Outer Approximation Algorithm Research Articles

Risk-averse distributionally robust optimization for construction waste reverse logistics with a joint chance constraint

The uncertainty of the amount of construction and demolition waste (CDW) generation affects the CDW reverse logistics network service level. We investigate a CDW reverse logistics network location-routing problem considering uncertainties. To minimize the total social cost, a two-stage risk-averse distributionally robust optimization model is developed, which aims to optimize the location and number of CDW disposal facilities and the CDW transportation scheme. We introduce the mean-conditional value at risk measure and a joint chance constraint into our model to consider the government’s risk aversion. The above model is approximated as a standard second-order cone programming model (SOCP) considering a special case. To exactly solve the SOCP, we design an outer approximation algorithm. Several performance tests and a case study are proposed, and sensitivity analysis provides helpful managerial insights.

Improve operational flexibility of distribution systems using transportable resources

With the continuous progress of carbon neutrality, the accelerating integration of distributed renewable energy sources (RES) poses tremendous pressure on distribution systems' safe and stable operation. The distribution systems that cannot be upgraded in time, e.g., those in border areas, are hard to adapt to the large-scale integration of RES and the growing energy demand, facing a severe lack of operational flexibility. In light of this, the integrated mobile energy station (IMES), which consists of a battery system and photovoltaic cells, is used to improve operational flexibility and alleviate the need for upgrades. An adaptive robust allocation and dispatch model is introduced to coordinate the operation of the distribution system and IMES under multiple uncertainties. The first stage of the proposed model optimizes the locations of IMES using the forecast information of RES and loads, while the second stage optimizes the dispatch plan that is adaptive to the realization of uncertainties. Since the uncertain photovoltaic power in the IMES depends on its deployment location, the proposed model is an adaptive robust optimization with decision-dependent uncertainty, bringing great challenges to the solution. To solve it efficiently, a warmstart-enhanced column-and-constraint generation algorithm is introduced, in which the bi-level subproblem is solved by the dual method and outer approximation algorithm. Finally, case studies demonstrate the effectiveness of the proposed method. The participation of six IMESs can save 5.23 % on the system's total cost and avoid the load shedding caused by transmission congestion, indicating IMES's potential for economical and flexible operation of the system.

Robust dynamic train regulation integrated with stop-skipping strategy in urban rail networks: An outer approximation based solution method

In dense urban rail networks with high passenger demands, uncertain disturbances occur frequently, and the resulting train delays will likely spread over the whole network rapidly, hence degrading the service quality offered to passengers. To cope with the uncertainties of frequent disturbances in urban rail networks, this paper proposes a robust train regulation strategy based on the information gap decision theory, which allows the operators to adjust the conservativeness of adjustment schemes flexibly by varying system performances but without the need for prior knowledge of uncertain disturbances. Specifically, considering the coupling relationship between train dynamic flows and passenger dynamic flows, a mixed integer quadratically constrained programming (MIQCP) model is constructed for the robust train regulation problem to generate solutions with immunity against disturbance uncertainties, in which the envelope-bound model is used to characterizing the uncertain sets of disturbances. To meet the real-time requirements of train operation adjustment, a tailored outer approximation algorithm incorporating a two-phase heuristics method is devised to effectively solve the developed robust train regulation model, thereby quickly generating high-quality solutions. Moreover, the warm start technique and domain reduction technique are carefully developed to accelerate the solving procedure. Numerical experiments based on the Beijing metro network illustrate the robustness of the proposed train regulation strategies and the effectiveness of the designed solution approach.

Impacts of capacity redundancy and process flexibility on risk mitigation in e-waste recycling supply chain management

Recently Waste Electrical and Electronic equipment (WEEE) has become urgent environmental issues around the world. Generally, the e-waste recycling reverse supply chain (RSC) features higher uncertainties and complicated network structures, making the operation more difficult. Since the study on impact of hybrid uncertainty mitigation strategy in RSC is rare, we study the coupling effect of two strategies — capacity redundancy and process flexibility through the design of two hybrid strategies. A distributionally robust optimization model is proposed to handle both waste returns and resource yields uncertainties, and solved by a proposed Outer Approximation (OA) algorithm after an adaptive robust formulation. With numerical experiments based on real data implemented, our research finds conflict situation of using capacity redundancy and process flexibility strategies together. We also give guidelines for the use of hybrid strategies in RSC. We find that the optimal strategy changes as required service level increases. We also suggest moderate investment in capacity redundancy with long chain flexibility and resource centralization when applying hybrid strategy with optimal flexibility.

