In this paper, we use the concept of convexificators to derive enhanced Fritz John optimality condition for nonsmooth optimization problems having equality, inequality and abstract set constraint, where involved functions admit convexificators. This necessary optimality condition provides some more information about the extremal point in terms of converging sequences towards it. Then we employ this optimality condition to study enhanced Karush–Kuhn–Tucker (KKT) condition and to define associated ∂*-pseudonormality and ∂*-quasinormality concepts in terms of convexificators. Later, sufficiency for ∂*-pseudonormality and some more results based on these concepts are investigated.
Accepté le :
Première publication :
Publié le :
DOI : 10.1051/ro/2019082
Keywords: Directional derivatives, convexificators, optimality conditions, constraint qualifications
@article{RO_2021__55_S1_S271_0,
author = {Khare, Abeka and Nath, Triloki},
title = {Improved enhanced {Fritz} {John} condition and constraints qualifications using convexificators},
journal = {RAIRO. Operations Research},
pages = {S271--S288},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
doi = {10.1051/ro/2019082},
mrnumber = {4237382},
zbl = {1475.90104},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ro/2019082/}
}
TY - JOUR AU - Khare, Abeka AU - Nath, Triloki TI - Improved enhanced Fritz John condition and constraints qualifications using convexificators JO - RAIRO. Operations Research PY - 2021 SP - S271 EP - S288 VL - 55 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/ro/2019082/ DO - 10.1051/ro/2019082 LA - en ID - RO_2021__55_S1_S271_0 ER -
%0 Journal Article %A Khare, Abeka %A Nath, Triloki %T Improved enhanced Fritz John condition and constraints qualifications using convexificators %J RAIRO. Operations Research %D 2021 %P S271-S288 %V 55 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/ro/2019082/ %R 10.1051/ro/2019082 %G en %F RO_2021__55_S1_S271_0
Khare, Abeka; Nath, Triloki. Improved enhanced Fritz John condition and constraints qualifications using convexificators. RAIRO. Operations Research, Tome 55 (2021), pp. S271-S288. doi: 10.1051/ro/2019082
[1] , and , Principle of Optimization Theory. Narosa Publishers, India and Alpha Science Publishers (2004).
[2] , Nonlinear Programming. Athena Scientific Publishers (1999). | MR | Zbl
[3] and , Pseudonormality and Lagrange multiplier theory for constrained optimization. J. Optim. Theory App. 114 (2002) 287–343. | MR | Zbl | DOI
[4] , and , Convex Analysis and Optimization. Athena Scientific, Belmont, MA (2003). | MR | Zbl
[5] , A new approach to Lagrange multipliers. Math. Oper. Res. 1 (1976) 165–174. | MR | Zbl | DOI
[6] , Optimization and Nonsmooth Analysis. Wiley Interscience (1983); reprinted as Vol. 5 of Classics Appl. Math. SIAM J. Optim., Philadelphia, PA (1990). | MR | Zbl
[7] , Convexification and Concavification of Positively Homogeneous Functions by the Same Family of Linear Functions. Technical Report, University of Pisa (1994) 1–11.
[8] , Generalized derivatives and nonsmooth optimization, a finite dimensional tour. Soc. Estadística Invest. Oper. 13 (2005) 185–314. | MR | Zbl
[9] and , Convexifactors, generalized convexity and optimality conditions. J. Optim. Theory App. 113 (2002) 41–64. | MR | Zbl | DOI
[10] , and , On the calmness of a class of multifunctions. SIAM J. Optim. 13 (2002) 603–618. | MR | Zbl | DOI
[11] , Optimization Theory: The Finite Dimensional Case. Wiley, New York, NY (1975). | MR | Zbl
[12] , Approximate subdifferentials and applications II. Mathematika 33 (1986) 111–128. | MR | Zbl | DOI
[13] and , Nonsmooth calculus, minimality, and monotonicity of convexificators. J. Optim. Theory App. 101 (1999) 599–621. | MR | Zbl | DOI
[14] , Extremum problems with inequalities as side constraints. In: Studies and Essays. Courant Anniversary Volume. Wiley (interscience) (1948) 187–204. | MR | Zbl
[15] , Constraint qualification and Lagrange multipliers in nondifferentiable programming problems. J. Optim. Theory App. 81 (1994) 533–548. | MR | Zbl | DOI
[16] and , Necessary optimality conditions in terms of convexificators in lipschitz optimization. J. Optim. Theory App. 131 (2006) 429–452. | MR | Zbl | DOI
[17] and , The Fritz john necessary optimality conditions in the presence of equality and inequality constraints. J. Math. Anal. Appl. 17 (1967) 37–47. | MR | Zbl | DOI
[18] , The Lagrange multiplier rule. Am. Math. Monthly. 80 (1973) 922–925. | MR | Zbl | DOI
[19] and , A generalized derivative for calm and stable functions. Differ. Integral Equ. 5 (1992) 433–454. | MR | Zbl
[20] , Metric approximation and necessary optimality condition for general classes of nonsmooth extremal problems. Sov. Math. Dokl. 22 (1980) 526–530. | MR | Zbl
[21] , Variational Analysis and Generalized Differentiation I: Basic Theory. Springer, New York (2006). | MR | Zbl
[22] and , On non-convex subdifferential calculus in Banach spaces. J. Convex Anal. 2 (1995) 211–228. | MR | Zbl
[23] , and , Fréchet subdifferential calculus and optimality conditions in nondifferentiable programming. Optimization 55 (2006) 685–708. | MR | Zbl | DOI
[24] and , The relation between pseudonormality and quasiregularity in constrained optimization. Optim. Methods Softw. 19 (2004) 493–506. | MR | Zbl | DOI
[25] , The Linear nonconvex generalized gradient and Lagrange multipliers. SIAM J. Optim. 5 (1995) 670–680. | MR | Zbl | DOI
[26] and , A sharp langrage multiplier rule for nonsmooth mathematical programming problems involving equality constraints. SIAM J. Optim. 10 (2000) 1136–1148. | MR | Zbl | DOI
[27] and , Enhanced Karush–Kuhn–Tucker condition and weaker constraint qualifications. Math. Program. Ser. B 139 (2013) 353–381. | MR | Zbl | DOI
Cité par Sources :
