共查询到20条相似文献,搜索用时 15 毫秒
1.
对于一类非光滑函数--光滑函数的有限次极大值复合函数,给出了计算它在一点处Clarke广义梯度呈一个元素的新方法,与以往方法比较,本文的方法不需判别线性不等式组的相容性,因而易于实现。 相似文献
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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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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
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J. S. Treiman 《Journal of Optimization Theory and Applications》1989,62(1):139-150
The B-gradients are a convex set of generalized gradients contained in Clarke's generalized gradients. These gradients retain many of the nice properties of Clarke's generalized gradients. In this paper, necessary conditions for optimality in finite-dimensional perturbed optimization problems are given. A calmness condition is used for a constraint qualification. 相似文献
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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
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带等式约束的光滑优化问题的一类新的精确罚函数 总被引:1,自引:0,他引:1
罚函数方法是将约束优化问题转化为无约束优化问题的主要方法之一. 不包含目标函数和约束函数梯度信息的罚函数, 称为简单罚函数. 对传统精确罚函数而言, 如果它是简单的就一定是非光滑的; 如果它是光滑的, 就一定不是简单的. 针对等式约束优化问题, 提出一类新的简单罚函数, 该罚函数通过增加一个新的变量来控制罚项. 证明了此罚函数的光滑性和精确性, 并给出了一种解决等式约束优化问题的罚函数算法. 数值结果表明, 该算法对于求解等式约束优化问题是可行的. 相似文献
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给出两种两个凸多面体差的表达式,利用这些表达式,可以具体计算这两种凸多面体的差,做为应用讨论了利用拟微分计算Penot微分和Clarke广义梯度,特别讨论了一类非光滑函数,极大值函数的光滑复合。 相似文献
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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
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Yu Xiao 《Applied mathematics and computation》2010,216(6):1868-1879
Aggregate function is a useful smoothing function to the max-function of some smooth functions and has been used to solve minimax problems, linear and nonlinear programming, generalized complementarity problems, etc. The aggregate function is a single smooth but complex function, its gradient and Hessian calculations are time-consuming. In this paper, a truncated aggregate smoothing stabilized Newton method for solving minimax problems is presented. At each iteration, only a small subset of the components in the max-function are aggregated, hence the number of gradient and Hessian calculations is reduced dramatically. The subset is adaptively updated with some truncating criterions, concerning only with computation of function values and not their gradients or Hessians, to guarantee the global convergence and, for the inner iteration, locally quadratic convergence with as few computational cost as possible. Numerical results show the efficiency of the proposed algorithm. 相似文献
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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
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In this paper we study conditions for optimality of a deterministic control problem where the state of the system is required to stop at the boundary. Using the Clarke generalized gradient, we refine the classical verification theorem and show that it is not only sufficient but also necessary for optimality. It is also shown that the solution to the generalized Bellman-Jacobi-Hamilton equation involving the Clarke generalized gradient is unique among the class of regular functions. 相似文献
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Daniel Azagra 《Journal of Mathematical Analysis and Applications》2003,283(1):180-191
We establish approximate Rolle's theorems for the proximal subgradient and for the generalized gradient. We also show that an exact Rolle's theorem for the generalized gradient is completely false in all infinite-dimensional Banach spaces (even when they do not possess smooth bump functions). 相似文献
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For twice smooth functions, the symmetry of the matrix of second partial derivatives is automatic and can be seen as the symmetry of the Jacobian matrix of the gradient mapping. For nonsmooth functions, possibly even extended-real-valued, the gradient mapping can be replaced by a subgradient mapping, and generalized second derivative objects can then be introduced through graphical differentiation of this mapping, but the question of what analog of symmetry might persist has remained open. An answer is provided here in terms of a derivative-coderivative inclusion. 相似文献
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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