A trust region gradient sampling algorithm that is applicable for solving nonsmooth unconstrained optimization problems in noisy environments is proposed, where a noisy environment is a scenario in which the function evaluation and gradient evaluation cannot be obtained precisely. The new algorithm constitutes a generalization of the basic trust region method. In the new approach, the subgradient used to formulate the trust region subproblem is computed through the gradient sampling approach. A novel reduction test is proposed to control the impact of noise on the performance of the developed algorithm. Under reasonable assumptions, the iterations of the algorithm ultimately enter one neighborhood and infinitely visit another neighborhood encompassed therein. The structures of these two neighborhoods are presented in this paper. Finally, the validity of the new algorithm is verified through numerical experiments.

In recent years, the application domain of equilibrium problems (EPs) has been significantly expanded; however, for nonsmooth EPs, the existing methods typically require solving a nonsmooth unconstrained optimization subproblem, which entails a high computational cost. In this paper, we propose a bundle-based trust region method that generates new iteration points by solving an approximate model defined as the sum of a piecewise linear function and a regularization term. Under mild assumptions, the global convergence of the proposed algorithm is established. Compared with the existing bundle methods, the proposed approach not only eliminates the need to select a proximal parameter that adversely affects the efficiency of the algorithm but is also theoretically proven to possess broader applicability. Numerical experimental results further demonstrate that the proposed algorithm exhibits excellent computational efficacy.

The parameter estimation problem is inevitable and necessary to address in numerous scientific domains. In this paper, we present a three-parameter scaled memoryless BFGS method applicable for solving the parameter estimation problem. This method enhances the efficiency and robustness of the algorithm through three adaptive parameters. Furthermore, a methodology for parameter selection is provided. Under reasonable conditions, the global convergence of the new algorithm is demonstrated. To validate the efficiency and robustness of the algorithm, a substantial number of academic problems originating from the real world are employed for numerical experiments. Moreover, two real-world problems, the Muskingum model and a machine learning problem, are also utilized to verify the effectiveness and practicability of the algorithm. The results of these experiments indicate that the newly proposed method in this paper is effective and robust and can be widely utilized.

In this paper, a new bundle trust region method for nonsmooth and nonconvex sem-infinite programming (SIP) problems is presented. This newly developed method imposes weaker assumptions on the structure of the objective function in comparison with the existing numerical methods for solving SIP. Furthermore, in contrast to most of the existing bundle methods, such as proximal bundle methods, level bundle methods, and bundle trust region methods, which all necessitate the solution of quadratic programming subproblems, the new algorithm proposed herein only requires the solution of linear programming subproblems. This substitution can reduce the computational cost. The new bundle trust region method addresses the challenge of SIP having infinitely many constraints by leveraging a relaxation approach to derive an upper bound problem of SIP. Based on this upper bound problem, a local approximation model is constructed. At each iteration, by virtue of the approximation model, the trust region technique, and the infinity norm, a linear subproblem is formulated. Under reasonable assumptions, the iteration sequence generated by the new algorithm converges to an epsilon-KKT point of the SIP. The numerical results confirm the effectiveness of the new algorithm.

In globalized and highly uncertain business environment, it is necessary to ensure the sustainable development of enterprises. Aiming at this problem, a multi-objective global robust optimization model for steel production is established to design a sustainable closed-loop supply chain network with fluctuation parameters. Firstly, for the sustainable development of enterprises, the model considers not only the recyclability of steel products, optimal production techniques and changes in retailers, but also the economic, environmental and social dimensions of sustainability. Secondly, to deal with the uncertainty of the raw material and demand parameters, a more relaxed global robust optimization method is adopted to convert the model into a tractable robust counterpart under the box and box-ellipsoid perturbation sets. Finally, for solving the model, the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon$$\end{document}-constraint method is used to transform the model into a single objective model. Meanwhile, a new sequential quadratic programming filter algorithm is constructed. Numerical results show: the robust model is effective in an uncertain environment and provides more reliable decision results than the deterministic model; the established model has certain reference value for the sustainable development of enterprises; the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1.37\%$$\end{document} reduction in economic profit, the adverse environmental impact can be reduced by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$18.9\%$$\end{document}.

