Gaining or Losing Perspective

  • Jon LeeEmail author
  • Daphne Skipper
  • Emily Speakman
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 991)


We study MINLO (mixed-integer nonlinear optimization) formulations of the disjunction \(x\in \{0\}\cup [l,u]\), where z is a binary indicator of \(x\in [l,u]\), and y “captures” \(x^p\), for \(p>1\). 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 strictly convex. One well-known concrete application (with \(p=2\)) is mean-variance optimization (in the style of Markowitz).

Using volume as a measure to compare convex bodies, we investigate a family of relaxations for this model, employing the inequality \(yz^q \ge x^p\), parameterized by the “lifting exponent” \(q\in [0,p-1]\). These models are higher-dimensional-power-cone representable, and hence tractable in theory. We analytically determine the behavior of these relaxations as functions of lup and q. We validate our results computationally, for the case of \(p=2\). Furthermore, for \(p=2\), we obtain results on asymptotic behavior and on optimal branching-point selection.


Mixed-integer nonlinear optimization Volume Integer Relaxation Polytope Perspective Higher-dimensional power cone 


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Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.University of MichiganAnn ArborUSA
  2. 2.U.S. Naval AcademyAnnapolisUSA
  3. 3.Otto-von-Guericke-UniversitätMagdeburgGermany

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