Coordinated risk-averse distributionally robust optimization for maintenance and generation schedules of offshore wind farm cluster

The coordinated optimization of maintenance and generation schedules of multiple offshore wind farms (OWFs) bring great benefits in reducing the maintenance and operation cost of OWF cluster. Firstly, a two-stage coordinated risk-averse distributionally robust optimization (CRDRO) model for maintenance and generation schedules of wind turbines (WTs) in OWF cluster is proposed. This model takes into account uncertainties related to the prediction errors in WTs’ maximum available active power outputs, the travel time of maintenance vessels, and the high operational cost risks associated with these uncertainties. A feasible route generating algorithm is designed to obtain all feasible maintenance combinations for WTs and the corresponding routes of maintenance vessels for the CRDRO model. The risk-averse term, which is a weighted sum of expected operation costs and GlueVaR value, is used to measure the tail risk of the worst probability distribution. To solve the two-stage CRDRO model, the risk-averse term is simplified into the form of expected value calculation, which can be further transformed into the tractable form by using the strong duality theorem. Finally, the two-stage CRDRO model is solved by the column-and-constraint generation algorithm, with the nonconvex sub-problem being addressed by the outer approximation algorithm. Case studies on an real-world cluster comprising three OWFs verify the correctness and efficiency of the proposed method, and demonstrate that the CRDRO model enhances the overall economic efficiency and reduces the high operational cost risk under extreme scenarios.

Parabolic Optimal Control Problems with Combinatorial Switching Constraints, Part II: Outer Approximation Algorithm

Parabolic Optimal Control Problems with Combinatorial Switching Constraints, Part II: Outer Approximation Algorithm

Efficient resource allocation for 5G/6G cognitive radio networks using probabilistic interference models

The Cognitive Radio Network, incorporating Device-to-Device communication and a heterogeneous network, has garnered significant attention due to its ability to address spectrum shortage issues and optimize the efficient utilization of spectrum resources. However, resource allocation considering stochastic behavior has not been considered in existing studies. In this paper, our work aimed to maximize the throughput of the overall network considering multiple users under the umbrella of the Cognitive radio network assisted by amplify and forward relay. The constraints are treated as chance constraints with a probability of satisfaction in them, which leads to a non-convex mixed integer nonlinear problem which is an NP-Hard problem. To solve this problem an exhaustive search solution for optimal results is required. However, the computational burden always increases with the user equipment. Therefore, to obtain an optimal solution with having low computational burden, the Outer Approximation Algorithm is utilized in this research. To evaluate the desired results, extensive simulations have been carried out. The effectiveness of the proposed algorithm is verified by results in terms of throughput maximization under the impact of chance constraint formulation in the cognitive radio networks.

An outer approximation algorithm for generating the Edgeworth–Pareto hull of multi-objective mixed-integer linear programming problems

In this paper, we present an outer approximation algorithm for computing the Edgeworth–Pareto hull of multi-objective mixed-integer linear programming problems (MOMILPs). It produces the extreme points (i.e., the vertices) as well as the facets of the Edgeworth–Pareto hull. We note that these extreme points are the extreme supported non-dominated points of a MOMILP. We also show how to extend the concept of geometric duality for multi-objective linear programming problems to the Edgeworth–Pareto hull of MOMILPs and use this extension to develop the algorithm. The algorithm relies on a novel oracle that solves single-objective weighted-sum problems and we show that the required number of oracle calls is polynomial in the number of facets of the convex hull of the extreme supported non-dominated points in the case of MOMILPs. Thus, for MOMILPs for which the weighted-sum problem is solvable in polynomial time, the facets can be computed with incremental-polynomial delay—a result that was formerly only known for the computation of extreme supported non-dominated points. Our algorithm can be an attractive option to compute lower bound sets within multi-objective branch-and-bound algorithms for solving MOMILPs. This is for several reasons as (i) the algorithm starts from a trivial valid lower bound set then iteratively improves it, thus at any iteration of the algorithm a lower bound set is available; (ii) the algorithm also produces efficient solutions (i.e., solutions in the decision space); (iii) in any iteration of the algorithm, a relaxation of the MOMILP can be solved, and the obtained points and facets still provide a valid lower bound set. Moreover, for the special case of multi-objective linear programming problems, the algorithm solves the problem to global optimality. A computational study on a set of benchmark instances from the literature is provided.

