Asynchronous relaxed iterative methods for solving linear systems of equations

I.AsynchronousRelaxedIterativeMethodsMoreandmorelargeorverylargescaleproblemsofscientificcomputationhavebeenproposedandarebeingproposedinmanyimportantfieldsofscienceandengineering.Manyoftheseproblemsresultinsolvinglargeorverylargelinearalgebraicsystemsofe…     

Numerical approximation of nonconvex optimal control problems defined by parabolic equations

In this paper, we consider an optimal control problem for distributed systems governed by parabolic equations. The state equations are nonlinear in the control variable; the constraints and the cost functional are generally nonconvex. Relaxed controls are used to prove existence and derive necessary conditions for optimality. To compute optimal controls, a descent method is applied to the resulting relaxed problem. A numerical method is also given for approximating a special class of relaxed controls, notably those obtained by the descent method. Convergence proofs are given for both methods, and a numerical example is provided.     

An exact penalty function algorithm for optimal control problems with control and terminal equality constraints,part 2

In Part 1 of this paper, implementable and conceptual versions of an algorithm for optimal control problems with control constraints and terminal equality constraints were presented. It was shown that anyL accumulation points of control sequences generated by the algorithms satisfy necessary conditions of optimality. Since such accumulation points need not exist, it is shown in this paper that control sequences generated by the algorithms always have accumulation points in the sense of control measure, and these accumulation points satisfy optimality conditions for the corresponding relaxed control problem.This work was supported by the United Kingdom Science Research Council, by the US Army Research Office, Contract No. DAA-29-73-C-0025, and by the National Science Foundation, Grant No. ENG-73-08214-A01.     

Equilibrium problems with generalized monotone mapping and its applications

In this paper, we deal with invex equilibrium problems (IEPs for short). When the constraint set is compact convex, an existence result of solutions is obtained by Schauder's fixed point theorem. If the constraint set is closed invex, we introduce the concept of relaxed α–η pseudomonotone mappings and prove some existence results of solutions for the (IEPs), which extend and generalize several well‐known results in many respects. Moreover, we present that if the IEPs are applied to hemivariational inequalities problems, the conditions can be weakened. Copyright © 2015 John Wiley & Sons, Ltd.     

A relaxed self-adaptive CQ algorithm for the multiple-sets split feasibility problem

The multiple-sets split feasibility problem (MSFP) is to find a point belongs to the intersection of a family of closed convex sets in one space, such that its image under a linear transformation belongs to the intersection of another family of closed convex sets in the image space. Many iterative methods can be employed to solve the MSFP. Jinling Zhao et al. proposed a modification for the CQ algorithm and a relaxation scheme for this modification to solve the MSFP. The strong convergence of these algorithms are guaranteed in finite-dimensional Hilbert spaces. Recently López et al. proposed a relaxed CQ algorithm for solving split feasibility problem, this algorithm can be implemented easily since it computes projections onto half-spaces and has no need to know a priori the norm of the bounded linear operator. However, this algorithm has only weak convergence in the setting of infinite-dimensional Hilbert spaces. In this paper, we introduce a new relaxed self-adaptive CQ algorithm for solving the MSFP where closed convex sets are level sets of some convex functions such that the strong convergence is guaranteed in the framework of infinite-dimensional Hilbert spaces. Our result extends and improves the corresponding results.     

The relaxed game chromatic number of outerplanar graphs

The (r,d)‐relaxed coloring game is played by two players, Alice and Bob, on a graph G with a set of r colors. The players take turns coloring uncolored vertices with legal colors. A color α is legal for an uncolored vertex u if u is adjacent to at most d vertices that have already been colored with α, and every neighbor of u that has already been colored with α is adjacent to at most d – 1 vertices that have already been colored with α. Alice wins the game if eventually all the vertices are legally colored; otherwise, Bob wins the game when there comes a time when there is no legal move left. We show that if G is outerplanar then Alice can win the (2,8)‐relaxed coloring game on G. It is known that there exists an outerplanar graph G such that Bob can win the (2,4)‐relaxed coloring game on G. © 2004 Wiley Periodicals, Inc. J Graph Theory 46:69–78, 2004     

Banach空间内一类含松弛-(H,η)-调算子的广义混合拟似变分包含组

在Banach空间内,引入和研究了一类新的含松弛-(H,η)-单调算子的广义混合拟似变分包含组.利用松弛-(H,η)-单调算子的豫解算子技巧,给出了求这类广义混合拟似变分包含组近似解的一个新逼近算法,并证明了由这个新算法生成的迭代序列强收敛于该变分包含组的精确解.文章结果改进与推广了近期文献中的相应结果.     

求解分裂可行问题的一种半空间投影算法

本文研究了分裂可行问题.利用松弛投影的方法,获得了分裂可行问题最优值点,推广了已有文献中的有关结果.     

Strong stability of nonlinear semigroups with weak dissipation and non-compact resolvent–applications to structural acoustics

We consider asymptotic stability, in the strong topology, of a nonlinear coupled system of partial differential equations (PDEs) arising in structural–acoustic interactions. The coupling involves parabolic and hyperbolic dynamics with interaction on an interface–a manifold of lower dimension. The distinctive feature of the model is that the resolvent associated with the generator governing the evolution is not compact and the dissipation considered is ‘weak’. Thus, strong stability is not to be generally expected. In linear problems this difficulty is circumvented by the use of Taubrien theorems and spectral analysis [W. Arendt and C.J.K. Batty, Tauberian theorems and stability of one-parameter semi-groups, Trans. Amer. Math. Soc. 306(8) (1988), pp. 837–852, Y.I. Lyubich and V.Q. Phong, Asymptotic stability of linear differential equations ain Banach spaces, Studia Math., LXXXXVII, (1988), pp. 37–42, G.M. Sklyar, On the maximal asymptotica for linear equations in Banach spaces, 2009]. However these methods are not applicable to nonlinear dynamics.

In this article, we present an approach to strong stability that is applicable to nonlinear semigroups governed by multivalued generators with non-compact resolvents. The method relies on a suitable relaxation of Lasalle invariance principle [J.P. LaSalle, Stability theory and invariance principles, in Dynamical Systems, Vol. 1, L. Cerasir, J.K. Hale, J.P. LaSalle, eds., Academic Press, New York, 1976, pp. 211–222] which then requires appropriate unique continuation theorems along with a string of a-priori PDE estimates specific to parabolic–hyperbolic coupled systems.