Search

Prior research on inventory control has been wide ranging, yet the environmental implications of an $ ({s,S}) $ (s , S) inventory policy remain uninvestigated. This paper seeks to bridge the gap by characterising a firm's voluntary environmental policy in the setup of an $ ({s,S}) $ (s , S) inventory control policy. We suggest a mixed model structure wherein, due to the presence of fixed production costs, the inventory is determined continuously by sales and impulsively with ordering decisions obeying an optimal stopping process, while the uncertain sales process is controlled by continuous-time environmental goodwill-related decisions. We show that a firm should successively use voluntary environmental efforts to stimulate its sales when there is inventory and to increase backlogging to improve its production efficiency. Given the recurrent pattern of this policy, we conclude that voluntary environmental efforts under an $ ({s,S}) $ (s , S) inventory control is not compatible with using these efforts as a means to generate ephemeral reputation insurance. [ABSTRACT FROM AUTHOR]

A novel generalization of Γ -convergence applicable to a class of equilibrium problems is studied. After the introduction of the latter, a variety of its applications is discussed. The existence of equilibria with emphasis on Nash equilibrium problems is investigated. Subsequently, our Γ -convergence notion for equilibrium problems is introduced and discussed as well as applied to a class of penalized generalized Nash equilibrium problems and quasi-variational inequalities. The work ends with a comparison of our results to previous generalizations in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the asymptotic properties of a particular class of state-dependent sweeping processes. While extensive research has been conducted on the existence and uniqueness of solutions for sweeping processes, there is a scarcity of studies addressing their behavior in the limit of large time. Additionally, we introduce novel algorithms designed for the resolution of quasi-variational inequalities. As a result, we introduce a new derivative-free algorithm to find zeros of nonsmooth Lipschitz continuous mappings with a linear convergence rate. This algorithm can be effectively used in nonsmooth and nonconvex optimization problems that do not possess necessarily second-order differentiability conditions of the data. [ABSTRACT FROM AUTHOR]

Impulse control problems pose significant challenges in decision making under uncertainty, particularly in scenarios where actions must be taken at irregular intervals or in response to abrupt changes in the environment. Ensuring both the existence and uniqueness of optimal control policies in such settings is paramount for practical implementation and theoretical understanding. In this study, we investigate the existence and uniqueness of solutions for impulse control problems utilizing asynchronous algorithms. This study presents a novel demonstration concerning the uniqueness and existence of Partially Quasi-Variational Inequalities (PQVIs). Our approach unfolds through a four-step methodology, leveraging distinctive features of a discrete iterative technique. This methodology integrates semi-implicit techniques with respect to the variable t and employs spatial approximation via finite element methods. We establish a pivotal connection between a fixed-point mapping and the discrete EQVIs system, utilizing it to formulate a discrete algorithm featuring a semi-implicit time scheme. Furthermore, we introduce a monotone iterative scheme inspired by Bensoussan’s algorithm and rigorously confirm several of its properties through mathematical proofs. This work contributes to advancing the understanding and application of iterative techniques in solving PQVIs, offering insights into their uniqueness and existence.

In this article, we study a class of differential quasi-variational–hemivariational inequalities involving time delays. We establish new systems and prove the solvability and the existence of decay solutions. Moreover, we are concerned with long-time behavior of solutions by showing the existence of a compact global attractor to m-semiflow associated with delay differential quasi-variational–hemivariational inequalities. An application to the contact problems driven by dynamic systems is discussed to demonstrate our theoretical results. [ABSTRACT FROM AUTHOR]

The objective of this manuscript is to study the convergence of three-step approximation methods for quasi-variational inequalities in the general case. First, we propose the three-step dynamical system and carry out an asymptotic analysis for the generated trajectories. The explicit time discretization of this system results into a three-step iterative method, which we prove to converge also when it is applied to strongly-monotone quasi-variational inequalities. In addition, we show that linear convergence is guaranteed under strong-monotonicity. [ABSTRACT FROM AUTHOR]

