Variational Inequality Formulation Research Articles

A variational inequality formulation for stochastic user equilibrium with a bounded choice set

This study develops a variational inequality (VI) formulation for the stochastic user equilibrium with a bounded choice set, known as the bounded choice model (BCM). A novel mapping is devised to facilitate the development of the VI formulation. The VI formulation offers a new avenue for analysing and solving the BCM. It is shown that all paths in the considered choice set should have an equal mapping at the optimal solution, which resembles the equilibrated travel cost in the user equilibrium (UE). The new formulation has two algorithmic benefits. First, an algorithm that solves a VI problem (i.e., the extragradient method) can outperform the method of successive averages (MSA) in large networks under a UE-type convergent measurement. Second, it is possible to avoid the column reduction procedure when finding the choice set that satisfies the BCM requirement because a projection-type algorithm that solves the VI formulation can assign zero flow to a path. This is more efficient than the MSA, which uses diminishing step sizes to update solutions over iterations, leading to a sublinear rate of convergence and small residual flows on paths that should carry zero flows. Numerical experiments are conducted to demonstrate the properties of the model and compare the performance of different algorithms using the Sioux Falls, Eastern Massachusetts, and Anaheim networks.

An efficient hyperpath-based algorithm for the capacitated transit equilibrium assignment problem

This paper studied the capacitated transit equilibrium assignment problem (CTEAP), which particularly accounts for the capacity effect that impacts passengers' route choices in a transit network. Specifically, we impose strict capacity constraints on line segments to ensure that the passenger flow does not exceed the line capacity. By introducing associated Lagrange multipliers, we derived the generalised hyperpath cost function and established an equivalent variational inequality formulation. A solution framework is developed to solve CTEAP based on the Method of Multipliers. To handle destroyed Cartesian product structures, we transformed the master problem into a sequence of uncapacitated subproblems, which can be tackled by two modified Newton-type hyperpath-based algorithms. Numerical analyses were performed for a small Gentile network and a large Shenzhen transit network to evaluate the impacts of the capacity constraint on passenger flows and computation costs. Our results demonstrate the superiority of the proposed CTEAP model to prevent over-loaded flows, and show the promise of applying the hyperpath-based algorithm in the implementation of real-world networks.

Range-constrained traffic assignment for electric vehicles under heterogeneous range anxiety

This paper studied the range-constrained traffic assignment problem (RTAP), where heterogeneous range anxiety is considered among the driving population by electric vehicles (EVs). In order not to get stranded en-route, each EV driver is assumed to have his/her own driving range limit for being able to complete the trip. As a result, two types of multi-class RTAP can be defined through discrete or continuous distributed range anxiety. Given path-based side constraint structures, we proposed two variational inequality (VI) formulations for modeling discrete and continuous RTAPs, where the former is finite-dimensional according to a discrete number of user classes and the latter is infinite-dimensional accounting for an infinite number of user classes. We reformulate the continuous RTAP into finite-dimensional by merging adjacent EV drivers into one group. A unified path-based solution framework is developed to solve the two RTAPs, built upon the gradient projection algorithm. We design column generation and dropping schemes to adaptively maintain compact path sets and an inner equilibration strategy to accelerate convergence. Finally, a small network is used to examine the correctness and effectiveness of proposed models, and a large Winnipeg network is adopted to evaluate the impacts of stochastic driving range on network flows and computation costs. Numerical results provide compelling evidence of the outstanding superiority of the proposed algorithm, and show that EV drivers with heightened sensitivity towards range anxiety may contribute to the emergence of critical links experiencing severe congestion.

The speed‐in‐action problem for the nonlinear hyperbolic equation with a nonlocal condition

AbstractIn the paper the speed‐in‐action problem is investigated for the second‐order nonlinear hyperbolic equation. The main feature of the paper is that for the first time the problem is considered with a nonlocal condition in differ from many works devoted the similar problems. A theorem on the existence of the optimal control is proved. Freshet differentiability of the corresponding functional is shown. The necessary optimality conditions are derived in the form of variational inequalities.

An efficient and provable sequential quadratic programming method for American and swing option pricing

A sequential quadratic programming numerical method is proposed for American option pricing based on the variational inequality formulation. The variational inequality is discretized using the θ-method in time and the finite element method in space. The resulting system of algebraic inequalities at each time step is solved through a sequence of box-constrained quadratic programming problems, with the latter being solved by a globally and quadratically convergent, large-scale suitable reflective Newton method. It is proved that the sequence of quadratic programming problems converges with a constant rate under a mild condition on the time step size. The method is general in solving the variational inequalities for the option pricing with many styles of optimal stopping and complex underlying asset models. In particular, swing options and stochastic volatility and jump diffusion models are studied. Numerical examples are presented to confirm the effectiveness of the method.

