Feasible Region Research Articles

Interactive niching-based two-stage evolutionary algorithm for constrained multiobjective optimization

When solving constrained multiobjective optimization problems (CMOPs), how to maintain diversity without losing convergence is a major challenge, because some small discrete feasible regions make the population hard to find a complete feasible Pareto Front. To this end, an interactive niching-based two-stage evolutionary algorithm for constrained multiobjective optimization, named INCMO, is proposed in this study. At each stage of INCMO, both two populations are employed to focus on different evolution purposes. Specifically, the main population optimizes the original CMOP, while the auxiliary population only considers the objectives by ignoring all constraints. INCMO uses two different niching mechanisms in two stages. In the early stage, each population independently uses the niching technique to generate offspring, to maintain diversity. In addition, two offspring populations are shared to realize migration between two populations. In the later stage, two populations cooperate to generate offspring based on an interactive niching technique. To be specific, two populations are merged and then divided into different niches. In this case, individuals from two populations will be assigned to the same niche. After that, individuals in the niche produce offspring through crossover and mutation. The process of generating offspring also promotes more effective migration between the two populations on the basis of increasing the diversity of population. Compared with other well-established methods, the proposed method in this study shows better performance on four test suites including 47 benchmark problems.

Multi-objective constrained black-box optimization algorithm based on feasible region localization and performance-improvement exploration

Over the past decade, surrogate-assisted evolutionary algorithms have demonstrated their effectiveness across various computationally expensive real-world domains. Nevertheless, the focus of surrogate-assisted multi-objective evolutionary algorithms has primarily centered on non-constrained optimization problems. There has been relatively limited exploration into addressing expensive constrained multi-objective optimization problems, which inherently require a delicate equilibrium between convergence, diversity, and feasibility. To bridge this gap, this paper concentrates on constrained multi-objective optimization problems where both objectives and constraints involve substantial computational costs. In response, a novel data-driven constrained multi-objective evolutionary algorithm is introduced, leveraging feasible region localization and performance-improvement exploration. For feasible region localization, a constraints-domain-search strategy is presented to locate the feasible region quickly. To enhance performance-improvement exploration, a progressive enhancement of convergence and diversity is achieved through the incorporation of constraint penalties. With the help of exploration and exploitation, the proposed algorithm can balance convergence, diversity, and feasibility while working within a limited number of function evaluations. By comparing the proposed algorithm with state-of-the-art algorithms on 66 mathematical problems and a resource-intensive black-box problem, its outstanding performance for solving multi-objective constrained black-box problems is validated.

Mission Profile Analysis of Point-to-Point Rocket Cargo Transportation System Using Trajectory Optimization

This paper proposes new concepts for the point-to-point rocket cargo transportation system (RCTS). The RCTS is a transportation system that can deliver cargo anywhere on Earth in an hour using reusable launch vehicle technologies. As a fundamental study of global transportation on Earth, we especially address the mission profiles for short-range transportation within hundreds of kilometers using small rocket engines, which are intended for logistics transportation within neighboring countries or domestic transportation in a country. Two mission profiles of the RCTS are introduced, and the flight phases of each concept are explained in detail. The trajectory optimization problem of the RCTS is formulated based on the mission profiles, incorporating the flight constraints, boundary conditions, and an objective function that aims to maximize the payload ratio. The explicit-guidance is employed during the landing burn phase to enhance the convergence of the optimization problem by expanding the feasible region. The coevolutionary augmented Lagrangian method, one of the evolutionary algorithms, is utilized to obtain the solution for the trajectory optimization problem. The optimal trajectory and state variables are presented through numerical simulations. Additionally, parametric studies are performed to determine the payload mass trend of the RCTS. The trajectory optimization results are summarized to provide an analysis and comparison of the proposed mission profiles.