Cluster based resource management using H-NOMA in heterogeneous networks beyond 5G

High throughput, secure transmission, and minimum interference are required in beyond-fifth-generation (B5G) networks. For this purpose, a mix of hybrid non-orthogonal multiple access (H-NOMA) and mobile user clustering (MUC) in a heterogeneous network (HetNet) can be explored. In this paper, we propose sum-secrecy throughput (SST) maximization problem based on a novel MUC-assisted downlink (DL) H-NOMA strategy in the presence of an eavesdropper in HetNet. We compare the performance of the proposed strategy for HetNet and a high-powered base station-only (HPS-only) network. We also compare the performance of the proposed strategy with the existing orthogonal multiple access only (OMA-only) and non-orthogonal multiple access only (NOMA-only) strategies. We have used an ϵ-optimal outer approximation algorithm (OAA) to solve the formulated problem based on the proposed strategy. We have evaluated the performance of the proposed strategy by considering key performance indicators (KPIs), i.e., the mobile user (MU)’s admission in the cluster, MU’s association with the HPS or low-powered base station (LPS), power allocation (PA) to MUs, quality of service (QoS), fairness in association with HPS or LPS, and SST. Extensive simulation results considering KPIs reveal that the proposed strategy performs better (1) in HetNet, (2) for HetNet compared to the HPS-only network, and (3) compared to the existing OMA-only and NOMA-only strategies in HetNet. Additionally, the proposed system model is compared with Amin et al. (2021) and Fu et al. (2022) in terms of SST and the simulation results reveal that the proposed system model’s performance is superior compared to Amin et al. (2021) and Fu et al. (2022) regarding SST.

A Risk-Averse Adaptive Stochastic Optimization Method for Transactive Energy Management of a Multi-Energy Microgrid

This paper proposes a new energy management method for a multi-energy microgrid (MEMG) which supplies both electrical and thermal energies. Based on the transactive energy (TE) concept, the problem is formulated as a Stackelberg game-theoretic bi-level optimization model. The MEMG operator optimizes the energy scheduling and pricing strategies at the upper level, and the industrial, commercial and residential agents optimize their energy trading strategies at the lower level. The equivalent single-level mixed-integer linear program (MILP) reformulation is then derived for computational tractability. To coordinate the strategies made in the day-ahead and intra-day energy markets, an adaptive stochastic optimization (SO) approach is adopted, by which a day-ahead stochastic MILP and an intra-day deterministic model are formed. Furthermore, the conditional value-at-risk (CVaR) measure is incorporated in the day-ahead stage for risk aversion of the MEMG operator towards uncertainties. To solve the models, an adaptive Progressive Hedging (PH) algorithm is developed to decompose the day-ahead stochastic MILP into multiple scenario-based subproblems which can be solved in parallel, and an outer approximation (OA) algorithm is adopted in the intra-day stage to linearize the bilinear objective function. Finally, the simulation results verify the effectiveness of the proposed energy management method and the efficiency of the solution algorithms.

Joint location and cost planning in maximum capture facility location under random utilities

We study a joint facility location and cost planning problem in a competitive market under random utility maximization (RUM) models. The objective is to locate new facilities and make decisions on the costs (or budgets) to spend on the new facilities, aiming to maximize an expected captured customer demand, assuming that customers choose a facility among all available facilities according to a RUM model. We examine two RUM frameworks in the discrete choice literature, namely, the additive and multiplicative RUM. While the former has been widely used in facility location problems, we are the first to explore the latter in the context. We numerically show that the two RUM frameworks can well approximate each other in the context of the cost optimization problem. In addition, we show that, under the additive RUM framework, the resultant cost optimization problem becomes highly non-convex and may have several local optima. In contrast, the use of the multiplicative RUM brings several advantages to the competitive facility location problem. For instance, the cost optimization problem under the multiplicative RUM can be solved efficiently by a general convex optimization solver, or can be reformulated as a conic quadratic program and handled by a conic solver available in some off-the-shelf solvers such as CPLEX or GUROBI. Furthermore, we consider a joint location and cost optimization problem under the multiplicative RUM and propose three approaches to solve the problem, namely, an equivalent conic reformulation, a multi-cut outer-approximation algorithm, and a local search heuristic. We provide numerical experiments based on synthetic instances of various sizes to evaluate the performances of the proposed algorithms in solving the cost optimization and the joint location and cost optimization problems.

