Multiobjective optimization problems arise in numerous real-world applications where multiple conflicting objectives must be optimized simultaneously. A key challenge in solving such problems is efficiently computing Pareto critical points without imposing restrictive assumptions, such as convexity, on the objective functions. While conjugate gradient methods have been widely studied for single-objective optimization, their extension to multiobjective settings remains an active research area, particularly with nonmonotone line search techniques that enhance robustness. This study proposes a Polak-Ribi & egrave;re-Polyak conjugate gradient method for unconstrained multiobjective optimization, where the objective functions are continuously differentiable. The method employs an average-type nonmonotone Armijo-like line search to determine the step-size, improving flexibility and convergence behavior. Under mild assumptions (without convexity requirements), we establish the asymptotic convergence of the method, proving that every limit point of the generated iterate sequence is Pareto critical. To validate the method's effectiveness, we apply it to benchmark test problems and compare its performance with existing multiobjective conjugate gradient methods, such as the Hager-Zhang and Liu-Storey conjugate gradient approaches. Furthermore, to isolate and assess the specific contribution of the nonmonotone line search strategy, we provide an additional comparison between the proposed method and its monotone counterpart, which employs a standard Armijo-type line search. The quality of the Pareto front approximations generated by the considered methods is evaluated using the Hypervolume and Inverted Generational Distance performance indicators on a set of representative convex and nonconvex test problems. The numerical results demonstrate that the proposed nonmonotone Polak-Ribi & egrave;re-Polyak conjugate gradient method consistently outperforms the competing methods in terms of both computational efficiency and the quality of the Pareto front approximation.

This paper proposes an inexact stochastic golden ratio algorithm with operator extrapolation for solving stochastic mixed variational inequalities. The new algorithm successfully incorporates both the golden ratio strategy and the operator extrapolation method. One distinctive feature of the proposed algorithm is that it only requires the calculation of one prox-subproblem and one stochastic approximation of the expectation mapping per iteration; hence, its computational cost is significantly cheap. Compared with some existing algorithms, the proposed algorithm still works when the computational cost of solving the prox-subproblem is very high since it employs a new inexact strategy to solve prox-subproblems. Under the assumption of monotonicity, we prove that our algorithm can reach the convergence rate of 𝒪(1/K) with respect to the expected gap function, where K denotes the maximum iteration. Furthermore, by introducing a new residual function, we show that the new algorithm can enjoy the convergence rate of 𝒪(1/K) with respect to the expected residual function when the mapping is generalized monotone. Some theoretical and practice numerical experiments are performed to show the competitiveness of our algorithm.

We propose a projected quasisubgradient method for constrained, nondifferentiable, quasi-convex multiobjective minimization problems. Unlike existing approaches that rely on Lipschitz continuity, our method only requires H\"older continuity of the objective components, thereby covering a broader class of quasi-convex functions. Under these assumptions, we establish convergence of the generated sequence to a Pareto optimal solution and derive a sublinear rate of convergence that explicitly depends on the H\"older parameters, recovering the Lipschitz case as a special instance. The method is simple to implement, robust to nondifferentiability, and theoretically well-defined. Numerical experiments including application on portfolio optimization, electric vehicle charging network optimization and smart grid energy management are provided. Dolan-Moré performance profiles indicate that the proposed method outperforms. Mathematics Subject Classification (2000) 49M37; 49J52; 90C29; 90C30

In this article, we propose a Newton-based method for solving multiobjective interval optimization problems (MIOPs). We first provide a connection between weakly Pareto optimal points and Pareto critical points in the context of MIOPs. Introducing this relationship, we develop an algorithm aimed at computing a Pareto critical point. The algorithm incorporates the computation of a descent direction at a non-Pareto critical point and employs an Armijo-like line search strategy to ensure sufficient decrease. Under suitable assumptions, we prove that the sequence generated by our proposed algorithm converges to a Pareto critical point. The effectiveness and performance of the proposed method are demonstrated through a series of numerical experiments on some test problems. Finally, we apply our proposed algorithm in a portfolio optimization problem with interval uncertainty.

