TY  - JOUR
T1  - ON THE WEAK SUBDIFFERENTIAL, AUGMENTED NORMAL CONES AND DUALITY IN NONCONVEX OPTIMIZATION
TT  - ON THE WEAK SUBDIFFERENTIAL, AUGMENTED NORMAL CONES AND DUALITY IN NONCONVEX OPTIMIZATION
AU  - Bila, Samet
AU  - Kasımbeyli, Refail
PY  - 2025
DA  - February
Y2  - 2025
DO  - 10.20290/estubtdb.1632350
JF  - Eskişehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B - Teorik Bilimler
JO  - Estuscience - Theory
PB  - Eskişehir Teknik Üniversitesi
WT  - DergiPark
SN  - 2667-419X
SP  - 67
EP  - 76
VL  - 13
IS  - 1
LA  - en
AB  - This article studies the properties of the weak subdifferential for nonsmooth and nonconvex analysis studied. This study presents a formulation that is directly involved in convex analysis carried out in the nonconvex case. In this work, we present a theory that applies epigraphs to obtain augmented normal cones. The perturbation function plays a crucial role in establishing optimality conditions. This study demonstrates that positively homogeneous and lower semicontinuous functions are weakly subdifferentiable. Moreover, under specific conditions related to the objective function, the constraint function, and the feasible set, we show that the perturbation function is positively homogeneous. Thus we obtain a zero duality gap condition by implementing conditions on the objective function, constraint functions, and the set S.
KW  - Operations Research
KW  - Nonconvex Optimization
KW  - Weak Subdifferential
KW  - Augmented Normal Cone
N2  - This article studies the properties of the weak subdifferential for nonsmooth and nonconvex analysis studied. This study presents a formulation that is directly involved in convex analysis carried out in the nonconvex case. In this work, we present a theory that applies epigraphs to obtain augmented normal cones. The perturbation function plays a crucial role in establishing optimality conditions. This study demonstrates that positively homogeneous and lower semicontinuous functions are weakly subdifferentiable. Moreover, under specific conditions related to the objective function, the constraint function, and the feasible set, we show that the perturbation function is positively homogeneous. Thus we obtain a zero duality gap condition by implementing conditions on the objective function, constraint functions, and the set S.
CR  - [1]	Azimov AY, Gasimov RN. On weak conjugacy, weak subdifferentials and duality with zero gap in nonconvex optimization, International Journal of Applied Mathematics, 1, 1999, pp. 171–192.
CR  - [2]	Bila S, Kasimbeyli R. On the some sum rule for the weak subdifferential and some properties of augmented normal cones, Journal of Nonlinear and Convex Analysis, 24(10), 2023, pp. 2239–2257.
CR  - [3]	Borwein JM, Lewis AS. Convex Analysis and Nonlinear Optimization, CMS Books in Mathematics, Springer Science+Business Media, Inc., New York, 2006.
CR  - [4]	Clarke FH. Necessary conditions for nonsmooth problems in optimal control and the calculus of variations, Thesis, University of Washington, Seattle, 1973.
CR  - [5 ]	Clarke FH. Generalized gradients and applications, Trans. Amer. Math. Soc., 205, pp. 247–262, 1975.
CR  - [6]	Ekeland I, Temam R. Convex Analysis and Variational Problems, Elsevier, 1976.
CR  - [7]	Gasimov RN. Duality in nonconvex optimization, Ph.D. Dissertation, Department of Operations Research and Mathematical Modeling, Baku State University, Baku, 1992.
CR  - [8]	Gasimov RN. Augmented Lagrangian duality and nondifferentiable optimization methods in nonconvex programming, J. Global Optimization, 24, 2002, pp. 187–203.
CR  - [9]	Kasimbeyli R, Mammadov M. Optimality conditions in nonconvex opti-mization via weak subdifferentials, Nonlinear Analysis: Theory, Methods and Applications, 74, 2011, pp. 2534–2547.
CR  - [10]	Rockafellar RT. Convex analysis, Princeton University Press, Princeton, 1970.
CR  - [11]	Rockafeller RT. Convex analysis and dual extremum problems. Thesis, Harvard, 1963
UR  - https://doi.org/10.20290/estubtdb.1632350
L1  - https://dergipark.org.tr/tr/download/article-file/4575818
ER  -