High-dimensional Optimization Problems Research Articles

Time-dependent Dirac Equation with Physics-Informed Neural Networks: Computation and Properties

In this paper, we are interested in the computation of the time-dependent Dirac equation using physics-informed neural networks (PINNs), a new powerful tool in scientific machine learning avoiding the use of approximate derivatives of differential operators. PINNs search solutions in the form of parameterized (deep) neural networks, whose derivatives (in time and space) are performed by automatic differentiation. The computational cost comes from the need to solve high-dimensional optimization problems using stochastic gradient methods in the training the network with a large number of points analogues of the discretization points for standard PDE solvers. Specifically, we derive a PINNs-based algorithm and present some key fundamental properties when applied to the Dirac equations in different physical frameworks.

Comparative Evaluation of Different Multi-Agent Reinforcement Learning Mechanisms in Condenser Water System Control

Model-free reinforcement learning (RL) techniques are currently drawing attention in the control of heating, ventilation, and air-conditioning (HVAC) systems due to their minor pre-conditions and fast online optimization. The simultaneous optimal control of multiple HVAC appliances is a high-dimensional optimization problem, which single-agent RL schemes can barely handle. Hence, it is necessary to investigate how to address high-dimensional control problems with multiple agents. To realize this, different multi-agent reinforcement learning (MARL) mechanisms are available. This study intends to compare and evaluate three MARL mechanisms: Division, Multiplication, and Interaction. For comparison, quantitative simulations are conducted based on a virtual environment established using measured data of a real condenser water system. The system operation simulation results indicate that (1) Multiplication is not effective for high-dimensional RL-based control problems in HVAC systems due to its low learning speed and high training cost; (2) the performance of Division is close to that of the Interaction mechanism during the initial stage, while Division’s neglect of agent mutual inference limits its performance upper bound; (3) compared to the other two, Interaction is more suitable for multi-equipment HVAC control problems given its performance in both short-term (10% annual energy conservation compared to baseline) and long-term scenarios (over 11% energy conservation).

An electronic transition-based bare bones particle swarm optimization algorithm for high dimensional optimization problems.

An electronic transition-based bare bones particle swarm optimization (ETBBPSO) algorithm is proposed in this paper. The ETBBPSO is designed to present high precision results for high dimensional single-objective optimization problems. Particles in the ETBBPSO are divided into different orbits. A transition operator is proposed to enhance the global search ability of ETBBPSO. The transition behavior of particles gives the swarm more chance to escape from local minimums. In addition, an orbit merge operator is proposed in this paper. An orbit with low search ability will be merged by an orbit with high search ability. Extensive experiments with CEC2014 and CEC2020 are evaluated with ETBBPSO. Four famous population-based algorithms are also selected in the control group. Experimental results prove that ETBBPSO can present high precision results for high dimensional single-objective optimization problems.

Solving high-dimensional multi-area economic dispatch problem by decoupled distributed crisscross optimization algorithm with population cross generation strategy

Multi-area economic dispatch (MAED) is a non-convex non-differentiable high-dimensional optimization problem, aiming to minimize the total fuel cost and meet the requirements of load balance. Current distributed mathematical programming methods are hard to solve the valve point effect in MAED. In addition, distributed meta-heuristic algorithms so far require a centralized controller. To address these issues, a decoupled distributed crisscross optimization with population cross generation (DDCSO-PCG) algorithm is proposed, which can 1) realize decentralization, 2) protect area data privacy, 3) reduce dimensions and improve convergence ability. First, the population cross generation (PCG) strategy is integrated into the crisscross optimization (CSO) algorithm to maintain population diversity and enhance exploitation ability. Second, the DDCSO-PCG algorithm is implemented to solve the MAED problem in a fully decentralized manner. Under the distributed framework, the proposed algorithm employs CSO-PCG independently to optimize the area dispatch in parallel. The total optimal cost is achieved by minimizing the cost of each area with no centralized controller required. The experimental results on multi-area static economic dispatch (MASED) and multi-area dynamic economic dispatch (MADED) problems show that the proposed DDCSO-PCG algorithm can not only provide a distributed solution, but also achieve the best economic cost compared with other state-of-the-art techniques.

