Computational Optimization and Applications

, Volume 63, Issue 3, pp 705–735 | Cite as

Approximated perspective relaxations: a project and lift approach

  • Antonio Frangioni
  • Fabio Furini
  • Claudio Gentile


The perspective reformulation (PR) of a Mixed-Integer NonLinear Program with semi-continuous variables is obtained by replacing each term in the (separable) objective function with its convex envelope. Solving the corresponding continuous relaxation requires appropriate techniques. Under some rather restrictive assumptions, the Projected PR (\(\mathrm{P}^2\mathrm{R}\)) can be defined where the integer variables are eliminated by projecting the solution set onto the space of the continuous variables only. This approach produces a simple piecewise-convex problem with the same structure as the original one; however, this prevents the use of general-purpose solvers, in that some variables are then only implicitly represented in the formulation. We show how to construct an Approximated Projected PR (\(\mathrm{AP}^2\mathrm{R}\)) whereby the projected formulation is “lifted” back to the original variable space, with each integer variable expressing one piece of the obtained piecewise-convex function. In some cases, this produces a reformulation of the original problem with exactly the same size and structure as the standard continuous relaxation, but providing substantially improved bounds. In the process we also substantially extend the approach beyond the original \(\mathrm{P}^2\mathrm{R}\) development by relaxing the requirement that the objective function be quadratic and the left endpoint of the domain of the variables be non-negative. While the \(\mathrm{AP}^2\mathrm{R}\) bound can be weaker than that of the PR, this approach can be applied in many more cases and allows direct use of off-the-shelf MINLP software; this is shown to be competitive with previously proposed approaches in some applications.


Mixed-integer nonlinear problems Semi-continuous variables Perspective reformulation Projection 

Mathematics Subject Classification

90C06 90C25 



The first and third authors gratefully acknowledge the contribution of the Italian Ministry for University and Research under the PRIN 2012 Project 2012JXB3YF “Mixed-Integer Nonlinear Optimization: Approaches and Applications”, as well as of the European Union under the 7FP Marie Curie Initial Training Network n. 316647 “MINO: Mixed-Integer Nonlinear Optimization”.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Antonio Frangioni
    • 1
  • Fabio Furini
    • 2
  • Claudio Gentile
    • 3
  1. 1.Dipartimento di InformaticaUniversità di PisaPisaItaly
  2. 2.LAMSADEUniversité Paris-DauphineParisFrance
  3. 3.Istituto di Analisi dei Sistemi ed Informatica “A. Ruberti” (IASI-CNR), Consiglio Nazionale delle RicercheRomeItaly

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