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1.
Hermitian matrices can be thought of as generalizations of real numbers. Many matrix inequalities, especially for Hermitian matrices, are derived from their scalar counterparts. In this paper, the Hardy-Littlewood-Pólya rearrangement inequality is extended to Hermitian matrices with respect to determinant, trace, Kronecker product, and Hadamard product. 相似文献
2.
It is shown that, given a lower semicontinuous convex integrandf satisfying a suitable integrability condition, there exists a sequence of Lipschitz simple integrands which Mosco converges tof and such that the sequence of conjugate integrands Mosco converges tof
*. Moreover, this sequence can be chosen so that the sequence of associated integral functionals, respectively defined onL
1(X) andL
(X
*), Mosco converges as well.We wish to thank Professor Erik J. Balder for interesting remarks on the first version of this work. 相似文献
3.
Given ann×n matrixM, a vectorq in
n
, a polyhedral convex setX={x|Axb, Bx=d}, whereA is anm×n matrix andB is ap×n matrix, the affinne variational inequality problem is to findxX such that (Mx+q)
T
(y–x)0 for allyX. IfM is positive semidefinite (not necessarily symmetric), the affine variational inequality can be transformeo to a generalized complementarity problem, which can be solved in polynomial time using interior-point algorithms due to Kojima et al. We develop interior-point algorithms that exploit the particular structure of the problem, rather than direictly reducing the problem to a standard linear complemntarity problem.This work was partially supported by the Air Force Office of Scientific Research, Grant AFOSR-89-0410 and the National Science Foundation, Grant CCR-91-57632.The authors acknowledge Professor Osman Güler for pointing out the valoidity of Theorem 2.1 without further assumptions and the proof to that effect. They are also grateful for his comments to improve the presentation of this paper. 相似文献
4.
Cases of equality in certain Hardy–Riesz–Rogers–Brascamp–Lieb–Luttinger rearrangement inequalities are characterized. 相似文献
5.
讨论变分不等式问题VIP(X,F),其中F是单调函数,约束集X为有界区域.利用摄动技术和一类光滑互补函数将问题等价转化为序列合两个参数的非线性方程组,然后据此建立VIP(X,F)的一个内点连续算法.分析和论证了方程组解的存在性和惟一性等重要性质,证明了算法很好的整体收敛性,最后对算法进行了初步的数值试验。 相似文献
6.
Improved Hardy inequalities, involving remainder terms, are established both in the classical and in the limiting case. The relevant remainders depend on a suitable distance from the families of the “virtual” extremals. 相似文献
7.
Convergence of stationary sequences for variational inequalities with maximal monotone operators 总被引:1,自引:0,他引:1
A. Auslender 《Applied Mathematics and Optimization》1993,28(2):161-172
LetT be a maximal monotone operator defined on
N
. In this paper we consider the associated variational inequality 0 T(x
*) and stationary sequences {x
k
*
for this operator, i.e., satisfyingT(x
k
*
0. The aim of this paper is to give sufficient conditions ensuring that these sequences converge to the solution setT
–1(0) especially when they are unbounded. For this we generalize and improve the directionally local boundedness theorem of Rockafellar to maximal monotone operatorsT defined on
N
. 相似文献
8.
M. J. Smith 《Journal of Optimization Theory and Applications》1984,44(3):485-496
The paper provides a descent algorithm for solving certain monotone variational inequalities and shows how this algorithm may be used for solving certain monotone complementarity problems. Convergence is proved under natural monotonicity and smoothness conditions; neither symmetry nor strict monotonicity is required.The author is grateful to two anonymous referees for their very valuable comments on an earlier draft of this paper. 相似文献
9.
《Optimization》2012,61(7):855-871
We introduce a fully explicit method for solving monotone variational inequalities in Hilbert spaces, where orthogonal projections onto the feasible set are replaced by projections onto suitable hyperplanes. We prove weak convergence of the whole generated sequence to a solution of the problem, under only the assumptions of continuity and monotonicity of the operator and existence of solutions. 相似文献
10.
To solve monotone variational inequalities, some existing APPA-based descent methods utilize the iterates generated by the well-known approximate proximal point algorithms (APPA) to construct descent directions. This paper aims at improving these APPA-based descent methods by incorporating optimal step-sizes in both the extra-gradient steps and the descent steps. Global convergence is proved under mild assumptions. The superiority to existing methods is verified both theoretically and computationally. 相似文献
11.
