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1.
1.IntroductionConsidersmoothcompositionsofmax-typefunctionsoftheform:f(x)=g(x,aestfij(x),'',,T?:fmj(x)),(1.1)wherexER",Ji,i~1,'',marefiniteindexsets,gandfij,jEJi,i=1,'',marecontinuouslydifferentiableonRill 71andR;'respectively.Thisclassofnonsmoothfunct… 相似文献
2.
Zsolt Páles 《Journal of Mathematical Analysis and Applications》2008,344(1):55-75
The extension to infinite dimensional domains of Clarke's generalized Jacobian is the focus of this paper. First, a generalization of a Fabian-Preiss theorem to the infinite dimensional setting is obtained. As a consequence, a new formula relating the Clarke's generalized Jacobians corresponding to finite dimensional spaces K, L with K⊆L is established. Furthermore, in the infinite dimensional case, basic properties pertaining the generalized Jacobian are developed and then an identification of this set-valued map is produced. Applications of these results in the form of chain rules including sum and product rules, and a computational formula for continuous selections are derived. 相似文献
3.
Norma Ortiz 《Journal of Mathematical Analysis and Applications》2005,305(2):513-527
This note uses Clarke's decoupling technique to obtain necessary conditions for the generalized problem of Bolza with Lipschitz continuously varying delay in both the state and velocity variables. 相似文献
4.
S. Carl 《Journal of Differential Equations》2003,191(1):206-233
In this paper we consider an initial boundary value problem for a parabolic inclusion whose multivalued nonlinearity is characterized by Clarke's generalized gradient of some locally Lipschitz function, and whose elliptic operator may be a general quasilinear operator of Leray-Lions type. Recently, extremality results have been obtained in case that the governing multivalued term is of special structure such as, multifunctions given by the usual subdifferential of convex functions or subgradients of so-called dc-functions. The main goal of this paper is to prove the existence of extremal solutions within a sector of appropriately defined upper and lower solutions for quasilinear parabolic inclusions with general Clarke's gradient. The main tools used in the proof are abstract results on nonlinear evolution equations, regularization, comparison, truncation, and special test function techniques as well as tools from nonsmooth analysis. 相似文献
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7.
Kazimierz Malanowski 《Mathematical Programming》1985,33(3):352-361
We consider a family of convex programming problems that depend on a vector parameter, characterizing those values of parameters at which solutions and associated Lagrange multipliers are Gâteaux differentiable.These results are specialized to the problem of the metric projection onto a convex set. At those points where the projection mapping is not differentiable the form of Clarke's generalized derivative of this mapping is derived. 相似文献
8.
本文以[7]的基本概念为基础,并根据Clarke的广义导数[1],以及Lasotra和Strauss[6]的多值函数f(x)的广义微分Df(x)的定义.从而建立了区域函数F(x)的广义导数DF=∪∩{G(x)?B(R),?x∈B(R);G(x)=Fx=F(x)}讨论了区域函数F(x)的广义导数的存在性;建立了区域函数的广义Fréchet导数存在的必要充分条件. 相似文献
9.
WANG Yun & ZHANG LiWei College of Information Sciences Engineering Sh ong Agricultural University Tai'an China 《中国科学 数学(英文版)》2010,(4)
Based on the differential properties of the smoothing metric projector onto the second-order cone,we prove that,for a locally optimal solution to a nonlinear second-order cone programming problem,the nonsingularity of the Clarke's generalized Jacobian of the smoothing Karush-Kuhn-Tucker system,constructed by the smoothing metric projector,is equivalent to the strong second-order sufficient condition and constraint nondegeneracy,which is in turn equivalent to the strong regularity of the Karush-Kuhn-Tucker p... 相似文献
10.
An optimal-control problem of a variational inequality of the elliptic type is investigated. The problem is approximated by a family of finite-dimensional problems and the convergence of the approximated optimal controls is shown. The finite-dimensional problems, being nonsmooth, are to be optimized by a bundle algorithm, which requires an element of Clarke's generalized gradient of the minimized function. A simple algorithm which yields this element is proposed. Some numerical experiments with a simple model problem have also been carried out. 相似文献
11.
In this paper generalized type-I, generalized quasi type-I, generalized pseudo type-I and other related functions over cones
are defined for a vector minimization problem. Sufficient optimality conditions are studied for this problem using Clarke’s
generalized gradients. A Mond-Weir type dual is formulated and weak and strong duality results are established. 相似文献
12.
《Optimization》2012,61(2):85-104
For nonlinear programs with non-Lipschitz. generalized con\ex data functions. we develop various explicit first-order sufficient and /or necessary optimality conditions. These involve a natural generalization of the well known Karush-Kuhn-Tucker conditions, but with the familiar gradient condition modified so as to involve asymptotic (i.e. singular), as well as ordinary, Clarke-Rockafellar generalized gradients. In this way we cover situations in which the sets of ordinary generalized gradients are empty or unbounded, which can occur even at points where the functions are finite everywhere nearby. Along wit the use of asymptotic gradients, the novelty here lies in the identification of weak hypotheses on the data functions suitable for deriving such optimality results. In particular. the notions of protoconvexity is found to play a central role. along with the more familiar notions of quasiconvexity and’ pseudoconvexity 相似文献
13.