Uncertain multiobjective optimization problems arise in various real-world scenarios where ob-jectives are affected by uncertainty. To address this, we propose a quasi-Newton method to solve the robust counterpart of an uncertain multiobjective optimization problem under an arbitrary finite uncertainty set. The robust counterpart is formulated as a nonsmooth deterministic multiobjective optimization problem, where we construct a sub-problem using Hessian approximation to determine a descent direction. An Armijo-type inexact line search technique is introduced to compute an appropriate step length, and a modified BFGS formula ensures positive definiteness of the Hessian matrix at each iteration. By incorporating these components, we develop a quasi-Newton descent algorithm for the robust counterpart and establish its convergence under standard assumptions, proving a superlinear convergence rate. Numerical experiments validate the e effectiveness of our method by comparing it with the weighted sum method through a performance profile, demonstrating its efficiency and robustness in solving uncertain multiobjective problems.

This work presents two different types of proximal gradient methods, with line search and without line search, for solving unconstrained set-valued optimization problems under the lower set-less ordering relation induced by a solid cone that is convex, pointed, and closed. The objective mapping of the problem involves finitely many functions, with each one being the sum of a continuously differentiable function and a convex function that is proper and closed. We present an approach to characterize weakly minimal points of the problem with the help of weakly efficient points of a family of vector optimization problems. Thereafter, we establish a stationarity condition along with its connection with weakly minimal points of the problem under study. Based on the stationary condition, the concept of a descent direction at a non-stationary point is discussed. In view of the line search-based method, we formulate an Armijo-type line search condition and establish the existence of such a step-size. For the proposed methods, global convergence is established under mild assumptions. The convergence analysis of the proximal gradient method with line search provides a theoretical advancement over the convergence results previously established for the steepest descent method in set-valued optimization problems. In addition, we analyze the computational complexity of the proposed methods and show that both methods achieve a convergence rate of 𝒪(1/√(k)). Numerical results are reported to test the performance of the methods in practice.

This paper addresses a class of uncertain multiobjective optimisation problems by reformulating them as deterministic objective-wise worst-case robust counterparts. To solve the resulting robust multiobjective optimisation problem, we develop a robust nonlinear conjugate gradient method in which a descent direction is first obtained by solving an auxiliary optimisation subproblem and then updated using classical conjugate gradient formulas, including Fletcher-Reeves, Conjugate Descent, Dai-Yuan, Polak-Ribi'ere-Polyak, and Hestenes-Stiefel. The proposed update strategy is designed to preserve the sufficient descent property, while employing an Armijo-type inexact line search to determine suitable step sizes. We establish the global convergence of the proposed algorithm under standard assumptions. Numerical experiments on a collection of benchmark test problems are conducted to evaluate the effectiveness of the proposed framework. The results are compared with those obtained from the classical weighted-sum approach and existing descent-based methods using performance profiles based on iteration counts, function evaluations, delta spread, and hypervolume metrics. The computational results demonstrate that the proposed robust nonlinear conjugate gradient framework is efficient, competitive, and capable of producing high-quality approximations of robust Pareto solutions for uncertain multiobjective optimisation problems.

In this article, we propose an algorithm for the nonlinear conjugate gradient method to find a Pareto critical point of unconstrained multiobjective interval optimization problems. In this algorithm, we use the Wolfe line search procedure to find the step length. After defining the standard Wolfe conditions and the strong Wolfe conditions, we prove that there exists an interval of the step length that satisfies the standard Wolfe conditions and the strong Wolfe conditions. Further, to study the convergence analysis of our proposed algorithm, we derive the result related to the Zoutendijk condition. In the convergence analysis, first, we prove the global convergence property of our proposed algorithm for a general conjugate gradient algorithmic parameter. Further, we consider four variants of the conjugate gradient algorithmic parameter, such as Fletcher-Reeves, conjugate descent, Dai-Yuan, and modified Dai-Yuan. For each variant of the algorithmic parameter, we prove the global convergence results of our proposed algorithm. Finally, we test our algorithm on some test problems and make a performance profile.

This work introduces a nonlinear Hager-Zhang conjugate gradient method for solving set optimization problems. The objective function under consideration is defined by a finite collection of continuously differentiable functions. Notably, the proposed approach imposes restrictions neither on the existence of a finite generator of the ordering cone nor on any regularity condition at the optimal solution. As a result, the proposed method holds considerable significance for both set optimization and vector optimization problems, with the latter serving as a special case of the former. The study begins by discussing Wolfe line search conditions using Drummond-Svaiter scalarization function. Thereafter, we establish the existence of a step length satisfying the Wolfe line search conditions along a descent direction. The Hager-Zhang scalar conjugate parameter is introduced to derive the search direction for the proposed method. It is established that the direction generated by the proposed method is a descent direction. The well-definedness of the proposed method is given. Furthermore, we discuss some important results and a Zoutendijk-like condition to ensure global convergence. Subsequently, the global convergence of the proposed method is established in an asymptotic manner. Finally, numerical experiments on various test problems validate the practical performance and effectiveness of the proposed technique.

