost at the expense of a noticeably higher constraint violation probability and vice versa. (2) The obtained total management cost is sensitive to small robustness coefficient values, but it hardly changes as the robustness coefficient increases to a specific value. (3) The obtained total management cost becomes increasingly sensitive to the perturbation factor with the decrease in constraint violation probability.

K-부분보형 부등식과 불확실성을 고려한 이산최적화 문제

We define k-submodular inequalities using the definition of the k-submodular set function. These inequalities can be applied to discrete robust optimization problems with mutually exclusive constraints. We define k-submodular polyhedron associated with the k-submodular function. Also we propose a polynomial-time separation algorithm for the most violated k-submodular inequality. The computational results show the effectiveness of the proposed inequalities when solving a robust discrete optimization problem by the branch-and-cut method.

Consensus Building for Uncertain Large-Scale Group Decision-Making Based on the Clustering Algorithm and Robust Discrete Optimization

Consensus Building for Uncertain Large-Scale Group Decision-Making Based on the Clustering Algorithm and Robust Discrete Optimization

Semantic Correspondence with Geometric Structure Analysis

This article studies the correspondence problem for semantically similar images, which is challenging due to the joint visual and geometric deformations. We introduce the Flip-aware Distance Ratio method (FDR) to solve this problem from the perspective of geometric structure analysis. First, a distance ratio constraint is introduced to enforce the geometric consistencies between images with large visual variations, whereas local geometric jitters are tolerated via a smoothness term. For challenging cases with symmetric structures, our proposed method exploits Curl to suppress the mismatches. Subsequently, image correspondence is formulated as a permutation problem, for which we propose a Gradient Guided Simulated Annealing (GGSA) algorithm to perform a robust discrete optimization. Experiments on simulated and real-world datasets, where both visual and geometric deformations are present, indicate that our method significantly improves the baselines for both visually and semantically similar images.

Multistage robust discrete optimization via quantified integer programming

Decision making needs to take an uncertain environment into account. Over the last decades, robust optimization has emerged as a preeminent method to produce solutions that are immunized against uncertainty. The main focus in robust discrete optimization has been on the analysis and solution of one- or two-stage problems, where the decision maker has limited options in reacting to additional knowledge gained after parts of the solution have been fixed. Due to its computational difficulty, multistage problems beyond two stages have received less attention.In this paper we argue that multistage robust discrete problems can be seen through the lens of quantified integer programs, where powerful tools to reduce the search tree size have been developed. By formulating both integer and quantified integer programming formulations, it is possible to compare the performance of state-of-the-art solvers from both worlds. Using selection, assignment, lot-sizing and knapsack problems as a testbed, we show that problems with up to nine stages can be solved to optimality in reasonable time.

Robust Mixed 0-1 Programming and Submodularity

The paper studies a robust mixed integer program with a single unrestricted continuous variable. The purpose of the paper is the polyhedral study of the robust solution set using submodularity. A submodular function is a set function with a diminishing returns property, and little work has been studied on the utilization of submodularity in the study of optimization problems considering data uncertainty. In this paper, we propose valid inequalities using submodularity. Valid inequalities for the robust mixed integer program are defined. A polynomial separation algorithm is proposed, and we show that the convex hull of the problem can be completely described using the proposed inequalities. In computational tests, we showed the proposed cuts are effective when they are applied to general robust discrete optimization problems with one or multiple constraints.

Two-stage self-scheduling of battery swapping station in day-ahead energy and frequency regulation markets

Battery swapping stations (BSS) face two major problems in the frequency regulation market: 1) battery degradation cost with the effect of regulation, and 2) the uncertainties in hourly cumulative regulation signals (HCRS), market prices, and demands. To address the problems, this paper proposes a novel battery model based on the historical data of regulation signals. Furthermore, the ambiguity sets of uncertainties are constructed based on the Wasserstein distance metric. A two-stage self-scheduling model of BSS in the day-ahead energy and regulation markets is presented. The first stage is to determine the batteries swapping plan with the uncertainty in EV demands by robust discrete optimization, while the second stage is formulated by the Wasserstein-distance-based distributionally robust chance-constrained program to optimize the bidding strategy with uncertainties in HCRS and market prices. The case studies verify the accuracy and effectiveness of the proposed battery model and the two-stage self-scheduling model of BSS.

Vehicles robust scheduling of hazardous materials based on hybrid particle swarm optimisation and genetic algorithm

This paper takes the vehicles scheduling of hazardous materials as the research object. First, considered the four objectives of risk minimization, cost minimization, risk equilibrium value minimization, and duration minimization, the vehicles robust scheduling of hazardous materials in multiple distribution centers with risk balance is constructed. Then, uncertain models are transformed into peer-to-peer models through the idea of robust discrete optimization. By introducing the crossover and mutation operators of genetic algorithm into particle swarm algorithm, a hybrid particle swarm algorithm is constructed, and three-stage coding rules are adopted. Finally, case study is exemplified to prove the feasibility of the model and algorithm. The results show that the obtained scheduling scheme for the vehicles of hazardous materials minimizes the risk value and makes the risk distribution more balanced for the government regulatory department, and the vehicle can avoid certain road sections with relatively large risk values. For transportation enterprises, the vehicle scheduling scheme for transporting hazardous materials minimizes the cost and shortens the task duration at the same time. In the case of risk uncertainty, different robust control parameters are introduced to make the vehicles robust scheduling of hazardous materials have different degrees of stability.

