Weak Fenchel and weak Fenchel-Lagrange conjugate duality for nonconvex scalar optimization problems. (English) Zbl 1269.90084
Summary: In this work, by using weak conjugate maps given in [A. Y. Azimov and R. N. Kasimov, Int. J. Appl. Math. 1, No. 2, 171–192 (1999; Zbl 1171.90514)], the weak Fenchel conjugate dual problem, \((D_F^w)\), and the weak Fenchel Lagrange conjugate dual problem \((D_{FL}^w)\) are constructed. Necessary and sufficient conditions for strong duality for the \((D_F^w)\), \((D_{FL}^w)\) and primal problem are given. Furthermore, relations among the optimal objective values of dual problems constructed by using augmented Lagrangian in [loc. cit], the \((D_F^w)\), \((D_{FL}^w)\) dual problems and primal problem are examined. Lastly, necessary and sufficient optimality conditions for the primal and the dual problems \((D_F^w)\) and \((D_{FL}^w)\) are established.
MSC:
| 90C26 | Nonconvex programming, global optimization |
| 90C46 | Optimality conditions and duality in mathematical programming |
| 49J52 | Nonsmooth analysis |
Keywords:
nonconvex analysis; nonsmooth analysis; weak subdifferentials; lower Lipschitz functions; nonconvex optimizationCitations:
Zbl 1171.90514References:
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