We establish Pontryagin optimality conditions for a generalized bilevel optimal control problem in which the leader is subject to a pure state inequality constraint, while the follower is governed by a non-convex quasi-variational inequality parameterized by the final state. To simplify the problem at hand, we convert it into a single-level optimal control problem by mapping the solution set of the quasi-variational inequality to a parametric optimization problem and employing the value function reformulation. Furthermore, we introduce certain regularity conditions to ensure that the derived maximum principle remains non-degenerate. Finally, we provide an illustrative example to elucidate our research findings. [ABSTRACT FROM AUTHOR]

In this paper, we provide necessary and sufficient conditions for the existence of solutions for variational inequalities and sufficient conditions for the existence of solutions for quasi-Ky Fan minimax inequality, quasi-variational inequalities, and generalized variational inequalities. As applications, we apply these results to derive existence results for strong Nash equilibria in generalized games which generalize some existence theorems for Nash equilibria in generalized games in the literature and the existence theorem for strong Nash equilibria in normal-form games by Nessah and Tian [J. Math. Anal. Appl. 414 (2014), 871-885]. We also provide sufficient conditions for the uniqueness of strong Nash equilibria in certain generalized games. [ABSTRACT FROM AUTHOR]

By applying the Brouwer fixed point theorem, we prove an existence result of solutions for strongly pseudomonotone quasi-variational inequalities which extends an analogous result in Kocvara and Outrata (Optim Methods Softw 5:275–295, 1995). The result is based on a new result concerning continuity property of solutions to a parametric variational inequality. Examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

AbstractWe study a new kind of nonzero-sum stochastic differential game with mixed impulse/switching controls, motivated by strategic competition in commodity markets. A representative upstream firm produces a commodity that is used by a representative downstream firm to produce a final consumption good. Both firms can influence the price of the commodity. By shutting down or increasing generation capacities, the upstream firm influences the price with impulses. By switching (or not) to a substitute, the downstream firm influences the drift of the commodity price process. We study the resulting impulse-regime switching game between the two firms, focusing on explicit threshold-type equilibria. Remarkably, this class of games naturally gives rise to multiple potential Nash equilibria, which we obtain thanks to a verification-based approach. We exhibit three candidate types of equilibria depending on the ultimate number of switches by the downstream firm (zero, one or an infinite number of switches). We illustrate the diversification effect provided by vertical integration in the specific case of the crude oil market. Our analysis shows that the diversification gains strongly depend on the pass-through from the crude price to the gasoline price.

In this paper, we study a class of reflected backward stochastic differential equations (RBSDEs) driven by the compensated random measure associated to a given pure jump Markov process X on a general state space U. The reflection keeps the solution above a given càdlàg process. We prove the uniqueness and existence both by a combination of the Snell envelope theory and a fixed-point argument. As a consequence of these results, the Markovian structure of X allows us to represent probabilistically the solution of some quasi-variational inequalities problem. [ABSTRACT FROM AUTHOR]

This paper is an extension and a generalization of the previous results, cf. [3, 6, 8, 11]. It is devoted to studying the finite element approximation of the non coercive system of parabolic quasi-variational inequalities related to the management of energy production problem. Specifically, we prove optimal L∞-asymptotic behavior of the system of evolutionary quasi-variational inequalities with nonlinear source terms using the finite element spatial approximation and the subsolutions method. [ABSTRACT FROM AUTHOR]

The goal of this paper is to study convergence of extragradient-like approximation methods for quasi-variational inequalities with the moving set. First, we propose the extragradient dynamical system and under strong monotonicity we show strong convergence with exponential rate of generated trajectory to the unique solution of quasi-variational inequality. Further, the explicit time discretization of this dynamical system leads to an extragradient algorithm with relaxation parameters. We prove the convergence of the generated iterative sequence to the unique solution of the quasi-variational inequality and derive the linear convergence rate under strong monotonicity. [ABSTRACT FROM AUTHOR]

