## Numerical methods for d-parametric nonlinear programming with chemical process control and optimization applications

##### Abstract

The problem of optimization often arises whenever we want to influence a
complex system, because we usually want our impact on such a system to be the
best it can be in some particular sense. Such complex optimization problems are
often accompanied by another problem, however: uncertainty. Specifically, there
may be uncertainty in the structure of our mathematical model, uncertainty in
various physical constants in our model or uncertainty in the weighting of various
conflicting objectives.
The purpose of this dissertation is to develop one particular approach to optimization
under uncertainty: parametric nonlinear programming (pNLP). Nonlinear
programming is the optimization of a scalar objective function subject to
a finite number of equality and inequality constraints over some finite number of
variables u ∈ R
n
:
min
u
f (u)
s.t.
h (u) = 0
g (u) ≤ 0.
(1)
v
Parametric nonlinear programming attempts to solve this problem as a function
of a finite number of parameters α ∈ R
d
:
u
∗
(α) = arg
{
min
u
f (u, α)
s.t.
h (u, α) = 0
g (u, α) ≤ 0
}
. (2)
This work approaches the pNLP problem numerically with a predictorcorrector
(continuation) method. Such methods have been developed for general
underdetermined nonlinear equations. Such equations are derived for this problem
using the Fritz-John necessary conditions for optimality.
Predictor-corrector methods specifically customized for the single parameter
nonlinear programming problem (1-pNLP) have already been developed quite
extensively [71, 112], but a comprehensive multi-parameter method has not yet
been developed. This work aims to combine some of the 1-pNLP work with
Rheinboldt and Brodzik’s work on multi-dimensional predictor-correctors in order
to allow for global exploration of the mpNLP problem [18, 149]. Specifically,
as in the previous work on 1-pNLP, where possible the particular structure of the
parametric nonlinear programming problem is exploited to both improve computation
times and to yield more optimization-specific information than a general
multi-parametric predictor-corrector algorithm would. The algorithm developed
also improves on Brodzik’s work by including auto-scaling, step-size adaptation
and, most importantly, singularity handling. Further, several verification examples,
and several more complex applications are presented and discussed. The
applications include implicit optimization model adequacy and multi-objective
optimization.