Improving relief operations via optimizing shelter location with uncertain covariates

Designing an efficient shelter location planning is crucial to the rapid implementation of relief operations under the uncertain number of casualties. In practice, the uncertain number of casualties is closely related to disaster severity, whereas previous studies ignore such correlations when modeling uncertainties. In this paper, the scenario-wise ambiguity set is adopted to capture the correlation between the uncertain number of casualties and uncertain covariates, i.e., disaster severity. We develop a two-stage scenario-wise distributionally robust (SDR) model, where the shelter location and capacity allocation decisions are made here-and-now, and recourse decisions to transport casualties are made after the uncertainties on the number of casualties and covariate information have been realized. We approximate the non-convex model into a tractable form, i.e., second-order cone programming (SOCP), which can be solved efficiently by an outer approximation (OA) algorithm for a large-scale computation case. The numerical results with real-world data show that covariate integration (CVI) can contribute to saving costs and improving relief efficiency, and illustrate the computational superiority of the proposed OA algorithm. The results further demonstrate that establishing shelters with large capacities and near the affected areas has indeed had a positive impact on improving relief efficiency.

Fill‐rate service level constrained distribution network design

AbstractThe design of distribution networks that simultaneously consider location and inventory decisions seeks to balance costs and product availability. The most commonly observed measure of product availability in practical settings is the fill‐rate service level. However, the optimal design of a distribution network that considers the fill rate to control shortages of fast‐moving consumer goods (FMCG) is considered intractable and has only been addressed by heuristic methods. This paper addresses the optimal design of a distribution network for FMCG able to provide high fill‐rate service level under a continuous review policy. Considering the exact formulation for the provided fill rate, we formulated a joint location–inventory model with fill‐rate service level constraints as a convex mixed integer nonlinear problem for which a novel decomposition‐based outer approximation algorithm is proposed. Numerical experiments have shown that our solution approach provides good‐quality solutions that are on average 0.15% and, at worst, 2.2% from the optimal solution.

Robust maximum capture facility location under random utility maximization models

We study a robust version of the maximum capture facility location problem in a competitive market, assuming that each customer chooses among all available facilities according to a random utility maximization (RUM) model. We employ the generalized extreme value (GEV) family of models and assume that the parameters of the RUM model are not given exactly but lie in convex uncertainty sets. The problem is to locate new facilities to maximize the worst-case captured user demand. We show that, interestingly, our robust model preserves the monotonicity and submodularity from its deterministic counterpart, implying that a simple greedy heuristic can guarantee a (1−1/e) approximation solution. We further show the concavity of the objective function under the classical multinomial logit (MNL) model, suggesting that an outer-approximation algorithm can be used to solve the robust model under MNL to optimality. We conduct experiments comparing our robust method to other deterministic and sampling approaches, using instances from different discrete choice models. Our results clearly demonstrate the advantages of our robust model in protecting the decision-maker from worst-case scenarios.

Data visibility, sourcing flexibility, and pricing decisions in supply chains

This article extends the supply chain planning problem under risk by including the visibility of purchasing information and sourcing flexibility. A novel geometric index is developed to measure purchasing visibility by including different attributes of the required data visibility level. The problem is formulated as a mixed integer non-linear optimisation model. We tailor an outer approximation (OA) algorithm as a decomposition-based algorithm to solve the resultant mathematical model. Our research experiments indicate how the basic OA algorithm could be applied in the non-convex models by utilising a reformulation-linearisation technique and a convexification scheme. Moreover, the superiority of our proposed OA algorithm is shown in terms of solution quality and CPU time.

Enabling the direct solution of challenging computer-aided molecular and process design problems: Chemical absorption of carbon dioxide

The search for improved CO2 capture solvents can be accelerated by deploying computer-aided molecular and process design (CAMPD) techniques to explore large molecular and process domains systematically. However, the direct solution of the integrated molecular-process design problem is very challenging as nonlinear interactions between physical properties and process performance render a large proportion of the search space infeasible. We develop a methodology that enables the direct and reliable solution of CAMPD for absorption–desorption processes, using the state-of-the-art SAFT-γ Mie group contribution approach to predict phase and chemical equilibria. We develop new feasibility tests and show them to be highly efficient at reducing the search space, integrating them in an outer-approximation algorithm. The framework is applied to design an aqueous solvent and CO2 chemical absorption–desorption process, with 150 CAMPD instances across three case studies solved successfully. The optimal solvents are more promising than those obtained with sequential molecular design approaches.

