We study MINLO (mixed-integer nonlinear optimization) formulations of the disjunction x∈{0}∪[l,u]\documentclass[12pt]{minimal}
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\begin{document}$$x\in \{0\}\cup [l,u]$$\end{document}, where z is a binary indicator of x∈[l,u]\documentclass[12pt]{minimal}
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\begin{document}$$x\in [l,u]$$\end{document} (u>ℓ>0\documentclass[12pt]{minimal}
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\begin{document}$$u> \ell > 0$$\end{document}), and y “captures” f(x), which is assumed to be convex on its domain [l, u], but otherwise y=0\documentclass[12pt]{minimal}
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\begin{document}$$y=0$$\end{document} when x=0\documentclass[12pt]{minimal}
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\begin{document}$$x=0$$\end{document}. 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≥zf(x/z)\documentclass[12pt]{minimal}
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\begin{document}$$y \ge zf(x/z)$$\end{document}. We compare this to various weaker relaxations, studying when they may be considered as viable alternatives. In the important special case when f(x):=xp\documentclass[12pt]{minimal}
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\begin{document}$$f(x) := x^p$$\end{document}, for p>1\documentclass[12pt]{minimal}
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\begin{document}$$p>1$$\end{document}, relaxations utilizing the inequality yzq≥xp\documentclass[12pt]{minimal}
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\begin{document}$$yz^q \ge x^p$$\end{document}, for q∈[0,p-1]\documentclass[12pt]{minimal}
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\begin{document}$$q \in [0,p-1]$$\end{document}, are higher-dimensional power-cone representable, and hence tractable in theory. One well-known concrete application (with f(x):=x2\documentclass[12pt]{minimal}
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\begin{document}$$f(x) := x^2$$\end{document}) is mean-variance optimization (in the style of Markowitz), and we carry out some experiments to illustrate our theory on this application.
