Generalized second-order directional derivatives and optimization with C<sup>1,1</sup> functions

Yang, Xiaoqi; Jeyakumar, V.

doi:10.1080/02331939208843851

Cited by 64 publications

(41 citation statements)

References 19 publications

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“…In another approach to second-order optimality conditions Yang and Jeyakumar studied convex composite minimization problems in [11]. Furthermore, in [19] the same authors introduced a new set-valued generalized Hessian for C 1,1 functions contained in the generalized Hessian of Cominetti and Correa that enabled them to sharpen the necessary optimality conditions given in [9] and [6]. More recently, in [16] Páles and Zeidan compared the second-order directional derivatives mentioned above and defined a new generalized Hessian ∂ 2 PZ f (x), which in the finitedimensional case reduces to a certain set of symmetric matrices.…”

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confidence: 98%

Stability in Generalized Differentiability Based on a Set Convergence Principle

Ovcharova

Gwinner

2007

Set-Valued Anal

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In this paper, we are concerned with epiconvergent sequences of nonsmooth functions. From a general principle of upper set convergence of set-valued maps we derive stability results for various objects in generalized differentiability. In particular, we establish stability results for the Clarke generalized gradient of locally Lipschitz functions, respectively for the generalized Hessian of C 1,1 functions.

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confidence: 98%

Stability in Generalized Differentiability Based on a Set Convergence Principle

Ovcharova

Gwinner

2007

Set-Valued Anal

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“…We note that the LC 1 problem is also known as C 1,1 data in [9], where second-order analysis of the underlying function is conducted. For further development along this line, see [30,31] and the references therein.…”

Section: Basic Concepts Consider the Mappingmentioning

confidence: 99%

“…Another interesting case is when f is an SC 1 function, i.e., f is not only an LC 1 function, but also its derivative function is semismooth. For both cases, we will show that (f • λ) is an LC and SC 1 functions is that they constitute a class of minimization problems which can be solved by Newton-type methods (see [6,20,22]) and by penalty-type methods (see [31,30]). …”

Section: Introductionmentioning

confidence: 99%

Semismoothness of Spectral Functions

Yang

2003

SIAM J. Matrix Anal. & Appl.

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Abstract. Any spectral function can be written as a composition of a symmetric function f : R n → R and the eigenvalue function λ(·) : S → R n , often denoted by (f • λ), where S is the subspace of n × n symmetric matrices. In this paper, we present some nonsmooth analysis for such spectral functions. Our main results are ( and only if f is LC 1 , and (c) (f • λ) is SC 1 if and only if f is SC 1 . Result (a) is complementary to a known (negative) fact that (f • λ) might not be directionally differentiable if f is directionally differentiable only. Results (b) and (c) are particularly useful for the solution of LC 1 and SC 1 minimization problems which often can be solved by fast (generalized) Newton methods. Our analysis makes use of recent results on continuously differentiable spectral functions as well as on nonsmooth symmetric-matrix-valued functions.

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“…Optimality conditions for C 1,1 scalar functions have been studied by many authors and numerical methods have been proposed too (see [7,9,10,15,17,18,21,22,23]). In [7], Guerraggio and Luc have given necessary and sufficient optimality conditions for vector optimization problems expressed by means of ∂ 2 f(x).…”

Section: )mentioning

confidence: 99%