In this paper, a hybrid strategy with optimal proportional reinsurance, dividends, and capital injections is discussed by a diffusion model with impulse process of discrete dividend events and discrete capital injections. Due to the presence of fixed transaction costs, the problem is formulated as a more complicated impulse stochastic control problem. By using methodologies from impulse control theory and the techniques of translation transformation, we obtain the explicit solutions of the value function and the corresponding optimal strategy. Firstly, the original problem is transformed into two types of suboptimal impulse control problems, one of which is "the case without capital injections" and the other one of which is "the case with compulsory capital injections." Then, by verification theorem and the expressions of value functions of these two suboptimal problems, we verify that the optimal solution of the original problem is either "keeping bankruptcy without injections" or "never bankruptcy with injections." It depends on the parameters of the model. [ABSTRACT FROM AUTHOR]

We consider quasi-variational inequalities (QVIs) with general non-local drivers and related systems of reflected backward stochastic differential equations (BSDEs) in a Brownian filtration. We show existence and uniqueness of viscosity solutions to the QVIs by first considering the standard (local) setting and then applying a contraction argument. In addition, the contraction argument yields existence and uniqueness of solutions to the related systems of reflected BSDEs and extends the theory of probabilistic representations of PDEs in terms of BSDEs to our specific setting. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new method for the standard finite element approximation in the L∞‐norm of the semilinear quasi‐variational inequality (QVI) arising in impulse control theory. We derive the optimal L∞‐error estimate by means of the concept of subsolutions for QVIs. [ABSTRACT FROM AUTHOR]

This paper establishes a general analytical framework for classical and impulse stochastic control problems in the presence of model uncertainty. We consider a set of dominated models, which are induced by the measures equivalent to that of a reference model. The state process under the reference model is a multidimensional Markov process with multidimensional Brownian motion, controlled by continuous and impulse control variates. We propose quasi-variational inequalities (QVI) associated with the value function of the control problem and prove a verification theorem for the solution to the QVI. With the relative entropy constraints and piecewise linear intervention penalty, we show that the QVI can be degenerated to the non-robust case and it can be solved via the solution to a free boundary problem. To illustrate the tractability of the proposed framework, we apply it to a linear-quadratic setting, which covers a broad class of problems including robust mean-reverting inventory controls. [ABSTRACT FROM AUTHOR]

We study the value and the optimal strategies for a two-player zero-sum optimal stopping game with incomplete and asymmetric information. In our Bayesian setup, the drift of the underlying diffusion process is unknown to one player (incomplete information feature), but known to the other one (asymmetric information feature). We formulate the problem and reduce it to a fully Markovian setup where the uninformed player optimises over stopping times and the informed one uses randomised stopping times in order to hide their informational advantage. Then we provide a general verification result that allows us to find the value of the game and players' optimal strategies by solving suitable quasi-variational inequalities with some nonstandard constraints. Finally, we study an example with linear payoffs, in which an explicit solution of the corresponding quasi-variational inequalities can be obtained. [ABSTRACT FROM AUTHOR]

We consider a general class of nonzero-sum N-player stochastic games with impulse controls, where players control the underlying dynamics with discrete interventions. We adopt a veri?cation approach and provide su?cient conditions for the Nash equilibria (NEs) of the game. We then consider the limiting situation when N goes to in?nity, that is, a suitable mean-?eld game (MFG) with impulse controls. We show that under appropriate technical conditions, there exists a unique NE solution to the MFG, which is an ϵ-NE approximation to the N-player game, with ϵ = O 1 N . As an example, we analyze in detail a class of two-player stochastic games which extends the classical cash management problem to the game setting. In particular, we present numerical analysis for the cases of the single player, the two-player game, and the MFG, showing the impact of competition on the player's optimal strategy, with sensitivity analysis of the model parameters. [ABSTRACT FROM AUTHOR]

