### Abstract

This paper develops second-order necessary conditions for nonsmooth infinite-dimensional optimization problems with Banach space-valued equality and real-valued inequality constraints. Another constraint in the form of a closed convex set is also present. The objective function is the maximum over a parameter of functions f(t, z) that are Lipschitz in z and upper semicontinuous in t. The inequality constraints g(s, z) depend on a parameter s. The technique we use is a generalization of that of Dubovitskii and Milyutin. The second-order conditions obtained here are in terms of a certain function σ that disappears when the parameters and a certain set that derives from the given convex set are absent. The presence of the function σ and that set is due to the envelope-like effect discovered by Kawasaki.

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

Pages (from-to) | 1476-1502 |

Number of pages | 27 |

Journal | SIAM Journal on Control and Optimization |

Volume | 32 |

Issue number | 5 |

Publication status | Published - Sep 1994 |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics
- Control and Optimization

### Cite this

*SIAM Journal on Control and Optimization*,

*32*(5), 1476-1502.

**Nonsmooth optimum problems with constraints.** / Páles, Z.; Zeidan, V. M.

Research output: Contribution to journal › Article

*SIAM Journal on Control and Optimization*, vol. 32, no. 5, pp. 1476-1502.

}

TY - JOUR

T1 - Nonsmooth optimum problems with constraints

AU - Páles, Z.

AU - Zeidan, V. M.

PY - 1994/9

Y1 - 1994/9

N2 - This paper develops second-order necessary conditions for nonsmooth infinite-dimensional optimization problems with Banach space-valued equality and real-valued inequality constraints. Another constraint in the form of a closed convex set is also present. The objective function is the maximum over a parameter of functions f(t, z) that are Lipschitz in z and upper semicontinuous in t. The inequality constraints g(s, z) depend on a parameter s. The technique we use is a generalization of that of Dubovitskii and Milyutin. The second-order conditions obtained here are in terms of a certain function σ that disappears when the parameters and a certain set that derives from the given convex set are absent. The presence of the function σ and that set is due to the envelope-like effect discovered by Kawasaki.

AB - This paper develops second-order necessary conditions for nonsmooth infinite-dimensional optimization problems with Banach space-valued equality and real-valued inequality constraints. Another constraint in the form of a closed convex set is also present. The objective function is the maximum over a parameter of functions f(t, z) that are Lipschitz in z and upper semicontinuous in t. The inequality constraints g(s, z) depend on a parameter s. The technique we use is a generalization of that of Dubovitskii and Milyutin. The second-order conditions obtained here are in terms of a certain function σ that disappears when the parameters and a certain set that derives from the given convex set are absent. The presence of the function σ and that set is due to the envelope-like effect discovered by Kawasaki.

UR - http://www.scopus.com/inward/record.url?scp=0028500399&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0028500399&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0028500399

VL - 32

SP - 1476

EP - 1502

JO - SIAM Journal on Control and Optimization

JF - SIAM Journal on Control and Optimization

SN - 0363-0129

IS - 5

ER -