### Abstract

In this paper we extend for a locally Lipschitz function the notion of Clarke's generalized Jacobian to the setting where the domain lies in an infinite dimensional normed space. When the function is real-valued this notion reduces to the Clarke's generalized gradient. Using this extension, we obtain an exact smooth-nonsmooth chain rule from which the sum rule and the product rule follow. Also an exact formula for the generalized Jacobian of piecewise differentiable functions will be provided.

Original language | English |
---|---|

Pages (from-to) | 433-454 |

Number of pages | 22 |

Journal | Journal of Convex Analysis |

Volume | 14 |

Issue number | 2 |

Publication status | Published - 2007 |

### Fingerprint

### Keywords

- Chain rule
- Generalized jacobian
- Piecewise smooth functions
- Sum rule

### ASJC Scopus subject areas

- Mathematics(all)
- Analysis

### Cite this

*Journal of Convex Analysis*,

*14*(2), 433-454.

**Infinite dimensional clarke generalized Jacobian.** / Páles, Z.; Zeidan, Vera.

Research output: Contribution to journal › Article

*Journal of Convex Analysis*, vol. 14, no. 2, pp. 433-454.

}

TY - JOUR

T1 - Infinite dimensional clarke generalized Jacobian

AU - Páles, Z.

AU - Zeidan, Vera

PY - 2007

Y1 - 2007

N2 - In this paper we extend for a locally Lipschitz function the notion of Clarke's generalized Jacobian to the setting where the domain lies in an infinite dimensional normed space. When the function is real-valued this notion reduces to the Clarke's generalized gradient. Using this extension, we obtain an exact smooth-nonsmooth chain rule from which the sum rule and the product rule follow. Also an exact formula for the generalized Jacobian of piecewise differentiable functions will be provided.

AB - In this paper we extend for a locally Lipschitz function the notion of Clarke's generalized Jacobian to the setting where the domain lies in an infinite dimensional normed space. When the function is real-valued this notion reduces to the Clarke's generalized gradient. Using this extension, we obtain an exact smooth-nonsmooth chain rule from which the sum rule and the product rule follow. Also an exact formula for the generalized Jacobian of piecewise differentiable functions will be provided.

KW - Chain rule

KW - Generalized jacobian

KW - Piecewise smooth functions

KW - Sum rule

UR - http://www.scopus.com/inward/record.url?scp=34250829019&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34250829019&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:34250829019

VL - 14

SP - 433

EP - 454

JO - Journal of Convex Analysis

JF - Journal of Convex Analysis

SN - 0944-6532

IS - 2

ER -