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In this article, we employ discontinuous Galerkin methods for the finite element approximation of the frictionless unilateral contact problem using quadratic finite elements over simplicial triangulation. We first develop a posteriori error estimates in the energy norm wherein, the reliability and efficiency of the proposed a posteriori error estimator is addressed. The suitable construction of the discrete Lagrange multiplier 흀 풉 and some intermediate operators play a key role in developing a posteriori error analysis. Further, we establish an optimal a priori error estimates under the appropriate regularity assumption on the exact solution 풖 . Numerical results presented on uniform and adaptive meshes illustrate and confirm the theoretical findings. [ABSTRACT FROM AUTHOR]

This paper investigates a new mathematical model of a static frictional contact for an ideally locking piezoelectric material with electrical and thermal conductivity conditions. A variational formulation of the model is derived, in the form of a coupled system for the displacements, the electric potential and temperature. The existence of a unique weak solution of the model is established and its proof is based on Tychonoff fixed point theorem for a multivalued operator. Moreover, a discrete scheme based on the finite element method is introduced. Under some regularity assumptions, optimal order error estimates are derived. [ABSTRACT FROM AUTHOR]

In this paper we propose a new finite element method for solving elliptic optimal control problems with pointwise state constraints, including the distributed controls and the Dirichlet or Neumann boundary controls. The main idea is to use energy space regularizations in the objective functional, while the equivalent representations of the energy space norms, i.e., the H - 1 (Ω) -norm for the distributed control, the H 1 / 2 (Γ) -norm for the Dirichlet control and the H - 1 / 2 (Γ) -norm for the Neumann control, enable us to transform the optimal control problem into an elliptic variational inequality involving only the state variable. The elliptic variational inequalities are second order for the three cases, and include additional equality constraints for Dirichlet or Neumann boundary control problems. Standard C 0 finite element methods can be used to solve the resulted variational inequalities. We provide preliminary a priori error estimates for the new method for solving distributed control problems. Extensive numerical experiments are carried out to validate the accuracy of the new method. [ABSTRACT FROM AUTHOR]

In this article, a posteriori error analysis of the elliptic obstacle problem is addressed using hybrid high-order methods. The method involve cell unknowns represented by degree-r polynomials and face unknowns represented by degree-s polynomials, where r = 0 and s is either 0 or 1. The discrete obstacle constraints are specifically applied to the cell unknowns. The analysis hinges on the construction of a suitable Lagrange multiplier, a residual functional and a linear averaging map. The reliability and the efficiency of the proposed a posteriori error estimator is discussed, and the study is concluded by numerical experiments supporting the theoretical results. [ABSTRACT FROM AUTHOR]

This paper explores the iterative approximation of solutions to equilibrium problems and proposes a simple proximal point method for addressing them. We incorporate the golden ratio technique as an extrapolation method, resulting in a two-step iterative process. This method is self-adaptive and does not require any Lipschitz-type conditions for implementation. We present and prove a weak convergence theorem along with a sublinear convergence rate for our method. The results extend some previously published findings from Hilbert spaces to 2-uniformly convex Banach spaces. To demonstrate the effectiveness of the method, we provide several numerical illustrations and compare the results with those from other methods available in the literature. [ABSTRACT FROM AUTHOR]

Variational inequality (VI) generalizes many mathematical programming problems and has a wide variety of applications. One class of VI solution methods is to reformulate a VI into a normal map nonsmooth equation system, which is then solved using nonsmooth equation-solving techniques. In this article, we propose a first practical approach for furnishing B-subdifferential elements of the normal map, which in turn enables solving the normal map equation system using variants of the B-subdifferential-based nonsmooth Newton method. It is shown that our new method requires less stringent conditions to achieve local convergence than some other established methods, and thus guarantees convergence in certain cases where other methods may fail. We compute a B-subdifferential element using the LD-derivative, which is a recently established generalized derivative concept. In our new approach, an LD-derivative is computed by solving a sequence of strictly convex quadratic programs, which can be terminated early under certain conditions. Numerical examples are provided to illustrate the convergence properties of our new method, based on a proof-of-concept implementation in Python. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two novel forward-backward splitting algorithms (FBSAs) for nonsmooth convex minimization. We provide a thorough convergence analysis, emphasizing the new algorithms and contrasting them with existing ones. Our findings are validated through a numerical example. The practical utility of these algorithms in real-world applications, including machine learning for tasks such as classification, regression, and image deblurring reveal that these algorithms consistently approach optimal solutions with fewer iterations, highlighting their efficiency in real-world scenarios. [ABSTRACT FROM AUTHOR]