Pontryagin's principle and variational inequality for optimal control problems governed by viscous Camassa–Holm equations

In this paper we prove the existence of optimal solutions as well as first-order necessary optimality conditions for a distributed optimal control problem governed by the three-dimensional viscous Camassa–Holm equations in bounded domains with the cost function of a quite general form. Particularly, in the box constraints case, the connection between the first-order necessary optimality conditions in the variational inequality form and the Pontryagin minimum principle is also established.

Mathematical program with equilibrium constraints approach with genetic algorithm for joint optimization of charging station location and discrete transport network design

ABSTRACT This paper focuses on the joint optimization of the charging station location problem (CSLP) and discrete network design problem (DNDP) in a transportation network. We present a variational inequality (VI) formulation to describe the user equilibrium (UE) state of gasoline vehicles (GVs) and electric vehicles (EVs). Based on the mixed-UE model, a mathematical program with equilibrium constraints (MPEC) model is formulated for integrating the decisions of deploying EV charging stations (EVCSs) and adding new links to minimize the total travel cost (TTC) of all vehicles. A modified genetic algorithm is developed to tackle the MPEC model with an adaptive path generation procedure to address the mixed-UE model. Finally, we conduct numerical experiments to identify the efficacy of the proposed models and algorithms. Specifically, we propose a two-step optimization model and explore a performance comparison between the joint and two-step optimization approaches, while the joint optimization exhibits superiority in minimizing the TTC.

Multi-Layer 5G Network Slicing with UAVs: An Optimization Model

In this paper, we present a network-based optimization model describing a closed-loop supply chain for the provision of 5G network slices on demand to users and devices on the ground. The three-tier supply chain network consists of a fleet of pre-existing UAVs, to which others can be added, managed by a fleet of UAV controllers, whose purpose is to perform services requested by users and devices on the ground. The aim of this paper is to provide a constrained optimization problem through which the providers’ profits are maximized, determining the global optimal distributions of request flows, the global optimal distributions of executed services and the optimal reliability level of pre-existing UAVs of the fleet. We also derive the associated Variational inequality formulation of the problem and, finally, a numerical simulation is performed to validate the effectiveness of the model.

Analysis of a new kind of elastic-rigid bilateral obstacle problem

This paper is devoted to a study of a new kind of bilateral obstacle problem. The obstacles are made of rigid bodies covered by soft layers which are deformable and allow penetration. A model of an elastic rigid bilateral obstacle problem is established, and three equivalent descriptions are derived: the energy form, the variational inequality form, and the differential equation form. We prove the solution existence and uniqueness of this model and provide an error estimate of the numerical solutions. The optimal order error estimate for the linear finite element method is derived under the proper regularity assumption. A penalty method is introduced to solve the finite element approximation problem, and the convergence results are obtained when the penalty parameter tends to infinity. Several numerical examples are reported, and the results are in good agreement with the theoretical analysis.

On implementing the implicit discrete‐time super‐twisting observer on mechanical systems

AbstractIn this paper, an extension of an algorithm for the implicit discretization of a super‐twisting sliding mode observer is presented. Implicit and explicit discretization algorithms for homogeneous differentiators, where no physical model information is considered, are investigated in literature. This article studies the behavior when considering models of a rather general class of nonlinear systems. The discrete equations of the super‐twisting observer are reformulated as generalized equation and an algorithm for the step‐by‐step solution is given. The uniqueness of the derived algorithm is investigated with an equivalent variational inequality formulation which is derived for a class of nonlinear systems. Furthermore, a semi‐implicit predictor‐corrector discretization is presented which is an approximation method for the presented algorithms and allows an explicit implementation in practical applications. Accuracy properties under noise and sampling are given. The algorithm is applied on two mechanical example systems taken from practice.

Hamilton–Jacobi equations for controlled gradient flows: The comparison principle

Motivated by recent developments in the fields of large deviations for interacting particle systems and mean field control, we establish a comparison principle for the Hamilton–Jacobi equation corresponding to linearly controlled gradient flows of an energy function E defined on a metric space (E,d). Our analysis is based on a systematic use of the regularizing properties of gradient flows in evolutional variational inequality (EVI) formulation, that we exploit for constructing rigorous upper and lower bounds for the formal Hamiltonian at hand and, in combination with the use of the Tataru's distance, for establishing the key estimates needed to bound the difference of the Hamiltonians in the proof of the comparison principle. Our abstract results apply to a large class of examples only partially covered by the existing theory, including gradient flows on Hilbert spaces and the Wasserstein space equipped with a displacement convex energy functional E satisfying McCann's condition.

