Variational Inequality Approach Research Articles

Responding to consumer environmental awareness: A variational inequality approach to green manufacturing optimization

In response to increasing consumer environmental awareness, our study introduces a sustainable supply chain optimization model that stimulates competition among multiple green manufacturers within an uncertain market context. Each manufacturer crafts a selection of products showcasing various degrees of greenness, purposefully designed to meet the expanding demand of environmentally conscious consumers. This strategic model emphasizes profit maximization for green manufacturers, necessitating the definition of ideal greenness levels and production quantities for an array of product types. By employing variational inequality theory, we transmute the preliminary optimization model into an interconnected variational inequality problem. We propose an improved Euler algorithm, a proficient tool for attaining equilibrium solutions. Empirical scenarios inspired by the electric vehicle industry lend credibility to our model, indicating that the surge in consumer environmental awareness compels manufacturers to diversify their offerings and accentuate their products’ greenness levels. Intriguingly, the research suggests that escalating competition can incentivize manufacturers to elevate their products’ greenness levels, strategically countering potential profit erosion. In addition, the study reveals that intensifying consumers’ green utility sensitivity doesn’t invariably augment demand for manufacturers, even with significant improvements in product greenness. This paradox highlights the necessity for manufacturers to strategically adjust their production approaches in light of varying consumer preferences. Moreover, when the level of consumer environmental awareness achieves a certain level, the fierce competition might not always be harmful to some manufacturers and the industry if they can adopt effective strategies to improve green utility sensitivity.

Ship allocation considering energy type and transportation preference: A variational inequality approach

When devising ship allocations, the holistic impact of ship energy supply types, cargo preferences, and Emission Control Areas (ECAs) is frequently disregarded. This paper introduces a variational inequality model to address the intrinsic correlation between time and economic costs in ship allocation along routes. It considers both the ship’s energy supply type and cargo transportation preferences within the transport fleet to optimize shipping costs across diverse routes. An improved diagonal algorithm is designed to precisely solve the model. The results show that factors such as energy prices, port handling capacity, and port service fees all influence the distribution of ships within the shipping network. Cargo transportation preferences further affect the allocation of ships to specific routes within the network. When transportation demand remains constant, disparate allocations are made for cost-preferred cargo and time-preferred cargo within the same type of energy-supply ship for the voyage segment. Ship allocation for time-preferred cargo tends to favor routes with fewer transit ports.

Existence of financial equilibria with real assets: a variational inequality approach

This paper presents the analysis of the existence of equilibria in a two-period general equilibrium model with incomplete financial markets and real assets by means of a variational inequality approach. A definition of fictitious equilibrium is presented and it is shown to be equivalent to the definition of equilibrium under a suitable rank condition of the return matrix. Then the fictitious equilibrium is characterized in terms of a generalized quasi-variational inequality involving Grassmannian manifolds. The chosen variational inequality is constituted by two parts: the former one relates to the individuals' maximization problems and the latter one exploits the market clearing conditions.

A novel variational inequality approach for modeling the optimal equilibrium in multi-tiered supply chain networks

We present a novel variational inequality model (VIM) to capture the complex real decision-making process in multi-tiered supply chain networks (MSCN) without strictly limiting the features of related functions. The VIM is formulated with the equilibrium conditions on links as the optimization goal and the flow conservation condition as the main constraints. We transform the VIM into a series of equivalent Non-Linear Programming Models (NLPMs) to solve. To address this challenge, we propose a novel population-based heuristic algorithm called the Multiscale Model Learning Algorithm (MMLA). The MMLA is inspired by the learning behavior of individuals in a group and can converge to an optimal equilibrium state of the MSCN. The MMLA has two key operations: zooming in on the search field and learning search in a learning stage. The excellent performers, called medalists, are imitated by other learners. With the increase in learning stages, the learning efficiency is improved, and the searching energy is concentrated in a more promising area. We employ sixteen benchmark optimization problems and two supply chain networks to demonstrate the effectiveness of the MMLA and the rationality of the equilibrium models. The results obtained by MMLA for the NLPM show that the MMLA can solve the equilibrium model effectively, and multiple optimal equilibrium states may exist for an MSCN. The flexibility of the NLPM makes it possible to consider more complicated decision-making mechanisms in the model.