In this paper, we introduce three new inertial subgradient extragradient methods for solving variational inequalities involving quasi-monotone operators in the setting of 2-uniformly convex and uniformly smooth Banach spaces. We dispense with the well-known requirement of the stepsizes of the subgradient extragradient method on the prior knowledge of the Lipschitz constant of the cost function in our proposed algorithms. Furthermore, we give many numerical examples to test the robustness of our proposed algorithms and compare their performance with several algorithms in the literature. In addition, we use our proposed algorithms in the restoration process of some degraded images and compare the quality of the restored images using our proposed algorithms and some recent algorithms in the literature. Finally, from the results of the numerical simulations, our proposed algorithms are competitive and promising. [ABSTRACT FROM AUTHOR]

The paper presents a detailed analysis of the principle of virtual work in its statics and dynamics aspects. Special attention is paid to not exact formulation and interpretation of mechanical system motion equations based upon this principle, what is presented quite often in text books. The general form of the principle of virtual work and variational formulations of the problems in statics are presented and analyzed. Additionally, the dynamic implications of the principle of virtual work and dynamics problem formulation of a system subjected to scleronomic constraints are detailed. The basis for the formulation is the dynamic implication of the principle of virtual work that specifies the relation between the constraint reactions and the body accelerations. This relation takes the form of the variation inequality. As the result, the Gauss principle which determines the body acceleration vector is presented. Furthermore, a description of impact formulation, based upon the Carnot model, caused by the constraints is provided. The theoretical development is illustrated with examples of applications of the principle of virtual work. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the existence of a finite stopping time for solutions in the form of variational inequality to fluid flows following a power law (or Ostwald–DeWaele law) in dimension N ∈ { 2 , 3 } . We first establish the existence of solutions for generalized Newtonian flows, valid for viscous stress tensors associated with the usual laws such as Ostwald–DeWaele, Carreau–Yasuda, Herschel–Bulkley and Bingham, but also for cases where the viscosity coefficient satisfies a more atypical (logarithmic) form. To demonstrate the existence of such solutions, we proceed by applying a nonlinear Galerkin method with a double regularization on the viscosity coefficient. We then establish the existence of a finite stopping time for threshold fluids or shear-thinning power-law fluids, i.e. formally such that the viscous stress tensor is represented by a p-Laplacian for the symmetrized gradient for p ∈ [ 1 , 2) . [ABSTRACT FROM AUTHOR]

We continue our study, started in [A. Češík, G. Gravina and M. Kampschulte, Inertial evolution of non-linear viscoelastic solids in the face of (self-)collision, Calc. Var. Partial Differential Equations 63 2024, 2, Paper No. 55], of (self-)collisions of viscoelastic solids in an inertial regime. We show existence of weak solutions with a corresponding contact force measure in the case of solids with only Lipschitz-regular boundaries. This necessitates a careful study of different concepts of tangent and normal cones and the role these play both in the proofs and in the formulation of the problem itself. Consistent with our previous approach, we study contact without resorting to penalization, i.e., by only relying on a strict non-interpenetration condition. Additionally, we improve the strategies of our previous proof, eliminating the need for regularization terms across all levels of approximation. [ABSTRACT FROM AUTHOR]

Inspired by the layered structure models in pavement mechanics research, in this study, a class of multilayer elastic contact systems with interlayer frictional contact conditions and deformable supporting frictional contact conditions on the foundation has been constructed. Based on the nonlinear elastic constitutive equations, the corresponding system of partial differential equations and variational inequalities are respectively introduced. Under the framework of variational inequalities, the existence and uniqueness of solutions for such models, along with the approximation properties of finite element numerical solutions, are proven and analyzed. The aforementioned conclusions provide fundamental and broadly applicable theoretical support for addressing mechanical problems in multilayer elastic contact systems within the framework of variational inequalities. Finally, the numerical experimental results based on the mixed finite element method also substantiate our theoretical conclusions. [ABSTRACT FROM AUTHOR]

