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

This paper presents an application of non-dominated sorting genetic algorithm-II (NSGA-II) technique for solving shortest path routing problems in computer networks. The problem is formulated as a non-linear constrained multi-objective optimisation problem. NSGA-II is applied to handle shortest path routing problem as a true multi-objective optimisation problem (MOOP) with competing and non-commensurable objectives. A priority-based encoding scheme is employed for population initialisation. Priorities are assigned to all the edges and NSGA-II is implemented to find the optimal solution. It is noted that this approach can find a diverse set of solutions and is converging near the true Pareto-optimal set. Results for a sample test network have been presented to demonstrate the capabilities of the NSGA-II algorithm to generate well-distributed Pareto-optimal solutions of shortest path routing problem in one single run. The results obtained by NSGA-II are compared with single objective weighting factor method for which genetic algorithm (GA) is applied.

The non-dominated sorting genetic algorithm II (NSGA-II) is the most intensively used multi-objective evolutionary algorithm (MOEA) in real-world applications. However, in contrast to several simple MOEAs analyzed also via mathematical means, no such study exists for the NSGA-II so far. In this work, we show that mathematical runtime analyses are feasible also for the NSGA-II. As particular results, we prove that with a population size larger than the Pareto front size by a constant factor, the NSGA-II with two classic mutation operators and three different ways to select the parents satisfies the same asymptotic runtime guarantees as the SEMO and GSEMO algorithms on the basic OneMinMax and LOTZ benchmark functions. However, if the population size is only equal to the size of the Pareto front, then the NSGA-II cannot efficiently compute the full Pareto front (for an exponential number of iterations, the population will always miss a constant fraction of the Pareto front). Our experiments confirm the above findings.

A Multiobjective Optimization Problem (MOP) requires the optimization of several objective functions simultaneously, usually in conflict with each other. One of the most efficient algorithms for solving MOPs is MOEA/D (Multiobjective Evolutionary Algorithm Based on Decomposition), which decomposes a MOP into single-objective optimization subproblems and solves them using information from neighboring subproblems. MOEA/D variants with other evolutionary operators have emerged over the years, improving their efficiency in various MOPs. Recently, the IWOA (Improved Whale Optimization Algorithm) was proposed, an optimization algorithm bioinspired by the whale hunting method hybridized with Differential Evolution, which presented excellent results in single-objective optimization problems. This work proposes the MOEA/D-IWOA algorithm, which associates characteristics of the evolutionary operators of the IWOA to MOEA/D. Computational experiments were accomplished to analyze the performance of the MOEA/D-IWOA in benchmark MOPs suites. The results were compared with those obtained by the MOEA/D, Non-dominated Sorting Genetic Algorithm II (NSGA-II), Third Evolution Step of Generalized Differential Evolution (GDE3), Improving the Strength Pareto Evolutionary Algorithm (SPEA2), and Indicator-Based Evolutionary Algorithm (IBEA) algorithms in the Hypervolume and Inverted Generational Distance Plus (IGD+) indicators. The MOEA/D-IWOA proved to be competitive, with a good performance profile, in addition to presenting the best results in some POMs.

Multi-objective optimization of the shell in autonomous intelligent argo profiling float

The number of cloud users and their aspiration for completion of tasks at less energy consumption and operating cost are rapidly increasing. Hence, the authors of this paper aim to minimize the makespan and operating cost by optimally scheduling the tasks and allocating the resources of cloud service. The optimum task scheduling and resource allocation are obtained for each objective function using the simple genetic algorithm. Further, the non-dominated solutions of the dual objectives are obtained using the non-dominated sorting genetic algorithm-II, the most successful multi-objective optimization technique. A complex cloud service problem consisting of ten tasks, fifteen subtasks and fifteen heterogeneous resources is considered to investigate the proposed method. The numerical results obtained in the single objective and multi objective optimization problems show that the makespan and the operating cost are significantly reduced using the simple genetic algorithm and a wide range of non-dominated solutions are obtained in the multi-objective optimization problem, by which the cloud users shall be benefitted to choose the most appropriate solution based on the other design constraints they have.

Multi-objective optimization of clustered wind farms based on potential game approach

Mathematical runtime analysis for the non-dominated sorting genetic algorithm II (NSGA-II)

Distribution system (DS) service restoration is a very complex multi-objective and multi-constraint optimization problem, which requires high-quality Pareto fronts for helping the DS operators' work. This paper proposes a multi-objective evolutionary algorithm (MOEA) that combines the advantages of MOEAs in Tables with the properties of the Non-dominated Sorting Genetic Algorithm-II (NSGA-II) to recover Pareto fronts of the service problem. Its main differentials are its ability of recovering high-quality Pareto fronts, even in large-scale DSs, and of prioritizing switching operations in Remotely Controlled Switches (RCSs), independently of amount of RCSs. In opposite of Manually Controlled Switches (MCSs), RCS can be faster and cheapest operated from operating center. Several tests were carried out to validate and to compare the proposed MOEA to three MOEAs published in literature. Four metrics were used for assessing the quality of Pareto fronts generated and a Welch's t hypothesis test enabled the comparison of MOEAs' performance. Test results indicate the proposed MOEA outperforms those three MOEAs published in literature.