Analysis of Syntactic Complexity and Semantic Coherence of Academic English Writing Based on Particle Swarm Optimization

Writing is an important part of testing language ability, and it is urgent to find some objective indicators to determine and evaluate the surface language structure, which will help language learners’ better master the target language. Complexity and semantic coherence are considered to be an important factor in the teaching of second language writing. In practice, due to the complexity of English writing syntax, such as a large number of high-dimensional nonlinear optimization problems, a new intelligent evaluation method is needed to solve them. At present, particle swarm optimization (PSO) has been widely used in function optimization, neural network training, combinatorial optimization, and other fields. This paper studies the syntactic complexity and semantic coherence of academic English writing based on PSO. The number of phrases is related to writing achievement. When the number of experiments reaches 25, the significant values of syntactic complexity and semantic coherence of data mining algorithm, artificial intelligence algorithm, decision tree algorithm, and PSO algorithm are 0.008, 0.003, 0.002, and 0.013, respectively, which shows that PSO algorithm is the best among them.

Proof of the satisfiability conjecture for large $k$

We establish the satisfiability threshold for random $k$-SAT for all $k\ge k_0$, with $k_0$ an absolute constant. That is, there exists a limiting density $\alpha_{\rm sat}(k)$ such that a random $k$-SAT formula of clause density $\alpha$ is with high probability satisfiable for $\alpha\lt \alpha_{\rm sat}$, and unsatisfiable for $\alpha>\alpha_{\rm sat}$. We show that the threshold $\alpha_{\rm sat}(k)$ is given explicitly by the one-step replica symmetry breaking prediction from statistical physics. The proof develops a new analytic method for moment calculations on random graphs, mapping a high-dimensional optimization problem to a more tractable problem of analyzing tree recursions. We believe that our method may apply to a range of random CSPs in the $1$-RSB universality class.

Configuration-based Optimization for Virtual Hand Haptic Simulation.

Interacting with virtual objects via haptic feedback using the user's hand directly (virtual hand haptic interaction) provides a natural and immersive way to explore the virtual world. It remains a challenging topic to achieve 1 kHz stable virtual hand haptic simulation with no penetration amid hundreds of hand-object contacts. In this paper, we advocate decoupling the high-dimensional optimization problem of computing the graphic-hand configuration, and progressively optimizing the configuration of the graphic palm and fingers, yielding a decoupled-and-progressive optimization framework. We also introduce a method for accurate and efficient hand-object contact simulation, which constructs a virtual hand consisting of a sphere-tree model and five articulated cone frustums, and adopts a configuration-based optimization algorithm to compute the graphic-hand configuration under non-penetration contact constraints. Experimental results show both high update rate and stability for a variety of manipulation behaviors. Non-penetration between the graphic hand and complex-shaped objects can be maintained under diverse contact distributions, and even for frequent contact switches. The update rate of the haptic simulation loop exceeds 1 kHz for the whole-hand interaction with about 250 contacts.

Coordinated Control Strategy of Electricity-Heat-Gas Integrated Energy System Considering Renewable Energy Uncertainty and Multi-Agent Mixed Game

The Integrated Energy System (IES) can promote social energy transformation and low-carbon development, which is also an effective means to make contributions to energy structure optimization, energy consumption reduction, and new energy consumption. However, the IES has the characteristics of complex energy flow, and strong uncertainty with multi-agents. Therefore, traditional mathematical optimization models are difficult to comprehensively and accurately reflect the interest needs of different entities in the integrated energy microgrid. Aiming at this problem, a two-level collaborative control strategy model of “electricity-heat-gas” IES based on multi-agent deep reinforcement learning is proposed in this paper. The upper layer of this model is a multi-agent hybrid game decision-making model based on the Multi-Agent Deep Deterministic Policy Gradient algorithm (MADDPG), and the lower layer contains the power and gas flow calculation model. The lower model provides the upper model with the energy flow data of the IES and the upper layer rewards the decision-making behavior of the agent based on the energy flow data provided by the lower layer. Effectively solving the high-dimensional nonlinear optimization problem existing in the complex coupling network, this method can improve the convergence and training speed of the model. In this paper, the IEEE 33-node distribution network and 20-node gas network coupling system are provided to verify the model. The simulation results show that the proposed collaborative control strategy method can provide effective decision-making for electric-agent and gas-agent and realize the efficient and economic operation of the integrated energy system.