In this paper, we discuss one of the most popular methods for multiobjective optimization problems, namely quasi-Newton method. We start with a brief survey on quasi-Newton methods for multiobjective optimization problems. In this, we focus on all three components, namely Hessian approximation, quasi-Newton direction, and step length of quasi-Newton algorithms for multiobjective optimization problems. It is commonly observed that the BFGS update formula is used to approximate the Hessian matrix in the quasi-Newton methods. However, we also mention the case in which self-scaling-BFGS and Huang-BFGS update formulae are also operated. It is also highlighted that the quasi-Newton direction has been calculated by solving the subproblems involving the Hessian approximation of the objectives. We mention an algorithm using a nonmonotone Armijo line search instead of a monotone Armijo line search. After the survey, an improved nonmonotone quasi-Newton method has been proposed in this paper. In this method, we use the conventional BFGS update formula to approximate the Hessian matrix of each component of the objective function of multiobjective optimization problems. The well-definedness of the proposed algorithm is also provided. Subsequently, global convergence has been established under some mild assumptions. Finally, the proposed algorithm is performed on some test problems to demonstrate the numerical performance of the proposed algorithm.

Uncertain optimization problems play a crucial role in fields that involve decision-making under uncertainty, such as finance, supply chain management, energy systems, healthcare, transportation, engineering design, risk management, telecommunications, and agriculture. This paper develops a trust region method for uncertain multiobjective optimization problems (UMOPs). To find the solution of UMOP, an objective-wise worst-case-type robust counterpart (OWRC) is considered, which transforms the UMOP into a deterministic multiobjective optimization problem (MOP). To solve the OWRC, a trust region algorithm is developed, and the global convergence of this algorithm is also presented. After that, the trust region algorithm is compared with existing methods (e.g., steepest descent method, Newton’s method, modified quasi-Newton method, weighted sum method) for UMOP. The algorithm’s effectiveness is validated through numerical test problems using performance profiles.

The main objective of this paper is to investigate the KKT optimality condition for fuzzy optimization problems with inequality constraints. To begin with, by proving that the intersection of the cone of descent directions and the cone of feasible directions at the optimal point is an empty set, we establish the first-order optimality condition for unconstrained fuzzy optimization problems. On this basis, the Fritz-John optimality condition for fuzzy optimization problems with inequality constraints is derived through the fuzzy Gordan’s theorem. Furthermore, in order to ensure that the Lagrangian multipliers must satisfy not all zero, we strengthen the assumptions to deduce the KKT optimality condition. Meanwhile, some numerical examples are created to verify the validity of theoretical results. It is particularly worth mentioning that the optimality conditions established in this paper are such that zero belongs to a certain interval, which makes our results computationally superior than in the previous literature, where the optimality conditions are equalities. Finally, the developed optimality conditions are employed to address a binary classification problem related to support vector machines with fuzzy data.

In this paper, we develop a quadratically convergent Newton method for uncertain vector optimization problems (UVOPs) under finite uncertainty sets. We use the min-max counterpart to convert the given UVOPs into a deterministic problem. The min-max counterpart is found to be a set-valued optimization problem. Using the upper set less order relation in set optimization, we define a robust weakly efficient solution for UVOPs. To derive a Newton method for solving UVOPs, we assume that the objective function corresponding to each scenario is twice continuously differentiable and locally K-convex. This method captures all robust weakly efficient solutions of UVOPs. In the proposed method, we use the concept of partition sets that helps in formulating a class of vector optimization problems to identify descent directions and step lengths that follow an Armijo-type rule. The proposed method is found to have a local quadratic convergence rate under standard hypotheses with regularity conditions and some mild assumptions. Finally, we show the performance of the proposed algorithm on some test problems. Also, we compare the proposed method with the existing Newton method for UVOPs using the Dolan-Mor & eacute; performance profiles. The proposed method is found to outperform.

