A continuous-time linear complementarity system for dynamic user equilibria in single bottleneck traffic flows

This paper formally introduces a linear complementarity system (LCS) formulation for a continuous-time, multi-user class, dynamic user equilibrium (DUE) model for the determination of trip timing decisions in a simplified single bottleneck model. Existence of a Lipschitz solution trajectory to the model is established by a constructive time-stepping method whose convergence is rigorously analyzed. The solvability of the time-discretized subproblems by Lemke’s algorithm is also proved. Combining linear complementarity with ordinary differential equations and being a new entry to the mathematical programming field, the LCS provides a computational tractable framework for the rigorous treatment of the DUE problem in continuous time; this paper makes a positive contribution in this promising research venue pertaining to the application of differential variational theory to dynamic traffic problems.     

Implicit solution function of P0 and Z matrix linear complementarity constraints

Using the least element solution of the P₀ and Z matrix linear complementarity problem (LCP), we define an implicit solution function for linear complementarity constraints (LCC). We show that the sequence of solution functions defined by the unique solution of the regularized LCP is monotonically increasing and converges to the implicit solution function as the regularization parameter goes down to zero. Moreover, each component of the implicit solution function is convex. We find that the solution set of the irreducible P₀ and Z matrix LCP can be represented by the least element solution and a Perron?CFrobenius eigenvector. These results are applied to convex reformulation of mathematical programs with P₀ and Z matrix LCC. Preliminary numerical results show the effectiveness and the efficiency of the reformulation.     

The composite Milstein methods for the numerical solution of Ito stochastic differential equations

In this paper, we present the composite Milstein methods for the strong solution of Ito stochastic differential equations. These methods are a combination of semi-implicit and implicit Milstein methods. We give a criterion for choosing either the implicit or the semi-implicit scheme at each step of our numerical solution. The stability and convergence properties are investigated and discussed for the linear test equation. The convergence properties for the nonlinear case are shown numerically to be the same as the linear case. The stability properties of the composite Milstein methods are found to be more superior compared to those of the Milstein, the Euler and even better than the composite Euler method. This superiority in stability makes the methods a better candidate for the solution of stiff SDEs.     

A Perturbation approach for an inverse quadratic programming problem

We consider an inverse quadratic programming (QP) problem in which the parameters in both the objective function and the constraint set of a given QP problem need to be adjusted as little as possible so that a known feasible solution becomes the optimal one. We formulate this problem as a linear complementarity constrained minimization problem with a positive semidefinite cone constraint. With the help of duality theory, we reformulate this problem as a linear complementarity constrained semismoothly differentiable (SC¹) optimization problem with fewer variables than the original one. We propose a perturbation approach to solve the reformulated problem and demonstrate its global convergence. An inexact Newton method is constructed to solve the perturbed problem and its global convergence and local quadratic convergence rate are shown. As the objective function of the problem is a SC¹ function involving the projection operator onto the cone of positively semi-definite symmetric matrices, the analysis requires an implicit function theorem for semismooth functions as well as properties of the projection operator in the symmetric-matrix space. Since an approximate proximal point is required in the inexact Newton method, we also give a Newton method to obtain it. Finally we report our numerical results showing that the proposed approach is quite effective.     

A Regularization Newton Method for Solving Nonlinear Complementarity Problems

In this paper we construct a regularization Newton method for solving the nonlinear complementarity problem (NCP(F )) and analyze its convergence properties under the assumption that F is a P ₀ -function. We prove that every accumulation point of the sequence of iterates is a solution of NCP(F ) and that the sequence of iterates is bounded if the solution set of NCP(F ) is nonempty and bounded. Moreover, if F is a monotone and Lipschitz continuous function, we prove that the sequence of iterates is bounded if and only if the solution set of NCP(F ) is nonempty by setting , where is a parameter. If NCP(F) has a locally unique solution and satisfies a nonsingularity condition, then the convergence rate is superlinear (quadratic) without strict complementarity conditions. At each step, we only solve a linear system of equations. Numerical results are provided and further applications to other problems are discussed. Accepted 25 March 1998     

Generalized equations and the generalized Newton method

We give some convergence results on the generalized Newton method (referred to by some authors as Newton's method) and the chord method when applied to generalized equations. The main results of the paper extend the classical Kantorovich results on Newton's method to (nonsmooth) generalized equations. Our results also extend earlier results on nonsmooth equations due to Eaves, Robinson, Josephy, Pang and Chan. We also propose inner-iterative schemes for the computation of the generalized Newton iterates. These schemes generalize popular iterative methods (Richardson's method, Jacobi's method and the Gauss-Seidel method) for the solution of linear equations and linear complementarity problems and are shown to be convergent under natural generalizations of classical convergence criteria. Our results are applicable to equations involving single-valued functions and also to a class of generalized equations which includes variational inequalities, nonlinear complementarity problems and some nonsmooth convex minimization problems.     

