# Problem Statement

MINLP

The mixed-integer nonlinear programming models to be solved can be expressed as follows.

Minimize:

$f(x,y)$

Subject to:

\begin{split}\begin{aligned} h(x,y) & = & 0 \\ g(x,y) & \leq & 0 \\ C y + D x & \leq & d \\ x \in {\cal X} & = & \{ x \in \mathbb{R}^n | x^L \leq x \leq x^U \} \\ y \in {\cal Y} & = & \mathbb{Z}^m\end{aligned}\end{split}

Usual assumption

The usual assumption is that the nonlinear subproblem (i.e. the model in which all integer variables are fixed) is convex. This assumption is to guarantee that each locally optimal solution of the nonlinear subproblem is also a globally optimal solution. In practice this assumption does not always hold, but the algorithm can still be applied. Convergence to a global optimum of the MINLP using the outer approximation algorithm is then no longer guaranteed.