Gaining or losing perspective

Abstract

We study MINLO (mixed-integer nonlinear optimization) formulations of the disjunction $$x\in \{0\}\cup [l,u]$$ , where z is a binary indicator of $$x\in [l,u]$$ ( $$u> \ell > 0$$ ), and y “captures” f(x), which is assumed to be convex on its domain [l, u], but otherwise $$y=0$$ when $$x=0$$ . 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 convex. Using volume as a measure to compare convex bodies, we investigate a variety of continuous relaxations of this model, one of which is the convex-hull, achieved via the “perspective reformulation” inequality $$y \ge zf(x/z)$$ . We compare this to various weaker relaxations, studying when they may be considered as viable alternatives. In the important special case when $$f(x) := x^p$$ , for $$p>1$$ , relaxations utilizing the inequality $$yz^q \ge x^p$$ , for $$q \in [0,p-1]$$ , are higher-dimensional power-cone representable, and hence tractable in theory. One well-known concrete application (with $$f(x) := x^2$$ ) is mean-variance optimization (in the style of Markowitz), and we carry out some experiments to illustrate our theory on this application.