Invex generalizations of some duality results (Q1111476)

Consider the pair of symmetric dual problems: \[ (P)\quad f(x,y)\quad \to \quad \min,\quad (x,y)\quad \in \quad P, \] \[ (D)\quad g(x,y)\quad \to \quad \max,\quad (x,y)\quad \in \quad D, \] \[ P=\{(x,y)\in C_ 1\times C_ 2:\quad K_ y'(x,y)\in C^*_ 2\},\quad D=\{(x,y)\in C_ 1\times C_ 2:\quad K_ x'(x,y)\in C^*_ 1\}, \] where \(C_ 1\), \(C_ 2\) are closed convex cones and K: \({\mathcal U}(C_ 1\times C_ s)\to R\) is a differentiable function. Under some generalized convexity conditions, the invexity of f and g, a weak duality theorem is proved. A similar theorem is stated for a pair of nonlinear mixed integer programming problems.