Nonlinear Multi-objective Optimization Problem Research Articles

Integrated LI-NSGA-II Approach for Solving the Non-linear Multi-objective Optimization Problem

Real-world engineering projects frequently involve complex, non-linear multi-objective optimization challenges. Traditional methods like PERT/CPM and basic heuristics often fail to provide optimal solutions in such scenarios. Multi-objective evolutionary algorithms, such as genetic algorithms and non-dominated sorting genetic algorithms, are more effective for identifying true Pareto solutions. Among these, the Non-dominated Sorting Genetic Algorithm-II (NSGA-II) is a well-known algorithm for solving multi-objective optimization problems. This study presents an integrated approach, called LI-NSGA-II, which combines Lagrange's Interpolation (LI) with NSGA-II to solve non-linear multi-objective optimization problems in real-world projects. In this LI-NSGA-II approach, LI is used to handle the non-linearity of competing objectives, whereas NSGA-II optimizes these objectives to find optimal solutions. The outcomes achieved through the LI-NSGA-II approach are ideal for real-time monitoring and control of non-linear multi-objective optimization problems.

An Activity Network Design and Charging Facility Planning Model Considering the Influence of Uncertain Activities in a Game Framework

In the planning of public charging facilities and the charging activity network of users, there is a decision-making conflict among three stakeholders: the government, charging station enterprises, and electric vehicle users. Previous studies have described the tripartite game relationship in a relatively simplistic manner, and when designing charging facility planning schemes, they did not consider scenarios where users’ choice preferences undergo continuous random changes. In order to reduce the impacts of queuing phenomenon and resource idleness on the three participants, we introduce a bilateral matching algorithm combined with the dynamic Huff model as a strategy for EV charging selection in the passenger flow problem based on the three-dimensional activity network of time–space–energy of users. Meanwhile, the Dirichlet distribution is utilized to control the selection preferences on the user side, constructing uncertain scenarios for the choice of user charging activities. In this study, we establish a bilevel programming model that takes into account the uncertainty in social responsibility and user charging selection behavior. Solutions for the activity network and facility planning schemes can be derived based on the collaborative relationships among the three parties. The model employs a robust optimization method to collaboratively design the charging activity network and facility planning scheme. For this mixed-integer nonlinear multi-objective multi-constraint optimization problem, the model is solved by the NSGA-II algorithm, and the optimal compromise scheme is determined by using the EWM-TOPSIS comprehensive evaluation method for the Pareto solution set. Finally, the efficacy of the model and the solution algorithm is illustrated by a simulation example in a real urban space.

Pareto-critical equilibrium condition for non-linear multiobjective optimization problems with applications

The new (necessary) Pareto-critical equilibrium condition (PCEC) discussed in this study is based on the Pareto-critical condition for non-linear optimization problems. The proposed condition is evaluated by defining a multiobjective optimization problems (MOOPs) subject to a set of constraints. Then this MOOPs is converted to a single objective optimization problem by using the available optimization techniques, for instance, the weighted sum method, and solved by the non-linear Nelder-Mead simplex method. The obtained solution points satisfy the proposed PCEC of an unconstrained multiobjective optimization problem (MOOPs) under the convexity assumption. This PCEC is evaluated in a search that either results in a point inside the feasible region or a point for which the PCEC holds. The resulting Pareto-critical equilibrium condition is globally valid for convex problems. Numerical experiments are carried out in order to clarify the proposed condition's validity. Moreover, the proposed PCEC is used to place a downlink transmit beamforming system base station in an ideal position as a first application and is also employed in situations where there are multiple agents working towards a common goal as a second application.

