An improved proximal method with quasi-distance for nonconvex multiobjective optimization problem

Multiobjective optimization is the optimization with several conflicting objective functions. However, it is generally tough to find an optimal solution that satisfies all objectives from a mathematical frame of reference. The main objective of this article is to present an improved proximal method involving quasi-distance for constrained multiobjective optimization problems under the locally Lipschitz condition of the cost function. An instigation to study the proximal method with quasi distances is due to its widespread applications of the quasi distances in computer theory. To study the convergence result, Fritz John’s necessary optimality condition for weak Pareto solution is used. The suitable conditions to guarantee that the cluster points of the generated sequences are Pareto–Clarke critical points are provided.

Semidefinite Multiobjective Mathematical Programming Problems with Vanishing Constraints Using Convexificators

In this paper, we establish Fritz John stationary conditions for nonsmooth, nonlinear, semidefinite, multiobjective programs with vanishing constraints in terms of convexificator and introduce generalized Cottle type and generalized Guignard type constraints qualification to achieve strong S—stationary conditions from Fritz John stationary conditions. Further, we establish strong S—stationary necessary and sufficient conditions, independently from Fritz John conditions. The optimality results for multiobjective semidefinite optimization problem in this paper is related to two recent articles by Treanta in 2021. Treanta in 2021 discussed duality theorems for special class of quasiinvex multiobjective optimization problems for interval-valued components. The study in our article can also be seen and extended for the interval-valued optimization motivated by Treanta (2021). Some examples are provided to validate our established results.

Optimality conditions for robust weak sharp efficient solutions of nonsmooth uncertain multiobjective optimization problems

In this paper, we investigate an uncertain multiobjective optimization problem involving nonsmooth and nonconvex functions. The notion of a (local/global) robust weak sharp efficient solution is introduced. Then, we establish necessary and sufficient optimality conditions for local and/or the robust weak sharp efficient solutions of the considered problem. These optimality conditions are presented in terms of multipliers and Mordukhovich/limiting subdifferentials of the related functions.

Optimization of the mathematical programming and applications

The present investigation responds to the need to solve optimization problems with optimality conditions. The KKT conditions are considered for multiobjective optimization problems with interval-valued objective functions.

From a Pareto Front to Pareto Regions: A Novel Standpoint for Multiobjective Optimization

This work presents a novel approach for multiobjective optimization problems, extending the concept of a Pareto front to a new idea of the Pareto region. This new concept provides all the points beyond the Pareto front, leading to the same optimal condition with statistical assurance. This region is built using a Fisher–Snedecor test over an augmented Lagragian function, for which deductions are proposed here. This test is meant to provide an approximated depiction of the feasible operation region while using meta-heuristic optimization results to extract this information. To do so, a Constrained Sliding Particle Swarm Optimizer (CSPSO) was applied to solve a series of four benchmarks and a case study. The proposed test analyzed the CSPSO results, and the novel Pareto regions were estimated. Over this Pareto region, a clustering strategy was also developed and applied to define sub-regions that prioritize one of the objectives and an intermediary region that provides a balance between objectives. This is a valuable tool in the context of process optimization, aiming at assertive decision-making purposes. As this is a novel concept, the only way to compare it was to draw the entire regions of the benchmark functions and compare them with the methodology result. The benchmark results demonstrated that the proposed method could efficiently portray the Pareto regions. Then, the optimization of a Pressure Swing Adsorption unit was performed using the proposed approach to provide a practical application of the methodology developed here. It was possible to build the Pareto region and its respective sub-regions, where each process performance parameter is prioritized. The results demonstrated that this methodology could be helpful in processes optimization and operation. It provides more flexibility and more profound knowledge of the system under evaluation.

Adaptive Strategies Based on Differential Evolutionary Algorithm for Many-Objective Optimization

The decomposition-based algorithm, for example, multiobjective evolutionary algorithm based on decomposition (MOEA/D), has been proved effective and useful in a variety of multiobjective optimization problems (MOPs). On the basis of MOEA/D, the MOEA/D-DE replaces the simulated binary crossover (SBX) operator with differential evolution (DE) operator, which is used to enhance the diversity of the solutions more effectively. However, the amplification factor and the crossover probability are fixed in MOEA/D-DE, which would lead to a low convergence rate and be more likely to fall into local optimum. To overcome such a prematurity problem, this paper proposes three different adaptive operators in DE with crossover probability and amplification factors to adjust the parameter settings adaptively. We incorporate these three adaptive operators in MOEA/D-DE and MOEA/D-PaS to solve MOPs and many-objective optimization problems (MaOPs), respectively. This paper also designs a sensitive experiment for the changeable parameter η in the proposed adaptive operators to explore how η would affect the convergence of the proposed algorithms. These adaptive algorithms are tested on many benchmark problems, including ZDT, DTLZ, WFG, and MaF test suites. The experimental results illustrate that the three proposed adaptive algorithms have better performance on most benchmark problems.

A multipopulation evolutionary framework with Steffensen’s method for dynamic multiobjective optimization problems

Dynamic multiobjective optimization problems (DMOPs) require the evolutionary algorithms that can track the moving Pareto-optimal fronts efficiently. This paper presents a dynamic multiobjective evolutionary framework (DMOEF-MS), which adopts a novel multipopulation structure and Steffensen’s method to solve DMOPs. In DMOEF-MS, only one population deals with the original DMOP, while the others focus on single-objective problems that are generated by the weighted summation of the original DMOP. Then, Steffensen’s method is used to control the evolving process in two ways: prediction and diversity-maintenance. Particularly, the prediction strategy is devised to predict the next promising positions for the individuals that handle single-objective problems, and the diversity-maintenance strategy is used to increase population diversity before the environment changes and reinitialize the multiple populations after the environment changes. This paper gives a comprehensive comparison of DMOEF-MS with some state-of-the-art DMOEAs on 14 DMOPs and the experimental results demonstrate the effectiveness of the proposed algorithm.