A variational inequality approach for the numerical simulation of steel solidification of two-phase continuous casting process

In this paper, we study a nonlinear parabolic type model with an inequality constraint on the solution describing the phase liquid-solid heat transfer during the solidification process in continuous casting. We consider a semi-discretization with respect to time (implicit Euler method in time) of the studied thermal problem. We have a sequence of stationary constrained problems. Reformulating the problem into a variational inequality problem and thanks to the approximation (linearization) of the nonlinear boundary condition (radiation condition) and appropriate assumptions, we establish a result of existence and uniqueness of the solution of the stationary problem. We also consider a multivalued formulation of the same problem allowing the analysis of the behavior of the iterative relaxation algorithm used for the numerical solution of the discretized problem. Finally, numerical simulations are displayed.

Identifying optimal capacity expansion and differentiated capacity payments under risk aversion and market power: A financial Stackelberg game approach

We investigate how capacity payments in combination with scarcity pricing of energy can ensure resource adequacy in electricity markets, defined as the ability of supply and other resources to provide enough energy and capacity to meet demand under steady-state operating conditions. This work generalizes models for determining capacity payments by deriving second-best discriminatory payments by resource type that account not only for the “missing money” market failure that arises from energy price caps, but also for market power in the capacity market and differences in risk tolerance among resource types that can arise from failures in risk and capital markets. A bi-level equilibrium-constrained optimization model is proposed to define second-best capacity payments in a static long-run setting, considering the impacts of those payments on the mix and cost of generation investment and energy outputs. The lower-level suppliers play a Nash game to determine the generation mix under a capacity payment scheme, while the upper-level regulator considers consumer welfare and resource adequacy. We introduce an equivalent formulation via a variational inequality approach, and find conditions for the solution to exist. Discriminatory payments are found to be second-best when there are market power in the investment game, price caps in energy markets and imperfections in risk markets that lead to diverse risk attitudes.

Collaborative Pricing in a Power-Transportation Coupled Network: A Variational Inequality Approach

This paper proposes a new collaborative pricing scheme for a power-transportation coupled network based on the variational inequality (VI) approach. In the proposed scheme, nodal electricity prices and congestion tolls on roads and at charging stations are considered to coordinate the coupled networks in order to minimize the operational cost of the whole system. The prices are determined by a second-order cone-based AC power flow model and a mixed user equilibrium model, respectively. A collaborative pricing model (CPM) is then built based on the two models and the interactions between them. In order to avoid the intractability of the developed non-convex model, the CPM is transformed into the VI formulation. With proven existence and uniqueness of solutions of the VI formulation, a new prediction-correction algorithm is proposed to accelerate the solution of the CPM problem, which is guaranteed to converge to the optimal solution. The proposed models and algorithm are verified using case studies on a real-world test system. The results show that the proposed pricing scheme can reduce the operational cost and the proposed algorithm shows improved convergence and higher computation efficiency compared with the existing algorithms.

The mean field optimal switching problem: Variational inequality approach

The mean field optimal switching problem: Variational inequality approach

A variational inequality approach with SVM optimization algorithm for identifying mineral lithology

In this paper, we describe a method for SVM optimization that converts SVM into a variational inequality issue and applies it to mineral lithology classification. To begin, we use the null space approach to convert the SVM constraint conditions into a linear optimization of inequality constraints. The optimization problem is then transformed into the equivalent variation inequality. We then solve the SVM algorithm using the IPPA technique and show that it is convergent. Finally, we demonstrate the method's benefits by using the logging data set and core sets.

