# On weak sharp minima for a special class of nonsmooth functions

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2000)

- Volume: 20, Issue: 2, page 195-207
- ISSN: 1509-9407

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topMarcin Studniarski. "On weak sharp minima for a special class of nonsmooth functions." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 20.2 (2000): 195-207. <http://eudml.org/doc/271556>.

@article{MarcinStudniarski2000,

abstract = {We present a characterization of weak sharp local minimizers of order one for a function f: ℝⁿ → ℝ defined by $f(x): = max\{f_i(x)| i = 1,...,p\}$, where the functions $f_i$ are strictly differentiable. It is given in terms of the gradients of $f_i$ and the Mordukhovich normal cone to a given set on which f is constant. Then we apply this result to a smooth nonlinear programming problem with constraints.},

author = {Marcin Studniarski},

journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},

keywords = {weak sharp minimizer of order one; maximum function; strictly differentiable function; normal cone; weak sharp local minimizers; strictly differentiable functions; Mordukhovich normal cone},

language = {eng},

number = {2},

pages = {195-207},

title = {On weak sharp minima for a special class of nonsmooth functions},

url = {http://eudml.org/doc/271556},

volume = {20},

year = {2000},

}

TY - JOUR

AU - Marcin Studniarski

TI - On weak sharp minima for a special class of nonsmooth functions

JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization

PY - 2000

VL - 20

IS - 2

SP - 195

EP - 207

AB - We present a characterization of weak sharp local minimizers of order one for a function f: ℝⁿ → ℝ defined by $f(x): = max{f_i(x)| i = 1,...,p}$, where the functions $f_i$ are strictly differentiable. It is given in terms of the gradients of $f_i$ and the Mordukhovich normal cone to a given set on which f is constant. Then we apply this result to a smooth nonlinear programming problem with constraints.

LA - eng

KW - weak sharp minimizer of order one; maximum function; strictly differentiable function; normal cone; weak sharp local minimizers; strictly differentiable functions; Mordukhovich normal cone

UR - http://eudml.org/doc/271556

ER -

## References

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- [8] M. Studniarski, Second-order necessary conditions for optimality in nonsmooth nonlinear programming, J. Math. Anal. Appl. 154 (1991), 303-317. Zbl0725.90085
- [9] M. Studniarski, Characterizations of strict local minima for some nonlinear programming problems, Nonlinear Anal. 30 (1997), 5363-5367 (Proc. 2nd World Congress of Nonlinear Analysts). Zbl0914.90243
- [10] M. Studniarski, Characterizations of weak sharp minima of order one in nonlinear programming, System Modelling and Optimization (Detroit, MI, 1997), 207-215, Chapman & Hall/CRC Res. Notes Math., 396, 1999. Zbl0955.90130
- [11] M. Studniarski and M. Studniarska, New characterizations of weak sharp and strict local minimizers in nonlinear programming, Preprint 1999/15, Faculty of Mathematics, University of ódź.
- [12] M. Studniarski and D.E. Ward, Weak sharp minima: characterizations and sufficient conditions, SIAM J. Control Optim. 38 (1999), 219-236. Zbl0946.49011

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