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Explicit convex and concave envelopes through polyhedral subdivisions

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Abstract

In this paper, we derive explicit characterizations of convex and concave envelopes of several nonlinear functions over various subsets of a hyper-rectangle. These envelopes are obtained by identifying polyhedral subdivisions of the hyper-rectangle over which the envelopes can be constructed easily. In particular, we use these techniques to derive, in closed-form, the concave envelopes of concave-extendable supermodular functions and the convex envelopes of disjunctive convex functions.

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Authors and Affiliations

  1. Krannert School of Management, Purdue University, West Lafayette, IN, USA

    Mohit Tawarmalani & Chuanhui Xiong

  2. Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, USA

    Jean-Philippe P. Richard

  3. School of Business, The University of North Carolina-Pembroke, Pembroke, NC, USA

    Chuanhui Xiong

Authors
  1. Mohit Tawarmalani
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  2. Jean-Philippe P. Richard
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  3. Chuanhui Xiong
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Correspondence to Mohit Tawarmalani.

Additional information

The work was supported by NSF CMMI 0900065 and 0856605.

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Tawarmalani, M., Richard, JP.P. & Xiong, C. Explicit convex and concave envelopes through polyhedral subdivisions. Math. Program. 138, 531–577 (2013). https://doi.org/10.1007/s10107-012-0581-4

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  • DOI: https://doi.org/10.1007/s10107-012-0581-4

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