Let \(M\subset {\mathbb{R}}^ n\times {\mathbb{R}}^ m\) be an open set containing \(S\times T\), where \(S=U\times C_ 1\), \(T=V\times C_ 2\) and K:M\(\to R\) is a twice continuously differentiable function. It is assumed that the first \(n_ 1\) components of n and the first \(m_ 1\) components of y \((0\leq n_ 1\leq n\); \(0\leq m_ 1\leq m)\) are arbitrary constrained to be integers and the following notations are used: \((x,y)=(x^ 1,x^ 2,y^ 1,y^ 2)\), \(x^ 1=(x_ 1,...,x_{n_ 1})\), \(y^ 1=(y_ 1,...,y_{m_ 1})\). The following two maximin and minimax (in duality sense) nonsymmetric nonlinear mixed integer programming problems are considered: \[ (P_ 0)\quad\text{Max}_{x^ 2}\text{Min}_{x^ 2,y}\{f=K(x,y)-\lambda (y^ 2)^ T\nabla_{y^ 2}K(x,y)\}, \] s.t. \(x^ 2\in U\), \((x^ 2,y)\in C_ 1xT\), \(\nabla_{y^ 2}K(x,y)\in C^*_ 2\), \(\lambda\geq 1\), and \[ (D_ 0)\quad\text{Min}_{y^ 2}\text{Max}_{x,y^ 2}\{g=K(x,y)-\mu (x^ 2)^ T\nabla_{x^ 2}K(x,y)\}, \] s.t. \(y^ 1\in V\), \((x,y^ 2)\in S\times C_ 2\), \(-\nabla_{x^ 2}K(x,y)\in C^*_ 1\), \(\mu\geq 1\), where \(C^*\) is the polar cone of C. Under weaker (pseudo- convex/pseudo-concave) assumptions a weak duality theorem is proved. As special case this result reduces to the weak duality theorem for minimax and symmetric dual nonlinear mixed integer programming problems. Then this is used to generalize available results on minimax and symmetric duality in nonlinear mixed integer programming.