In many real-life problems, decision-making gets complicated due to dual sources of uncertainty, known as randomness and fuzziness or imprecision, which can be challenging for traditional optimization methods. Most of the existing fuzzy optimization techniques that optimize fuzzy-valued objective functions ignore fuzziness, while the probabilistic optimization techniques ignore randomness. To handle this dual source of uncertainty, Kwakernaak introduced the concept of a fuzzy random variable as ``random variables whose values are not real, but fuzzy numbers". This work aims to derive a theoretical background for the Gaussian fuzzy process and fuzzy acquisition functions, which will be used to develop a novel \emph{Bayesian fuzzy optimization} (BFO) technique that optimizes a fuzzy-values objective function. Based on fuzzy random variables, the Gaussian fuzzy process is developed, which is used as a prior belief about the fuzzy-valued objective function in the BFO. Fuzzy acquisition functions are defined to act as a guide for the search process of BFO with the help of posterior fuzzy mean and fuzzy variance. The proposed method demonstrated effective performance in both fuzzy mean-variance portfolio allocation and Indian temperature data analysis, showing robust predictive accuracy and adaptability. The proposed method can have broader applications in various fields like healthcare, material science, agriculture, etc.

In this article, we develop a trust-region technique to find critical points of unconstrained set optimization problems with the objective set-valued map defined by finitely many twice continuously differentiable functions. The technique is globally convergent and has the descent property. To ensure the descent property, a new rule of trust-region reduction ratio is introduced for the considered set-valued maps. In the derived method, to find the sequence of iteration points, we need to perform one iteration of a different vector optimization problem at each iteration. Thus, the derived technique is found to be not a straight extension of that for vector optimization. The effectiveness of the proposed algorithm is reported through performance profiles of the proposed approach with the existing methods on various test examples. A list of test problems for set optimization is also provided.

Numerous real-world applications of uncertain multiobjective optimization problems (UMOPs) can be found in science, engineering, business, and management. To handle the solution of uncertain optimization problems, robust optimization is a relatively new field. An extended version of the projected gradient method (PGM) for a deterministic smooth multiobjective optimization problem (MOP) is presented in the current study as a PGM for UMOP. An objective-wise worst-case cost (OWWC) type robust counterpart is considered, and the PGM is used to solve a UMOP by using OWWC. A projected gradient descent algorithm is created using theoretical findings. It is demonstrated that the projected gradient descent algorithm's generated sequence converges to the robust counterpart's weak Pareto optimal solution, which will be the robust weak Pareto optimal solution for UMOP. Under a few reasonable presumptions, the projected gradient descent algorithm's full convergent behavior is also justified. Finally, numerical tests are presented to validate the proposed method.

This paper investigates the bilevel split pseudomonotone variational inequality problem (BSPVIP) and the split common fixed point problem (SCFPP) involving demimetric mappings in real Hilbert spaces. We propose a novel composite Tseng-type extragradient method that incorporates an adaptive inertial correction term to effectively address the BSPVIP under the SCFPP constraints. Our approach combines an inertial technique with a self-adaptive step size strategy to enhance algorithmic efficiency. The BSPVIP consists of an upper-level variational inequality problem for a strongly monotone operator and a lower-level split variational inequality problem for two pseudomonotone operators. Under mild assumptions, we establish the strong convergence of the proposed algorithm. To demonstrate the practical applicability of the method, we apply it to a BSPVIP under split fixed point problem constraints. A numerical example is provided to illustrate the algorithm's performance and examine the influence of the involved parameters on its behavior.

. In this paper, we propose a projection-type hybrid conjugate gradient method for solving multiobjective optimization problems. It is an extension of the Hestenes-Stiefel and Dai-Yuan projection-type hybrid conjugate gradient method for vector-valued cases. We show that the proposed method generates the directions that satisfy the sufficient descent condition under the strong Wolfe line search for vector-valued functions. The global convergence of the proposed scheme is studied without any convexity assumption. To demonstrate the strength of the method and its practical applicability, we apply it to a set of commonly used test problems. The efficiency of the proposed method is evaluated through empirical analysis, which includes the computation of relative efficiency and the generation of performance profiles using the methodology developed by Dolan and More. As an application, the proposed method is applied to an SIR epidemiological model with vaccination and treatment as their controls. This study explores a multiobjective optimization framework designed to tackle two key challenges simultaneously: minimizing the spread of infection within a population and reducing the economic impact of control strategy implementation.