Multicommodity trade enables the production, consumption, and flow of commodities across the globe from agricultural ones to precious metals. Mathematical formalisms to model, analyze, and solve such problems have advanced and are also relevant to policy and decision-making. In this paper, we construct a variational inequality trade network model in price and quantity variables, which captures possible losses on transportation routes, which can occur because of perishability of commodities, as in the case of agricultural ones, or outright thefts. The equilibrium conditions are stated and the variational inequality formulation derived. Qualitative properties of existence and uniqueness of the equilibrium supply price, commodity shipment, and demand price pattern are provided under reasonable conditions. Illustrative examples help to demonstrate the model. An algorithm that is proposed yields closed form expressions at each iteration and can also be interpreted as a discrete time adjustment process for the evolution of the economic variables. A spectrum of algorithmically solved numerical examples, with full input and output data provided, yields insights into the impacts of commodity losses, increased congestion, as well as enhanced marketing on producers as well as consumers. This new model expands the scope of spatial price equilibrium modeling under commodity losses. [ABSTRACT FROM AUTHOR]

We study zero-sum stochastic games between a singular controller and a stopper when the (state-dependent) diffusion matrix of the underlying controlled diffusion process is degenerate. In particular, we show the existence of a value for the game and determine an optimal strategy for the stopper. The degeneracy of the dynamics prevents the use of analytical methods based on solution in Sobolev spaces of suitable variational problems. Therefore we adopt a probabilistic approach based on a perturbation of the underlying diffusion modulated by a parameter γ > 0 . For each γ > 0 the approximating game is non-degenerate and admits a value u γ and an optimal strategy τ ∗ γ for the stopper. Letting γ → 0 we prove convergence of u γ to a function v, which identifies the value of the original game. We also construct explicitly optimal stopping times θ ∗ γ for u γ , related but not equal to τ ∗ γ , which converge almost surely to an optimal stopping time θ ∗ for the game with degenerate dynamics. [ABSTRACT FROM AUTHOR]

This paper studies a system of $ m $ variational inequalities with interconnected obstacles in infinite horizon associated to optimal multi-modes switching problems. Our main result is the existence and uniqueness of a continuous solution in viscosity sense, for that system. The proof of the main result strongly relies on the connection between the systems of variational inequalities and reflected backward stochastic differential equations (RBSDEs) with oblique reflection, which will be characterized through a Feynman-Kac's formula. The main feature of our system of infinite horizon RBSDEs is that its components are interconnected through both the generators and the obstacles. [ABSTRACT FROM AUTHOR]

In this paper, we present a relatively complete analysis of the regularization method of the bilateral obstacle problem with a non-homogeneous condition on the boundary. We provide both the convergence result and a priori estimates. Then by the duality theory from the convex analysis, we determine the dual problem associated with the bilateral obstacle problem. Based on this dual problem, we provide a posteriori error estimates for the continuous and the discrete problems. The a posteriori error estimates are needed for the practical implementation of the regularized problem. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a novel dual inertial Tseng's extragradient method for solving variational inequality problems in real Hilbert spaces, particularly those involving pseudomonotone and Lipschitz continuous operators. Our secondary method incorporates variable step‐size, updated at each iteration based on some previous iterates. A notable advantage of these algorithms is their ability to operate without prior knowledge of Lipschitz‐type constants and without the need for any line‐search procedure. We establish the convergence theorem of the proposed algorithms under mild assumptions. To illustrate the numerical behavior of the algorithms and to make comparisons with other methods, we conduct several numerical experiments. The results of these evaluations are showcased and thoroughly examined to exemplify the practical significance and effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, we first characterize generalized convex functions introduced by Linh and Penot Optimization (62: 943–959, 2013) by using generalized monotonicity of the generalized subdifferentials. We use vector variational inequalities in terms of generalized subdifferentials to identify efficient solutions of a multiobjective optimization problem involving quasiconvex functions. We also establish the Minty variational principle by utilizing the mean value theorem established by Kabgani and Soleimani-damaneh (Numer. Funct. Anal. Optim 38: 1548–1563, 2017) for quasiconvex functions in terms of Greenberg–Pierskalla subdifferentials. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a modified Tseng's extragradient method with a new step-length rule to solve pseudo-monotone variational inequalities in real Hilbert spaces. Under suitable conditions, the sequence generated by this algorithm strongly converges to the common elements of the solution set of pseudo-monotone variational inequality problems and the fixed point set of k-demicontractive mappings. Finally, we give some numerical experiments to illustrate the effectiveness of the proposed algorithm. The main results of this paper generalize and improve some known results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the problem of solving strongly monotone variational inequalities over the solution sets of split feasibility problems with multiple output sets in real Hilbert spaces. We present two new iterative algorithms when the involved subsets are given as the level sets of convex functions. In our algorithms, the projection to the half-space is replaced by the one to the intersection of two half-spaces. The algorithms are accelerated using the inertial technique and eliminate the need for calculating or estimating the norms of linear operators by employing self-adaptive step size criteria. We give convergence of the sequence generated by our algorithms under some suitable assumptions. Some applications to monotone variational inequalities over the solution set of the split feasibility problem are also reported. Finally, we present some numerical examples to illustrate the efficiency and implementation of our algorithms in comparison with existing algorithms in the literature. [ABSTRACT FROM AUTHOR]