Improving Particle Swarm Optimization Analysis Using Differential Models

This paper employs the approach of the differential model to effectively improve the analysis of particle swarm optimization. This research uses a unified model to analyze four typical particle swarm optimization (PSO) algorithms. On this basis, the proposed approach further starts from the conversion between the differential equation model and the difference equation model and proposes a differential evolution PSO model. The simulation results of high-dimensional numerical optimization problems show that the algorithm’s performance can be greatly improved by increasing the step size parameter and using different transformation methods. This analytical method improves the performance of the PSO algorithm, and it is a feasible idea. This paper uses simple analysis to find that many algorithms are improved by using the difference model. Through simple analysis, this paper finds that many AI-related algorithms have been improved by using differential models. The PSO algorithm can be regarded as the social behavior of biological groups such as birds foraging and fish swimming. Therefore, these behaviors described above are an ongoing process and are more suitable for using differential models to improve the analysis of PSO. The simulation results of the experiment show that the differential evolution PSO algorithm based on the Runge–Kutta method can effectively avoid premature results and improve the computational efficiency of the algorithm. This research analyzes the influence of the differential model on the performance of PSO under different differenced conditions. Finally, the analytical results of the differential equation model of this paper also provide a new analytical solution.

A Metaheuristic Algorithm for Coverage Enhancement of Wireless Sensor Networks

When wireless sensors are randomly deployed in natural environments such as ecological monitoring, military monitoring, and disaster monitoring, the initial position of sensors is generally formed through deployment methods such as air-drop, and then, the second deployment is carried out through the existing optimization methods, but these methods will still lead to serious coverage holes. In order to solve this problem, this paper proposes an algorithm to improve the coverage rate for wireless sensor networks based on an improved metaheuristic algorithm. The sensor deployment coverage model was firstly established, and the sensor network coverage problem was transformed into a high-dimensional multimodal function optimization problem. Secondly, the global searching ability and searching range of the algorithm are enhanced by the reverse expansion of the initial populations. Finally, the firefly principle is introduced to reduce the local binding force of sparrows and avoid the local optimization problem of the population in the search process. The experimental results showed that compared with ALO, GWO, BES, RK, and SSA algorithms, the EFSSA algorithm is better than other algorithms in benchmark function tests, especially in the test of high-dimensional multimodal function. In the tests of different monitoring ranges and number of nodes, the coverage of EFSSA algorithm is higher than other algorithms. The result can tell that EFSSA algorithm can effectively enhance the coverage of sensor deployment.

Elite Directed Particle Swarm Optimization with Historical Information for High-Dimensional Problems

High-dimensional optimization problems are ubiquitous in every field nowadays, which seriously challenge the optimization ability of existing optimizers. To solve this kind of optimization problems effectively, this paper proposes an elite-directed particle swarm optimization (EDPSO) with historical information to explore and exploit the high-dimensional solution space efficiently. Specifically, in EDPSO, the swarm is first separated into two exclusive sets based on the Pareto principle (80-20 rule), namely the elite set containing the top best 20% of particles and the non-elite set consisting of the remaining 80% of particles. Then, the non-elite set is further separated into two layers with the same size from the best to the worst. As a result, the swarm is divided into three layers. Subsequently, particles in the third layer learn from those in the first two layers, while particles in the second layer learn from those in the first layer, on the condition that particles in the first layer remain unchanged. In this way, the learning effectiveness and the learning diversity of particles could be largely promoted. To further enhance the learning diversity of particles, we maintain an additional archive to store obsolete elites, and use the predominant elites in the archive along with particles in the first two layers to direct the update of particles in the third layer. With these two mechanisms, the proposed EDPSO is expected to compromise search intensification and diversification well at the swarm level and the particle level, to explore and exploit the solution space. Extensive experiments are conducted on the widely used CEC’2010 and CEC’2013 high-dimensional benchmark problem sets to validate the effectiveness of the proposed EDPSO. Compared with several state-of-the-art large-scale algorithms, EDPSO is demonstrated to achieve highly competitive or even much better performance in tackling high-dimensional problems.

A velocity-based butterfly optimization algorithm for high-dimensional optimization and feature selection

Throughout the last decade, high-dimensional function optimization problems have received substantial research interest in the field of intelligence computation. Butterfly optimization algorithm is a new meta-heuristic technique that has been proven to be more competitive than other optimization methods on low- dimensional problems. The basic butterfly optimization algorithm and its variants are not used to solve high-dimensional problems. Therefore, this paper develops an improved version of butterfly optimization algorithm to deal with the high- dimensional optimization problems. Inspired by particle swarm optimization, a modified position updated equation is designed by introducing velocity item and memory item in the local search phase to guide the search for candidate individuals. Moreover, a novel refraction-based learning strategy is introduced into butterfly optimization algorithm to effectively enhance diversity and exploration. The effectiveness of the proposed algorithm is tested on 40 benchmark problems with 100, 1000, 10,000 dimensions. The statistical results show that our algorithm has better performance than the basic butterfly optimization algorithm, its variants, and other population-based approaches to deal with the high-dimensional optimization problems. Finally, the high-dimensional feature selection and fault identification of wind turbine problems are solved and the comparisons show that the proposed algorithm outperforms better than most methods in terms of classification accuracy and number of the optimal feature subset.