Almost Robust Discrete Optimization

The main goal of this paper is to present a simple and tractable methodology for incorporating data uncertainty into optimization models in the presence of binary variables. We introduce the Almost Robust Discrete Optimization (ARDO). ARDO extends the Integrated Chance-Constrained approach, developed for linear programs, to include binary integer variables. Both models trade off the objective function value with robustness and find optimal solutions that are almost robust (feasible under most realizations). These models are attractive due to their simplicity, ability to capture dependency among uncertain parameters, and that they incorporate the decision maker’s attitude towards risk by controlling the degree of conservatism of the optimal solution. To solve the ARDO model efficiently, we decompose it into a deterministic master problem and a single subproblem that checks the master problem solution under different realizations and generates cuts if needed. In contrast to other robust optimization models that are less tractable with binary decision variables, we demonstrate that with these cuts, the ARDO remains tractable. Computational experiments for the capacitated single-source facility location problem where demands in each node are uncertain demonstrate the effectiveness of our approach.

Distribution path robust optimization of electric vehicle with multiple distribution centers.

To identify electrical vehicle (EV) distribution paths with high robustness, insensitivity to uncertainty factors, and detailed road-by-road schemes, optimization of the distribution path problem of EV with multiple distribution centers and considering the charging facilities is necessary. With the minimum transport time as the goal, a robust optimization model of EV distribution path with adjustable robustness is established based on Bertsimas’ theory of robust discrete optimization. An enhanced three-segment genetic algorithm is also developed to solve the model, such that the optimal distribution scheme initially contains all road-by-road path data using the three-segment mixed coding and decoding method. During genetic manipulation, different interlacing and mutation operations are carried out on different chromosomes, while, during population evolution, the infeasible solution is naturally avoided. A part of the road network of Xifeng District in Qingyang City is taken as an example to test the model and the algorithm in this study, and the concrete transportation paths are utilized in the final distribution scheme. Therefore, more robust EV distribution paths with multiple distribution centers can be obtained using the robust optimization model.

Research on robust optimization of emergency logistics network considering the time dependence characteristic

Considering the time dependence of emergency logistic network and complexity of the environment that the network exists in, in this paper the time dependent network optimization theory and robust discrete optimization theory are combined, and the emergency logistics dynamic network optimization model with characteristics of robustness is built to maximize the timeliness of emergency logistics. On this basis, considering the complexity of dynamic network and the time dependence of edge weight, an improved ant colony algorithm is proposed to realize the coupling of the optimization algorithm and the network time dependence and robustness. Finally, a case study has been carried out in order to testify validity of this robustness optimization model and its algorithm, and the value of different regulation factors was analyzed considering the importance of the value of the control factor in solving the optimal path. Analysis results show that this model and its algorithm above-mentioned have good timeliness and strong robustness.

Bad-scenario-set robust optimization framework with two objectives for uncertain scheduling systems

This paper proposes a robust optimization framework generally for scheduling systems subject to uncertain input data, which is described by discrete scenarios. The goal of robust optimization is to hedge against the risk of system performance degradation on a set of bad scenarios while maintaining an excellent expected system performance. The robustness is evaluated by a penalty function on the bad-scenario set. The bad-scenario set is identified for current solution by a threshold, which is restricted on a reasonable-value interval. The robust optimization framework is formulated by an optimization problem with two conflicting objectives. One objective is to minimize the reasonable value of threshold, and another is to minimize the measured penalty on the bad-scenario set. An approximate solution framework with two dependent stages is developed to surrogate the biobjective robust optimization problem. The approximation degree of the surrogate framework is analyzed. Finally, the proposed bad-scenario-set robust optimization framework is applied to a scenario job-shop scheduling system. An extensive computational experiment was conducted to demonstrate the effectiveness and the approximation degree of the framework. The computational results testified that the robust optimization framework can provide multiple selections of robust solutions for the decision maker. The robust scheduling framework studied in this paper can provide a unique paradigm for formulating and solving robust discrete optimization problems.

Approximate robust optimization for the Connected Facility Location problem

In this paper we consider the Robust Connected Facility Location (ConFL) problem within the robust discrete optimization framework introduced by Bertsimas and Sim (2003). We propose an Approximate Robust Optimization (ARO) method that uses a heuristic and a lower bounding mechanism to rapidly find high-quality solutions. The use of a heuristic and a lower bounding mechanism–as opposed to solving the robust optimization (RO) problem exactly–within this ARO approach significantly decreases its computational time. This enables one to obtain high quality solutions to large-scale robust optimization problems and thus broadens the scope and applicability of robust optimization (from a computational perspective) to other NP-hard problems. Our computational results attest to the efficacy of the approach.