The stiffeners and piezoelectric actuators are used in many aerospace structures as an auxiliary layer with laminated composites. A question then arises as to whether we can estimate the percentage of these materials in an efficient design. Due to the high computational cost, it is not easy to answer through numerical solutions. The objective of this paper is concurrently to maximize the buckling load and minimize the weight of the cylindrical shell. To reach this aim, a multi-objective optimization problem is developed based on the closed-form solutions of thermal/mechanical buckling and weight of the piezolaminated shell with eccentric/concentric stiffener. The Non-dominated Sorting Genetic Algorithm II (NSGA-II) is used for solving multi-criteria optimization. Shannon’s entropy-based TOPSIS decision-making algorithm is employed to select the best design from Pareto fronts. To illustrate the potential of lightweight optimal design in structural stability, the obtained optimal and conventional designs are compared.

Multi-objective optimization problems (MOPs), which involve the simultaneous optimization of multiple conflicting objectives, arise in various fields. A persistent challenge in this domain is achieving high computational efficiency while preserving a diverse and high-quality approximation of the Pareto front. Non-dominated sorting, as implemented in algorithms such as the Non-dominated Sorting Genetic Algorithm II (NSGA-II), is a fundamental operation in evolutionary multi-objective optimization. However, the standard Fast Non-dominated Sorting (FNDS) suffers from a computational bottleneck of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O\left(M N^{2}\right)$</tex>, where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$M$</tex> is the number of objectives and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N$</tex> is the population size, making it impractical for large-scale problems. To address this limitation, we introduce the Specialty Student Mechanism (SSM), a novel framework we propose for accelerating non-dominated sorting. Building on SSM, we design a heuristic approach, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$H$</tex>-SSM, which first identifies extreme individuals (“specialty students”) and then employs a pre-sorting strategy to rapidly eliminate dominated solutions, reducing the time complexity to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O\left(M^{2} N \log N\right)$</tex> for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$M \ll N$</tex>. This mechanism significantly improves scalability while preserving sorting accuracy. Extensive experiments on benchmark large-scale MOPs show that H-SSM consistently achieves substantial runtime reductions compared to conventional FNDS-based methods, without compromising solution quality or Pareto front diversity. The proposed method offers a practical and scalable solution for large-scale multi-objective optimization, striking a balance between computational efficiency and optimization performance.

The multiobjective simulation optimization (MOSO) problem is a nonlinear multiobjective optimization problem in which multiple simultaneous and conflicting objective functions can only be observed with stochastic error. We provide an introduction to MOSO at the advanced tutorial level, aimed at researchers and practitioners who wish to begin working in this emerging area. Our focus is exclusively on MOSO methods that characterize the entire efficient or Pareto-optimal set as the solution to the MOSO problem; later, this set may be used as input to the broader multicriteria decision-making process. Our introduction to MOSO includes an overview of existing theory, methods, and provably convergent algorithms that explicitly control sampling error for (1) MOSO on finite sets, called multiobjective ranking and selection; (2) MOSO with integer-ordered decision variables; and (3) MOSO with continuous decision variables. In the context of integer-ordered and continuous decision variables, we focus on methods that provably converge to a local efficient set under the natural ordering. We also discuss key open questions that remain in this emerging field.

This study compared the performances of three state-of-theart evolutionary multi-objective optimization algorithms: the Non-Dominated Sorted Genetic Algorithm II (NSGA-II), the Epsilon-Dominance Non-Dominated Sorted Genetic Algorithm II (e -NSGA-II) and the Epsilon-Dominance Multi-Objective Evolutionary Algorithm ( e MOEA), on a constrained biobjective crop-mix planning problem (BCP). The BCP test case objectives were to: (i) maximize crop production and (ii) minimize planting area. The BCP problem was enumerated to provide the true Pareto-optimal solution set to facilitate rigorous testing of the evolutionary multi-objective optimization algorithms. The performances of the three algorithms were assessed and compared using three performance metrics (additive epsilon indicator, inverted generational distance and spacing). For the comparative study, a graphical comparison with respect to true Pareto front of optimal crop planning problem was provided. Results of the analyses indicated that the e MOEA greatly outperforms the NSGAII and the e -NSGA-II.

Reinforcement learning-based multi-objective differential evolution algorithm for feature selection

Runtime analysis of a multi-objective evolutionary algorithm for obtaining finite approximations of Pareto fronts

This paper provides a novel framework for solving multiobjective discrete optimization problems with an arbitrary number of objectives. Our framework represents these problems as network models, in that enumerating the Pareto frontier amounts to solving a multicriteria shortest-path problem in an auxiliary network. We design techniques for exploiting network models in order to accelerate the identification of the Pareto frontier, most notably a number of operations to simplify the network by removing nodes and arcs while preserving the set of nondominated solutions. We show that the proposed framework yields orders-of-magnitude performance improvements over existing state-of-the-art algorithms on five problem classes containing both linear and nonlinear objective functions. Summary of Contribution: Multiobjective optimization has a long history of research with applications in several domains. Our paper provides an alternative modeling and solution approach for multiobjective discrete optimization problems by leveraging graphical structures. Specifically, we encode the decision space of a problem as a layered network and propose graph reduction operators to preserve only solutions whose image are part of the Pareto frontier. The nondominated solutions can then be extracted through shortest-path algorithms on such a network. Numerical results comparing our method with state-of-the-art approaches on several problem classes, including the knapsack, set covering, and the traveling salesperson problem (TSP), suggest orders-of-magnitude runtime speed-ups for exactly enumerating the Pareto frontier, especially when the number of objective functions grows.