Structure-driven methods for large-scale optimization

http://www.mat.univie.ac.at/~neum/structure.html

Project goals

This project aims at the reliable solution of large-scale optimization problems with multiple solutions, coming from chemical engineering. This includes continuous constraint satisfaction problems (nonlinear systems of equations with additional inequality constraints), as these are typically solved by optimization techniques.

Note that the techniques are expected to transfer without difficulties to other large-scale optimization problems.

This project is funded by a grant of the Austrian Science Fund FWF under contract number P23554.

A wide variety of chemical engineering problems give rise to optimization problems or continuous constraint satisfaction problems that may have multiple solutions:

  • analyzing distillation columns;
  • computation of fluid phase equilibrium from activity coefficient models and cubic equation-of-state models;
  • modeling liquid-liquid equilibrium of ionic liquid systems;
  • computation of solid-liquid equilibrium;
  • location of homogeneous azeotropes and reactive azeotropes;
  • calculation of critical points from cubic equation-of-state models;
  • parameter estimation using standard least squares and error-in-variables (EIV) approach;
  • global optimization of molecular structures;
  • computing high-pressure chemical and multiphase equilibrium;
  • process simulation;
  • automated controller synthesis in quantitative feedback theory.


    Because of the possibility of multiple solutions, these problems are computationally troublesome, hence the need for reliable methods. Current solver technology only covers problems that are small-scale, having less than about one hundred variables.

    The goal of the project is to extend these methods to work reliably in higher dimensions, by exploiting the problem structure.

    We are pursuing primarily two different approaches:

  • Ensemble methods for stage-by-stage calculations;
  • Branch and bound methods using interval techniques, in particular affine arithmetic and linear or quadratic relaxations.


    In each case, exploiting the connectivity structure of the variables and equations plays a crucial role in improving the efficiency in the large-scale case.

    First results

    A. Baharev and A. Neumaier,
    A globally convergent method for finding all steady-state solutions of distillation columns, Manuscript, 2013.
    supplementary material, including source code

    A method is proposed that either returns all solutions to steady-state models of distillation columns or proves infeasibility. No initial estimates are required. The computational effort grows linearly with the number of stages. Successful solution of a numerically challenging reactive distillation column with 7 steady-states show the robustness of the method. The authors do not know any published software that could compute all solutions to this difficult model without expert tuning.

    A. Baharev and A. Neumaier,
    Chemical Process Modeling in Modelica, Proceedings of the 9th International Modelica Conference, pages 955--962. Munich, Germany; Sep 3-5, 2012, DOI
    One-page abstract
    supplementary material, including source code

    Chemical process models are highly structured. Information on how the hierarchical components are connected helps to solve the model efficiently. The structural information retrieved from the JModelica environment will play an important role in the development of our novel optimization methods.
    Foundations of a Modelica library for general-purpose chemical process modeling have been built. Multiple steady-states in ideal two-product distillation were computed as a proof of concept. The Modelica source code is available at the project homepage. The issues encountered during modeling may be valuable to the Modelica language designers.

    my home page (http://www.mat.univie.ac.at/~neum/)

    Arnold Neumaier (Arnold.Neumaier@univie.ac.at)