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Sequence of polyhedral relaxations for nonlinear univariate functions

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Abstract

Given a nonlinear, univariate, bounded, and differentiable function f(x), this article develops a sequence of Mixed Integer Linear Programming (MILP) and Linear Programming (LP) relaxations that converge to the graph of f(x) and its convex hull, respectively. Theoretical convergence of the sequence of relaxations to the graph of the function and its convex hull is established. For nonlinear non-convex optimization problems, the relaxations presented in this article can be used to construct tight MILP and LP relaxations. These MILP and the LP relaxations can also be used with MILP-based and spatial branch-and-bound based global optimization algorithms, respectively.

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Acknowledgements

The work was funded LANL’s Directed Research and Development (LDRD) projects, “20170201ER: POD: A Polyhedral Outer-approximation, Dynamic-discretization optimization solver” and “20200603ECR: Distributed Algorithms for Large-Scale Ordinary Differential/Partial Differential Equation (ODE/PDE) Constrained Optimization Problems on Graphs”. This work was carried out under the U.S. DOE Contract No. DE-AC52-06NA25396.

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Authors and Affiliations

  1. Los Alamos National Laboratory, Los Alamos, NM, USA

    Kaarthik Sundar & Harsha Nagarajan

  2. Sabre Labs, Southlake, TX, USA

    Sujeevraja Sanjeevi

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  1. Kaarthik Sundar
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  2. Sujeevraja Sanjeevi
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  3. Harsha Nagarajan
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Correspondence to Kaarthik Sundar.

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Sundar, K., Sanjeevi, S. & Nagarajan, H. Sequence of polyhedral relaxations for nonlinear univariate functions. Optim Eng 23, 877–894 (2022). https://doi.org/10.1007/s11081-021-09609-z

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  • DOI: https://doi.org/10.1007/s11081-021-09609-z

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