A Newton method for a class of quasi-variational inequalities

A variant of the Newton method for nonsmooth equations is applied to solve numerically quasivariational inequalities with monotone operators. For this purpose, we investigate the semismoothness of a certain locally Lipschitz operator coming from the quasi-variational inequality, and analyse the generalized Jacobian of this operator to ensure local convergence of the method. A simplified variant of this approach, applicable to implicit complementarity problems, is also studied. Small test examples have been computed.This work has been supported in parts by a grant from the German Scientific Foundation and by a grant from the Czech Academy of Sciences.     

An algorithm for a class of nonlinear complementarity problems with non-Lipschitzian functions

In this paper, we focus on solving a class of nonlinear complementarity problems with non-Lipschitzian functions. We first introduce a generalized class of smoothing functions for the plus function. By combining it with Robinson's normal equation, we reformulate the complementarity problem as a family of parameterized smoothing equations. Then, a smoothing Newton method combined with a new nonmonotone line search scheme is employed to compute a solution of the smoothing equations. The global and local superlinear convergence of the proposed method is proved under mild assumptions. Preliminary numerical results obtained applying the proposed approach to nonlinear complementarity problems arising in free boundary problems are reported. They show that the smoothing function and the nonmonotone line search scheme proposed in this paper are effective.     

Optimization methods for Dirichlet control problems

We discuss several optimization procedures to solve finite element approximations of linear-quadratic Dirichlet optimal control problems governed by an elliptic partial differential equation posed on a 2D or 3D Lipschitz domain. The control is discretized explicitly using continuous piecewise linear approximations. Unconstrained, control-constrained, state-constrained and control-and-state constrained problems are analysed. A preconditioned conjugate method for a reduced problem in the control variable is proposed to solve the unconstrained problem, whereas semismooth Newton methods are discussed for the solution of constrained problems. State constraints are treated via a Moreau–Yosida penalization. Convergence is studied for both the continuous problems and the finite dimensional approximations. In the finite dimensional case, we are able to show convergence of the optimization procedures even in the absence of Tikhonov regularization parameter. Computational aspects are also treated and several numerical examples are included to illustrate the theoretical results.     

Convergence analysis of a projection semi-implicit coupling scheme for fluid–structure interaction problems

In this paper, we provide a convergence analysis of a projection semi-implicit scheme for the simulation of fluid–structure systems involving an incompressible viscous fluid. The error analysis is performed on a fully discretized linear coupled problem: a finite element approximation and a semi-implicit time-stepping strategy are respectively used for space and time discretization. The fluid is described by the Stokes equations, the structure by the classical linear elastodynamics equations (linearized elasticity, plate or shell models) and all changes of geometry are neglected. We derive an error estimate in finite time and we prove that the time discretization error for the coupling scheme is at least ${\sqrt{\delta t}}In this paper, we provide a convergence analysis of a projection semi-implicit scheme for the simulation of fluid–structure systems involving an incompressible viscous fluid. The error analysis is performed on a fully discretized linear coupled problem: a finite element approximation and a semi-implicit time-stepping strategy are respectively used for space and time discretization. The fluid is described by the Stokes equations, the structure by the classical linear elastodynamics equations (linearized elasticity, plate or shell models) and all changes of geometry are neglected. We derive an error estimate in finite time and we prove that the time discretization error for the coupling scheme is at least ?{dt}{\sqrt{\delta t}}. Finally, some numerical experiments that confirm the theoretical analysis are presented.     

A non-standard-Euler–Maruyama scheme

We construct a non-standard finite difference numerical scheme to approximate stochastic differential equations (SDEs) using the idea of weighed step introduced by R.E. Mickens. We prove the strong convergence of our scheme under locally Lipschitz conditions of a SDE and linear growth condition. We prove the preservation of domain invariance by our scheme under a minimal condition depending on a discretization parameter and unconditionally for the expectation of the approximate solution. The results are illustrated through the geometric Brownian motion. The new scheme shows a greater behaviour compared with the Euler–Maruyama scheme and balanced implicit methods which are widely used in the literature and applications.     