In this article, we study the numerical corroboration of a variational model governed by a fourth-order elliptic operator that describes the deformation of a linearly elastic flexural shell subjected not to cross a prescribed flat obstacle. The problem under consideration is modelled by means of a set of variational inequalities posed over a non-empty, closed and convex subset of a suitable Sobolev space and is known to admit a unique solution. Qualitative and quantitative numerical experiments corroborating the validity of the model and its asymptotic similarity with Koiter's model are also presented. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'. [ABSTRACT FROM AUTHOR]

This paper introduces a schematic approximation method for a solution to a variational inequality problem in the framework of Hadamard manifold. The method is a combination of the subgradient extragradient technique and Popov extragradient method. Using this method, some convergence algorithms were proved when the cost operators are pseudomonotone and strongly pseudomonotone, respectively. In the construction of this method, the dependence on Lipschitz constants of the operators is dispensed with by the use of a monotone decreasing step size. We give an application of our main result to the constrained convex minimization problem. Finally, we report some numerical examples to illustrate the efficiency and applicability of the method. [ABSTRACT FROM AUTHOR]

This paper investigates, without any regularization procedure, the sensitivity analysis of a mechanical friction problem involving the (nonsmooth) Tresca friction law in the linear elastic model. To this aim a recent methodology based on advanced tools from convex and variational analyses is used. Precisely we express the solution to the so-called Tresca friction problem thanks to the proximal operator associated with the corresponding Tresca friction functional. Then, using an extended version of twice epi-differentiability, we prove the differentiability of the solution to the parameterized Tresca friction problem, characterizing its derivative as the solution to a boundary value problem involving tangential Signorini’s unilateral conditions. Finally our result is used to investigate and numerically solve an optimal control problem associated with the Tresca friction model. [ABSTRACT FROM AUTHOR]

We examine the last-iterate convergence rate of Bregman proximal methods--from mirror descent to mirror-prox and its optimistic variants--as a function of the local geometry induced by the prox-mapping defining the method. For generality, we focus on local solutions of constrained, nonmonotone variational inequalities, and we show that the convergence rate of a given method depends sharply on its associated Legendre exponent, a notion that measures the growth rate of the underlying Bregman function (Euclidean, entropic, or other) near a solution. In particular, we show that boundary solutions exhibit a stark separation of regimes between methods with a zero and nonzero Legendre exponent: The former converge at a linear rate, while the latter converge, in general, sublinearly. This dichotomy becomes even more pronounced in linearly constrained problems where methods with entropic regularization achieve a linear convergence rate along sharp directions, compared to convergence in a finite number of steps under Euclidean regularization. [ABSTRACT FROM AUTHOR]

We present the full approximation scheme constraint decomposition (FASCD) multilevel method for solving variational inequalities (VIs). FASCD is a joint extension of both the full approximation scheme multigrid technique for nonlinear partial differential equations, due to A. Brandt, and the constraint decomposition (CD) method introduced by X.-C. Tai for VIs arising in optimization. We extend the CD idea by exploiting the telescoping nature of certain subset decompositions arising from multilevel mesh hierarchies. When a reduced-space (active set) Newton method is applied as a smoother, with work proportional to the number of unknowns on a given mesh level, FASCD V-cycles exhibit nearly mesh-independent convergence rates. The full multigrid cycle version is an optimal solver. The example problems include differential operators which are symmetric linear, nonsymmetric linear, and nonlinear, in unilateral and bilateral VI problems. [ABSTRACT FROM AUTHOR]

