Single-objective Optimization Problem Research Articles

Multiobjectivization of Single-Objective Optimization in Evolutionary Computation: A Survey.

Multiobjectivization has emerged as a new promising paradigm to solve single-objective optimization problems (SOPs) in evolutionary computation, where an SOP is transformed into a multiobjective optimization problem (MOP) and solved by an evolutionary algorithm to find the optimal solutions of the original SOP. The transformation of an SOP into an MOP can be done by adding helper-objective(s) into the original objective, decomposing the original objective into multiple subobjectives, or aggregating subobjectives of the original objective into multiple scalar objectives. Multiobjectivization bridges the gap between SOPs and MOPs by transforming an SOP into the counterpart MOP, through which multiobjective optimization methods manage to attain superior solutions of the original SOP. Particularly, using multiobjectivization to solve SOPs can reduce the number of local optima, create new search paths from local optima to global optima, attain more incomparability solutions, and/or improve solution diversity. Since the term "multiobjectivization" was coined by Knowles et al. in 2001, this subject has accumulated plenty of works in the last two decades, yet there is a lack of systematic and comprehensive survey of these efforts. This article presents a comprehensive multifacet survey of the state-of-the-art multiobjectivization methods. Particularly, a new taxonomy of the methods is provided in this article and the advantages, limitations, challenges, theoretical analyses, benchmarks, applications, as well as future directions of the multiobjectivization methods are discussed.

Multi-objective optimization of PEMFC performance based on grey correlation analysis and response surface method

This paper aims to combine grey correlation analysis and response surface method to propose a fast and effective performance optimization method for PEMFC. First, based on orthogonal test data, grey correlation analysis method is used to select four variables that have significant influence on PEMFC’s comprehensive performance from eight common parameters. Secondly, based on grey correlation analysis, the multi-objective optimization problem is transformed into a single objective optimization problem about correlation degree, and applying the response surface method to build the key parameters and the correlation between the second order prediction model. Therefore, the current density, system efficiency and oxygen distribution uniformity on cathode catalyst layer of PEMFC were optimized as a whole. Finally, the optimal parameter combination was obtained by optimizing the prediction model. The simulation results show that the optimized operating conditions are significantly improved in the three performance indexes compared with the basic model, which confirms the feasibility of this method in solving the multi-objective optimization problem, and can provide some reference for the optimal design of hydrogen fuel cells.

Multi-agent deep reinforcement learning for user association and resource allocation in integrated terrestrial and non-terrestrial networks

Integrating the terrestrial network with non-terrestrial networks to provide radio access as anticipated in the beyond 5G networks calls for efficient user association and resource allocation strategies. In this work, a weighted sum single objective optimization problem that maximizes the total network data rate while minimizing the mobility-induced handoffs and prioritizing service provisioning of mission-critical users in the integrated terrestrial and non-terrestrial network is formulated. The problem’s complexity is reduced by solving the defined problem in two phases; the user association followed by the resource distribution phase. Several proposed approaches to the user association sub-problem utilize a central node that requires nearly-complete information, which may not be available in real time. On the contrary, this paper proposes a centralized training and distributed execution multi-agent dueling double deep Q network (MA3DQN) solution. Each user collects the channel state and access node loading information in this approach and makes an association decision that considers its quality of service requirements. The algorithm’s performance is validated through comparison with the genetic algorithm (GA), the integer linear programming (ILP), a heuristic approximation-based solution, the greedy approach, and the random user association (RUA) algorithm. Moreover, the paper simulates the multi-agent deep Q network solution as an additional benchmark algorithm. Simulation results reveal that as the number of users in the network increases, the acquired data rate of the MA3DQN is within 0.48% and 0.4% of that achieved by the GA and ILP, respectively, and outperforms all other algorithms. Notably, the proposed MA3DQN algorithm presents the best running time, attaining a gain of 99.9% over the GA algorithm, which performs the poorest among the algorithms characterized by polynomial worst-case time complexity. Besides, the MA3DQN approach maintains a handoff probability of zero, unlike the ILP, the approximation-based, greedy, and RUA solutions.

