Constrained Extremum Problems with Infinite-Dimensional Image: Selection and Necessary Conditions

This paper deals with image space analysis for constrained extremum problems having an infinite-dimensional image. It is shown that the introduction of selection for point-to-set maps and of quasi multipliers allows one to establish optimality conditions for problems where the classical approach fails.     

Extragradient and extrapolation methods with generalized Bregman distances for saddle point problems

In this paper, we introduce two Bregman-type algorithmic frameworks to generalize the extragradient and extrapolation methods. With the help of relative Lipschitzness and the Bregman distance tool, the iteration properties of the proposed frameworks are analyzed. As applied to smooth convex-concave saddle point problems, our theory rediscovers the main results in Mokhtari et al. (2020) [14] for wider frameworks under weaker assumptions via a conceptually different approach.     

An efficient numerical method for preconditioned saddle point problems

In this paper, we consider the solution of linear systems of saddle point type by a preconditioned numerical method. We first transform the original linear system into two sub-systems with small size by a preconditioning strategy, then employ the conjugate gradient (CG) method to solve the linear system with a SPD coefficient matrix, and a splitting iteration method to solve the other sub-system, respectively. Numerical experiments show that the new method can achieve faster convergence than several effective preconditioners published in the recent literature in terms of total runtime and iteration steps.     

An alternating preconditioner for saddle point problems

Based on matrix splittings, a new alternating preconditioner with two parameters is proposed for solving saddle point problems. Some theoretical analyses for the eigenvalues of the associated preconditioned matrix are given. The choice of the parameters is considered and the quasi-optimal parameters are obtained. The new preconditioner with these quasi-optimal parameters significantly improves the convergence rate of the generalized minimal residual (GMRES) iteration. Numerical experiments from the linearized Navier-Stokes equations demonstrate the efficiency of the new preconditioner, especially on the larger viscosity parameter ν. Further extensions of the preconditioner to generalized saddle point matrices are also checked.     

Convergence analysis of the corrected Uzawa algorithmfor symmetric saddle point problems简

For the large sparse saddle point problems, Pan and Li recently proposed in [H. K. Pan, W. Li, Math. Numer. Sinica, 2009, 31（3）： 231-242] a corrected Uzawa algorithm based on a nonlinear Uzawa algorithm with two nonlinear approximate inverses, and gave the detailed convergence analysis. In this paper, we focus on the convergence analysis of this corrected Uzawa algorithm, some inaccuracies in [H. K. Pan, W. Li, Math. Numer. Sinica, 2009, 31（3）： 231-242] are pointed out, and a corrected convergence theorem is presented. A special case of this modified Uzawa algorithm is also discussed.     

Spectral properties of primal-based penalty preconditioners for saddle point problems

For large and sparse saddle point linear systems, this paper gives further spectral properties of the primal-based penalty preconditioners introduced in [C.R. Dohrmann, R.B. Lehoucq, A primal-based penalty preconditioner for elliptic saddle point systems, SIAM J. Numer. Anal. 44 (2006) 270-282]. The regions containing the real and non-real eigenvalues of the preconditioned matrix are obtained. The model of the Stokes problem is supplemented to illustrate the theoretical results and to test the quality of the primal-based penalty preconditioner.     

Coercivity and image of constrained extremum problems

The concept of coercivity is extended to the image space, and it is exploited to obtain a lower bound for the minimum of a constrained extremum problem and a sufficient condition for the optimality. Problems which are coercive in the image space turn out to have zero duality gap. A possible application to variational inequalities is illustrated by means of an example.     

The alternating-direction iterative method for saddle point problems

In the paper, a new alternating-direction iterative method is proposed based on matrix splittings for solving saddle point problems. The convergence analysis for the new method is given. When the better values of parameters are employed, the proposed method has faster convergence rate and less time cost than the Uzawa algorithm with the optimal parameter and the Hermitian and skew-Hermitian splitting iterative method. Numerical examples further show the effectiveness of the method.     

On the optimal parameters of GMSSOR method for saddle point problems

For solving saddle point problems, SOR-type methods are investigated by many researchers in the literature. In this short note, we study the GMSSOR method for solving saddle point problems and obtain the optimal parameters which minimize the spectral (or pseudo-spectral) radii of the iteration matrices.     

On general infinite dimensional complementarity problems

The aim of this paper is to present a way to study directly generalized complementarity problems in normed spaces. By means of new results on infinite dimensional Lagrange theory we show some optimality conditions which reduce the study of the problems to the one of suitable systems of equalities and inequalities.     