We consider a class of zero-sum stopper versus singular-controller games in which the controller can only act on a subset d0 < d of the d coordinates of a controlled diffusion. Due to the constraint on the control directions these games fall outside the framework of recently studied variational methods. In this paper we develop an approximation procedure, based on L1-stability estimates for the controlled diffusion process and almost sure convergence of suitable stopping times. That allows us to prove existence of the game's value and to obtain an optimal strategy for the stopper under continuity and growth conditions on the payoff functions. This class of games is a natural extension of (single-agent) singular control problems, studied in the literature, with similar constraints on the admissible controls. [ABSTRACT FROM AUTHOR]

In this paper, we present an inertial projection and contraction algorithm (IPCA for short) for variational inequality problems (VIP for short) without Lipschitz continuity of the underlying mapping in Euclidean space. To obtain a larger step, the next iterate point of IPCA is generated by projecting the current iterate point onto a selected half-space, which is selected from finite half-spaces (one half-space can strictly separate the current iterate point from the solution set of VIP and other half-spaces are all contain the solution set of VIP) and has the largest distance to the current iterate point. Moreover, a new initial step-size strategy is used to accelerate IPCA. The global convergence of the sequence generated by IPCA is established. Numerical experiments show that IPCA can accelerate PCA both from the number of iterations point of view and the number of projections point of view. [ABSTRACT FROM AUTHOR]

This work is devoted to the mathematical and numerical study of a framework handling a system of coupled variational inequalities. We prove both the existence and the uniqueness of a weak solution to the problem. Then, we introduce a convergent iterative scheme. Using this latter, we decouple the problem into further subproblems and derive their corresponding minimization problems. As a practical application of this class of coupled abstract variational inequalities, we consider a class of problems that model an electroelastic body coming into frictional contact with a rigid electrically conductive foundation. Both electrical and mechanical contacts are of Signorini type. In other words, our model prescribes the mechanical response produced by the foundation and the outflow of the free charges across the contact zone. The last part of this paper is mainly reserved for the numerical resolution of the problem at hand. For this purpose, we have developed an alternating direction method of multipliers and convex dualities to compute and illustrate the solutions. [ABSTRACT FROM AUTHOR]

In this paper we use the penalization method to prove the existence of solution for variational inequalities of Leray–Lions type, in the setting of Musielak spaces and where the Musielak function doesn't satisfy the Δ 2 -condition. Here the right-hand side is in L 1. [ABSTRACT FROM AUTHOR]

In this paper, we discuss how to use the Dantzig–Wolfe (DW) decomposition method to solve a class of constrained variational inequality (VI) problems. These problems include multi-regional energy equilibrium models with linking constraints or nonlinear multicommodity network flow problems with asymmetric cost functions and side constraints. The decomposed VI problem has a subproblem which is a constrained optimisation problem consisting of all structural constraints. The resulting master problem is a VI problem with dummy linking constraints. The size of the master problem is much smaller than that of the original constrained VI problem. If the subproblem comprises the constraint set with a special structure, such as block-angular structure, it can be further decomposed by the DW decomposition method (nested DW). We find that by performing an iteration of the nested DW decomposition on the subproblem (truncated DW), we can obtain an equilibrium solution. The efficiency of this truncated DW may depend on the VI problems. Theoretical analysis indicated that the algorithm is guaranteed to converge under some assumptions. Illustrative examples are given. From the results of the examples, we find that moving the linking constraints of the structural constraints back from the subproblem to the master problem may worsen the computational performance. [ABSTRACT FROM AUTHOR]

The projection is often used in solving variational inequalities. When projection onto the feasible set is not easy to calculate, the projection algorithms are replaced by the relaxed projection algorithms. However, these relaxed projection algorithms are not feasible, and to ensure the convergence of these relaxed projection algorithms, in addition to assuming some basic conditions, such as the Slater condition holds for the feasible set, the mapping is pseudomonotone and Lipschitz continuous, but also need to assume some additional conditions, which require some relationship between the mapping and the feasible set. In this paper, by replacing the projection onto the feasible set with the projection onto a ball (which changes from iteration) contained in the feasible set, a new feasible moving ball projection algorithm for pseudomonotone variational inequalities is obtained. Since the projection onto a ball has an explicit expression, this algorithm is easy to implement. At the same time, all the balls are contained in the feasible set, so the iteration points generated by this algorithm are all in the feasible set, which ensures the feasibility of this algorithm. The convergence of this algorithm is proved when the Slater condition holds for the feasible set, and the mapping is pseudomonotone and Lipschitz continuous. The fundamental difference between this moving ball projection algorithm and the previous relaxed projection algorithms lie in that the previous relaxed projection algorithms are all projected onto the half-space containing the feasible set, and this moving ball projection algorithm is projected onto a ball contained in the feasible set. In particular, this algorithm does not need to assume any additional conditions between the mapping and the feasible set. Finally, some numerical examples are given to illustrate the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