Numerical algorithms for generating an almost even approximation of the Pareto front in nonlinear multi-objective optimization problems

A multiobjective optimization problem (MOP) returns a set of non-dominated points, the so-called Pareto front. Since this set is usually infinite, it is impossible to generate it completely in practice. Therefore, a discrete approximation of the Pareto front is created. One of the most important features of this approximation is a uniform distribution of points on the full Pareto front in order to present a wide variety of solutions to the decision maker who chooses a final solution. While a few algorithms consider this property, two algorithms based on the Pascoletti-Serafini (PS) scalarization approach are proposed. In addition, six well-known test problems with convex and non-convex Pareto fronts are considered to show the effectiveness of the proposed algorithms. Their results are compared with some algorithms including Normal Constraint (NC), Benson type, Non-Dominated Sorting Genetic Algorithm-II (NSGA-II), S-Metric Selection Evolutionary Multiobjective Algorithm (SMS-EMOA), Differential Evolution (DE) with Binomial Crossover and MOEA/D-DE. The computational results on CPU time and reasonable distribution of points obtained on the Pareto front show that the presented algorithms perform better than other algorithms on these criteria. In addition, although the proposed algorithms compete closely with some algorithms in terms of CPU time, they have more non-dominated solutions and more appropriate distribution than they do in most problems.

Determination of Optimal Locations and Parameters of Passive Harmonic Filters in Unbalanced Systems Using the Multiobjective Genetic Algorithm

This paper discusses the problem of optimal placement and sizing of passive harmonic filters to mitigate harmonics in unbalanced distribution systems. The problem is formulated as a nonlinear multiobjective optimisation problem and solved using the multiobjective genetic algorithm. The performance of the proposed algorithm is tested on unbalanced IEEE 13- and 37-bus three-phase systems. The optimal solutions are obtained based on the following objective functions: 1) minimisation of total harmonic distortion in voltage, 2) minimisation of costs of filters, 3) minimisation of voltage unbalances, and 4) a simultaneous minimisation of total harmonic distortion in voltage, costs of filters, and voltage unbalances. Finally, an analysis of the influence of uncertainties of load powers and changes in system frequency and filter parameters on filter efficiency was performed.

An optimal approach of pure/mixed CO2 sequestration with producing hydrated natural gas: Physical insights and validation

In this contribution, an optimal approach is developed to sequestrate pure/mixed CO2 by dislodging the hydrated natural gas, the cleanest fossil fuel, thereby catching two birds by one stone. For depicting its intricacies, a transient thermo-kinetic swapping model is developed with considering first time the effect of permeability and water saturation, along with porosity involved in sediments of hydrate reservoir. The formulation takes into account various fundamental issues having practical relevance, like porous particles with uneven size and shape, surface dynamics involved in phase transformation, non-unity reaction order (i.e., nth order), among others. A mixed-integer nonlinear multi-objective sequential optimization (MOSO) problem is articulated within the framework of a robust global optimizer (i.e., nondominated sorting genetic algorithm II) to deal with two conflicting objectives, concerning minimization of absolute average relative deviation and maximization of swapping efficiency. This optimal approach is validated with laboratory and field data, showing its outperformance over the latest models at diverse geological conditions. The proposed model framework is employed for lab-scale cases to identify the optimal guest gas (CO2/N2) ratio that secures the maximum swapping efficiency. Simulation results suggest that the presented theoretical model is rigorous in interpreting these swapping dynamics in hydrates.

Multi-objectives transmission expansion planning considering energy storage systems and high penetration of renewables and electric vehicles under uncertain conditions