P. Recht 《Mathematical Methods of Operations Research》1992,36(3):201-210
In this paper we present a concept of the construction of generalized gradients by considering a development of directional derivatives into spherical harmonics. This leads to a derivation system as a system of generalized partial derivatives. Necessary conditions for local extrema for a broad class of not necessarily differentiable function can be given and a characterization of points of differentiability can be proved by using generalized gradients. 相似文献
14.
特征值问题(1.1)的敏度分析,主要是指研究特征值λ(p)对于矩阵所含变数p_1,…,p_N的偏导数;这一研究,在结构动力优化等应用中,具有重要意义(见[2]、[4]、[5]、[14]).对于单特征值,以及对于矩阵A(p)与B(p)只含1个变数(即N=1) 相似文献
15.
We consider the Dirichlet boundary value problem for an elliptic inclusion governed by a quasilinear elliptic operator of Leray-Lions type and a multivalued term which is given by the difference of Clarke's generalized gradient of some locally Lipschitz function and the subdifferential of some convex function. Problems of this kind arise, e.g., in mechanical models described by nonconvex and nonsmooth energy functionals that result from nonmonotone, multivalued constitutive laws. Our main goal is to characterize the solution set of the problem under consideration. In particular we are going to prove that the solution set possesses extremal elements with respect to the underlying natural partial ordering of functions, and that the solution set is compact. The main tools used in the proofs are abstract results on pseudomonotone operators, truncation, and special test function techniques, Zorn's lemma as well as tools from nonsmooth analysis. 相似文献
16.
Convergence of Newton's method for convex best interpolation 总被引:7,自引:0,他引:7
Summary. In this paper, we consider the problem of finding a convex function which interpolates given points and has a minimal norm of the second derivative. This problem reduces to a system of equations involving semismooth functions. We study a Newton-type
method utilizing Clarke's generalized Jacobian and prove that its local convergence is superlinear. For a special choice of
a matrix in the generalized Jacobian, we obtain the Newton method proposed by Irvine et al. [17] and settle the question of
its convergence. By using a line search strategy, we present a global extension of the Newton method considered. The efficiency
of the proposed global strategy is confirmed with numerical experiments.
Received October 26, 1998 / Revised version received October 20, 1999 / Published online August 2, 2000 相似文献
17.
A. Bihain 《Journal of Optimization Theory and Applications》1984,44(4):545-568
In this paper, we present an implementable algorithm to minimize a nonconvex, nondifferentiable function in
m
. The method generalizes Wolfe's algorithm for convex functions and Mifflin's algorithm for semismooth functions to a broader class of functions, so-called upper semidifferentiable. With this objective, we define a new enlargement of Clarke's generalized gradient that recovers, in special cases, the enlargement proposed by Goldstein. We analyze the convergence of the method and discuss some numerical experiments.The author would like to thank J. B. Hiriart-Urruty (Toulouse) for having provided him with Definition 2.1 and the referees for their constructive remarks about a first version of the paper. 相似文献
18.
In this paper, a method to approximate the directions of Clarke's generalized gradient of the upper level function for the
demand adjustment problem on traffic networks is presented. Its consistency is analyzed in detail. The theoretical background
on which this method relies is the known property of proximal subgradients of approximating subgradients of proximal bounded
and lower semicountinuous functions using the Moreau envelopes. A double penalty approach is employed to approximate the proximal
subgradients provided by these envelopes. An algorithm based on partial linearization is used to solve the resulting nonconvex
problem that approximates the Moreau envelopes, and a method to verify the accuracy of the approximation to the steepest descent
direction at points of differentiability is developed, so it may be used as a suitable stopping criterion. Finally, a set
of experiments with test problems are presented, illustrating the approximation of the solutions to a steepest descent direction
evaluated numerically.
Research supported under Spanish CICYT project TRA99-1156-C02-02. 相似文献
19.
V. V. Tokarchuk 《Journal of Applied Mathematics and Mechanics》1989,53(6):708-715
The Langevin dynamics and fluctuational-dissipative relationships for the hydrodynamic fluctuations for systems which are described in the third Barnett order with respect to the gradients of the hydrodynamic variables are generalized on the basis of a kinetic approach. 相似文献
20.
讨论了不等式约束优化问题中拟微分形式下Fritz John必要条件与 Clarke广义梯度形式下Fritz John必要条件的关系.在较弱条件下给出了具有等式与不等式约束条件的两个Lagrange乘子形式的最优性必要条件,在这两个条件中等式约束函数的拟微分和Clarke广义梯度分别被使用。 相似文献