Using the WOWA operator in robust discrete optimization problems

In this paper a class of discrete optimization problems with uncertain costs is discussed. The uncertainty is modeled by introducing a scenario set containing a finite number of cost scenarios. A probability distribution over the set of scenarios is available. In order to choose a solution the weighted OWA criterion (WOWA) is applied. This criterion allows decision makers to take into account both probabilities for scenarios and the degree of pessimism/optimism. In this paper the complexity of the considered class of discrete optimization problems is described and some exact and approximation algorithms for solving it are proposed. Applications to the selection and the assignment problems, together with results of computational tests are shown.

Sensor Network Deployment under Distance Uncertainty with Robust Optimization

We consider the sensor deployment problem in the context of distance uncertainty. It is characterized by differentiated arrangement of specific detection probability thresholds at different locations. The problem is formulated as an integer linear programming (ILP) model firstly, aiming at optimizing the number of sensors and their locations. Based on the robust discrete optimization methodology, the uncertain model is transformed into an equivalent ILP problem considering distance uncertainty. The proposed approach can control the tradeoff between optimality and robustness by varying the parameters named protection levels. Uniform and non-uniform event detection probabiliy distributions are considered in the experiment. The results show that, as the distance uncertainty increases, the constraint violation can be avoided in the robust model and the robust solution can provide a significant improvement at the expense of a small loss in optimality when compared to the optimal solution of a deterministic scenario.

A Topology Control Algorithm for Sensor Networks Based on Robust Optimization

In recent years, wireless sensor networks have been widely used in data acquisition, surveillance, event monitoring, and so forth. Topology control is an important issue in designing sensor networks. Considering the uncertainty of distance between nodes, a distributed topology control algorithm named as LRMST, which is based on the local minimum spanning tree (LMST) algorithm, is proposed by applying the 0-1 robust discrete optimization theory. Firstly, when only the cost coefficients are subject to uncertainty, it is proved that the robust counterpart of the 0-1 discrete optimization problem on n variables can be solved by solving at most [Formula: see text] deterministic problems, where Γ denotes the number of cost coefficients which change in an uncertainty set. Then we present a robust model for the MST problem under distance uncertainty. According to the proved conclusion, an algorithm is proposed to obtain the robust solution of uncertain MST problem by solving only one deterministic MST problem, after which LRMST algorithm is designed when the distance between nodes is affected by uncertainty. Simulation results show that LRMST algorithm tends to select some edges whose estimated distance is slightly longer and obtains the robustness when the distance is uncertain at the expense of less optimal value compared with LMST algorithm.

Distributionally robust discrete optimization with Entropic Value-at-Risk

We study the discrete optimization problem under the distributionally robust framework. We optimize the Entropic Value-at-Risk, which is a coherent risk measure and is also known as Bernstein approximation for the chance constraint. We propose an efficient approximation algorithm to resolve the problem via solving a sequence of nominal problems. The computational results show that the number of nominal problems required to be solved is small under various distributional information sets.

Robust Optimization for the Connected Facility Location Problem

In this paper we present a robust optimization (RO) model for the Connected Facility Location (ConFL) problem within the framework introduced by Bertsimas and Sim [Bertsimas, D. and M. Sim, Robust discrete optimization and network flows, Mathematical Programming 98 (2003), pp. 49–71], and show how to use a heuristic in conjunction with a lower bounding mechanism to rapidly find high-quality solutions. The use of a heuristic and a lower bound mechanism within this RO approach decreases significantly its computational time and broadens its applicability to other NP-hard problems. Here we present some of our computational results that attest to the efficiency of the approach, particularly on the Robust ConFL problem.

Semidefinite and Conic Programming for Robust wireless OFDMA networks

In this paper, we study three robust optimization approaches [Adasme Pablo, Abdel Lisser and Ismael Soto, Robust Semidefinite Relaxations for a New Quadratic OFDMA Resource Allocation Approach, Working Paper Number 1522, LRI, University of Paris Sud, France]. The first one is based on the worst case scenario approach from Kouvelis and Yu [Kouvelis, and G. Yu, Robust discrete optimization and its applications, Kluwer Academic Publishers, 1997]. The second, corresponds to a scaled simplex polyhedral approach due to Bertsimas and Sim [Bertsimas Dimitris, and Melvyn Sim, The Price of Robustness, Operations Research, 52 (2004)] whilst the third, correspond to an ellipsoidal uncertainty approach proposed by Ben Tal and Nemirovski [Ben-Tal Aharon, and Arkadi Nemirovski, Robust solutions of Linear Programming problems contaminated with uncertain data, Mathematical Programming, Springer Berlin-Heidelberg, 88 (2000), 411–424]. The study of the different approaches is made on the basis of a binary quadratic constrained program (BQCP). We derive two semidefinite programming (SDP) relaxations for the first two approaches whilst we use a second order conic program for the last one. Numerical results are given for a resource allocation of OFDMA wireless networks.