Privacy-Preserving Personalized Revenue Management

This paper examines how data-driven personalized decisions can be made while preserving consumer privacy. Our setting is one in which the firm chooses a personalized price based on each new customer’s vector of individual features; the true set of individual demand-generating parameters is unknown to the firm and so must be estimated from historical data. We extend the existing personalized pricing framework by requiring also that the firm’s pricing policy preserve consumer privacy, or (formally) that it be differentially private: an industry standard for privacy preservation. We develop privacy-preserving personalized pricing algorithms and show that they achieve near-optimal revenue by deriving theoretical (upper and lower) performance bounds. Our analyses further suggest that, if the firm possesses a sufficient amount of historical data, then it can achieve a certain level of differential privacy almost “for free.” That is, the revenue loss due to privacy preservation is of smaller order than that due to estimation. We confirm our theoretical findings in a series of numerical experiments based on synthetically generated and online auto lending (CPRM-12-001) data sets. Finally, motivated by practical considerations, we also extend our algorithms and findings to a variety of alternative settings, including multiproduct pricing with substitution effect, discrete feasible price set, categorical sensitive features, and personalized assortment optimization. This paper was accepted by Vishal Gaur, operations management. Funding: R. Momot acknowledges financial support from the HEC Paris Foundation and the Agence Nationale de la Recherche (French National Research Agency) “Investissements d’Avenir” [Grant LabEx Ecodec/ANR-11-LABX-0047] during the initial stages of this project. Supplemental Material: The data files and online appendices are available at https://doi.org/10.1287/mnsc.2023.4925 .

A constrained multi-objective evolutionary algorithm with Pareto estimation via neural network

The main challenge in addressing constrained multi-objective optimization problems (CMOPs) lies in achieving a balance among convergence, diversity, and feasibility. To address this issue, this paper proposes a constrained multi-objective evolutionary algorithm with Pareto estimation via neural network named CMOEA-PeNN. In order to exploit and explore the decision space, the proposed algorithm employs a dual-population mechanism, which is trained with a self-organizing map (SOM). Firstly, the population distribution structure in decision space is mapped to objective space while preserving neighborhood information, and then the neuron weight is utilized to estimate the Pareto front (PF). Secondly, a novel approach is devised to preserve the feasibility of the population and enhance the estimation of the Pareto front by SOM. The achievement scalarizing function (ASF) is employed to choose promising solutions. This strategy could guide the population toward the optimal solution while exploring the small feasible regions. Finally, the performance of CMOEA-PeNN is compared with five state-of-the-art constrained multi-objective evolutionary algorithms (CMOEAs) on three widely used benchmark problems and a real-world problem. The experimental results show that CMOEA-PeNN could archive competitive performance in solving CMOPs.

Plant‐wide control for processes with recycle and narrow feasible sets

AbstractRecycling mass and energy is a feature in almost all plants and industries. It is reported as an inevitable characteristic in actual industries due to differences between the designed processes and implementation. When a process is set, engineers identify inefficiencies in the planned performance and utilize recycles to try to reduce this difference. However, although recycles help in improving a plant's overall performance compared with a serial arrangement, its effect on dynamic plant behaviour is seldom considered, and these changes begin to impact the features of processes like controllability and stability. Similarly, even though recycles affect the feasible region of individual equipment, its effect is not verified. Based on the aforementioned background, several studies have demonstrated that advanced control strategies allow for the improvement of the dynamic behaviour of complex plants. Different techniques, such as model‐based predictive control and plant‐wide control (PWC), also show that their implementation reduces the effects of recycles. However, there is no explicit methodology to design these controllers and address the impacts of recycles on dynamic behaviour and feasibility. Additionally, there is no tuning procedure to design controllers that optimize the dynamic performance of plants, including the control system. In this context, this work presents an analysis to characterize the effects of recycles on the dynamic behaviour of plants, including how they affect the feasible operating region. Based on the previous analysis, a control methodology is also proposed to design the PWC strategy, explicitly addressing the dynamical phenomena of recycles in processes.

Dependence in constrained Bayesian optimization

AbstractConstrained Bayesian optimization optimizes a black-box objective function subject to black-box constraints. For simplicity, most existing works assume that multiple constraints are independent. To ask, when and how does dependence between constraints help?, we remove this assumption and implement probability of feasibility with dependence (Dep-PoF) by applying multiple output Gaussian processes (MOGPs) as surrogate models and using expectation propagation to approximate the probabilities. We compare Dep-PoF and the independent version PoF. We propose two new acquisition functions incorporating Dep-PoF and test them on synthetic and practical benchmarks. Our results are largely negative: incorporating dependence between the constraints does not help much. Empirically, incorporating dependence between constraints may be useful if: (i) the solution is on the boundary of the feasible region(s) or (ii) the feasible set is very small. When these conditions are satisfied, the predictive covariance matrix from the MOGP may be poorly approximated by a diagonal matrix and the off-diagonal matrix elements may become important. Dep-PoF may apply to settings where (i) the constraints and their dependence are totally unknown and (ii) experiments are so expensive that any slightly better Bayesian optimization procedure is preferred. But, in most cases, Dep-PoF is indistinguishable from PoF.