Transmission expansion planning (TEP) integrating electric vehicles (EVs) and renewable energy sources (RESs) is pivotal for the transition toward cleaner and more sustainable energy systems. One of the biggest challenges in TEP with EVs and RESs is the uncertainty inherent in their behaviors. Managing this uncertainty is critical for ensuring the grid's reliability and resilience, and for facilitating a seamless transition to EVs. A potent method for addressing uncertainties in power systems is the application of a scenario-based approach. However, the efficiency of this approach is contingent upon an increase in the number of representative scenarios, which, in turn, escalates computational time, making solving the TEP problem a crucial task. To tackle these challenges, this paper proposes a multi-objective resilience model for TEP, to advance towards more sustainable energy systems. The model seeks to strike a balance between economic, environmental, and technical considerations by minimizing planning costs, carbon dioxide emissions, and enhancing the voltage profile. Furthermore, the paper proposes a cascaded intelligent strategy to handle uncertainties and reduce computational complexity by reducing the number of representative scenarios without compromising system reliability. The adaptive neuro-inference system is designed to manage long-term uncertainties, while two fuzzy systems are developed to address short-term uncertainties. The problem is formulated as a complex non-linear multi-objective optimization problem. To address this, a hybrid approach combining the mountain gazelle optimizer and the multi-objective salp swarm optimizer is developed to solve the problem. The efficacy of the proposed method is validated on two separate test systems, demonstrating its superiority in maintaining system reliability while reducing computing time by at least 92 % in comparison with the considered traditional methods. The results also highlight the superior performance of the proposed hybrid algorithm compared to existing meta-heuristic algorithms in solving the TEP.

Bio-Inspired Jumping Spider Optimization for Controller Tuning/ Parameter Estimation of an Uncertain Aerodynamic MIMO System

The practical near impossibility of empirical attempts in estimating optimal controller gains makes the use of metaheuristics strategies inevitable to automatically obtain these gains by an iterative, heuristic simulation procedure. The convergence of the gains values to the local or global solutions occur with ease. In designing controllers for the Twin-Rotor MIMO System (TRMS), Jumping Spider Optimization Algorithm (JSOA), a novel neoteric population-based bio-inspired metaheuristic approach is used to obtain optimum values for the Proportional, Integral and Derivative (PID) controllers. With the k,p,i controller gains as the decision variables, the JSOA solution to a nonlinear multi-objective optimization problem subject to some intrinsic constraints spawned optimal values for the controllers’ variables. Counter to other algorithms (deterministic and stochastic) that get caught in local minima, JSOA evolved a solution after searchingly rummaging the entire solution search space in a vectorized fashion for an optimal value. Compared with several other versatile controllers (using GA, PSO, Pattern Search and Simulated Annealing), statistical results obtained showed JSOA technique provided a unique solution and found the gains of the PID controllers, marginally in relation to the others (optimization methods).

Optimal load management of autonomous power systems in conditions of water shortage

The issues of optimizing the operation of micro hydropower plants in conditions of water scarcity, performed by additional connection to the grid of an energy storage system and wind power turbine, as well as optimal load management, are considered. It is assumed that the load of the system is a concentrated autonomous power facility that consumes only active power. The paper presents a rigorous mathematical formulation of the problem, the solution of which corresponds to the minimum cost of an energy storage system and a wind turbine, which allows for uninterrupted supply of electricity to power facilities in conditions of water shortage necessary for the operation of micro hydropower plants (under unfavorable hydrological conditions). The problem is formulated as a nonlinear multi-objective optimization problem to apply metaheuristic stochastic algorithms. At the same time, a significant part of the problem is taken out and framed as a subproblem of linear programming which will make it possible to solve it by a deterministic simplex method that guarantees to find the exact global optimum. This approach will significantly increase the efficiency of solving the entire problem by combining metaheuristic algorithms and taking into account expert knowledge about the problem being solved.

Highway Alignment Optimization Design Method Based on Multi-Objective Particle Swarm Optimization

Highway alignment design is a nonlinear multi-objective optimization problem. Due to the uncertainty and complexity of the constraints, it is challenging to develop a suitable model for 3D road alignment design. To comprehensively consider the constraints and realize the intelligent design, this study puts forward a novel highway alignment model to optimize the horizontal and vertical alignment design, in which the length utility cost and structure construction cost are projected into the optimization space. Finally, the numerical solution is carried out by MOPSO on MATLAB platform. The results show that this method can effectively solve the problem of highway alignment design under the condition of realizing the highway design specifications, shortening the design cycle, and improving the design quality, which verifies the feasibility of the optimization algorithm for the horizontal and vertical alignment optimization design of highway.