Lattice Approximations for Stochastic Quasi-Linear Parabolic Partial Differential Equations driven by Space-Time White Noise II

We approximate quasi-linear parabolic SPDEs substituting the derivatives with finite differences. We investigate the resulting implicit and explicit schemes. For the implicit scheme we estimate the rate of L^p convergence of the approximations and we also prove their almost sure convergence when the nonlinear terms are Lipschitz continuous. When the nonlinear terms are not Lipschitz continuous we obtain convergence in probability provided pathwise uniqueness for the equation holds. For the explicit scheme we get these results under an additional condition on the mesh sizes in time and space.     

Differential variational inequalities

This paper introduces and studies the class of differential variational inequalities (DVIs) in a finite-dimensional Euclidean space. The DVI provides a powerful modeling paradigm for many applied problems in which dynamics, inequalities, and discontinuities are present; examples of such problems include constrained time-dependent physical systems with unilateral constraints, differential Nash games, and hybrid engineering systems with variable structures. The DVI unifies several mathematical problem classes that include ordinary differential equations (ODEs) with smooth and discontinuous right-hand sides, differential algebraic equations (DAEs), dynamic complementarity systems, and evolutionary variational inequalities. Conditions are presented under which the DVI can be converted, either locally or globally, to an equivalent ODE with a Lipschitz continuous right-hand function. For DVIs that cannot be so converted, we consider their numerical resolution via an Euler time-stepping procedure, which involves the solution of a sequence of finite-dimensional variational inequalities. Borrowing results from differential inclusions (DIs) with upper semicontinuous, closed and convex valued multifunctions, we establish the convergence of such a procedure for solving initial-value DVIs. We also present a class of DVIs for which the theory of DIs is not directly applicable, and yet similar convergence can be established. Finally, we extend the method to a boundary-value DVI and provide conditions for the convergence of the method. The results in this paper pertain exclusively to systems with “index” not exceeding two and which have absolutely continuous solutions. The work of J.-S. Pang is supported by the National Science Foundation under grants CCR-0098013 CCR-0353074, and DMS-0508986, by a Focused Research Group Grant DMS-0139715 to the Johns Hopkins University and DMS-0353016 to Rensselaer Polytechnic Institute, and by the Office of Naval Research under grant N00014-02-1-0286. The work of D. E. Stewart is supported by the National Science Foundation under a Focused Research Group grant DMS-0138708.     

Interior-point methods for nonlinear complementarity problems

We present a potential reduction interior-point algorithm for monotone nonlinear complementarity problems. At each iteration, one has to compute an approximate solution of a nonlinear system such that a certain accuracy requirement is satisfied. For problems satisfying a scaled Lipschitz condition, this requirement is satisfied by the approximate solution obtained by applying one Newton step to that nonlinear system. We discuss the global and local convergence rates of the algorithm, convergence toward a maximal complementarity solution, a criterion for switching from the interior-point algorithm to a pure Newton method, and the complexity of the resulting hybrid algorithm.This research was supported in part by NSF Grant DDM-89-22636.The authors would like to thank Rongqin Sheng and three anonymous referees for their comments leading to a better presentation of the results.     

On sensitivity of central solutions in semidefinite programming

In this paper we study the properties of the analytic central path of a semidefinite programming problem under perturbation of the right hand side of the constraints, including the limiting behavior when the central optimal solution, namely the analytic center of the optimal set, is approached. Our analysis assumes the primal-dual Slater condition and the strict complementarity condition. Our findings are as follows. First, on the negative side, if we view the central optimal solution as a function of the right hand side of the constraints, then this function is not continuous in general, whereas in the linear programming case this function is known to be Lipschitz continuous. On the positive side, compared with the previous conclusion we obtain a (seemingly) paradoxical result: on the central path any directional derivative with respect to the right hand side of the constraints is bounded, and even converges as the central optimal solution is approached. This phenomenon is possible due to the lack of a uniform bound on the derivatives with respect to the right hand side parameters. All these results are based on the strict complementarity assumption. Concerning this last property we give an example. In that example the set of right hand side parameters for which the strict complementarity condition holds is neither open nor closed. This is remarkable since a similar set for which the primal-dual Slater condition holds is always open. Received: April 2, 1998 / Accepted: January 16, 2001?Published online March 22, 2001     