An accelerated subgradient extragradient algorithm for solving bilevel variational inequality problems involving non-Lipschitz operator

In this paper, an accelerated subgradient extragradient algorithm with a new non-monotonic step size is proposed to solve bilevel variational inequality problems involving non-Lipschitz continuous operator in Hilbert spaces. The proposed algorithm with a new non-monotonic step size has the advantage of requiring only one projection onto the feasible set during each iteration and does not require prior knowledge of the Lipschitz constant of the mapping involved. Under suitable and weaker conditions, the proposed algorithm achieves strong convergence. Some numerical tests are provided to demonstrate the efficiency and advantages of the proposed algorithm against existing related algorithms.

A novel framework for the identification of complex feasible space

Feasibility analysis is suitable to determine the combinations of input parameters that satisfy all operating and quality constraints, i.e., the process design space. In the presence of disjoint feasible regions and for computationally expensive/nonconvex problems, the use of surrogate-based approaches has been increasingly adopted. However, the choice and prediction accuracy of a suitable surrogate depend on the process of interest, and on the available dataset. In this study, we propose a novel workflow which combines different mathematical tools to (i) investigate how to identify the process feasible space relying on the available training dataset, (ii) assess whether there is a minimum number of sampling points that are necessary to uncover the complexity of the original feasibility function, and (iii) support proper surrogate model selection, while improving accuracy. The proposed methodology is tested on analytical problems and a pharmaceutical process, showing its effectiveness in the identification of complex feasible regions.

Quantitative Stability of Optimization Problems with Stochastic Constraints

In this paper, we consider optimization problems with stochastic constraints. We derive quantitative stability results for the optimal value function, the optimal solution set and the feasible solution set of optimization models in which the underlying stochastic constraints involve the mathematical expectation of random single-valued and set-valued mappings, respectively. New primal sufficient conditions are developed for the uniform error bound property of the stochastic constraint system for the single-valued case.

Parameterization of minimal order strongly stabilizing controllers for two-link underactuated planar robot with single sensor

This paper investigates the minimal order of strongly stabilizing controllers for a two-link planar robot moving in a vertical plane with a single actuator and sensor, focusing on stabilizing the robot around its upright equilibrium point where both links are upright. By designing an output as the horizontal displacement of a point along the second link’s straight line, existing literature demonstrates that the robot’s strong stabilization problem is equivalent to that of a fourth-order linear plant, featuring two pairs of symmetric real poles and adjustable zeros. In contrast to an existing second-order strongly stabilizing controller tailored to a pair of symmetric real zeros around the origin within a given range, this paper achieves four main contributions. First, this paper obtains a parameterization of second-order strongly stabilizing controllers allowing the positive real zero in an expanded range. Second, this paper considers the plant with a pair of symmetric real zeros around the origin, a double origin zero, and a pair of conjugate imaginary zeros in a unified manner, and presents a more general parameterization of second-order strongly stabilizing controllers accommodating these three types of zeros in a more expanded range. Third, this paper provides a necessary and sufficient condition on these zeros (equivalently, all ranges of the observation point) for the existence of a second-order strongly stabilizing controller. Fourth, this paper demonstrates that a first-order stabilizing controller does not exist for the plant, regardless of its types of zeros, confirming that the minimal order of the strongly stabilizing controllers is two. This paper obtains all feasible regions for the parameters in each parameterization by explicitly expressing the range of each parameter using inequalities in a cascade structure, which makes the inequalities easy to check and verify. The results are applicable to both the Acrobot and Pendubot. The paper validates the theoretical findings through numerical investigations and simulations.