PAHSS-PTS alternating splitting iterative methods for nonsingular saddle point problems

In this paper, we propose the PAHSS-PTS alternating splitting iterative methods for nonsingular saddle point problems. Convergence properties of the proposed methods are studied and corresponding convergence results are given under some suitable conditions. Numerical experiments are presented to confirm the theoretical results, which impliy that PAHSS-PTS iterative methods are effective and feasible.     

Lagrange multipliers theorem and saddle point optimality criteria in mathematical programming

We prove a version of Lagrange multipliers theorem for nonsmooth functionals defined on normed spaces. Applying these results, we extend some results about saddle point optimality criteria in mathematical programming.     

Regularity conditions for constrained extremum problems via image space

Exploiting the image-space approach, we give an overview of regularity conditions. A notion of regularity for the image of a constrained extremum problem is given. The relationship between image regularity and other concepts is also discussed. It turns out that image regularity is among the weakest conditions for the existence of normal Lagrange multipliers.     

Convergence analysis of generalized nonlinear inexact Uzawa algorithm for stabilized saddle point problems

This paper deals with a modified nonlinear inexact Uzawa (MNIU) method for solving the stabilized saddle point problem. The modified Uzawa method is an inexact inner-outer iteration with a variable relaxation parameter and has been discussed in the literature for uniform inner accuracy. This paper focuses on the general case when the accuracy of inner iteration can be variable and the convergence of MNIU with variable inner accuracy, based on a simple energy norm. Sufficient conditions for the convergence of MNIU are proposed. The convergence analysis not only greatly improves the existing convergence results for uniform inner accuracy in the literature, but also extends the convergence to the variable inner accuracy that has not been touched in literature. Numerical experiments are given to show the efficiency of the MNIU algorithm.     

A condition for the nonsymmetric saddle point matrix being diagonalizable and having real and positive eigenvalues

This paper discusses the spectral properties of the nonsymmetric saddle point matrices of the form with A symmetric positive definite, B full rank, and C symmetric positive semidefinite. A new sufficient condition is obtained so that is diagonalizable with all its eigenvalues real and positive. This condition is weaker than that stated in the recent paper [J. Liesen, A note on the eigenvalues of saddle point matrices, Technical Report 10-2006, Institute of Mathematics, TU Berlin, 2006].     

A systematization of the saddle point method. Application to the Airy and Hankel functions

The standard saddle point method of asymptotic expansions of integrals requires to show the existence of the steepest descent paths of the phase function and the computation of the coefficients of the expansion from a function implicitly defined by solving an inversion problem. This means that the method is not systematic because the steepest descent paths depend on the phase function on hand and there is not a general and explicit formula for the coefficients of the expansion (like in Watson's Lemma for example). We propose a more systematic variant of the method in which the computation of the steepest descent paths is trivial and almost universal: it only depends on the location and the order of the saddle points of the phase function. Moreover, this variant of the method generates an asymptotic expansion given in terms of a generalized (and universal) asymptotic sequence that avoids the computation of the standard coefficients, giving an explicit and systematic formula for the expansion that may be easily implemented on a symbolic manipulation program. As an illustrative example, the well-known asymptotic expansion of the Airy function is rederived almost trivially using this method. New asymptotic expansions of the Hankel function Hn(z) for large n and z are given as non-trivial examples.     

Modified block preconditioner for generalized saddle point matrices with highly singular(1,1) blocks

ABSTRACT

In this paper, based on the preconditioners presented by Zhang [A new preconditioner for generalized saddle matrices with highly singular(1,1) blocks. Int J Comput Maths. 2014;91(9):2091-2101], we consider a modified block preconditioner for generalized saddle point matrices whose coefficient matrices have singular (1,1) blocks. Moreover, theoretical analysis gives the eigenvalue distribution, forms of the eigenvectors and the minimal polynomial. Finally, numerical examples show the eigenvalue distribution with the presented preconditioner and confirm our analysis.     

Existence and multiplicity of nonzero periodic solution with saddle point character for some nonautonomous second order systems

The purpose of this paper is to study the existence and multiplicity of nonzero periodic solutions with saddle point character for the following nonautonomous second order systems:

     

A note on block preconditioner for generalized saddle point matrices with highly singular (1,1) block

In this paper, we present a block triangular preconditioner for generalized saddle point matrices whose coefficient matrices have singular (1,1) blocks. Theoretical analysis shows that all the eigenvalues of the preconditioned matrix are strongly clustered when choosing an optimal parameter. Numerical experiments are given to demonstrate the efficiency of the presented preconditioner.     

Generalized saddle point theorem and asymptotically linear problems with periodic potential

We prove a critical point theorem, which is an infinite dimensional generalization of the classical saddle point theorem of P. H. Rabinowitz. As an application we obtain solution of asymptotically linear Schrödinger equations with periodic potential.