Page 53 - Sustainability in the process industry

30 Cha p te r T h r ee

one, repeating this action until no further solution improvement is
possible. This strategy means that the methods find only local
optima, so there is no guarantee of global optimality. Although
modern optimization software packages are capable of solving
constrained problems, they are mostly based on the search techniques
developed for unconstrained optimization.

3.7.1 Search Algorithms for Nonlinear Unconstrained
Problems
Many methods have been developed for performing local search for
the minimum of an objective function (Luenberger and Ye, 2008).
Deterministic search algorithms mainly assume that the objective
function is minimized and use the concept of descent: a series of
iterations are performed that aim reducing the objective function
value at each iteration. A thorough review of the deterministic search
algorithms for optimization and their performance, as applied to
chemical process models has been provided elsewhere, for example,
Klemeš and Vašek (1973). Such methods include pattern search, the
Rosenbrock method, and conjugate gradients.
All descent-based algorithms rely on a common general strategy.
First, an initial solution point is chosen. Then, a rule-based direction
of movement and a step size are determined and executed. At the
new point, a new pair of direction and step size are determined, and
the process is repeated until the desired convergence is achieved.
The main difference between the various search algorithms is in the
rule applied for determining the iteration steps.
When the search domain is a given line, the process of determining
the function minimum is known as a line search. Higher-dimensional
problems are solved by executing a sequence of successive line
searches. The two simplest line search algorithms are the Fibonacci
method (e.g., Avriel and Wilde, 1966) and the golden section method
(e.g., Kiefer, 1953). Both algorithms search over a fixed interval and
assume that the function is unimodal within it. The golden section
algorithm has a linear convergence rate of d /d = 0.618. Another line
k+1 k
search algorithm is the Newton method, which is based on the
additional assumption of function smoothness; this technique
achieves even faster convergence than the golden section one. There
are also other line search algorithms that are grouped together as
curve-fitting methods.
Another important search algorithm—one that is applicable to
multidimensional function domains—is the method of steepest
descent or gradient method. Its update rule derives the new step
based on the gradient information in the current point, identifying
the direction of the fastest decrease of the objective function. This is
one of the oldest and best-known function minimization methods.
Hence it is often used as a benchmark against which the performance