Enhancing the performance of radial distribution systems via optimal integration of electric vehicles

<span lang="EN-US">Electric vehicles (EVs) and their associated charging stations (CSs) are a promising avenue for greenhouse gas reduction and energy security. However, improper location and size of the EV charging station (CSs/EVs) negatively affect the power system. It results in some technical challenges such as increased real power-loss, decreased voltage stability and increased voltage deviation. This paper suggests optimum planning of CSs/EVs locations to conserve voltage, improve power-loss and reduce the effect of CSs/</span><span lang="EN-US">EVs in electrical distribution systems. The multi objective genetic algorithm method (MOGA) is utilized to solve the optimization problem and reduce the impact of the random location of CSs/EVs. <span>The simulations are</span> performed on IEEE 33-bus and IEEE 69-bus radial distribution systems (RDS). In addition, the optimal EV allocation is carried out with two fixed levels of loading from 100% to 150% of the candidate EV load. A comparison between the proposed MOGA technique and other optimization techniques is carried out. The results demonstrated the capability of the proposed technique for optimal placement of CSs/EVs in RDS. </span><span lang="EN-US">Moreover, it is providing an objective value to solve a complex multi-objective nonlinear optimization problem to reduce the total power-loss, improve the voltage profile and extend the voltage stability.</span>

Adaptive surrogate assisted multi-objective optimization approach for highly nonlinear and complex engineering design problems

Despite enormous advances in computer power, computationally costly models impede the use of traditional optimization approaches that must be invoked repeatedly during the optimization process in practical engineering applications. Surrogate models have been found to be a promising endeavor in multi-objective optimization problems involving expensive analysis and simulation processes such as multi-physics modeling and simulation, finite element analysis (FEA), and computational fluid dynamics (CFD. Developing an optimization algorithm that can easily identify the Pareto frontier of highly nonlinear multi-objective optimization problems with less computation cost is the aim of this work. In this paper, an Adaptive Multi-Objective Optimization approach based Surrogate models (AMOS) is developed to reduce computation cost of fitness evaluations and discover the Pareto optima for multi-objective optimization problems with comparable high accuracy. AMOS explores the design space by sampling using LHD to identify promising regions. Then, AMOS exploits the identified promising region by adaptively constructing the most suitable surrogate model, which could be response surface, radial basis, or Kriging surrogates, in the feasible design space based on root mean square error values (RMSE). AMOS stops iterating when a termination criterion is met, and a Pareto frontier is identified based on developed guidance and fitness functions. AMOS has successfully identified the pareto frontier of practical engineering optimization problems with expensive black box functions and significantly reduced the computation cost. The novel method was put to the test utilizing real-world challenges and engineering design examples such vehicle magnetorheological design and wind turbine airfoil geometry.

Fuzzy prediction of the mine's ventilation structure's tunnel air volume

For the problem that it is difficult to accurately measure the air volume in the tunnel where the mine ventilation structure is located. Firstly, the tunnel air volume value is expressed by the interval number, and the air volume value of the tunnel where the ventilation structure is located is obtained by the law of nodal air balance. Then the interval number is transformed into a triangular fuzzy number to represent the tunnel air volume, the generalized induced ordered weighted average operator is introduced, the minimum-maximum closeness is used as the optimal quasi-measure, and the preference coefficient is introduced to transform the multi-objective nonlinear optimization problem into a single-objective optimization problem. And consider the temporal characteristics of the error series and the trend of the error value change for the error correction of air volume prediction error, and construct the interval combination type air volume fuzzy prediction model. Using the air volume data of a mine tunnel in Inner Mongolia, the example calculation of the model shows that the interval combination type prediction method can effectively improve the accuracy of air volume prediction in the tunnel of ventilation structures. Keywods: Error correction, Interval combination type, Interval number, Preference coefficient, Triangular fuzzy number

