Search

Conjugate gradient (CG) methods are a popular class of iterative methods for solving linear systems of equations and nonlinear optimization problems. In this paper, a new hybrid conjugate gradient (CG) method is presented and analyzed for solving unconstrained optimization problems, where the parameter β k is a convex combination of β k WYL and β k CD . Under the strong Wolfe line search, the new method possesses the sufficient descent condition and the global convergence properties. The preliminary numerical results show the efficiency of our method in comparison with other CG methods. Furthermore, the proposed algorithm HWYLCD was extended to solve the problem of a mode function. [ABSTRACT FROM AUTHOR]

This paper is concerned with the convergence of a series associated with a certain version of the convexification method.That version has been recently developed by the research group of the first author for solving coefficient inverse problems.The convexification method aims to construct a globally convex Tikhonov-like functional with a Carleman weight function in it.In the previous works, the construction of the strictly convex weighted Tikhonov-like functional assumes a truncated Fourier series (i.e. a finite series instead of an infinite one) for a function generated by the total wave field.In this paper, we prove a convergence property for this truncated Fourier series approximation.More precisely, we show that the residual of the approximate PDE obtained by using the truncated Fourier series tends to zero in L 2 L^{2} as the truncation index in the truncated Fourier series tends to infinity.The proof relies on a convergence result in the H 1 H^{1} -norm for a sequence of L 2 L^{2} -orthogonal projections on finite-dimensional subspaces spanned by elements of a special Fourier basis.However, due to the ill-posed nature of coefficient inverse problems, we cannot prove that the solution of that approximate PDE, which results from the minimization of that Tikhonov-like functional, converges to the correct solution. [ABSTRACT FROM AUTHOR]

The extended linear complementarity problem (denoted by XLCP), of which the linear and horizontal linear complementarity problems are two special cases, can be reformulated as the solution of a nonsmooth system of equations. By the symmetrically perturbed smoothing Fischer-Burmeister function, the XLCP is approximated by a family of parameterized smoothness optimization problems. A smoothing damped Gauss-Newton method is designed for solving the XLCP. The proposed algorithm is proved to be convergent globally under suitable assumptions. Some numerical results are reported in the paper. [ABSTRACT FROM AUTHOR]

- We propose a new smoothing function in this paper by smoothing the symmetric perturbed Fischer-Burmeister function. Based on this new function, we present a new noninterior continuation method for solving the second-order cone programming. We adopt a variant merit function. Our algorithm needs to solve only one system of linear equations and to perform only one line search at each iteration. It can start from an arbitrary point and does not require the iteration points to be in the set of strictly feasible solutions. We prove the global and local quadratic convergence of the algorithm under suitable assumptions. Numerical results indicate that our algorithm performs well. [ABSTRACT FROM AUTHOR]

In this paper, nonlinear complementarity problem with P0-function is studied. Based on a new parametric nonlinear complementarity function, the problem is approximated by a family of parameterized smooth equations, and a nonmonotone inexact smoothing Newton-type method is presented. At each iteration, the proposed algorithm only needs to solve one system of linear equations inexactly and performs only one nonmonotone line search. It is proved to be convergent globally and superlinearly without strict complementarity at the solution. Moreover, the algorithm has locally quadratic convergence under mild conditions. Numerical results are also reported for the tested problems, which show the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

This is a review paper of the role of Carleman estimates in the theory of multidimensional coefficient inverse problems since the first inception of this idea in 1981. [ABSTRACT FROM AUTHOR]

For solving linear ill-posed problems regularization methods are required when the right-hand side is with some noise. In the present paper regularized solutions are obtained by multi-parameter regularization and the regularization parameters are chosen by a multi-parameter discrepancy principle. Under certain smoothness assumptions we provide order optimal error bounds that characterize the accuracy of the regularized solutions. For the computation of the regularization parameters fast algorithms of Newton type are applied which are based on special transformations. These algorithms are globally and monotonically convergent. Some of our theoretical results are illustrated by numerical experiments. We also show how the proposed approach may be employed for multi-task approximation. [ABSTRACT FROM AUTHOR]

Conjugate gradient methods are very popular for solving large scale unconstrained optimization problems because of their simplicity to implement and low memory requirements. In this paper, we present a hybrid three-term conjugate gradient method with a direction that always satisfies the sufficient descent condition. We establish global convergence of the new method under the weak Wolfe line search conditions. We also report some numerical results of the proposed method compared to relevant methods in the literature.

In this paper, a new improved smoothing Newton algorithm for the nonlinear complementarity problem was proposed. This method has two-fold advantages. First, compared with the classical smoothing Newton method, our proposed method needn’t nonsingular of the smoothing approximation function; second, the method also inherits the advantage of the classical smoothing Newton method, it only needs to solve one linear system of equations at each iteration. Without the need of strict complementarity conditions and the assumption of P0 property, we get the global and local quadratic convergence properties of the proposed method. Numerical experiments show that the efficiency of the proposed method.

We propose a new approach to constructing globally strictly convex objective functional in a 1-D inverse medium scattering problem using multi-frequency backscattering data. The global convexity of the proposed objective functional is proved. We also prove the global convergence of the gradient projection algorithm and derive an error estimate. Numerical examples are presented to illustrate the performance of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this study, we examine dynamic allocation of resources and consumption over an infinite time horizon in an overlapping-generations model with two-sided altruism. Each generation cares not only about its own consumption and the utility of its successor, but also about the utility of its parental generation. The consumption decisions of different generations could display time-inconsistency in a model of two-sided altruism. Consequently, the outcome solutions could differ depending on whether members of the current generation are naïve and falsely believe that the future generation will comply with their, or are sophisticated and know exactly how the future generation will choose to behave. We assume that the young generation is naïve and behaves as a central planner in allocating capital resources. The capital stocks inherited by each generation from the parental generation are actually realized and used in the joint production of capital stocks and consumption goods. We refer to a sequence of such capital stocks as a two-sided altruistic path. This study examines a more general production function than earlier models of two-sided altruism, and our results prove that any two-sided altruistic path will eventually converge to the two-sided altruistic stationary path. [ABSTRACT FROM AUTHOR]

Numerical issues for the 3D travel time tomography problem with non-overdetemined data are considered. Truncated Fourier series with respect to a special orthonormal basis of functions depending on the source position is used. In addition, truncated trigonometric Fourier series with respect to two out of three spatial variables are used. First, the Lipschitz stability estimate is obtained. Next, a globally convergent numerical method is constructed using a Carleman estimate for an integral operator. [ABSTRACT FROM AUTHOR]

- We prove global convergence of an inexact extended polyhedral Gauß-Seidel method for the minimization of strictly convex functionals that are continuously differentiable on each polyhedron of a polyhedral decomposition of their domains of definition. While pure Gauß-Seidel methods are known to be very slow for problems governed by partial differential equations, the presented convergence result also covers multilevel methods that extend the Gauß-Seidel step by coarse level corrections. Our result generalizes the proof of [10] for differentiable functionals on the Gibbs simplex. Example applications are given that require the generality of our approach. [ABSTRACT FROM AUTHOR]

Computational results for the convexification algorithm of the authors are presented in the 2-Dimensional case. Convexification is a numerical method for some multidimensional coefficient inverse problems with rigorously guaranteed global convergence. [ABSTRACT FROM AUTHOR]