In the field of supply chain management, supplier selection and order allocation (SS/OA) are critical strategic decisions that significantly impact product pricing and quality. This paper addresses a sustainable SS/OA problem under parameter uncertainty, aiming to balance four conflicting objectives: cost, $ {\rm CO}_2 $ CO2 emissions, social impact, and suppliers' comprehensive value. To tackle the challenges posed by uncertain parameters (e.g. transportation cost, $ {\rm CO}_2 $ CO2 emissions, and demand), we propose a novel globalised robust goal programming model. The model incorporates priority levels to reflect decision-makers' preferences and employs inner-outer uncertainty sets to handle multiple uncertainties effectively. The proposed model is transformed into a computationally tractable mixed-integer linear programming (MILP) formulation, ensuring practical applicability. Through a case study in the steel industry, we demonstrate the model's efficacy in achieving robust and sustainable solutions. The results highlight the model's ability to balance conflicting objectives while maintaining resilience to uncertainty, offering significant value for sustainable supply chain management.

Accepted by: M. Zied BABAIIn order to provide consumers with seamless shopping experiences, retailers integrate different sales channels to form omnichannel sales. We design a dynamic pricing model for an omnichannel retailer with ship-from-store that manages an online channel and multiple physical stores, and consider the joint decision problems of dynamic pricing, inventory control and online orders realization, so that the retailer can obtain the optimal price and inventory holding at different stages and maximize the profit. Since demand from different sales channels is uncertain in the omnichannel environment, we adopt the globalized robust optimization approach to deal with the uncertainty in the constraints due to the demand function and make the model computationally tractable through a series of transformations. Then, numerical experiments are conducted on the constructed model using the filter algorithm based on the trust region method. The experimental results demonstrate the correctness and effectiveness of the model. Compared with the deterministic model, the results of the robust optimization model are more secure and can effectively cope with uncertainty. Furthermore, we conducted sensitivity analyses and supplementary experiments. Based on the empirical findings, we put forward reasonable management suggestions for omnichannel retailers according to the experimental results. For example, with the increase in channel consumers, prices can be appropriately raised to obtain higher profits.

Novel classes of generalized convexity, specifically including G–V-semipreinvexity and semistrictly G–V-semipreinvexity, are introduced in this paper. Under these newly defined convexity, necessary and sufficient optimality conditions for semi-infinite minimax optimization problems are established. Furthermore, Mond–Weir type and Wolfe type dual models are developed for the primal problem. Rigorous proofs demonstrate that duality theorems hold consistently within the same framework. The developed theory can be applied across multiple domains. It can be employed in uncertainty and minimax optimization, particularly in aerodynamic design and financial risk modeling. Extensions of the theory can also be made to robust optimization frameworks and machine learning fields.

In globalised and highly uncertain business environment, it is necessary to design a supply chain that is not only efficient but also resilient, with continuity to operate and meet demand in the face of disruption. Aiming at this problem, a two-stage distributionally robust optimisation model based on data-driven is established to design a closed-loop supply chain network that can be flexibly executed in case of disruption. To analyse the resilience of the supply chain, the model considers the possible random disruptions of two types of facilities, and aims to deal with them through active and passive strategies such as supplier fortifying, recovery, signing with backup suppliers, and lateral transshipment. In addition, in view of the uncertainty of disruption scenarios and the limited disruption historical data available, distributionally robust optimisation method with the Wasserstein ambiguity set is used. In solving, the established robust model is transformed into a tractable model form using duality and linearisation technology, and solved by Gurobi solver. The numerical results show: Considering resilient measures effectively mitigate disruption hazards; Adding the recycling strategy can significantly reduce the production costs; Comparing with stochastic and classical robust optimisation models, the performance of the model established in this paper is highlighted.

Considering carbon emission in inventory management model is a hot topic in current academic circles. Studying inventory pricing can provide managers with better managerial insights. Economic order quantity model mainly solves the inventory management problem of demand determination. However, due to the complex market environment in real life, it is difficult to meet the conditions of constant demand. In addition, with the enhancement of low-carbon awareness of enterprises, the study of pricing model based on low-carbon policies has also attracted the attention of a large number of scholars. The robust optimization theory is applied, the inventory pricing model of demand fluctuation under carbon cap, carbon tax and carbon cap-and-trade policies is considered, the robust equivalence form of the problem is studied. The profit comparison under different policies is given. Numerical experiments show that the carbon cap-and-trade policy can better control the carbon emission of enterprises.

In this paper, we mainly study the dual problem of semi-infinite programming problem with mixed constraints. we introduce the concept of higher-order (φ,ρ)-V-invexity and construct Wolfe and Mond-Weir type dual models. Weak, strong and strict converse duality theorems are discussed under the assumptions of higher-order (φ,ρ)-V-invexity.