Machine learning with knowledge constraints for process optimization of open-air perovskite solar cell manufacturing

Machine learning with knowledge constraints for process optimization of open-air perovskite solar cell manufacturing

An Improved particle swarm optimization based on lévy flight and simulated annealing for high dimensional optimization problem

Particle swarm optimization (PSO) is a simple metaheuristic method to implement with robust performance. PSO is regarded as one of the numerous researchers' most well-studied algorithms. However, two of its most fundamental problems remain unresolved. PSO converges onto the local optimum for high-dimensional optimization problems, and it has slow convergence speeds. This paper introduces a new variant of a particle swarm optimization algorithm utilizing Lévy flight-McCulloch, and fast simulated annealing (PSOLFS). The proposed algorithm uses two strategies to address high-dimensional problems: hybrid PSO to define the global search area and fast simulated annealing to refine the visited search region. In this paper, PSOLFS is designed based on a balance between exploration and exploitation. We evaluated the algorithm on 16 benchmark functions for 500 and 1,000 dimension experiments. On 500 dimensions, the algorithm obtains the optimal value on 14 out of 16 functions. On 1,000 dimensions, the algorithm obtains the optimal value on eight benchmark functions and is close to optimal on four others. We also compared PSOLFS with another five PSO variants regarding convergence accuracy and speed. The results demonstrated higher accuracy and faster convergence speed than other PSO variants. Moreover, the results of the Wilcoxon test show a significant difference between PSOLFS and the other PSO variants. Our experiments' findings show that the proposed method enhances the standard PSO by avoiding the local optimum and improving the convergence speed.

A Dimension Group-Based Comprehensive Elite Learning Swarm Optimizer for Large-Scale Optimization

High-dimensional optimization problems are more and more common in the era of big data and the Internet of things (IoT), which seriously challenge the optimization performance of existing optimizers. To solve these kinds of problems effectively, this paper devises a dimension group-based comprehensive elite learning swarm optimizer (DGCELSO) by integrating valuable evolutionary information in different elite particles in the swarm to guide the updating of inferior ones. Specifically, the swarm is first separated into two exclusive sets, namely the elite set (ES) containing the top best individuals, and the non-elite set (NES), consisting of the remaining individuals. Then, the dimensions of each particle in NES are randomly divided into several groups with equal sizes. Subsequently, each dimension group of each non-elite particle is guided by two different elites randomly selected from ES. In this way, each non-elite particle in NES is comprehensively guided by multiple elite particles in ES. Therefore, not only could high diversity be maintained, but fast convergence is also likely guaranteed. To alleviate the sensitivity of DGCELSO to the associated parameters, we further devise dynamic adjustment strategies to change the parameter settings during the evolution. With the above mechanisms, DGCELSO is expected to explore and exploit the solution space properly to find the optimum solutions for optimization problems. Extensive experiments conducted on two commonly used large-scale benchmark problem sets demonstrate that DGCELSO achieves highly competitive or even much better performance than several state-of-the-art large-scale optimizers.

Research on Forecast of Macroeconomic Indicators Based on Multiobjective Optimization

In recent years, with China’s economic development entering a new normal, the total scale and complexity of economic development have increased unprecedentedly, and the requirements of government management and regulation of the economy are getting higher and higher. Traditional management concepts, methods, and tools are difficult to meet the needs of the new situation. Therefore, using computer science and technology to fully consider the correlation between various industries and explore new laws of economic operation under the new normal can provide new technical support for promoting the precision of government governance. In previous studies, Most of them take statistics and econometrics as the theoretical basis of macroeconomic decision-making research. This study is based on the analysis of the relationship between the macroeconomic indicators, the establishment of forecasting models to predict the trend of the macroeconomic indicators, and according to the relationship between various economic indicators to build a multiobjective optimization problem, through the multiobjective optimization algorithm to obtain optimal data for the government decision-making basis. In the research process, we focus on solving the problem of prediction overfitting caused by small samples of economic index data and high feature dimension of samples and how to balance the convergence and diversity of solution set in high-dimensional multiobjective optimization problems. In view of the above problems, we propose two economic index forecasting models to solve a variety of fitting problems. On the premise of optimizing space decomposition, an adaptive generation strategy based on inflection point and center point is proposed, and an adaptive solution set selection strategy based on included angle, inflection point, and strong dominance relationship is proposed to balance the convergence and diversity of the multiobjective optimization algorithm.