Joint dynamic routing and resource allocation in satellite-terrestrial integrated networks

The Satellite-Terrestrial Integrated Networks (STINs) is considered as a reliable and agile next-generation communication network scheme, because it is flexible to deploy and robust to disasters (e.g., earthquakes, floods, and volcanic eruptions). The routing and resource allocation directly affect the power consumption of satellites in the STINs. However, most of existing research works study these two issues separately, and neglect the impact of path selection on the satellite power consumption. This paper investigates the joint optimization problem of routing, bandwidth allocation, user association, channel allocation, and power allocation of the STINs. We aim at minimizing satellites’ power consumption while satisfying user's Quality of Service (QoS). Because the power consumption of satellites for routing in the space segment is involved with the data rate through satellite gateways in the user segment, this optimization problem is a Multi-Objective Optimization (MOO) problem. To this end, we convert the MOO problem to a Single-Objective Optimization (SOO) problem by linear weighted method. Considering that the SOO problem is a Mixed Integer Nonlinear Programming (MINLP) problem, we decompose it into the resource allocation sub-problem in the user segment and the routing optimization sub-problem in the space segment. And we solve these two sub-problems iteratively. Specifically, we convert the first sub-problem into a convex optimization problem and obtain the current user association and channel allocation. Then we formulate the second sub-problem into a multi-commodity flow problem and solve it based on the obtained user association and channel allocation. The SOO problem is solved by iteratively optimizing these two sub-problems. Experimental results demonstrate that our proposed algorithm is able to get an approximate optimal solution quickly and effectively reduce the power consumption of satellites compared with benchmark algorithms in the STINs.

Multiobjective bilevel programming model for multilayer perceptron neural networks

The architecture of multilayer perceptron (MLP) neural networks dictates the network's performance. However, aiming at the specific classification problems, suitable architectures of MLPs must be determined beforehand. More often, the design decisions rely on a trial and error learning and the experience knowledge. To automatically design an MLP's network architecture and optimize network parameters, a multiobjective bilevel programming model is built. In this model, a multiobjective optimization problem is constructed in the upper level for obtaining a set of Pareto optimal architectures of the MLPs, considering network complexity, training error rate, and validation error rate, while a single-objective optimization problem is established in the lower level to search for the optimum network parameters for a given network architecture. For dealing with this model efficiently, a novel multiobjective hierarchical learning algorithm (MOHLA) is proposed, in which an integer-coding NSGA-II is developed as the upper-level optimizer for a set of Pareto optimal network structures of the MLPs, while a non-iterative method is regarded as the lower-level solver for the MLPs' connection parameters. After a set of trained MLPs is obtained finally by using MOHLA, a selective ensemble strategy is adopted for improving identification accuracy. Three types of multiobjective bilevel programming models are investigated and compared in the experiments. Moreover, the proposed MOHLA is compared with several state-of-the-art learning approaches on various classification problems. The experimental results confirm that MOHLA performs well.

A multi-objective model for optimizing hydrogen injected-high pressure natural gas pipeline networks

The paper develops a statistical model for optimizing the Hydrogen-injected Natural gas (H-NG) high-pressure pipeline network. Gas hydrodynamic principles are utilized to construct the pipeline and compressor station model. The model developed is implemented on a pipeline grid that is supposed to carry Hydrogen as an energy carrier in a natural gas-carrying pipeline. The paper aims to optimize different objectives using ant colony optimization (ACO). The first objective includes a single objective optimization problem that evaluates the maximum permissible hydrogen amounts blended with natural gas (NG) for a set of pipeline constraints. We also evaluated the variations in operational variables on injecting Hydrogen into the natural gas pipeline networks at varying fractions. The study further develops a multi-objective optimization model that includes bi-objective and tri-objective problems and is optimized using ACO. Traditional studies have focused on single-objective optimization with minimal bi-objective issues. In addition, none of the earlier research has shown the effect of introducing Hydrogen to the NG network using tri-objective function evaluations. The bi-objective and tri-objective functions help evaluate the effect of injecting Hydrogen on different operational parameters. The study further attempts to fill the gap by detailing the modelling equations implemented through a bi-objective and tri-objective function for the H-NG pipeline network and optimized through ACO. Pareto fronts that show the tradeoff between the different objectives for the multi-objective problem have been generated. The primary objective of the bi-objective and tri-objective optimization problems is maximizing hydrogen mole percent in natural gas. The other objective chosen is minimizing compressor fuel consumption and maximizing delivery pressure, throughput, and power delivered at the delivery station. The findings will serve as a roadmap for pipeline operators interested in repurposing natural gas pipeline networks to transport hydrogen and natural gas blend (H-NG) and seeking to reduce carbon intensity per unit of energy-delivered fuel.

