Methods for solving nonlinear equations used in evaluating emergency vehicle busy probabilities

In this paper we present two iterative methods for solving a model to evaluate busy probabilities for Emergency Medical Service (EMS) vehicles. The model considers location dependent service times and is an alternative to the mean service calibration method; a procedure, used with the Hypercube Model, to accommodate travel times and location-dependent service times. We use monotonicity arguments to prove that one iterative method always converges to a solution. A large computational experiment suggests that both methods work satisfactorily in EMS systems with low ambulance busy probabilities and the method that always converges to a solution performs significantly better in EMS systems with high busy probabilities.     

Modified projection method for pseudomonotone variational inequalities

We consider and analyze a new projection method for solving pseudomonotone variational inequalities by modifying the extragradient method. The modified method converges for pseudomonotone Lipschitz continuous operators, which is a much weaker condition than monotonicity. The new iterative method differs from the existing projection methods. Our proof of convergence is very simple as compared with other methods.     

Solving Linear Systems Whose Elements Are Nonlinear Functions of Intervals

The paper addresses the problem of solving linear algebraic systems the elements of which are, in the general case, nonlinear functions of a given set of independent parameters taking on their values within prescribed intervals. Three kinds of solutions are considered: (i) outer solution, (ii) interval hull solution, and (iii) inner solution. A simple direct method for computing a tight outer solution to such systems is suggested. It reduces, essentially, to inverting a real matrix and solving a system of real linear equations whose size n is the size of the original system. The interval hull solution (which is a NP-hard problem) can be easily determined if certain monotonicity conditions are fulfilled. The resulting method involves solving n+1 interval outer solution problems as well as 2n real linear systems of size n. A simple iterative method for computing an inner solution is also given. A numerical example illustrating the applicability of the methods suggested is solved.     

On an Implicit Method for Nonconvex Variational Inequalities

In this paper, we suggest and analyze an implicit iterative method for solving nonconvex variational inequalities using the technique of the projection operator. We also discuss the convergence of the iterative method under partially relaxed strongly monotonicity, which is a weaker condition than cocoerciveness. Our method of proof is very simple.     

A Cutting Plane Method for Solving Quasimonotone Variational Inequalities

We present an iterative algorithm for solving variational inequalities under the weakest monotonicity condition proposed so far. The method relies on a new cutting plane and on analytic centers.     

A convergence analysis of nonlinear implicit iterative method for nonlinear ill-posed problems

Implicit iterative method acquires good effect in solving linear ill-posed problems. We have ever applied the idea of implicit iterative method to solve nonlinear ill-posed problems, under the restriction that α is appropriate large, we proved the monotonicity of iterative error and obtained the convergence and stability of iterative sequence, numerical results show that the implicit iterative method for nonlinear ill-posed problems is efficient. In this paper, we analyze the convergence and stability of the corresponding nonlinear implicit iterative method when α_k are determined by Hanke criterion.     

An iterative back substitution algorithm for the solution of tridiagonal matrix systems with fringes

For tridiagonal matrix systems, a simple direct algorithm giving the solution exists, but in the most general case of tridiagonal matrix with fringes, the direct solving algorithms are more complicated. For big systems, direct methods are not well fitted and iterative algorithms are preferable. In this paper a relaxation type iterative algorithm is presented. It is an extension of the backward substitution method used for simple tridiagonal matrix systems. The performances show that this algorithm is a good compromise between a direct method and other iterative methods as block SOR. Its nature suggests its use as inner solver in the solution of problems derived by application of a decomposition domain method. A special emphasis is done on the programming aspect. The solving Fortran subroutines implementing the algorithm have been generated automatically from their specification by using a computer algebra system technique.     

Über die näherungsweise Lösung nichtlinearer Integralgleichungen

Summary In this paper we study the iterative solvability of nonlinear systems of equations which arise from the discretization of Hammerstein integral equations. It is shown that, for a large class of equations satisfying monotonicity assumptions, it is possible to solve these systems by means of a linearly convergent iteration method. Moreover, for general monotone operators on a Hilbert space a globally convergent variant of Newton's method is given. Finally it is shown that this method effectively can be applied in a natural way to the systems of equations under consideration.     

Projection algorithms for the system of generalized mixed variational inequalities in Banach spaces

Some projection algorithms are suggested for solving the system of generalized mixed variational inequalities, and the convergence of the proposed iterative methods are proved without any monotonicity assumption for the mappings in Banach spaces. Our theorems generalize some known results.     

Iterative algorithms for generalized mixed equilibrium problems

In this paper, we consider a generalized mixed equilibrium problem in real Hilbert space. Using the auxiliary principle, we define a class of resolvent mappings. Further, using fixed point and resolvent methods, we give some iterative algorithms for solving generalized mixed equilibrium problem. Furthermore, we prove that the sequences generated by iterative algorithms converge weakly to the solution of generalized mixed equilibrium problem. These results require monotonicity (θ-pseudo monotonicity) and continuity (Lipschitz continuity) for mappings.     

Projection method for solving a singular system of linear equations and its applications

Summary The iterative method for solving system of linear equations, due to Kaczmarz [2], is investigated. It is shown that the method works well for both singular and non-singular systems and it determines the affine space formed by the solutions if they exist. The method also provides an iterative procedure for computing a generalized inverse of a matrix.     