METHODS OF MATHEMATICAL SIMULATION AND MACHINE IDENTIFICATION OF ANOMALOUS DIFFUSION PROCESSES

For the class of anomalous diffusion processes, the mathematical models of which are formalized in the form of variational inequalities in partial derivatives, a method of mathematical modeling based on the optimization procedure is proposed. The method is considered in relation to the generalized mathematical model of the studied class of anomalous diffusion processes. Which made it possible to ensure the principle of unification and typification in the application of this method, as well as the correctness of using the generalized mathematical model in applied problems of mathematical modeling of known industrial and practically important natural cases of anomalous diffusion processes. At the same time, the task of implementing mathematical models of anomalous diffusion processes based on the proposed method is reduced to finding the maximum of the Hamiltonian function defined in the state space of the processes under consideration. A method of parametric identification of mathematical models of anomalous diffusion processes in the formulation of the problem of optimal control is also proposed. The method is reduced to the use of the optimization procedure of the gradient projection method. The possibility of solving the problem of parametric identification in cases of both linear and non-linear mathematical models of anomalous diffusion processes is proved. Moreover, the nonlinear formulation of the parametric identification problem does not lead to computational implementation complications, since the solution is based only on finding the gradient projection of the state function of the anomalous diffusion process. The proposed methods are presented in strict compliance with the provisions of functional analysis, which ensures their correctness and adequacy in solving a wide range of applied problems.

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.

A variational approach for supply chain networks with environmental interests

Nowadays, the supply chain networks, consisting of different tiers of decision-makers, provide an effective framework for the production, the distribution, and the consumption of goods. In this paper we propose a supply chain network optimization model where manufacturers, retailers and consumers in the demand markets have a degree of interest in environmental sustainability. The manufacturers can improve their energy level (assumed as variables), aim to minimize their environmental emissions (for production and transport) and can also establish the amount of quantity of the production waste to dispose in a eco-sustainable way. The retailers, who are also profit-maximizers, aim to minimize their environmental emissions (which depend on the chosen shipping methods). The consumers at demand markets make their own choices according to the prices and to their degree of aversion to the environmental emissions. We describe the behavior of each decision-maker and we present the mathematical model for each of them, deriving the variational inequality problems. Furthermore, we derive a unique variational inequality formulation for the entire network for whose solution an existence and uniqueness result is obtained. Finally, we illustrate some numerical simulations that highlight how the use of UAVs and the presence of waste sorting centers in the supply chain reduce environmental emissions and related costs.

Alternating direction method for the fixed point problem of set-valued mappings with second-order cone double constraints

<abstract><p>This paper focuses on solving a class of equilibrium problems, namely, the fixed point problem of set-valued mappings with second-order cone double constraints. Under certain conditions, the variational inequality form of the fixed point problem of set-valued mappings with second-order cone double constraints is obtained by using the generalized saddle point theory three times. The alternating direction method is used to solve the fixed point problem of set-valued mappings with second-order cone double constraints, and the global convergence of the algorithm is proved. Finally, numerical results of solving five examples with an inexact alternating direction method are given, and the feasibility and effectiveness of the algorithm are demonstrated by comparing with other algorithms.</p></abstract>

Equilibrium Optimization with Multi-Energy-Efficiency-Grade Products: Government and Market Perspective

This paper studies the multi-level supply chain network equilibrium optimization problem of multi-energy-efficiency products under different government subsidies and demand scales. In the equilibrium optimization problem, manufacturers determine the production volume of the energy-saving products; retailers decide the transaction volume with manufacturers, distribution volume for markets, and marketing efforts of energy-saving products; markets determine the transaction price. Firstly, the optimal decision-making behaviors of manufacturers, retailers, and markets are described. Simultaneously, the global optimization problem is transformed into a finite-dimensional variational inequality formulation. Then, the equilibrium conditions of the whole supply chain network are derived by the Euler method. Finally, a case study verifies the effectiveness of the proposed method. Interestingly, we found that energy-saving subsidies and demand scales were negatively correlated with the marketing efforts of the subsidized retailers for high energy-efficient products and positively correlated with the marketing efforts of non-subsidized retailers for high energy-efficient products in the same market; the development of retailers in the same market tended to be consistent, and the differentiation of the demand scale eliminated the retailers without a competitive advantage.

