Gaining or Losing Perspective

We study MINLO (mixed-integer nonlinear optimization) formulations of the disjunction \(x\in \{0\}\cup [l,u]\), where z is a binary indicator of \(x\in [l,u]\), and y “captures” \(x^p\), for \(p>1\). This model is useful when activities have operating ranges, we pay a fixed cost for carrying out each activity, and costs on the levels of activities are strictly convex. One well-known concrete application (with \(p=2\)) is mean-variance optimization (in the style of Markowitz).