A dynamic stochastic search algorithm for high-dimensional optimization problems and its application to feature selection

As the number of dimensions of an optimization problem increases, the process of deriving the solution becomes more complicated and difficult. Metaheuristic algorithms are effective methods for solving many optimization problems, whereas most of these algorithms perform poorly on high-dimensional problems. In this paper, a novel metaheuristic algorithm called Dynamic Stochastic Search (DSS) is proposed for high-dimensional optimization problems. To effectively execute the exploration (diversification) and exploitation (intensification) processes of DSS, a dynamic stochastic search process is designed by defining a stochastic search control factor, a search process based on the Gaussian distribution, two shrink modes inspired by the Whale Optimization Algorithm, and a balance mechanism derived from the Bat Algorithm. The proposed algorithm has the following advantages: no specific control parameters other than the population size and the maximum number of iterations, a simple structure, and less computational effort in the implementation. Analyzing the computational complexity of DSS demonstrates its simplicity. To evaluate the performance of DSS, twenty 300-dimensional and 3000-dimensional classical benchmark functions as well as thirty 100-dimensional CEC2014 benchmark functions are utilized. The statistical results prove the effectiveness and feasibility of DSS for high-dimensional optimization problems because it shows better convergence performance and higher efficiency than various advanced optimization algorithms. Moreover, the superiority of DSS in solving high-dimensional real-world optimization problems is validated by applying it to feature selection for ten real-world benchmark classification datasets from the UCI machine learning repository.

Common defects in diamond lattices as instances of the general T ⊗ (e+t2) Jahn-Teller effect

Finding the ground-state configuration of point defects in semiconductors is in general a high-dimensional optimization problem. However, their adiabatic potential energy surfaces (APES) can be described with few effective coordinates by exploiting symmetry arguments within the Jahn-Teller (JT) theoretical framework. In this paper, we propose a general framework, that combines group theory and ab initio calculations, to build a full picture of common defects in diamond lattices (vacancies and substitutional impurities) as instances of the general T $\ensuremath{\bigotimes}\phantom{\rule{4pt}{0ex}}(e+{t}_{2})$ APES. Starting from the original tetrahedral symmetry, we consider all possible JT symmetries and we numerically investigate the shape of the APES by targeting the minimum energy path between equivalent JT distortions taking as examples the vacancy (V) and the metallic impurities Pd, Pt, and Au in crystalline silicon.

A Multipopulation Dynamic Adaptive Coevolutionary Strategy for Large-Scale Complex Optimization Problems.

In this paper, a multipopulation dynamic adaptive coevolutionary strategy is proposed for large-scale optimization problems, which can dynamically and adaptively adjust the connection between population particles according to the optimization problem characteristics. Based on analysis of the network evolution characteristics of collaborative search between particles, a dynamic adaptive evolutionary network (DAEN) model with multiple interconnection couplings is established in this algorithm. In the model, the swarm type is divided according to the judgment threshold of particle types, and the dynamic evolution of collaborative topology in the evolutionary process is adaptively completed according to the coupling connection strength between different particle types, which enhances the algorithm’s global and local searching capability and optimization accuracy. Based on that, the evolution rules of the particle swarm dynamic cooperative search network were established, the search algorithm was designed, and the adaptive coevolution between particles in different optimization environments was achieved. Simulation results revealed that the proposed algorithm exhibited a high optimization accuracy and converging rate for high-dimensional and large-scale complex optimization problems.

Adaptive backtracking search optimization algorithm with a dual-learning strategy for dynamic economic dispatch with valve-point effects

Dynamic economic dispatch with valve-point effect (DED_vpe) is a dynamic nonlinear high-dimensional optimization problem with non-smooth and non-convex characteristics. Meta-heuristic methods have become the mainstream for solving the DED_vpe problem. However, most of these methods only focus on minimizing the generation costs and ignore the algorithmic robustness. In this paper, an adaptive backtracking search optimization algorithm with a dual-learning strategy (DABSA) is proposed for solving the DED_vpe problem. In DABSA, a dual-learning strategy (DL) based on the current and historical optimal individuals is developed to update each individual. This updating strategy helps DABSA improve solution accuracy and overcome premature convergence. In addition, an adaptive parameter control mechanism (APC), which can automatically adjust parameter ‘mixrate’ value according to the current iteration number, is presented. To handle the system constraints, a ‘repair + penalty’ constraints-handling approach is employed to lead non-feasible solutions towards the feasible region quickly. The performance of DABSA is assessed by testing on four DED problems containing 5, 10 and 30 units. The experimental results show that DABSA is very competitive compared with reported representative methods in yielding low fuel costs along with high robustness.