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 ( $$u> \ell > 0$$ ), and y “captures” f(x), which assumed to be convex on its domain [l, u], but otherwise $$y=0$$ when $$x=0$$ . This model useful activities have operating ranges, we pay fixed cost for carrying out each activity, costs levels are convex. Using volume as measure compare bodies, investigate variety continuous relaxations this model, one convex-hull, achieved via “perspective reformulation” inequality $$y \ge zf(x/z)$$ various weaker relaxations, studying they may considered viable alternatives. In important special case $$f(x) := x^p$$ $$p>1$$ utilizing $$yz^q $$q \in [0,p-1]$$ higher-dimensional power-cone representable, hence tractable in theory. One well-known concrete application (with x^2$$ ) mean-variance optimization (in style Markowitz), carry some experiments illustrate our theory application.