Gaining or losing perspective

We study MINLO (mixed-integer nonlinear optimization) formulations of the disjunction $$x\in \{0\}\cup [l,u]$$ x ∈ { 0 } ∪ [ l , u ] , where z is a binary indicator of $$x\in [l,u]$$ x ∈ [ l , u ] ( $$u> \ell > 0$$ u > ℓ > 0 ), and y “captures” f(x), which is assumed to be convex on its domain [l, u], but otherwise $$y=0$$ y = 0 when $$x=0$$ 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 $$y \ge zf(x/z)$$ y ≥ z f ( x / z ) . We compare this to various weaker relaxations, studying when they may be considered as viable alternatives. In the important special case when $$f(x) := x^p$$ f ( x ) : = x p , for $$p>1$$ p > 1 , relaxations utilizing the inequality $$yz^q \ge x^p$$ y z q ≥ x p , for $$q \in [0,p-1]$$ q ∈ [ 0 , p - 1 ] , are higher-dimensional power-cone representable, and hence tractable in theory. One well-known concrete application (with $$f(x) := x^2$$ 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.