On weak sharp minima for a special class of nonsmooth functions

Marcin Studniarski

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

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

Abstract

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We present a characterization of weak sharp local minimizers of order one for a function f: ℝⁿ → ℝ defined by f ( x ) : = m a x 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.

How to cite

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Marcin 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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  1. [1] J.F. Bonnans and A. Ioffe, Second-order sufficiency and quadratic growth for nonisolated minima, Math. Oper. Res. 20 (1995), 801-817. Zbl0846.90095
  2. [2] J.M. Borwein, Stability and regular points of inequality systems, J. Optim. Theory Appl. 48 (1986), 9-52. Zbl0557.49020
  3. [3] J.V. Burke and M.C. Ferris, Weak sharp minima in mathematical programming, SIAM J. Control Optim. 31 (1993), 1340-1359. Zbl0791.90040
  4. [4] F.H. Clarke, Yu.S. Ledyaev, R.J. Stern and P.R. Wolenski, Nonsmooth Analysis and Control Theory, Springer-Verlag, New York 1998. Zbl1047.49500
  5. [5] D. Pallaschke and S. Rolewicz, Foundations of Mathematical Optimization. Convex Analysis without Linearity, Kluwer Academic Publishers, Dordrecht 1997. Zbl0887.49001
  6. [6] R.T. Rockafellar and R.J-B. Wets, Variational Analysis, Springer-Verlag, Berlin 1998. Zbl0888.49001
  7. [7] M. Studniarski, Necessary and sufficient conditions for isolated local minima of nonsmooth functions, SIAM J. Control Optim. 24 (1986), 1044-1049. Zbl0604.49017
  8. [8] M. Studniarski, Second-order necessary conditions for optimality in nonsmooth nonlinear programming, J. Math. Anal. Appl. 154 (1991), 303-317. Zbl0725.90085
  9. [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. [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. [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. [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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