The purpose of this article is to study convergence analysis of quasi-variational inequalities using a projection-type method coupled with inertial extrapolation step. First, we give strong convergence analysis of the sequence of iterates generated by our proposed method to the unique solution of quasi-variational inequality under some mild assumptions. Later, we show that the sequence converges linearly to the unique solution in a special case of choice of parameters. Another contribution in this article is that the inertial factor in our proposed method is allowed to be equal to 1 unlike other previously proposed inertial projection-type method for solving quasi-variational inequalities in the literature where inertial factor is assumed to be bounded away from 1. [ABSTRACT FROM AUTHOR]

In this paper, we deal with a class of ill-posed quasi-variational inequalities with contaminated data by employing the elliptic regularization in Banach spaces. Firstly, we get an existence result for quasi-variational inequalities and show that the sequence of bounded regularized solutions converges strongly to a solution of the original quasi-variational inequalities under mild coercivity conditions. Next, we give the boundedness of regularized solutions for quasi-variational inequalities. Finally, an example is presented to illustrate our main results. [ABSTRACT FROM AUTHOR]

This paper investigates a stochastic optimal control problem by the impulse dividend model with stochastic returns. To search for its candidate solution, we propose a series of quasi-variational inequalities (QVI), for which an analytic solution composed of a power series is provided. Moreover, some properties, such as the uniqueness of uncertain parameters and partition points, of the solution are also verified under some conditions. The procedure on how to calculate unknown parameters is also presented. Theorematic analysis verifies that the policy based on the proposed solution is just the optimal dividend policy. [ABSTRACT FROM AUTHOR]

In this article, we design a projection type iterative algorithm with two inertial steps for solving quasi-variational inequalities with Lipschitz continuous and strongly monotone mappings in real Hilbert spaces. We establish different strong convergence results through this algorithm. We give a non-trivial example to validate one of our results and to illustrate the efficiency of the proposed algorithm compared with an already existing one. We also present some numerical experiments to demonstrate the potential applicability and computing performance of our algorithm compared with some other algorithms existing in the literature. The results obtained herein are generalizations and substantial improvements of some earlier results. [ABSTRACT FROM AUTHOR]

We study the existence of projected solutions for generalized Nash equilibrium problems defined in Banach spaces, under mild convexity assumptions for each loss function and without lower semicontinuity assumptions on the constraint maps. Our approach is based on Himmelberg's fixed point theorem. As a consequence, we also obtain existence of projected solutions for quasi-equilibrium problems and quasi-variational inequalities. Finally, we show the existence of projected solutions for Single-Leader–Multi-Follower games. [ABSTRACT FROM AUTHOR]

In this paper, we present a maximum norm analysis of an overlapping Schwartz method on non matching grids for a quasi-variational inequality, where the obstacle and the second member depend on the solution. Our result improves and generalizes some previous results. [ABSTRACT FROM AUTHOR]

We propose a new projection algorithm with alternated inertial extrapolation steps, and obtain the linear convergence of the algorithm for a quasi-variational inequality problem in Hilbert spaces. We investigate the cases of under-relaxation and over-relaxation of the proposed algorithm, and give linear convergence results in each case. We also obtain some priori and apriori error bounds of our proposed algorithm. [ABSTRACT FROM AUTHOR]

We consider an electroelastic-locking material coming in frictional contact with a conductive foundation. The locking character makes the solution belongs to convex set and the contact is modelled with Coulomb friction which leads to more constraints. Existence and uniqueness of the solution is proved using results obtained in elliptic quasi-variational inequality theory and Banach fixed point theorem. The term coupling the mechanical field and the electric potential in the problem is non symmetric which makes us unable to get constrained minimization problem. To overcome this difficulty, we use a convergent fixed point iterative scheme and alternating directions method of multipliers (ADMM). The solutions are computed basing variational inequalities and both Lagrange (KKT conditions) and Fenchel convex dualities. Two numerical examples are implemented to show solutions and the efficiency of our approach. [ABSTRACT FROM AUTHOR]

The main goal of this article is to study a general augmented Lagrangian method for optimal control problems governed by mixed (quasi‐)equilibrium problems. We establish zero duality gap properties between a primal problem and its augmented Lagrangian dual problem. As a consequence, we give some existence results for optimal control problem governed by evolutionary quasi‐variational inequalities. An application to optimal control of obstacle problems described by quasi‐hemivariational inequalities is studied. The results obtained in this article are new and improves considerably many recent results in literature. [ABSTRACT FROM AUTHOR]