Three-Dimensional Path Planning of Deep-Sea Mining Vehicle Based on Improved Particle Swarm Optimization

Three-dimensional path planning is instrumental in path decision making and obstacle avoidance for deep-sea mining vehicles (DSMV). However, conventional particle swarm algorithms have been prone to trapping in local optima and have slow convergence rates when applied to underwater robot path planning. In order to secure a safe and economical three-dimensional path for the DSMV from the mining area to the storage base in connection with innovative mining system, this paper proposes a multi-objective optimization algorithm based on improved particle swarm optimization (IPSO) path planning. Firstly, we construct an unstructured seabed mining area terrain model with hazardous obstacles. Consequently, by considering optimization objectives such as the path length, terrain undulation, comprehensive energy consumption, and crawler slippage rate, we convert the path planning problem into a multi-objective optimization problem, constructing a multi-objective optimization mathematical model. Following that, we propose an IPSO algorithm to tackle the multi-objective non-linear optimization problem, which enables global optimization for DSMV path planning. Finally, we conduct a comprehensive set of experiments using the MATLAB simulation platform and compare the proposed method with existing advanced methods. Experimental results indicate that the path planned by the IPSO exhibits superior performance in terms of path length, terrain undulation, energy consumption, and safety.

Stochastic Multi-Objective Optimal Reactive Power Dispatch with the Integration of Wind and Solar Generation

The exponential growth of unpredictable renewable power production sources in the power grid results in difficult-to-regulate reactive power. The ultimate goal of optimal reactive power dispatch (ORPD) is to find the optimal voltage level of all the generators, the transformer tap ratio, and the MVAR injection of shunt VAR compensators (SVC). More realistically, the ORPD problem is a nonlinear multi-objective optimization problem. Therefore, in this paper, the multi-objective ORPD problem is formulated and solved considering the simultaneous minimization of the active power loss, voltage deviation, emission, and the operating cost of renewable and thermal generators. Usually, renewable power generators such as wind and solar are uncertain; therefore, Weibull and lognormal probability distribution functions are considered to model wind and solar power, respectively. Due to the unavailability and uncertainty of wind and solar power, appropriate PDFs have been used to generate 1000 scenarios with the help of Monte Carlo simulation techniques. Practically, it is not possible to solve the problem considering all the scenarios. Therefore, the scenario reduction technique based on the distance metric is applied to select the 24 representative scenarios to reduce the size of the problem. Moreover, the efficient non-dominated sorting genetic algorithm II-based bidirectional co-evolutionary algorithm (BiCo), along with the constraint domination principle, is adopted to solve the multi-objective ORPD problem. Furthermore, a modified IEEE standard 30-bus system is employed to show the performance and superiority of the proposed algorithm. Simulation results indicate that the proposed algorithm finds uniformly distributed and near-global final non-dominated solutions compared to the recently available state-of-the-art multi-objective algorithms in the literature.

Three-dimensional path planning for AUVs in ocean currents environment based on an improved compression factor particle swarm optimization algorithm

Three-dimensional path planning for autonomous underwater vehicles (AUVs) in underwater environments is the key to ensuring safe navigation and reliable mission completion. To obtain a safe and smooth three-dimensional path for an AUV in ocean currents and seabed obstacle environments, an improved compression factor particle swarm optimization algorithm is proposed. First, a three-dimensional seabed environment model and Lamb vortex current environment model are constructed. Second, by considering optimization objectives such as travel distance cost, seabed terrain constraints and ocean current constraints, a three-dimensional path planning mathematical model is constructed. Finally, an improved compression factor particle swarm optimization algorithm is proposed and applied to solve the multi-objective nonlinear optimization problem. To verify the optimization performance of the new algorithm, its optimization results are compared with those of other algorithms by minimizing the fitness value. The experimental results reveal that the improved compressed factor particle swarm optimal algorithm has better planning efficiency, path quality, and shorter planning time, which provides a new effective method for path planning of autonomous underwater vehicle.