Local Feasible QP-Free Algorithms for the Constrained Minimization of SC1 Functions

We consider the problem of minimizing an SC¹ function subject to inequality constraints. We propose a local algorithm whose distinguishing features are that: (a) a fast convergence rate is achieved under reasonable assumptions that do not include strict complementarity at the solution; (b) the solution of only linear systems is required at each iteration; (c) all the points generated are feasible. After analyzing a basic Newton algorithm, we propose some variants aimed at reducing the computational costs and, in particular, we consider a quasi-Newton version of the algorithm.     

Evolution of phase transitions in methane hydrate

We consider a simplified model of methane hydrates which we cast as a nonlinear evolution problem. For its well-posedness we extend the existing theory to cover the case in which the problem involves a measurable family of graphs. We represent the nonlinearity as a subgradient and prove a useful comparison principle, thus optimal regularity results follow. For the numerical solution we apply a fully implicit scheme without regularization and use the semismooth Newton algorithm for a solver, and the graph is realized as a complementarity constraint (CC). The algorithm is very robust and we extend it to define an easy and superlinearly convergent fully implicit scheme for the Stefan problem and other multivalued examples.     

Convergence of the semi-implicit Euler method for neutral stochastic delay differential equations with phase semi-Markovian switching

Recently, in the numerical analysis for stochastic differential equations (SDEs), it is a new topic to study the numerical schemes of neutral stochastic functional differential equations (NSFDEs) (see Wu and Mao [1]). Especially when Markovian switchings are taken into consideration, these problems will be more complicated. Although Zhou and Wu [2] develop a numerical scheme to neutral stochastic delay differential equations with Markovian switching (short for NSDDEwMSs), their method belongs to explicit Euler–Maruyama methods which are in general much less accurate in approximation than their implicit or semi-implicit counterparts. Therefore, to propose an implicit method becomes imperative to fill the gap. In this paper we will extend Zhou and Wu [2] to the case of the semi-implicit Euler–Maruyama methods and equations with phase semi-Markovian switching rather than Markovian switching. The employment of phase semi-Markovian chains can avoid the restriction of the negative exponential distribution of the sojourn time at a state. We prove the semi-implicit Euler solution will converge to the exact solution to NSDDEwMS under local Lipschitz condition. More precise inequalities and new techniques are put forward to overcome the difficulties for the existence of the neutral part.     

Smoothing Newton method for nonsmooth second-order cone complementarity problems with application to electric power markets

This paper is dedicated to solving a nonsmooth second-order cone complementarity problem, in which the mapping is assumed to be locally Lipschitz continuous, but not necessarily to be continuously differentiable everywhere. With the help of the vector-valued Fischer-Burmeister function associated with second-order cones, the nonsmooth second-order cone complementarity problem can be equivalently transformed into a system of nonsmooth equations. To deal with this reformulated nonsmooth system, we present an approximation function by smoothing the inner mapping and the outer Fischer-Burmeister function simultaneously. Different from traditional smoothing methods, the smoothing parameter introduced is treated as an independent variable. We give some conditions under which the Jacobian of the smoothing approximation function is guaranteed to be nonsingular. Based on these results, we propose a smoothing Newton method for solving the nonsmooth second-order cone complementarity problem and show that the proposed method achieves globally superlinear or quadratic convergence under suitable assumptions. Finally, we apply the smoothing Newton method to a network Nash-Cournot game in oligopolistic electric power markets and report some numerical results to demonstrate its effectiveness.

     

The composite Milstein methods for the numerical solution of Stratonovich stochastic differential equations

In this paper, we present two composite Milstein methods for the strong solution of Stratonovich stochastic differential equations driven by d-dimensional Wiener processes. The composite Milstein methods are a combination of semi-implicit and implicit Milstein methods. The criterion for choosing either the implicit or the semi-implicit method at each step of the numerical solution is given. The stability and convergence properties of the proposed methods are analyzed for the linear test equation. It is shown that the proposed methods converge to the exact solution in Stratonovich sense. In addition, the stability properties of our methods are found to be superior to those of the Milstein and the composite Euler methods. The convergence properties for the nonlinear case are shown numerically to be the same as the linear case. Hence, the proposed methods are a good candidate for the solution of stiff SDEs.