Abstract
We are interested in a nonsmooth minimax programming Problem (SIP). Firstly, we establish the necessary optimality conditions theorems for Problem (SIP) when using the well-known Caratheodory's theorem. Under the Lipschitz(Φ,ρ)-invexity assumptions, we derive the sufficiency of the necessary optimality conditions for the same problem. We also formulate dual and establish weak, strong, and strict converse duality theorems for Problem (SIP) and its dual. These results extend several known results to a wider class of problems.
Highlights
Convexity plays a central role in many aspects of mathematical programming including analysis of stability, sufficient optimality conditions, and duality
Making use of the optimality conditions of the preceding section, we present dual Problem (DI) to the primal one (SIP) and establish weak, strong, and strict converse duality theorems
We have extended the necessary optimality conditions for Problem (SIP) considered in [17] to the nonsmooth case; we have extended the sufficient optimality conditions and dual results of Problem (SIP) addressed by M
Summary
Convexity plays a central role in many aspects of mathematical programming including analysis of stability, sufficient optimality conditions, and duality. Many other concepts of generalized convexity such as (p, r)-invexity [4], (F, ρ)-convexity [5], (F, α, ρ, d)-convexity [6], (C, α, ρ, d)convexity [7], and V-r-invexity [8] have been introduced With these definitions of generalized invexity on the hand, several authors have been interested recently in the optimality conditions and duality results for different classes of minimax programming problems; see [9,10,11,12] for details.
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