Application of the Plant Propagation Algorithm and NSGA-II to Multiple Objective Linear Programming

Multiple Objective Linear Programming (MOLP) problems are usually solved by exact methods. However, nature-inspired population based stochastic algorithms such as the plant propagation algorithm are becoming more and more prominent. This paper applies the multiple objective plant propagation algorithm (MOPPA) and nondominated sorting genetic algorithm II (NSGA-II) for the first time to MOLP and compares their outcomes with those of prominent exact methods. Computational results from a collection of 51 existing MOLP instances suggests that MOPPA compares favourably with four of the most prominent exact methods namely extended multiple objective simplex algorithm (EMSA), affine scaling interior MOLP algorithm (ASIMOLP), Benson’s outer-approximation algorithm (BOA) and parametric simplex algorithm (PSA), and returns best nondominated points which are of higher quality than those returned by NSGA-II. However, the nondominated points approximated by NSGA-II are evenly distributed across the nondominated front. The methods compare well with the four exact methods especially on the large instances which the exact methods failed to solve even when given generous amounts of computation times.

Designing a multi-objective closed-loop supply chain for multi-period multi-generational products with social impacts considerations

This paper assesses a multi-period closed-loop supply chain (CLSC) for making strategic and operational decisions in multi-generation applications. For this purpose, joint decisions on network design, inventory control, production planning, pricing, supplier selection, and service level requirements are determined along with the forward flows of the supply chain. Besides, a segmentation policy is proposed for disassembling and recovering the different quality grades of returned products. All the raised issues have become more challengeable when an inner competition on coexistence goes up between the incumbent and new generations. Accordingly, this paper attempts to bridge the gaps in the literature by proposing a bi-objective mathematical model that involves a demand function with the integration of price drop effects, marketing efforts, and technology desirability to tackle trade-offs decisions between multiple generations. This research aims to simultaneously optimize the total profit and social benefits. To solve the problem, first, the ε-constraint method has been applied to transform the bi-objective problem into a single objective. Then, as a great deal to solve the complex mixed-integer non-linear program model, an outer approximation algorithm is applied based on approximation, relaxation, and decomposition. Finally, a case example in a mobile phone CLSC problem has been investigated to validate the proposed model and provide managerial insights.

Radio resource management in energy harvesting cooperative cognitive UAV assisted IoT networks: A multi-objective approach

Cooperative communication through energy harvested relays in Cognitive Internet of Things (CIoT) has been envisioned as a promising solution to support massive connectivity of Cognitive Radio (CR) based IoT devices and to achieve maximal energy and spectral efficiency in upcoming wireless systems. In this work, a cooperative CIoT system is contemplated, in which a source acts as a satellite, communicating with multiple CIoT devices over numerous relays. Unmanned Aerial Vehicles (UAVs) are used as relays, which are equipped with onboard Energy Harvesting (EH) facility. We adopted a Power Splitting (PS) method for EH at relays, which are harvested from the Radio frequency (RF) signals. In conjunction with this, the Decode and Forward (DF) relaying strategy is used at UAV relays to transmit the messages from the satellite source to the CIoT devices. We developed a Multi-Objective Optimization (MOO) framework for joint optimization of source power allocation, CIoT device selection, UAV relay assignment, and PS ratio determination. We formulated three objectives: maximizing the sum rate and the number of admitted CIoT in the network and minimizing the carbon dioxide emission. The MOO formulation is a Mixed-Integer Non-Linear Programming (MINLP) problem, which is challenging to solve. To address the joint optimization problem for an epsilon optimal solution, an Outer Approximation Algorithm (OAA) is proposed with reduced complexity. The simulation results show that the proposed OAA is superior in terms of CIoT device selection and network utility maximization when compared to those obtained using the Nonlinear Optimization with Mesh Adaptive Direct-search (NOMAD) algorithm.

Communication capacity maximization in drone swarms

Employment of unmanned aerial vehicles (UAVs) or drones as swarms of coordinating nodes offers multiple advantages for commercial as well as military applications. However, the complex communication requirements of these swarms, coupled with high data rates of advanced UAV payloads, require innovative techniques for optimizing data throughput. Channel capacity being the key resource, optimum communication architecture and network topology is critical to ensure quality of service while remaining within transmission power constraints. This paper proposes a capacity maximization approach for swarm communication architectures using mixed-integer nonlinear programming (MINLP). These techniques are designed to tackle optimization applications involving both discrete variables and nonlinear system dynamics. Mathematical model formulated considering system constraints and desired objective function establishes applicability of MINLP. Since MINLP problems are NP-hard in general, computational overheads and search space exponentially grow with number of nodes in the swarm. Therefore, outer approximation algorithm has been applied that achieves near-optimal solutions with reduced convergence time and complexity compared with exhaustive search. Applicability of algorithm regardless of selected communication architecture has been established through realistic simulations.