I have a problem where I want to optimize a path while performing obstacle avoidance in a known map like the following:
Representing the hiperplanes as $h_i x\leq k_i$ but I need to put into the optimizer the valid region, but this one is non-convex and I don't know how to represent it to put it into an optimizer. In
Prodan, Ionela; Stoican, Florian; Olaru, Sorin; Niculescu, Silviu-Iulian, Mixed-integer representations in control design. Mathematical foundations and applications, SpringerBriefs in Electrical and Computer Engineering. Control, Automation and Robotics. Cham: Springer (ISBN 978-3-319-26993-1/pbk; 978-3-319-26995-5/ebook). xii, 107 p. (2016). ZBL1337.93001.
The authors present the following simple example, but the problem is that I do not know which value corresponds to $\alpha_i \quad \forall i$ because they work as activators depending on my current but I cannot provide this condition in a NLP solver like ACADO.
This is a non-convex problem so you will likely need binary variables to model it. In the model you are referencing in the text, the $\alpha$ variables are binary and represent the conditions that the point $x, y$ is outside the boundary defined by a side of a convex obstacle.
If a point $x, y$ is outside a set of rectangles bounded by bottom, left, top, right $(b_i, l_i, t_i, r_i)$ it must satisfy at least one of the following. $$ y \le b_i \\ y \ge t_i \\ x \le l_i \\ x \ge r_i$$
Adopting the notation in the book, we define $\alpha_{ib}, \alpha_{it}, \alpha_{il}, \alpha_{ir}$ to indicate that $x, y$ is either below, above, to the left or to the right of object $i$. We enforce the definition with the following constraints $$y \le b_i + (1 - \alpha_{ib}) M \\ x \le l_i + (1 - \alpha_{il}) M \\ y \ge \alpha_{it} t_i\\ x \ge \alpha_{ir} r_i \\ \alpha_{ik} \in \{0, 1\}, \forall {k \in \{ b,l,t,r \}} $$
Where $M$ is the maximum of the upper or right coordinate ($M$ could be reduced for each rectangle and side). These constraints work because if a given $\alpha$ is 0, the associated constraint is always satisfied and if $\alpha$ has value $1$, then it is equivalent to the conditions defining below, to the left, above or to the right.
The point is outside the rectangle if $$\sum_{k \in \{ b,l,t,r \}} \alpha_{ik} \ge 1$$
and outside of all rectangles if the condition holds for all rectangles $i$.
One issue you now have is that binary restrictions may not be supported in ACADO, however there are many solvers that do.