A Collaborative Transit Signal Priority Method Considering Gaming among Traffic Participants

How to realize the game equilibrium between bus and nontransit vehicle is a hot topic in the field of transit signal priority (TSP). To this end, a collaborative transit signal priority (Co-TSP) method is proposed. The core of Co-TSP is a two-objective optimization problem which takes the expected delays of buses and the average delays of nontransit vehicles as the objectives. Different from previous studies, Co-TSP uses game theory to realize collaborative optimization, instead of transforming the problem into a single objective optimization problem by weighting. A finite state machine-based algorithm is developed to estimate the average delays of nontransit vehicles. The stochasticity of bus arrival time is also considered in the estimation of bus delays to improve the robustness. Candidate timing plans obtained by the nondominated sorting genetic algorithm (NSGA) are divided into three priority levels based on the delays of buses. The final timing plans can be picked intuitively from the candidates by rules representing expert knowledge and demands to control the priority level. Co-TSP guarantees theoretically by preliminary screening that the expected delays of bus after optimization must be no higher than that before optimization. Simulation experiments are conducted in Shanghai, China, to verify the performance. Results show that Co-TSP reduces the delays of buses by 27.7%∼41.0% and still performs well under low and high congestion levels, while the conventional TSP (CTSP) fails in some cases. Priority control proves to be effective at last. The research provides a new idea for the benefit allocation among participants at intersections.

Study of a hull form optimization system based on a Gaussian process regression algorithm and an adaptive sampling strategy, Part I: Single-objective optimization

Simulation-based design (SBD) is recognized as a hull form optimization method with great advantages, but its development process has exposed two obvious shortcomings: difficulty in uncertainty quantification (UQ) of the surrogate model and inefficiencies when solving large-scale optimization problems. To mitigate these deficiencies, this paper first proposes a UQ method for the surrogate model based on a Gaussian process regression (GPR) algorithm. Mathematical optimization results show that when the number of samples is insufficient, coupling this UQ method into the deterministic optimization process can, to some extent, reduce the uncertainty of the optimization results. The drawback of the SBD over-reliance on the surrogate model for global prediction accuracy is addressed by designing a hull form optimization system that performs the sequential sampling specifically for the region of interest. This method is referred to as sequential sampling-based optimization (SBO). It was combined with the basic properties of a GPR surrogate model, and a variety of adaptive sampling strategies suitable for the SBO optimization system were designed. Mathematical optimization results show that a hybrid optimization system, SBO-MSE + LCB, which uses the adaptive sampling strategy with the maximum squared error (MSE) and the minimum lower confidence bound (LCB) of the GPR surrogate model has the best efficiency when solving single-objective optimization problems. This method was also used for optimization design of the total resistance coefficient of a 130,000-ton deep-sea aquaculture vessel. The results show that the optimization efficiency of the SBO-MSE + LCB optimization system was improved by 46.67% with respect to that of a conventional SBD optimization system when the optimization solutions were similar, hence demonstrating the efficiency advantage of this system in practical applications.