We introduce a new concept of approximate solutions to quasi-equilibrium problems (QEP) conceived as approximate fixed points of the implicit selection map associated with this problem. As an application, we consider quasivariational inequalities (QVI) for which we discuss several types of approximations from regularization perspectives to devise a consequential term of approximate solutions. Then we present sensitivity analysis results for parametric versions of (QEP) and (QVI) wherein a recent approximate version of the well-known Lim's Lemma is employed to obtain quantitative stability of approximate solutions to these problems. We finally emphasize that the proposed approximate quasi-equilibrium points converge, from both of the qualitative and quantitative stability aspects, to exact solutions. As a further consequence, we improve some previous stability results of solutions to traffic networks problems with respect to feasible flows under perturbation. [ABSTRACT FROM AUTHOR]

In this paper, we consider the dividend optimization problem for a financial corporation with fixed transaction costs. Besides the dividend control, the financial corporation takes proportional reinsurance to reduce risk and invests its reserve in a financial market consisting of a risk-free asset (bond) and a risky asset (stock). Because of the presence of the fixed transaction costs, the problem becomes a mixed classical-impulse stochastic control problem. We solve this problem explicitly and construct the value function together with the optimal policy. [ABSTRACT FROM AUTHOR]

Abstract The Schwarz algorithm for a class of elliptic quasi-variational inequalities with nonlinear source terms is studied in this work. The authors prove a new error estimate in uniform norm, making use of a stability property of the discrete solution. The domain is split into two sub-domains with overlapping non-matching grids. This approach combines the geometrical convergence of solutions and the uniform convergence of variational inequalities.

In this paper, strongly pseudomonotone quasi-variational inequalities are investigated. We provide sufficient conditions for existence and uniqueness of solutions of strongly pseudomonotone quasi-variational inequalities. We present some error bounds in terms of residual and regularized gap functions. The global exponential stability of equilibrium solutions of a projected dynamical system for strongly pseudomonotone quasi-variational inequalities is investigated. Strong convergence and error estimates for sequence generated by the (modified) gradient projection method with suitable choices of stepsizes are also studied. Some examples and numerical experiments are provided to support our main results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an inertial projection-type method with different updating strategies for solving quasi-variational inequalities with strongly monotone and Lipschitz continuous operators in real Hilbert spaces. Under standard assumptions, we establish different strong convergence results for the proposed algorithm. Primary numerical experiments demonstrate the potential applicability of our scheme compared with some related methods in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we extend the approach developed by the author for the standard finite element method in the L∞‐norm of the noncoercive variational inequalities (VI) (Numer Funct Anal Optim.2015;36:1107‐1121.) to impulse control quasi‐variational inequality (QVI). We derive the optimal error estimate, combining the so‐called Bensoussan‐Lions Algorithm and the concept of subsolutions for VIs. [ABSTRACT FROM AUTHOR]

We study methods for solving quasi-variational inequalities which are a notable generalization of the variational inequalities. Solving quasi-variational inequality requires that the corresponding variational inequality be solved concurrently with the calculation of a fixed point of the set-valued mapping. For this reason, the literature on quasi-variational inequalities is not very extensive in what concerns solution methods. In this paper we suggest and analyze a new continuous and iterative variants of some generalizations of the gradient-type projection method for solving quasi-variational inequalities. Using the technique of Noor, we also propose a new two-step iterative scheme. We also establish sufficient conditions for the convergence of the proposed methods and estimate the rates of convergence. [ABSTRACT FROM AUTHOR]