A Variational Inequality-Based Location-Allocation Algorithm for Locating Multiple Interactive Facilities

Multi-source Weber problem (MWP) is an important model in facility location, which has wide applications in various areas such as health service management, transportation system management, urban planning, etc. The location-allocation algorithm is a well-known method for solving MWP, which consists of a location phase and an allocation phase at each iteration. In this paper, we consider more general and practical case of MWP–the constrained multi-source location problem (CMSLP), i.e., the location of multiple facilities with considering interactive transportation between facilities, locational constraints on facilities and the gauge for measuring distances. A variational inequality approach is contributed to solving the location subproblem called the constrained multi-facility location problem (CMFLP) in location phase, which leads to an efficient projection-type method. Then a new location-allocation algorithm is developed for CMSLP. Global convergence of the projection-type method as well as local convergence of new location-allocation algorithm are proved. The efficiency of proposed methods is verified by some preliminary numerical results.

Generalized Nash equilibrium analysis of transmission and distribution coordination in coexistence of centralized and local markets

The increasing scale of distributed energy resources connected to distribution networks can not only solve the operational problems in the distribution system, but it can also supply the transmission system with flexibility and ancillary services. Thus, the design of a market framework to suit this situation is required. This paper builds a generalized Nash game model with shared constraints for the joint day-ahead market where the centralized market coexists with the local market. In this model, the transmission system operator clears the centralized market assuming that resources of distribution systems are fixed, the distribution system operator clears the local market assuming that resources of the transmission system are fixed, and the clearing of the two markets are conducted simultaneously. Subsequently, the generalized Nash game model can be reformulated to a variational inequality problem that can be solved by the Goldstein–Levitin–Polyak projection method, and more than one generalized Nash equilibrium solution is obtained by the parametrized variational inequality approach. Finally, case studies on two systems, which consist of a 30-bus transmission system and 33-bus distribution system, and a practical 343-bus transmission system and 110-bus distribution system, respectively, are used to verify the effectiveness of the proposed method.

Generalized Nash equilibrium problem over a fuzzy strategy set

In this paper, the generalized Nash equilibrium problem over a fuzzy domain is considered. An existence result of the solution is given. Then, on the basis of the variational inequality approach, an algorithm is developed to solve the fuzzy Nash equilibrium problem. In addition, a stability analysis of the solution set is provided while the membership function is perturbed by a parameter. Finally, the fuzzy Nash equilibrium formulation is applied to the study of the conflicts faced by the fashion and textile industry: production and pollution.

A stochastic variational approach to study economic equilibrium problems under uncertainty

This paper focuses on the analysis of an economic equilibrium model under time and uncertainty by using a stochastic variational inequality approach. Such an approach allows to capture, in a finite set of stages, the evolutionary aspects of the problem in response to an increasing level of information.

Dynamics of on-board rotors on finite-length journal bearings subject to multi-axial and multi-frequency excitations: numerical and experimental investigations

On-board rotating machinery subject to multi-axial excitations is encountered in a wide variety of high-technology applications. Such excitations combined with mass unbalance forces play a considerable role in their integrity because they can cause parametric instability and rotor–stator interactions. Consequently, predicting the rotordynamics of such machines is crucial to avoid triggering undesirable phenomena or at least limiting their impacts. In this context, the present paper proposes an experimental validation of a numerical model of a rotor-shaft-hydrodynamic bearings system mounted on a moving base. The model is based on a finite element approach with Timoshenko beam elements having six degrees of freedom (DOF) per node to account for the bending, torsion and axial motions. Classical 2D rectangular finite elements are also employed to obtain the pressure field acting inside the hydrodynamic bearing. The finite element formulation is based on a variational inequality approach leading to the Reynolds boundary conditions. The experimental validation of the model is carried out with a rotor test rig, designed, built, instrumented and mounted on a 6-DOF hydraulic shaker. The rotor’s dynamic behavior in bending, torsion and axial motions is assessed with base motions consisting of mono- and multi-axial translations and rotations with harmonic, random and chirp sine profiles. The comparison of the predicted and measured results achieved in terms of shaft orbits, full spectrums, transient history responses and power spectral densities is very satisfactory, permitting the experimental validation of the model proposed.