Scalable semismooth Newton methods with multilevel domain decomposition for subsurface flow and reactive transport in porous media

Large-scale modeling and predictive simulations of the subsurface flow and reactive transport system in porous media is significantly challenging, due to the high nonlinearity of the governing equations and the strong heterogeneity of material coefficients. The design of novel mathematical models and state-of-the-art methods for the flow simulation through porous media typically needs to satisfy the so-called bound-preserving property, i.e., the computed solution should stay inside a physically meaningful range. This paper presents a robust, scalable numerical framework based on the variational inequality formulation and the semismooth Newton method in a fully implicit manner, to model and simulate highly nonlinear flows without violating the boundedness requirement of the solution. Rigorous theoretical analysis for the variational inequality formulation of the problem is provided for facilitating the design of algorithms. Specifically, our approach further enhances the numerical formulation by utilizing a family of multilevel monolithic overlapping Schwarz methods for efficiently preconditioning, and the parallel implementation of the bound-preserving solver is based on the fast and robust domain decomposition technique. Numerical experiments are presented to demonstrate the efficiency and parallel scalability of the solution strategy for both standard benchmarks as well as realistic flow problems involving strong heterogeneity and high nonlinearity. We also show that the proposed framework is more robust and efficient than the commonly used inexact Newton algorithm in terms of the bound-preserving property.

The Sustainable Supply Chain Network Competition Based on Non-Cooperative Equilibrium under Carbon Emission Permits

Under the background of a circular economy, this paper examines multi-tiered closed-loop supply chain network competition under carbon emission permits and discusses how stringent carbon regulations influence the network performance. We derive the governing equilibrium conditions for carbon-capped mathematical gaming models of each player and provide the equivalent variational inequality formulations, which are then solved by modified projection and contraction algorithms. The numerical examples empower us to investigate the effects of diverse carbon emission regulations (cap-and-trade regulation, mandatory cap policy, and cap-sharing scheme) on enterprises’ decisions. The results reveal that the cap-sharing scheme is effective in coordinating the relationship between system profit and carbon emission abatement, while cap-and-trade regulation loses efficiency compared with the cap-sharing scheme. The government should allocate caps scientifically and encourage enterprises to adopt green production technologies, especially allowing large enterprises to share carbon quotas. This study can also contribute to the enterprises’ decision-making and revenue management under different carbon emissions reduction regulations.

Solving a Multi-Class Traffic Assignment Model with Mixed Modes

In comparison to conventional human-driven vehicles (HVs), connected and automated vehicles (CAVs) provide benefits (e.g., reducing travel time and improving safety). However, before the period of fully CAVs appears, there will be a situation in which both HVs and CAVs are present, and the traffic flow pattern may differ from that of a single class (e.g., HV or CAV). In this study, we developed a multi-class traffic assignment problem (TAP) for a transportation network that explicitly considered mixed modes (e.g., HV and CAV). As a link’s travel time is dependent on the degree of mixed flows, each mode required an asymmetric interaction cost function. For TAP, the multi-class user equilibrium (UE) model was used for the route choice model. A route-based variational inequality (VI) formulation was used to represent the multi-class TAP and solve it using the gradient projection (GP) algorithm. It has been demonstrated that the GP algorithm is an effective route-based solution for solving the single-class user equilibrium (UE) problem. However, it has rarely been applied to solving asymmetric UE problems. In this study, the single-class GP algorithm was extended to solve the multi-class TAP. The numerical results indicated the model’s efficacy in capturing the features of the proposed TAP utilizing a set of simple networks and real transportation networks. Additionally, it demonstrated the computational effectiveness of the GP algorithm in solving the multi-class TAP.

Sensitivity analysis for transit equilibrium assignment and applications to uncertainty analysis

Systematic uncertainty analysis can be used to quantitatively evaluate variation in model outputs and identify the critical sources of uncertainty to improve the reliability and stability of a system. To analyze the effects of uncertainties in transit networks that may be caused by probabilistic travel demand, congestion, or vehicle frequencies, this study develops a sensitivity-based uncertainty analysis approach as a post-analysis tool for equilibrium transit systems. The congestion effect is considered in the waiting time and in-vehicle travel time of a passengers’ route-choice model. The hyperpath concept is used to manage passengers’ riding strategies due to the common-line problem at transit stops. A hyperpath-based gradient projection (GP) solution algorithm is developed for the solution of the variational inequality formulation of the transit equilibrium assignment problem (TEAP). A restricted sensitivity analysis approach originally developed for road networks is re-developed for the TEAP in transit networks. An analytical sensitivity-based approach is derived to conduct uncertainty analysis for the TEAP, which enables the simultaneous propagation of uncertainties from different input sources to the model outputs. Numerical examples are provided for the following purposes. (1) To demonstrate three applications of the sensitivity analysis of the TEAP, namely the perturbed solution estimation problem, the critical parameter identification problem, and the paradox analysis problem. (2) To illustrate the use of uncertainty analysis of the TEAP, such as estimating the variance and confidence level of model outputs with respect to various model inputs/parameters as random variables, and ranking the importance of arcs using sensitivity-based uncertainty analysis. (3) To demonstrate the applicability of the proposed approach to real transit networks. The findings demonstrate not only the importance of the analytical sensitivity analysis development for the TEAP, but also for the practical applications of sensitivity and uncertainty analyses.