This study addresses the impact of government subsidy policies and farmers' environmental safety preferences on bio-pesticides from a supply chain network perspective with consideration of economic benefits and research and development (R&D) efficiency. We formulate a supply chain network equilibrium model to characterize the competition and cooperation among the various entities in the supply chain. To solve the model, a self-adaptive projection-based prediction correction algorithm is introduced. An avermectin supply chain case is used to analyze the impacts of different subsidy strategies and farmers' preferences on the equilibrium decisions. The numerical results show that: (1) While increased R&D subsidies can improve the quality of bio-pesticides, the excessive subsidies could lead to a lower actual profitability in the supply chain; (2) farmers' environmental safety preference has a strong impetus to manufacturers' R&D investment decisions; and (3) the combinations with a high R&D subsidy ratio are more conducive to the co-development of economic and ecological benefits. [ABSTRACT FROM AUTHOR]

In this paper, an inertial subgradient extragradient algorithm is proposed to solve the pseudomonotone variational inequality problems in Banach space. This iterative scheme employs a new line-search rule. Strong convergence theorems for the proposed algorithms are established under the assumptions that the operators are non-Lipschitz continuous. Furthermore, several numerical experiments are given to show that our method has better convergence performance than the known ones in the literatures. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of bilevel programming problems (BLPP) in the framework of Euclidean space. We derive relationships among the solutions of approximate Minty-type variational inequalities (AMTVI), approximate Stampacchia-type variational inequalities (ASTVI), and local ϵ -quasi solutions of the BLPP, under generalized approximate convexity assumptions, via limiting subdifferentials. Moreover, by employing the generalized Knaster–Kuratowski–Mazurkiewicz (KKM)-Fan's lemma, we derive some existence results for the solutions of AMTVI and ASTVI. We have furnished suitable, non-trivial, illustrative examples to demonstrate the importance of the established results. To the best of our knowledge, there is no research paper available in the literature that explores relationships between the approximate variational inequalities and BLPP under the assumptions of generalized approximate convexity by employing the powerful tool of limiting subdifferentials. [ABSTRACT FROM AUTHOR]

In this paper, we study some Krasnoselskii-Mann type dynamical systems in solving fixed point problems. The first one can be regarded as a continuous version of the Krasnoselskii-Mann iterations. We prove that the solution of this dynamical system converges weakly to a fixed point of the involving mapping. Next, we focus our attention on a regularized Krasnoselskii-Mann type dynamical system. Besides proving existence and uniqueness of strong global solutions, we show that the generated trajectories converge strongly to a unique solution of a variational inequality over the fixed point set. Also, we provide a convergence rate analysis for the regularized dynamical system. [ABSTRACT FROM AUTHOR]

With the enhancement of people's awareness of green development, the supply chain management of circular agriculture with integrated planting and breeding that can effectively utilize resources becomes particularly important. In order to promote the sustainable development of agriculture, the supply chain network equilibrium methodology is adopted to construct a planting and breeding integration supply chain model with considering economic and environmental objectives under the influence of market consumers' organic preferences. The result shows that the profit and carbon sequestration of the planting and breeding integration supply chain are increasing with the increase of market size and organic preference, and the performance is better than single-production mode. Strikingly, farmers marginal profit does not always increase monotonically and retailer profits could be compromised by the fact that the growth of market price sales is less than that of production costs. This study provides effective organic manure decisions in planting and breeding integration supply chain for farmers under different market conditions, which could promote green production and sustainable agricultural development. [ABSTRACT FROM AUTHOR]

An infeasible projection algorithm (IPA) for solving nonmonotone variational inequality problems was proposed by Ye (2022). Without needing any monotonicity condition of the underlying mapping, the global convergence of the sequence generated by IPA is established whenever the underlying mapping is continuous and the solution set of the dual variational inequality is nonempty. In this paper, we present an inertial IPA for solving nonmonotone variational inequalities. The global convergence of this new algorithm is proved under the same assumptions in IPA. Numerical experiments show that the inertial technique can accelerate IPA. [ABSTRACT FROM AUTHOR]

