# How to deal with mixed linear integral optimization problem with extra nonconvex but differentiable constraints?

I got a model of MILP but I want to add extra constraints.

$$\min\{c^T x + d^T y:Ax+By=a,Cx+Dy \leq b,x,y\geq0,y \in \{0,1\}\}$$

More specificlly, I got extra original constraints with indicator function in the following form

$$\frac{1}{N}\sum_{i=1}^{N}\mathbb{I}(z\mu_i-1) \geq 1-\epsilon,Ex+Fy+Gz\geq0$$ where $$z$$ is extra matrix variable to be optimized.

The indicator function can be approximated by logistic function $$g(x)=\frac{e^{-x/{\epsilon}}}{1+e^{-x/{\epsilon}}}$$, transforming the above constrints to

$$\frac{1}{N}\sum_{i=1}^{N}g(z\mu_i-1) \geq 1-\epsilon$$

So how could the resulting problem be solved?

• Please give your definition of $\mathbb{I}(x).$
– prubin
Commented Apr 8 at 15:25
• @prubin $\mathbb{I}(x)$ is the indicator function which takes 0 when $x$ is greater than 0 and 1 otherwise. Commented Apr 17 at 1:23
• What are $z$ and $\mu_i$? On the one hand, you say $z$ a matrix variable. On the other hand, the constraint $Ex + Fy + Gz \geq 0$ indicates it's a vectorial variable, i.e. $z \in \mathbb{R}^n$.
– joni
Commented Apr 17 at 9:30
• @KaimingZhang what are you modelling with the indicator constraint? Should it be interpreted as something along the lines of "$z\mu_i\leq 1$ for $1-\epsilon$ percent of the $N$ constraints?
– Sune
Commented Apr 19 at 9:15
• @Sune Yes, that is the case. Commented Apr 25 at 3:22

I think you can achieve what you want as follows: Let $$u_i$$ be upper bounds on $$z\mu_i$$, for all $$i=1,...,N$$. Let furthermore $$\alpha_i$$ be binary variables equalling one, if $$z\mu_i>1$$ is allowed. Then you may formulate your requirement as \begin{align} &z\mu_i\leq 1+(u_i-1)\alpha_i,&&\text{for } i=1,..,N&(1)\\ &\sum_{i=1}^N\alpha_i\leq \lfloor N\epsilon\rfloor&&&(2) \end{align} This way, you allow violation of only $$N\epsilon$$ of the constraints ($$\alpha_i=1$$), stating that at least $$N(1-\epsilon)$$ of the constraints must be satisfied. You can round down the right hand side of (2) as the left hand side is an integer.