An optimal joint planning of DGs and electric vehicle charging stations in grid‐connected and islanded microgrids

AbstractThis paper proposes a new framework for the planning of both distributed generators (DGs) and electric vehicle charging stations (EVCSs). The proposed method efficiently produces a unified solution for the joint planning of DGs and EVCSs for both grid‐connected and islanded scenarios. The problem is formulated as a novel two‐stage planning problem. The first stage determines the locations and sizes of the DGs with locations of EVCSs in grid‐connected scenario, whereas the second stage planning identifies the optimal islands under the islanded microgrid scenario. A non‐dominated sorting genetic algorithm (NSGA‐II) is applied to solve the first stage planning problem; in this stage, the algorithm minimizes two objective functions: the system‐losses and total cost. In the second stage, another single objective optimization problem is designed which minimizes supply voltage variations to find optimal islands for the DGs and EVCSs to ensure a secure supply of power for EVs. The proposed framework is implemented on the IEEE 33‐bus system and verified with four test cases. The results demonstrate the effectiveness of the proposed method and show that the sizes and locations of DGs, and locations of EVCSs are adequate for both grid‐connected and islanded microgrids.

Automatic detection of semantic primitives using optimization based on genetic algorithm.

In this article, we propose a method for the automatic retrieval of a set of semantic primitive words from an explanatory dictionary and a novel evaluation procedure for the obtained set of primitives. The approach is based on the representation of the dictionary as a directed graph with a single-objective constrained optimization problem via a genetic algorithm with the PageRank scoring model. The problem is defined as a subset selection. The algorithm is fit to search for the sets of words that should fulfil several requirements: the cardinality of the set should not exceed empirically selected limits and the PageRank word importance score is minimized with cycle prevention thresholding. In the experiments, we used the WordNet dictionary for English. The proposed method is an improvement over the previous state-of-the-art solutions.

Multi-Objective Neural Evolutionary Algorithm for Combinatorial Optimization Problems.

There has been a recent surge of success in optimizing deep reinforcement learning (DRL) models with neural evolutionary algorithms. This type of method is inspired by biological evolution and uses different genetic operations to evolve neural networks. Previous neural evolutionary algorithms mainly focused on single-objective optimization problems (SOPs). In this article, we present an end-to-end multi-objective neural evolutionary algorithm based on decomposition and dominance (MONEADD) for combinatorial optimization problems. The proposed MONEADD is an end-to-end algorithm that utilizes genetic operations and rewards signals to evolve neural networks for different combinatorial optimization problems without further engineering. To accelerate convergence, a set of nondominated neural networks is maintained based on the notion of dominance and decomposition in each generation. In inference time, the trained model can be directly utilized to solve similar problems efficiently, while the conventional heuristic methods need to learn from scratch for every given test problem. To further enhance the model performance in inference time, three multi-objective search strategies are introduced in this work. Our experimental results clearly show that the proposed MONEADD has a competitive and robust performance on a bi-objective of the classic travel salesman problem (TSP), as well as Knapsack problem up to 200 instances. We also empirically show that the designed MONEADD has good scalability when distributed on multiple graphics processing units (GPUs).

Differential Evolution-Based Three Stage Dynamic Cyber-Attack of Cyber-Physical Power Systems

With the rapid development of communication, control, and computer technology, traditional power systems have evolved into cyber-physical power system (CPPS). However, CPPS not only affords convenience but also introduces more cyber-attacks. Among many types of cyber-attacks, false data injection attacks (FDIAs) have drawn much attention in the CPPS security domain due to their stealthiness. Most FDIAs are based on the static model on a single snapshot and often ignore the reality of dynamically time-evolving CPPS. Accordingly, this article proposes a novel three-stage dynamic false data injection attack (DFDIA) model in CPPS by considering potential dynamic behaviors. To consider both attack location and attack amplitude, the designing DFDIA is formulated as two constrained single-objective optimization problems. Two versions of constrained differential evolution are presented as the solver to determine the attack location and optimize the attack vector for collaboratively altering the meter measurements. Then, an interval state forecasting-based countermeasure is proposed to detect the established DFDIA. In this detector, the variation bounds of state values are determined via ensemble deep learning-based state forecasting method. Finally, extensive simulation results on several IEEE bus systems demonstrate the feasibility of DFDIA and the effectiveness of the defense mechanism.

