### Abstract

Many interesting real functions on Euclidean space are differentiable almost everywhere. All Lipschitz functions have this property, but so, for example, does the spectral abscissa of a matrix (a non-Lipschitz function). In practice, the gradient is often easy to compute. We investigate to what extent we can approximate the Clarke subdifferential of such a function at some point by calculating the convex hull of some gradients sampled at random nearby points.

Original language | English (US) |
---|---|

Pages (from-to) | 567-584 |

Number of pages | 18 |

Journal | Mathematics of Operations Research |

Volume | 27 |

Issue number | 3 |

State | Published - Aug 2002 |

### Fingerprint

### Keywords

- Bundle method
- Clarke subdifferential
- Eigenvalue optimization
- Generalized gradient
- Nonsmooth analysis
- Spectral abscissa
- Stochastic gradient

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics
- Management Science and Operations Research

### Cite this

*Mathematics of Operations Research*,

*27*(3), 567-584.

**Approximating subdifferentials by random sampling of gradients.** / Burke, J. V.; Lewis, A. S.; Overton, M. L.

Research output: Contribution to journal › Article

*Mathematics of Operations Research*, vol. 27, no. 3, pp. 567-584.

}

TY - JOUR

T1 - Approximating subdifferentials by random sampling of gradients

AU - Burke, J. V.

AU - Lewis, A. S.

AU - Overton, M. L.

PY - 2002/8

Y1 - 2002/8

N2 - Many interesting real functions on Euclidean space are differentiable almost everywhere. All Lipschitz functions have this property, but so, for example, does the spectral abscissa of a matrix (a non-Lipschitz function). In practice, the gradient is often easy to compute. We investigate to what extent we can approximate the Clarke subdifferential of such a function at some point by calculating the convex hull of some gradients sampled at random nearby points.

AB - Many interesting real functions on Euclidean space are differentiable almost everywhere. All Lipschitz functions have this property, but so, for example, does the spectral abscissa of a matrix (a non-Lipschitz function). In practice, the gradient is often easy to compute. We investigate to what extent we can approximate the Clarke subdifferential of such a function at some point by calculating the convex hull of some gradients sampled at random nearby points.

KW - Bundle method

KW - Clarke subdifferential

KW - Eigenvalue optimization

KW - Generalized gradient

KW - Nonsmooth analysis

KW - Spectral abscissa

KW - Stochastic gradient

UR - http://www.scopus.com/inward/record.url?scp=0036672502&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036672502&partnerID=8YFLogxK

M3 - Article

VL - 27

SP - 567

EP - 584

JO - Mathematics of Operations Research

JF - Mathematics of Operations Research

SN - 0364-765X

IS - 3

ER -