Simplicial decomposition (SD) of variational inequalities experiences the long-tail convergence property. That is, the equilibrium solution rapidly progresses at first but then tails off, making only a tiny amount of progress per column generation iteration, which is a drawback of SDVI. In the context of Dantzig-Wolfe of LP, it is reported that the more proposals are used to initialize the algorithm, the faster the solution can be found by reducing the number of decomposition steps. Therefore, I proposed to solve multiple nonlinear column generation (mNCG) subproblems in each SD-VI iteration (SD-VI-mNCG) instead of solving only one subproblem as in SD-VI. Generating multiple column generation subproblem solutions in each SD-VI iteration enabled the corresponding convex hull to be rapidly enlarged. Consequently, the number of SD-VI iterations could be greatly reduced. A transportation network equilibrium problem was used to study the performance of the SDVI-mNCG. [ABSTRACT FROM AUTHOR]

In 2012, Censor et al. (Extensions of Korpelevich's extragradient method for the variational inequality problem in Euclidean space. Optimization 61(9):1119–1132, 2012b) proposed the two-subgradient extragradient method (TSEGM). This method does not require computing projection onto the feasible (closed and convex) set, but rather the two projections are made onto some half-space. However, the convergence of the TSEGM was puzzling and hence posted as open question. Very recently, some authors were able to provide a partial answer to the open question by establishing weak convergence result for the TSEGM though under some stringent conditions. In this paper, we propose and study an inertial two-subgradient extragradient method (ITSEGM) for solving monotone variational inequality problems (VIPs). Under more relaxed conditions than the existing results in the literature, we prove that proposed method converges strongly to a minimum-norm solution of monotone VIPs in Hilbert spaces. Unlike several of the existing methods in the literature for solving VIPs, our method does not require any linesearch technique, which could be time-consuming to implement. Rather, we employ a simple but very efficient self-adaptive step size method that generates a non-monotonic sequence of step sizes. Moreover, we present several numerical experiments to demonstrate the efficiency of our proposed method in comparison with related results in the literature. Finally, we apply our result to image restoration problem. Our result in this paper improves and generalizes several of the existing results in the literature in this direction. [ABSTRACT FROM AUTHOR]

We study two Bregman projection methods for solving variational inequality problems in real reflexive Banach spaces. Our methods have simple and elegant structures, and they require only one Bregman projection onto the feasible set and one evaluation of the cost operator at each iteration. We prove that these methods converge weakly when the cost operator is pseudomonotone on the entire space and that they converge strongly when the cost operator is strongly pseudomonotone only on the feasible set. Extensions of our proposed methods to mixed variational inequality problems are also given. Finally, we consider some examples regarding the implementation of our methods in comparisons with known methods in the literature. [ABSTRACT FROM AUTHOR]

In this paper we deal with the optimal bankruptcy problem for agents who can optimally allocate their consumption rate, the amount of capital invested in the risky asset, as well as their leisure time. In our framework, the agents are endowed by an initial debt, and they are required to repay their debt continuously. Declaring bankruptcy, the debt repayment is exempted at the cost of a wealth shrinkage. We implement the duality method to solve the problem analytically and conduct a sensitivity analysis to the bankruptcy cost and benefit parameters. Introducing the flexible leisure/working rate, and therefore the labour income, into the bankruptcy model, we investigate its effect on the optimal strategies. [ABSTRACT FROM AUTHOR]

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Hundreds of millions of people are now facing food insecurity as challenges from climate change, the aftereffects of the COVID‐19 pandemic, and strife and conflicts make availability of food at reasonable prices challenging. International trade has enabled the reallocation of agricultural products, essential for nutrition, from countries with supply markets to other points of demand and has been the subject of an increasing number of policy interventions by governments. In this paper, a multicommodity international trade network equilibrium model is constructed with the inclusion of nutritional minimal standards to support food security, accompanied by consumer subsidies, for which explicit formulae are provided. The theoretical and computational framework is based on variational inequalities. Numerical examples focusing on Ukraine and MENA (Middle Eastern and North African) countries and a staple commodity of wheat demonstrate the modeling and policy framework. [ABSTRACT FROM AUTHOR]