Insight into single- and bi-objective optimization of industrial problems

ABSTRACT Evolutionary single- and bi-objective optimization of industrial problems, namely, naphtha cracking and styrene reactor are considered. Bi-objective optimization is solved using the HMODE-DLS algorithm, whereas an improved black hole optimizer (BHO) is employed for the single-objective optimization (SOO) problem. For the naphtha cracking process, ethylene selectivity (SE) and severity index (SI) are selected as objectives. Similarly, styrene selectivity (SST) and styrene flow rates (FST) are considered objectives for styrene reactor. Pareto ranking is carried out by using the net flow method (NFM) and the best solution is compared with the single- and multi-objective optimization results. In a single objective optimization study, the optimum solution corresponds to the lowest value of the SI (1.571) at the cost of the worst value of SE (0.269). Similarly, the highest values of FST (16.642 kmol/h) and SST (96.158%) are resulted in the SOO study of styrene reactor.

Low-Carbon Path Transformation for Different Types of Enterprises under the Dual-Carbon Target

Under pressure from the environment and resources, emission peak and carbon neutrality have rapidly become a global issue. The optimization of the ecological goal should be in line with the energy target. In most instances, however, the economic and the ecological goals cannot be unified. This paper establishes a multi-objective optimization model that maximizes the economic benefit of enterprises and the ecosystem activity of the government in the meantime. The idea point method is used in solving this multi-objective optimization problem in the form of a single-objective optimization problem. The numerical experiment documents four types of Chinese enterprises, which are primary resources, industrial manufacturing, public services and commercial consumption. Some management insights are summarized at the end, such as the cores of achieving high-quality and low-carbon development are industrial manufacturing and public services in China.

A novel graph-theoretical clustering approach to find a reduced set with extreme solutions of Pareto optimal solutions for multi-objective optimization problems

Multi-objective optimization problems and their solution algorithms are of great importance as single-objective optimization problems are not usually a true representation of many real-world problems. In general, multi-objective optimization problems result in a large set of Pareto optimal solutions. Each solution in this set is optimal with some trade-offs. Therefore, it is difficult for the decision-maker to select a solution, especially in the absence of subjective or judgmental information. Moreover, an analysis of all the solutions is computationally expensive and, hence, not practical. Thus, researchers have proposed several techniques such as clustering and ranking of Pareto optimal solutions to reduce the number of solutions. The ranking methods are often used to obtain a single solution, which is not a good representation of the entire Pareto set. This paper deviates from the common approach and proposes a novel graph-theoretical clustering method. The quality of the clustering based on the Silhouette score is used to determine the number of clusters. The connectivity in the objective space is used to find representative solutions for clusters. One step forward, we identify ‘extreme solutions’. Hence, the reduced set contains both extreme solutions and representative solutions. We demonstrate the performance of the proposed method by using different 3D and 8D benchmark Pareto fronts as well as Pareto fronts from a case study in Royal Australian Navy. Results revealed that the reduced set obtained from the proposed method outperforms that from the K-means clustering, which is the most popular traditional clustering approach in Pareto pruning.

Multiobjective Dynamic Optimization of Nonlinear Systems With Path Constraints

In this article, two algorithms are proposed to solve multiobjective path-constrained dynamic optimization problems. In each algorithm, an adaptive <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\varepsilon $ </tex-math></inline-formula> -constraint method is employed to solve the multiobjective dynamic optimization problems (MODOPs) with path constraints in two iterative loops. In the outer loop, the adaptive <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\varepsilon $ </tex-math></inline-formula> -constraint method adaptively adjusts the choice of the parameters <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\varepsilon $ </tex-math></inline-formula> , which transfers MODOP into a sequence of single-objective dynamic optimization problems (SODOPs) with extra inequality constraints. In the inner loop, two different algorithms are used to solve the single-objective optimization problems. The first algorithm guarantees that the path constraints can be satisfied with any finite prescribed tolerance by replacing path constraints with a finite number of point constraints. Furthermore, the second algorithm guarantees that the path constraints are rigorously satisfied by enforcing the path constraints at a limited number of time points and by restricting the right-hand side of the path constraints. The proposed algorithms are proven to converge within finite iterations. The effectiveness of the algorithms is verified via numerical studies, along with a comparison to a state-of-the-art algorithm.

