Dennis Michaels

The Convex Envelope of (

The question of determining strong convex underestimators for nonlinear functions is theoretically and practically of major interest. Unfortunately, results along these lines are quite limited as very few general procedures are at hand that can be applied to general classes of functions. In this paper we show how to reduce the question of determining a convex envelope to lower-dimensional optimization problems when the underlying function is indefinite and (

n

n

-1)-Convex Functions

-1)-convex. Our structural result about this reduction technique enables us to give descriptions for the convex envelope of a variety of two-dimensional functions.

A theoretical study of continuous counter-current chromatography for adsorption isotherms with inflection points

Continuous counter-current chromatographic processes have been increasingly applied in the last years. For chromatographic systems characterized by linear or Langmuir adsorption equilibria, there are nowadays reliable design rules available to decide whether a separation is feasible and which operating parameters should be used. For more complex equilibrium functions, theoretical methods are less developed. More realistic adsorption isotherms are frequently characterized by inflection points in their courses. A flexible model capable to quantify single solute and competitive adsorption isotherms is provided by statistical thermodynamics. In this work, second-order truncations of a statistical model involving inflection points are investigated. The classical scanning technique is used to identify the region of applicable operating parameters. Then an alternative approach is suggested, which verifies the existence and shape of the suitable parameter region by infeasibility certificates. Using the True Moving Bed model, both approaches are compared for different feed concentrations, purity requirements and column efficiencies.

A method to evaluate the feasibility of TMB chromatography for reduced efficiency and purity requirements based on discrete optimization

This paper introduces an exact mathematical approach based on combinatorial optimization to analyze continuous linear countercurrent chromatographic processes. The analysis is based on a given set of input parameters including, in particular, purity requirements and number of plates per zone. As an outcome of the analysis either a proof of infeasibility is derived for a given set of parameters, or a feasible solution is found that is in correspondence with the input requirements. Details regarding the mathematical combinatorial subproblems are presented.

Global analysis of combined reaction distillation processes

Abstract Polyhedral relaxation is a powerful tool for determining global bounds on optimal solutions in chemical process synthesis. Combined reaction distillation processes are considered as a challenging application example. To reduce complexity of the resulting mixed integer linear optimization problems, model reduction by means of wave functions is proposed, and polyhedral relaxations of sigmoidal wave functions in two variables are derived. It is shown that these relaxations provide better approximation quality than approximating the composed functions individually. Further, the concave envelope of such functions is characterized and (nonlinear) convex underestimators are derived. The approximation results are employed in the computation of lower bounds on the vapor flow of a combined reaction distillation process with a metathesis reaction 2 B ⇋ A + C . We restrict computations to a domain around a known local optimum, trading computation time for some of the globality. This still proves a bound on the vapor flow, but for restricted operating conditions of the column.

A Reformulation Strategy for Deterministic Global Optimization of Ideal Multi-component Distillation Processes

Abstract This paper deals with deterministic global optimization of ideal multi-component distillation. A model reformulation based on a linear transformation of concentration variables is discussed, and the impact of the reformulation on the computational effort for solving the underlying mixed-integer non-linear optimization problems is demonstrated by several numerical examples.

Towards global optimization of combined distillation-crystallization processes for the separation of closely boiling mixtures

This paper deals with global optimization of hybrid distillation/melt crystallization processes. For this purpose, improved algorithms are presented taking the problem specific structure into account. We use recent results to derive improved convex relaxations for terms occurring in the material balance equations and propose a constraint propagation strategy that enables us to exclude several process configurations very fast. Computations are presented demonstrating the usefulness of our methods. It is shown that the speed of global optimization can be improved by orders of magnitude.

Polyhedra related to integer-convex polynomial systems

This paper deals with the reformulation of a polynomial integer program. For deducing a linear integer relaxation of such a program a class of polyhedra that are associated with nonlinear functions is introduced. A characterization of the family of polynomials for which our approach leads to an equivalent linear integer program is given. Finally the family of so-called integer-convex polynomials is defined, and polyhedra related to such a polynomial are investigated.

Optimal Process Design for the Synthesis of 2,3-Dimethylbutene-1

Abstract This paper is concerned with the computer-aided optimal design of reaction-distillation processes. The production of solvent 2,3-dimethylbutene-1 by isomerization of 2,3-dimethylbutene-2 is considered as an innovative benchmark problem. Possible process candidates are a reactive distillation column, a reactor coupled to a nonreactive distillation column or a reactive reboiler with a nonreactive distillation column on top. Local MINLP optimization indicates that the reactive distillation has the lowest total annualized costs. However, due to the non-convexity of the underlying mathematical problem better solutions for the other process candidates cannot be excluded. For this purpose a new global approach is proposed which is based on discrete optimization of the underlying model equations and which proves globally that reactive distillation is the best option.