Necessary conditions for optimal controls have been obtained for strongly monotone variational inequalities by the penalty method, Ekeland's Variational Principle, and lower-semicontinuity of set-valued mappings. It has been shown that these conditions are easy to apply and can imply some known necessary conditions. They also yield new optimality conditions. 相似文献
12.
The Douglas–Peaceman–Rachford–Varga operator splitting methods (DPRV methods) are attractive methods for monotone variational
inequalities. He et al. [Numer. Math. 94, 715–737 (2003)] proposed an inexact self-adaptive operator splitting method based on DPRV. This paper relaxes the inexactness
restriction further. And numerical experiments indicate the improvement of this relaxation.
相似文献
13.
Some block iterative methods for solving variational inequalities with nonlinear operators are proposed. Monotone convergence of the algorithms is obtained. Some comparison theorems are also established. Compared with the research work in given by Pao in 1995 for nonlinear equations and research work in given by Zeng and Zhou in 2002 for elliptic variational inequalities, the algorithms proposed in this paper are independent of the boundedness of the derivatives of the nonlinear operator. 相似文献
14.
《Optimization》2012,61(12):2247-2258
ABSTRACTIn this paper, we introduce two new algorithms for solving classical variational inequalities problem with Lipschitz continuous and monotone mapping in real Hilbert space. We modify the subgradient extragradient methods with a new step size, the convergence of algorithms are established without the knowledge of the Lipschitz constant of the mapping. Finally, some numerical experiments are presented to show the efficiency and advantage of the proposed algorithms. 相似文献
15.
Combining rearrangement techniques with Gromov’s proof (via optimal mass transportation) of the 1-Sobolev inequality, we prove a sharp quantitative version of the anisotropic Sobolev inequality on BV(Rn). We also deduce, as a corollary of this result, a sharp stability estimate for the anisotropic 1-log-Sobolev inequality. 相似文献
16.
Parallel splitting augmented Lagrangian methods for monotone structured variational inequalities 总被引:2,自引:0,他引:2
Bing-Sheng He 《Computational Optimization and Applications》2009,42(2):195-212
The typical structured variational inequalities can be interpreted as a system of equilibrium problems with a leader and two
cooperative followers. Assume that, based on the instruction given by the leader, each follower can solve the individual equilibrium
sub-problems in his own way. The responsibility of the leader is to give a more reasonable instruction for the next iteration
loop based on the feedback information from the followers. This consideration leads us to present a parallel splitting augmented
Lagrangian method (abbreviated to PSALM). The proposed method can be extended to solve the system of equilibrium problems
with three separable operators. Finally, it is explained why we cannot use the same technique to develop similar methods for
problems with more than three separable operators. 相似文献
17.
This paper addresses the question of global convergence of descent processes for solving monotone variational inequalities defined on compact subsets ofR
n
. The approach applies to a large class of methods that includes Newton, Jacobi and linearized Jacobi methods as special cases. Furthermore, strict monotonicity of the cost mapping is not required.Research supported by NSERC grant A5789. 相似文献
18.
We propose new interior projection type methods for solving monotone variational inequalities. The methods can be viewed as a natural extension of the extragradient and hyperplane projection algorithms, and are based on using non Euclidean projection-like maps. We prove global convergence results and establish rate of convergence estimates. The projection-like maps are given by analytical formulas for standard constraints such as box, simplex, and conic type constraints, and generate interior trajectories. We then demonstrate that within an appropriate primal-dual variational inequality framework, the proposed algorithms can be applied to general convex constraints resulting in methods which at each iteration entail only explicit formulas and do not require the solution of any convex optimization problem. As a consequence, the algorithms are easy to implement, with low computational cost, and naturally lead to decomposition schemes for problems with a separable structure. This is illustrated through examples for convex programming, convex-concave saddle point problems and semidefinite programming.The work of this author was partially supported by the United States–Israel Binational Science Foundation, BSF Grant No. 2002-2010. 相似文献
19.
Global error bounds for possibly degenerate or nondegenerate monotone affine variational inequality problems are given. The error bounds are on an arbitrary point and are in terms of the distance between the given point and a solution to a convex quadratic program. For the monotone linear complementarity problem the convex program is that of minimizing a quadratic function on the nonnegative orthant. These bounds may form the basis of an iterative quadratic programming procedure for solving affine variational inequality problems. A strong upper semicontinuity result is also obtained which may be useful for finitely terminating any convergent algorithm by periodically solving a linear program.This material is based on research supported by Air Force Office of Scientific Research Grant AFOSR-89-0410 and National Science Foundation Grants CCR-9101801 and CCR-9157632. 相似文献
20.