Energy-Delay Tradeoff for Dynamic Trajectory Planning in Priority-Oriented UAV-Aided IoT Networks

Unmanned aerial vehicles (UAVs) play a crucial role in emergency-oriented applications. However, in UAV-aided Internet of Things (IoT) networks, the sensor nodes (SNs) would be mobile which poses a big challenge for trajectory planning of the UAV. In this paper, we investigate priority-oriented UAV-aided time-sensitive data collection problems in an IoT network with movable SNs. By defining different levels of delay sensitivities for each SN, we jointly minimize the energy consumed by a UAV and the average delay of different SNs through optimizing the trajectory of the UAV. The problem is formulated as a multi-objective optimization problem (MOP). To solve the formulated problem, we first transform the MOP into a single-objective optimization problem based on the weighted sum method. Then, we propose a novel autofocusing heuristic trajectory planning algorithm based on reinforcement learning (AHTP-RL) which can be operated in an online manner. The proposed algorithm can well extract the network dynamic topology and the delay-priority of SN through an attention mechanism, hence can structure the UAV’s trajectory efficiently. Extensive simulations results demonstrate that the proposed online AHTP-RL algorithm can achieve a superior balance between the communication delay and energy consumption for both low and high SN mobilities.

X-MODE: Extended Multi-operator Differential Evolution algorithm

During past years, multi-operator optimization techniques gained vast attention due to their flexible structure and self-adaptation. They yielded promising results when compared with the single-operator based methods. However, the existing multi-operator models can be further improved regarding exploration–exploitation equilibrium. Therefore, we propose a novel multi-operator algorithm for solving single-objective optimization problems that categorize differential evolution mutation strategies according to the purpose each of them fulfills. A new mutation scheme is also presented to promote the mutation across search space boundaries. A novel extended crossover technique is proposed in this study to retain the properties of applied mutation strategies, as well as develop the potential of the standard crossover function. The model is afterward evaluated on the CEC2020 benchmark test-suite. The proposed algorithm has overwhelming results while comparing single-operator and multi-operator algorithms for several classes of functions. It has been observed from the study that proposed mutation and crossover techniques help improve the efficiency of MODE variants.

Outer approximation algorithms for convex vector optimization problems

In this study, we present a general framework of outer approximation algorithms to solve convex vector optimization problems, in which the Pascoletti-Serafini (PS) scalarization is solved iteratively. This scalarization finds the minimum ‘distance’ from a reference point, which is usually taken as a vertex of the current outer approximation, to the upper image through a given direction. We propose efficient methods to select the parameters (the reference point and direction vector) of the PS scalarization and analyse the effects of these on the overall performance of the algorithm. Different from the existing vertex selection rules from the literature, the proposed methods do not require solving additional single-objective optimization problems. Using some test problems, we conduct an extensive computational study where three different measures are set as the stopping criteria: the approximation error, the runtime, and the cardinality of the solution set. We observe that the proposed variants have satisfactory results, especially in terms of runtime compared to the existing variants from the literature.

Nash Equilibria Strategies and Equivalent Single-Objective Optimization Problems. The Case of Linear Partial Differential Equations

In this paper we study the existence and uniqueness of Nash equilibria (solution to competition-wise problems, with several controls trying to reach possibly different goals) associated to linear partial differential equations and show that, in some cases, they are also the solution of suitable single-objective optimization problems (i.e. cooperative-wise problems, where all the controls cooperate to reach a common goal). We use cost functions associated with a particular linear parabolic partial differential equation and distributed controls, but the results are also valid for more general linear differential equations (including elliptic and hyperbolic cases) and controls (e.g. boundary controls, initial value controls,...).