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Mixed-Type Duality on Nonsmooth Minimax Fractional Programming Involving Exponential (p, r)-Invexity

Considering a nonsmooth minimax fractional programming problem involving exponential (p, r)-invexity, we construct a mixed-type dual problem, which is performed by an incomplete Lagrangian dual model. This mixed-type dual model involves the Wolfe type dual and Mond-Weir type dual as the special cases under exponential (p, r)-invexity. We establish the mixed-type duality problem with conditions for exponential (p, r)-invexity and prove that the optimal values of the primary problem and the mixed-type duality problem have no duality gap under the framwork of exponential (p, r)-invexity.