A game theory application in the copper market

Mongolia has a small open economy that is growing due to its status as one of the major copper exporters in the Asian copper market. We used game theory to analyze the Chinese copper market and determine the competitive strategies used by Mongolian exporters. We show that a variational inequality approach is one option to prove that game theory is applicable in the analysis of the Chinese copper market, as it is spatial and concentrated. To solve the profit maximization problem with a concave objective function for each player and linear strategy set, it can be reduced to variational inequalities. Numerical calculations allow us to predict the responses of certain parameters against external factors or so-called non-economic shocks, such as changes in Chinese import policies, strikes in Chile, etc. In the future, if Mongolia builds a new copper smelter, it will be a competitor within a more concentrated market than that of the current copper concentrate trade. Thus, the importance of developing such a model is increasing.

A Variational Inequality Approach for One Dimensional Stefan Problem

A Variational Inequality Approach for One Dimensional Stefan Problem

Restricted Participation on Financial Markets: A General Equilibrium Approach Using Variational Inequality Methods

We deal with the analysis of a general equilibrium model with restricted participation in financial markets and with numeraire assets. We consider an exchange economy and assume that there are two periods of time and S possible states of nature in the second period. Markets may in principle be complete, but each household has her own specific restricted way to access to it. In particular, we assume that households are allowed to choose portfolios in a closed and convex set containing zero. Our main goal in this work is to provide a proof of existence of equilibria under relatively general assumptions, by assuming that the households may have non-complete or non-transitive preferences, and by using a variational inequality approach. More precisely, we introduce a sequence of generalized quasi-variational inequalities and we show that an associated sequence of solutions converges to an equilibrium.

A parametrized variational inequality approach to track the solution set of a generalized nash equilibrium problem

In this paper, we present a numerical method to describe the solution set of a generalized Nash equilibrium problem (GNEP). Previous approaches show how to reformulate the GNEP as a family of parametric variational inequalities in the special case where the game has shared constraints. We extend this result to generalized Nash problems by means of an umbrella shared constraint approximation of the game. We show the validity of our approach on numerical examples from the literature, and we provide new results that pinpoint the handling of the algorithm’s parameters for its implementation. Last but not least, we extend, solve, and discuss an applied example of a generalized Nash equilibrium problem of environmental accords between countries.

Verification theorems for models of optimal consumption and investment with annuitization

One can find it challenging to deal with verification theorems for optimal consumption and investment problems with annuitization. I pose a tractable framework to prove verification theorems for the problems. I revisit an annuitization model of Park (2015) and prove the verification theorem for the model. A key idea of the proof is the application of the variational inequality approach of Bensoussan and Lions (1982) to the annuitization problem solved by the suggested value function. Further, I obtain analytic comparative statics for the optimal consumption and investment strategies with annuitization.

Carbon emission regulation and operations in the supply chain supernetwork under stringent carbon policy

We consider a supply chain supernetwork constrained by stringent carbon policy of mandatory cap, the purpose of which is to examine how stringent carbon regulations and operational decision modes jointly influnce the profitability and emission control of the system. We show that two key aspects of caps sharing schemes and decision modes eventually constitute four basic configurations of operations environment. Drawing on these configurations, this paper formulates novel carbon-capped mathematical gaming models to portray the supply chain supernetwork operations, which is then transformed into equivalent ones based on variational inequality approach and solved by devising augmented algorithms. The algorithms empower us to conduct typical computational experiments to compare supply chain performances resulting from diverse scenarios. The results reveal that only when the mandatory cap is set in intermediate level rather than excessively mild or tight can it be effective to balance system profitability and emission control well. Furthermore, regardless of decision modes, caps sharing scheme can always incur more system profit than the nonsharing scheme, while this rule does not apply to total carbon emission. Specifically, it is illustrated that properly changing the combination of decision mode and cap scheme can achieve Pareto improvement on low-carbon supernetwork performances (i.e. increasing profit and reducing emission), which is however dependent on the emission intensities contrast of all emitting firms. This study provides the advanced analytical framework and new insights for better understanding the joint impact of stringent (mandatory) cap policies and decision modes on supply chain performances. We shed light on that properly adjusting the cap tightness and the combination of decision mode and cap-sharing scheme may achieve Paretoly improvement for individual profits and particularly attain win-win result for system profit and emission abatement.