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  Relaxations of functions AVT.SVT Relaxations of functions

Deterministic global optimization (DGO) is concerned with the estimation of an optimal global solution to nonconvex problems. It is widely used in the field of chemical engineering for modeling of polymerization processes, heat-exchanger networks and even in parameter estimation. In order to solve global optimization models, valid over- and underestimating relaxations of the objective function and the limiting constraints are constructed. McCormick proposed a technique for a systematic construction of valid convex and concave relaxations for factorable functions of the form

F1(f1)+F2(f2)·F3(f3)

with univariate outer functions Fi and multivariate inner functions fi . We have been involved in the development of novel relaxation methods of functions for the deterministic global optimization, partly in collaboration with the Massachusetts Institute of Technology (MIT) and the Imperial College London (ICL). In particular, we have shown that relaxation of programs is possible with the McCormick technique. We also extended the original method to multivariate outer functions and developed the Taylor model arithmetic. This resulted in the open-source code MC++, led by Dr. Benoît Chachuat. The open-source implementation is available at COIN-OR and includes our multivariate relaxations and the Taylor model arithmetic.

If you are interested in using this software please feel free to contact us.

 

Literature

  1. Agustin Bompadre, Alexander Mitsos, and Benoît Chachuat. Convergence analysis of Taylor models and McCormick-Taylor models. Journal of Global Optimization, 57(1):75–114, 2013.
  2. Benoît Chachuat. MC++: A versatile library for bounding and relaxation of factorable functions, 2013.
  3. Garth P. McCormick. Computability of global solutions to factorable nonconvex programs: Part I-Convex underestimating problems. Mathematical Programming, 10:147–175, 1976.
  4. Alexander Mitsos, Benoît Chachuat, and Paul I Barton. McCormick-based relaxations of algorithms. SIAM Journal on Optimization, 20(2):573–601, 2009.
  5. Angelos Tsoukalas and Alexander Mitsos. Multivariate McCormick relaxations. Journal of Global Optimization, 59:633–662, 2014. (recognized with best paper award).