Modelling the dynamics of acute and chronic hepatitis B with optimal control

This article examines hepatitis B dynamics under distinct infection phases and multiple transmissions. We formulate the epidemic problem based on the characteristics of the disease. It is shown that the epidemiological model is mathematically and biologically meaningful of its well-posedness (positivity, boundedness, and biologically feasible region). The reproductive number is then calculated to find the equilibria and the stability analysis of the epidemic model is performed. A backward bifurcation is also investigated in the proposed epidemic problem. With the help of two control measures (treatment and vaccination), we develop control strategies to minimize the infected population (acute and chronic). To solve the proposed control problem, we utilize Pontryagin’s Maximum Principle. Some simulations are conducted to illustrate the investigation of the analytical work and the effect of control analysis.

Dealing with infeasibility in multi-parametric programming for application to explicit model predictive control

Motivated by explicit model predictive control, we address infeasibility in multi-parametric quadratic programming according to the exact penalty function approach, where some user-chosen parameter-dependent constraints are relaxed and the 1-norm of their violation is penalized in the cost function. We characterize the relation between the resulting multi-parametric quadratic program and the original one and show that, as the penalty coefficient grows to infinity, the solution to the former provides a piecewise affine continuous function, which is an optimal solution for the latter over the feasibility region, while it minimizes the 1-norm of the relaxed constraints violation over the infeasibility region.

Constructing extremal triangle-free graphs using integer programming

The maximum number of edges in a graph with matching number m and maximum degree d has been determined in Chvátal and Hanson (1976) and Balachandran and Khare (2009), where some extremal graphs have also been provided. Then, a new question has emerged: how the maximum edge count is affected by forbidding some subgraphs occurring in these extremal graphs? In Ahanjideh et al. (2022), the problem is solved in triangle-free graphs for d≥m, and for d<m with either Z(d)≤m<2d or d≤6, where Z(d) is approximately 5d/4. The authors derived structural properties of triangle-free extremal graphs, which allows us to focus on constructing small extremal components to form an extremal graph. Based on these findings, in this paper, we develop an integer programming formulation for constructing extremal graphs. Since our formulation is highly symmetric, we use our own implementation of Orbital Branching to reduce symmetry. We also implement our integer programming formulation so that the feasible region is restricted iteratively. Using a combination of the two approaches, we expand the solution into d≤10 instead of d≤6 for m>d. Our results endorse the formula for the number of edges in all extremal triangle-free graphs conjectured in Ahanjideh et al. (2022).

A Novel Projection Neural Network for Solving a Class of Monotone Variational Inequalities

This article provides a novel projection neural network (PNN) for a category of monotone variational inequality (MVI). For simplifying calculation, the feasible region of MVI is separated into two parts: one is the set on which projection has quantitative expression, and the rest part can be described by inequalities. A novel continuous-time PNN is well designed in this article based on the division and its states are verified to exist and ultimately converge to a solution of MVI. The proposed PNN differs from conventional approaches since it only depends on the projection operator on the sets with projection-quantifiable structure instead of the whole feasible region. The feature determines the PNN here owns the advantage of low amount of calculation and flexibility when encountering diverse MVIs. Compared with existing numerical algorithms, model convergence in this article is established on weaker assumptions and one can monitor its state at every moment. Finally, the effectiveness and the efficiency of PNN are testified by several simulation examples and an application in image fusion.

A constraint equivalent model of heat network for combined heat and power systems cooperation considering uncertainties

The cooperation of combined heat and power (CHP) systems can improve efficiency in energy utilization by fully utilizing the complementarity between heat and power. However, heat and power systems are operated by different companies where data sharing is restricted for privacy reasons, thus posing significant obstacles to collaboration. This paper proposes a coordinated operation method for CHP systems based on the constraint equivalent model. To preserve the operational constraints of heat networks and protect data privacy, a boundary point search method is firstly developed, where the operational constraints of heat networks are mapped to the feasible region of CHP units. Then, a multi-period cooperation model considering uncertainties of both networks is established based on the proposed constraint equivalent model and the rolling horizon strategy. Moreover, the impact of heat storage tanks and uncertainties of heat loads on the boundaries of the feasible region for heat networks is further analyzed. Simulation results demonstrate the effectiveness of the proposed constraint equivalent model and the cooperation method.

