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.
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This paper is an extended version of the paper by the same name appearing in the proceedings WCGO 2019 (Metz, France).
J. Lee was supported in part by ONR grant N00014-17-1-2296 and LIX, l’École Polytechnique. D. Skipper was supported in part by ONR grant N00014-18-W-X00709. E. Speakman was supported in part by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 314838170, GRK 2297 MathCoRe.
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Lee, J., Skipper, D. & Speakman, E. Gaining or losing perspective. J Glob Optim 82, 835–862 (2022). https://doi.org/10.1007/s10898-021-01055-6
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DOI: https://doi.org/10.1007/s10898-021-01055-6