An infeasible interior-point technique to generate the nondominated set for multiobjective optimization problems

In this paper, an infeasible interior-point technique is proposed to generate the nondominated set of nonlinear multi-objective optimization problems with the help of the direction-based cone method. We derive the proposed method for both convex and nonconvex problems. In order to solve the parametric optimization problems of the cone method, the infeasible interior-point method starts with an initial iterate outside the feasible region, and then gradually reduces the primal and dual infeasibility measures and the objective function value across the iterations with the help of a merit function. Estimates of the reduction of primal and dual infeasibility parameters per iteration are given. The convergence analysis of the method and an estimate of the number of iterations to reach an ϵ-precise solution are also provided. We provide the performance of the proposed methods on a variety of convex and nonconvex multi-objective test problems. Performance comparison between the proposed method and popular existing solvers is provided with respect to two performance measures and the corresponding relative efficiency measures. The reduction of a combined infeasibility measure, as the iterations progress, on the test problems is also shown graphically.

A Newton-type proximal gradient method for nonlinear multi-objective optimization problems

In this paper, a globally convergent Newton-type proximal gradient method is developed for composite multi-objective optimization problems where each objective function can be represented as the sum of a smooth function and a nonsmooth function. The proposed method deals with unconstrained convex multi-objective optimization problems. This method is free from any kind of priori chosen parameters or ordering information of objective functions. At every iteration of the proposed method, a subproblem is solved to find a suitable descent direction. The subproblem uses a quadratic approximation of each smooth function. An Armijo type line search is conducted to find a suitable step length. A sequence is generated using the descent direction and the step length. The global convergence of this method is justified under some mild assumptions. The proposed method is verified and compared with some existing methods using a set of test problems.

A combined variational encoding and optimization framework for design of the water–energy–food nexus

Nowadays, water, energy and food supplies are among the main problems faced by humanity. These issues are strongly linked and must be addressed by considering their interactions. The linkage is formulated as a nonlinear multiobjective optimization problem. Moreover, the solution can benefit from large databases that lead to the development of surrogate models. In this work, surrogate models are used to incorporate recorded input information. To avoid large-scale models, we use generative deep learning strategies to build low-order models. The surrogate models are coupled to the optimization problem to render optimal operation and structure of the nexus. The results show that the reduced model benefits the performance of the optimization problem. Furthermore, we implemented a clustering algorithm to reduce the number of solutions found by the multiobjective approach. The proposed methodology establishes a contribution for the integration of traditional optimization and deep learning tools to address the complex interactions of the water–energy–food nexus.

NOMA-Enabled Optimization Framework for Next-Generation Small-Cell IoV Networks Under Imperfect SIC Decoding

To meet the demands of massive connections, diverse quality of services (QoS), ultra-reliable and low latency in the future sixth-generation (6G) Internet-of-vehicle (IoV) communications, we propose non-orthogonal multiple access (NOMA)-enabled small-cell IoV network (SVNet). We aim to investigate the trade-off between system capacity and energy efficiency through a joint power optimization framework. In particular, we formulate a nonlinear multi-objective optimization problem under imperfect successive interference cancellation (SIC) detecting. Thus, the objective is to simultaneously maximize the sum-capacity and minimize the total transmit power of NOMA-enabled SVNet subject to individual IoV QoS, maximum transmit power and efficient signal detecting. To solve the nonlinear problem, we first exploit a weighted-sum method to handle the multi-objective optimization and then adopt a new iterative Sequential Quadratic Programming (SQP)-based approach to obtain the optimal solution. The proposed optimization framework is compared with Karush-Kuhn-Tucker (KKT)-based NOMA framework, average power NOMA framework, and conventional OMA framework. Monte Carlo simulation results unveil the validness of our derivations. The presented results also show the superiority of the proposed optimization framework over other benchmark frameworks in terms of system sum-capacity and total energy efficiency.