Deep Reinforcement Learning Empowers Wireless Powered Mobile Edge Computing: Towards Energy-Aware Online Offloading

Deep integration of wireless power transmission and mobile edge computing (MEC) promotes wireless powered MEC to become a new research hotspot in the field of Internet of Things. In this paper, we focus on the joint optimization problem of online offloading decision and charging resource allocation for minimizing task accomplishing time in dynamic time-varying wireless channel scenarios. The optimal solution involves addressing a mixed integer programming problem in real time, which is proved to be NP-hard, and imposes nontrivial challenges to design with conventional optimization methods. To efficiently address this problem, we leverage the deep reinforcement learning (DRL) technology to propose an energy-aware online offloading algorithm called EAOO. EAOO algorithm learns empirically the online offloading decision policies via a well-designed DRL framework, and adopts the feasible solution region analysis method to implement the charging resource allocation. We further propose a novel feasible decision vector generation method, and incorporate the crossover and mutation technology to expand the offloading vector search space with the provable feasibility guarantee. Extensive experimental results show that, our EAOO algorithm outperforms existing baseline algorithms, and achieves near-optimal performance with low CPU execution latency, which well satisfies the practical requirements of real-time and efficiency.

A Novel Model-Free Deep Reinforcement Learning Framework for Energy Management of a PV Integrated Energy Hub

This paper utilizes a fully model-free and data-driven deep reinforcement learning (DRL) framework to develop an intelligent controller that can exploit information to optimally schedule the energy hub with the aim of minimizing energy costs and emissions. By posing the energy hub scheduling problem as a multi-dimensional continuous state and action space, the proposed deep deterministic policy gradient (DDPG) method enables more cost-effective control strategies. The method can lead to a more efficient operation by considering nonlinear physical characteristics of the energy hub components like nonconvex feasible operating regions of combined heat and power (CHP) units, valve-point effects of power-only units, and fuel cell dynamic efficiency. Moreover, to provide great potential for the DDPG agent to learn an optimal policy in an efficient way, a hybrid forecasting model based on convolutional neural networks (CNNs) and bidirectional long short-term memories (BLSTMs) is developed to overcome the risk associated with PV power generation that can be highly intermittent, particularly on cloudy days. The effectiveness and applicability of the proposed scheduling framework in reducing energy costs and emissions while coping with uncertainties are demonstrated by comparing it against conventional robust optimization and stochastic programming approaches as well as state-of-the-art DRL methods in different case studies.

Energy-Efficient Base Station Switching-Off With Guaranteed Cooperative Profit Gain of Mobile Network Operators

This paper studies the energy-efficient cooperative base station (BS) switching-off in multi-input single-output (MISO) cellular networks, where the roaming-cost-based cooperation and the cooperative beamforming among multiple mobile network operators (MNOs) are jointly designed. To make the cooperation among MNOs practical, we introduce a new constraint to make sure the cooperative profit gain of MNOs is non-negative and also formulate an optimization problem to maximize the system energy efficiency (EE) by jointly considering multiple factors, including the operation modes of BSs, the connection states between users and BSs, the beamforming vectors of multi-antenna BSs under constraints of users’ quality of service, BSs’s power budget, and the non-negative profit gain of MNOs, etc. To solve the non-convex problem, we propose an energy-efficient beamforming-aware BS switching off method (EBBSOM), in which we first design a roaming-cost-based BS switching-off algorithm (EE-BSOA) to search the feasible inactive BSs and then propose a successive convex approximation (SCA) and Dinkelbach’s method-based EE maximization beamforming algorithm (EE-BFA) to optimize beamforming vectors. In order to keep the cooperative profit gain non-negative for MNOs and also to balance the load and power consumption of BS from the profit perspective, we present a profit-wise BS switching-off priority method. We theoretically prove that the proposed EBBSOM is able to resolve the inconsistency between BS switching-off and EE maximization. Simulations show that the proposed EBBSOM enhances the system EE without degrading the economic profit gain of MNOs. Also, the proposed profit-wise BS switching-off priority enables the EBBSOM to achieve higher EE than traditional load-wise one. Furthermore, it is also observed that there exists a feasible rate price region, which is able to clearly characterize network EE and the average profit gain of MNOs simultaneously.

A novel probabilistic feasible region method for reliability-based design optimization with varying standard deviation

A novel probabilistic feasible region method for reliability-based design optimization with varying standard deviation