Semi-infinite minimax problems are widely utilized in various fields; however, there is a scarcity of algorithms that can directly tackle convex-convex and convex-concave semi-infinite minimax problems. An inexact algorithm based on the bundle method is introduced in this paper, which can be directly applied to solve both types of semi-infinite minimax problems. The novel algorithm offers the advantage of not requiring exact solutions for the inner maximization problem but only necessitates optimal solution with a certain level of precision. Additionally, the augmentation function method is employed to address nonconvergence issues encountered in traditional bundle method when dealing with convex-convex minimax problems. Global convergence of our algorithm is proven under reasonable assumptions. Numerical results from several examples demonstrate the effectiveness and practicality of our proposed approach.

In this paper, we proposed an adaptive QP-free method without a penalty function or a filter for minimax optimization. In each iteration, solved two linear systems of equations constructed from Lagrange multipliers and KKT-conditioned NCP functions. Based on the work set, the computational scale is further reduced. Instead of the filter structure, we adopt a nonmonotonic equilibrium mechanism with an adaptive parameter adjusted according to the result of each iteration. Feasibility of the algorithm are given, and the convergence under some assumptions is demonstrated. Numerical results and practical application are reported at the end.

With the development of social economy and the improvement of people living standards, the distribution of cold chain products has become increasingly prominent. In the process of distribution of cold chain products, due to its perishable nature, it is not only necessary to consider the efficiency and economy of transportation, but also to pay attention to risk aversion factors. Therefore, the stochastic programming of cold chain logistics based on risk aversion has become an urgent problem to be solved. Aiming to minimize the economic cost and transform the carbon emission level of cold chain transportation into carbon emission cost, a distributionally robust mean-conditional value-at-risk optimization model for the perishable goods distribution management problem is established, in which the partial distribution information of uncertain demand and transportation environment temperature was known. Then a computationally tractable equivalence model is obtained under the box ambiguity set. For the proposed nonlinear model, a trust region sequential quadratic programming method with filter is proposed to solve it. Finally, through a case study, the relationship between the stochastic programming model and the distributionally robust model is analysed.

Based on the definition of approximate solution (also known as epsilon -solution), we propose the idea of approximate generalized convexity of semi -infinite optimization in this research. The nonsmooth sufficient and necessary criteria of the resilient approximate solutions are then provided. Additionally, it is demonstrated that the approximative generalized convexity assump-tion holds for both the strong and weak robust dual theorems. The efficient cardinality/mean-variance portfolio is where we finally implement this idea.

In this paper, we construct a modified gradient sampling method for solving a type of nonsmooth semi-infinite optimization problem. The algorithm is grounded in the modified ideal direction, a subgradient computed in the convex hull of some sampling points. In addition, we discretize the semi-infinite optimization problem as a finite constraint problem based on the modified adaptive discretization method, ensure the convergence of the algorithm with respect to the discretization problem, and diminish the number of evaluations of the constraint function. Moreover, we establish the theoretical convergence of the algorithm under suitable assumptions. Finally, we establish numerical results by applying algorithms and demonstrating that the new algorithm has advantages over the others.

In this paper, an infeasible QP-free method without penalty function is proposed for inequality constrained optimization. We first compute a fundamental direction and then bend the search direction based on the constraint function and the Lagrange multiplier. Based on the modified nonmonotone filter technique, the acceptable criterion of trial points is relaxed and Maratos effects are avoided to a certain degree. At each iteration, only two or three systems of linear equations with the same coefficient are needed to solve to obtain the search direction. Under suitable conditions, the global convergence of the algorithm is proved without the strict complementarity conditions. In the end, some numerical results are reported.

In this paper, we propose a new adaptive method for solving nonlinear semi-infinite programming(SIP). In the presented method, the continuous infinite inequality constraints are transformed into equivalent equality constraints in integral form. Based on penalty method and trust region strategy, we propose a modified quadratic subproblem, in which an adaptive parameter is considered. The acceptable criterion of the trial point is adjustable according to the value of this adaptive parameter and the improvements that made by the current iteration. Compared with the existing methods, our method is more flexible. Under some reasonable conditions, the convergent properties of the proposed algorithm are proved. The numerical results are reported in the end.

In this paper, we define a new area-type filter algorithm based on the trust-region method. A relaxed trust-region quadratic correction subproblem is proposed to compute the trial direction at the current point. Consider the objective function and the constraint violation function at the current point as a point pair. We divide the point pairs into different partitions by the dominant region of the filter and calculate the contributions of the point pairs to the area of the filter separately. Different from the conventional filter, we define the contribution as the filter acceptance criterion for the trial point. The nonmonotone area-average form is also adopted in the filter mechanism. In this paper, monotone and nonmonotone methods are proposed and compared with the numerical values. Furthermore, the algorithm is proved to be convergent under some reasonable assumptions. The numerical experiment shows the effectiveness of the algorithm.