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Vietnam Journal of Mathematics 40:4(2012)
407-437
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Qualification Conditions and Farkas-Type
Results for Systems Involving Composite Functions
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Nguyen Dinh1 and Tran Hong Mo2
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1International University, VNU-HCMC, Linh Trung
ward, Thu Duc district,
Ho Chi
Minh City, Vietnam
2Tien Giang University, 119 Ap Bac street, My Tho
city,
Tien
Giang Province, Vietnam
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Dedicated to Professor
Phan Quoc Khanh on the occasion of his 65th birthday
Received June 17, 2012
Revised September 26,
2012
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Abstract. We are interested in establishing
necessary and sufficient conditions for the validity of the functional
inequality:
f(x) + g(x)+ (k\circ H)(x) \geq h(x)\quad
\forall x \in X.
Necessary and sufficient conditions for
such an inequality give rise to Farkas-type results. However,
non-asymptotic Farkas-type results often hold under some kind of
qual-ification conditions. In this paper, we firstly propose variants of
such conditions associ-ated to the mentioned inequality in the absence of
convexity and lower semi-continuity. Secondly, variants of necessary and
sufficient conditions of this inequality under our new qualification
conditions are established, which lead to new Farkas-type results in
general setting (without convexity nor lower semi-continuity). It turns out
that these qualification conditions are necessary and sufficient conditions
for these Farkas-type results. The results extend or cover many known
results of Farkas-type for convex systems or systems involving DC functions
in the literature. Alternative-type theorems, set containment results, and
generalized Fenchel-Rockafellar duality formula are obtained as
consequences of the main results.
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2000 Mathematics
Subject Classification. 90C48, 90C46, 49N15, 90C25.
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Keywords.
Functional inequalities, alternative theorems, Farkas-type results for
non-convex systems, set containments.
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Established
by Vietnam Academy of Science and Technology &
Vietnam Mathematical Society
Published by
Springer since January 2013
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