Solving a non-linear optimisation problem with min type constraint.

Question

The problem is $\min_{\boldsymbol{\mu}}f(\boldsymbol{\mu})$ given that $\min|\mu_i-\mu_j|\geq\delta>0$, where $\boldsymbol{\mu} = (\mu_1,\ldots,\mu_n)^{\rm T}$. I am having difficulty to convert $\min|\mu_i-\mu_j|\geq\delta>0$ to a set of linear constraints so that I can get the solution (at least numerically). Any help will be appreciated. Thanks in advance.

Answer 1

If the $\min|\mu_ - \mu_|$ were in the objective function, you could write a smooth reformulation by introducing additional continuous variables. However, since it is a constraint, I guess you need to introduce binary variables to model the disjunction.
Unless you can order your $\mu$s by increasing order?

Answer 2

Your desired constraint $$\min_{i<j} |\mu_i-\mu_j| \ge \delta$$ is equivalent to the set of constraints $$|\mu_i-\mu_j| \ge \delta \quad\text{for all $i<j$} \tag1$$ You can linearize $(1)$ by introducing binary variable $z_{i,j}$ and linear big-M constraints \begin{align} \mu_j - \mu_i + \delta &\le Mz_{i,j} &&\text{for all $i<j$} \tag2 \\ \mu_i - \mu_j + \delta &\le M(1-z_{i,j}) &&\text{for all $i<j$} \tag3 \\ \end{align} Constraint $(2)$ enforces $z_{i,j}=0 \implies \mu_i-\mu_j \ge \delta$. Constraint $(3)$ enforces $z_{i,j}=1 \implies \mu_j-\mu_i \ge \delta$.