On nondifferentiable fractional minimax programming

Necessary and sufficient optimality conditions are derived for a nondifferentiable fractional minimax programming problem: Minimizex∈Rⁿ [supy∈Y {(f(x,y) + (x^TBx)^1/2)/(h(x,y) − (x^TDx)^1/2)}] subject to g(x) ⩽ 0. where Y is a compact subset of Rⁿ; f(·,·) : Rⁿ × R^m → R and h(·,·) : Rⁿ × R^m → R are C¹ on Rⁿ × R^m; g(·) : Rⁿ → R^r is C¹ on Rⁿ; B and D are n × n symmetric positive semidefinite matrices. For this class of problems, two duals are proposed and weak, strong and strict converse duality theorems are established for each dual problem.