Some algorithms for hemiequilibrium problems

In this paper, we suggest and analyze a class of iterative methods for solving hemiequilibrium problems using the auxiliary principle technique. We prove that the convergence of these new methods either requires partially relaxed strongly monotonicity or pseudomonotonicity, which is a weaker condition than monotonicity. Results obtained in this paper include several new and known results as special cases.     

Iterative schemes for quasimonotone mixed variational inequalities

We consider some new iterative methods for solving quasimonotone mixed variational inequalities by updating the solution. These algorithms are based on combining extrapolation and splitting techniques. The convergence analysis of these new methods is considered. These new methods are versatile and are easy to implement. Our method of proof of convergence is very simple and uses either monotonicity or quasimonotonicity of the operator.     

Two algorithms for solving systems of inclusion problems

The goal of this paper is to present two algorithms for solving systems of inclusion problems, with all components of the systems being a sum of two maximal monotone operators. The algorithms are variants of the forward-backward splitting method and one being a hybrid with the alternating projection method. They consist of approximating the solution sets involved in the problem by separating half-spaces which is a well-studied strategy. The schemes contain two parts, the first one is an explicit Armijo-type search in the spirit of the extragradient-like methods for variational inequalities. The second part is the projection step, this being the main difference between the algorithms. While the first algorithm computes the projection onto the intersection of the separating half-spaces, the second chooses one component of the system and projects onto the separating half-space of this case. In the iterative process, the forward-backward operator is computed once per inclusion problem, representing a relevant computational saving if compared with similar algorithms in the literature. The convergence analysis of the proposed methods is given assuming monotonicity of all operators, without Lipschitz continuity assumption. We also present some numerical experiments.     

Generalized Invex Monotonicity and Its Role in Solving Variational-Like Inequalities

The paper stresses the role of new classes of generalized invex monotonicity in the convergence of iterative schemes for solving a variational-like inequality problem on a closed convex set. This work was supported by Grant NSFC 70432001.     

Semiconvergence of nonnegative splittings for singular matrices

Summary. In this paper, we discuss semiconvergence of the matrix splitting methods for solving singular linear systems. The concepts that a splitting of a matrix is regular or nonnegative are generalized and we introduce the terminologies that a splitting is quasi-regular or quasi-nonnegative. The equivalent conditions for the semiconvergence are proved. Comparison theorem on convergence factors for two different quasi-nonnegative splittings is presented. As an application, the semiconvergence of the power method for solving the Markov chain is derived. The monotone convergence of the quasi-nonnegative splittings is proved. That is, for some initial guess, the iterative sequence generated by the iterative method introduced by a quasi-nonnegative splitting converges towards a solution of the system from below or from above. Received August 19, 1997 / Revised version received August 20, 1998 / Published online January 27, 2000     

Extragradient methods for solving nonconvex variational inequalities

In this paper, we introduce and consider a new class of variational inequalities, which are called the nonconvex variational inequalities. Using the projection technique, we suggest and analyze an extragradient method for solving the nonconvex variational inequalities. We show that the extragradient method is equivalent to an implicit iterative method, the convergence of which requires only pseudo-monotonicity, a weaker condition than monotonicity. This clearly improves on the previously known result. Our method of proof is very simple as compared with other techniques.     

Iterative schemes for trifunction hemivariational inequalities

In this paper, we consider and study a new class of hemivariational inequalities, which is called trifunction hemivariational inequality. We suggest and analyze a class of iterative methods for solving trifunction hemivariational inequalities using the auxiliary principle technique. We prove that the convergence of these new methods either requires partially relaxed strongly monotonicity or pseudomonotonicity, which is a weaker condition than monotonicity. Results obtained in this paper include several new and known results as special cases.     

On General Mixed Quasivariational Inequalities

In this paper, we suggest and analyze several iterative methods for solving general mixed quasivariational inequalities by using the technique of updating the solution and the auxiliary principle. It is shown that the convergence of these methods requires either the pseudomonotonicity or the partially relaxed strong monotonicity of the operator. Proofs of convergence is very simple. Our new methods differ from the existing methods for solving various classes of variational inequalities and related optimization problems. Various special cases are also discussed.     

Output Feedback Model Predictive Tracking Control Using a Slope Bounded Nonlinear Model

In this paper, an output feedback model predictive tracking control method is proposed for constrained nonlinear systems, which are described by a slope bounded model. In order to solve the problem, we consider the finite horizon cost function for an off-set free tracking control of the system. For reference tracking, the steady state is calculated by solving by quadratic programming and a nonlinear estimator is designed to predict the state from output measurements. The optimized control input sequences are obtained by minimizing the upper bound of the cost function with a terminal weighting matrix. The cost monotonicity guarantees that tracking and estimation errors go to zero. The proposed control law can easily be obtained by solving a convex optimization problem satisfying several linear matrix inequalities. In order to show the effectiveness of the proposed method, a novel slope bounded nonlinear model-based predictive control method is applied to the set-point tracking problem of solid oxide fuel cell systems. Simulations are also given to demonstrate the tracking performance of the proposed method.