We study the inverse problem of identifying a variable parameter in variational and quasi-variational inequalities. We consider a quasi-variational inequality involving a multi-valued monotone map and give a new existence result. We then formulate the inverse problem as an optimization problem and prove its solvability. We also conduct a thorough study of the inverse problem of parameter identification in noncoercive variational inequalities which appear commonly in applied models. We study the inverse problem by posing optimization problems using the output least-squares and the modified output least-squares. Using regularization, penalization, and smoothing, we obtain a single-valued parameter-to-selection map and study its differentiability. We consider optimization problems using the output least-squares and the modified output least-squares for the regularized, penalized and smoothened variational inequality. We give existence results, convergence analysis, and optimality conditions. We provide applications and numerical examples to justify the proposed framework. [ABSTRACT FROM AUTHOR]

We introduce an extended mathematical programming framework for specifying equilibrium problems and their variational representations, such as generalized Nash equilibrium, multiple optimization problems with equilibrium constraints, and (quasi-) variational inequalities, and computing solutions of them from modeling languages. We define a new set of constructs with which users annotate variables and equations of the model to describe equilibrium and variational problems. Our constructs enable a natural translation of the model from one formulation to another more computationally tractable form without requiring the modeler to supply derivatives. In the context of many independent agents in the equilibrium, we facilitate expression of sophisticated structures such as shared constraints and additional constraints on their solutions. We define shared variables and demonstrate their uses for sparse reformulation, economic equilibrium problems sharing economic states, mixed pricing behavior of agents, and so on. We give some equilibrium and variational examples from the literature and describe how to formulate them using our framework. Experimental results comparing performance of various complementarity formulations for shared variables are provided. Our framework has been implemented and is available within GAMS/EMP. [ABSTRACT FROM AUTHOR]

We present a novel penalty approach for a class of quasi-variational inequalities (QVIs) involving monotone systems and interconnected obstacles.We show that for any given positive switching cost, the solutions of the penalized equations converge monotonically to those of the QVIs. We estimate the penalization errors and are able to deduce that the optimal switching regions are constructed exactly. We further demonstrate that as the switching cost tends to zero, the QVI degenerates into an equation of HJB type, which is approximated by the penalized equation at the same order (up to a log factor) as that for positive switching cost. Numerical experiments on optimal switching problems are presented to illustrate the theoretical results and to demonstrate the effectiveness of the method. [ABSTRACT FROM AUTHOR]

We study a zero-sum stochastic differential game (SDG) in which one controller plays an impulse control while their opponent plays a stochastic control. We consider an asymmetric setting in which the impulse player commits to, at the start of the game, performing less than q impulses (q can be chosen arbitrarily large). In order to obtain the uniform continuity of the value functions, previous works involving SDGs with impulses assume the cost of an impulse to be decreasing in time. Our work avoids such restrictions by requiring impulses to occur at rational times. We establish that the resulting game admits a value, and in turn, the existence and uniqueness of viscosity solutions to an associated Hamilton-Jacobi-Bellman-Isaacs quasi-variational inequality. [ABSTRACT FROM AUTHOR]

Abstract We prove the existence of weak solution for a system of quasi-variational inequalities related to a switching problem with dynamic driven by operator associated with a semi-Dirichlet form and with measure data. We give a stochastic representation of solutions in terms of solutions of a system of reflected BSDEs with oblique reflection. As a by-product, we prove the existence of an optimal strategy in the switching problem and show regularity of the payoff function. [ABSTRACT FROM AUTHOR]

In this paper, we consider the optimal impulse control problem of a currency with exchange rate dynamics whose state follows a geometric Lévy process. The objective of the Central Bank is to keep the exchange rate as close as possible to a given target. Running costs associating with the difference between the actual exchange rate and the target are continuously incurred to the system. We suppose that, when the Central Bank intervenes in the system, it requires the fixed and proportional costs. Applying the theory of stochastic impulse controls, we find the optimal exchange rate at which interventions should be performed and the optimal sizes of the interventions, so as to minimize the expected total discounted sum of the intervention costs and running costs incurred over the infinite time horizon. Furthermore, numerical comparisons for the optimal intervention strategy between the geometric Lévy process model and the geometric Brownian motion model are investigated in details. [ABSTRACT FROM AUTHOR]