Page 51 - Sustainability in the process industry

28 Cha p te r T h r ee

factors are linearity and the existence of integer variables. There are
two main reasons for the significance of these factors:

1. Continuous problems are solved by using simpler methods
that incorporate gradient or other search mechanisms. The
presence of integer variables introduces combinatorial
complexity caused by the need to build search trees that
branch through the integer variables.
2. Nonlinearity introduces a different type of complexity.
Whereas linear problems (LPR and MILP) have been shown
to be convex (Floudas, 1995; Williams, 1999), the convexity of
nonlinear problems (NLP and MINLP) must be evaluated on
a case-by-case basis. Such problems are generally assumed to
be nonconvex until proven otherwise.

3.5 Conditions for Optimality

One of the most popular methods for handling optimization
formulations is the use of MPR solvers. They usually take
standardized input in the form of matrices of variables, parameters,
and equations, after which they explore the search space, using local
search and gradients, to reach an optimal solution.

3.5.1 Conditions for Local Optimality
When employing local search (e.g., gradient-based) algorithms, the
desired extremum (minimum or maximum) needs to be located and
proven. The optimality condition usually employed by solver
algorithms is based on the mathematical definition of an extremum.
For finding a minimum, the definition requires the existence of a
point in the search space such that any small deviation from that
point, in any direction within the search space, will result in an
increase of the objective function value or in keeping it the same:

* * * *
, F x y d F xy , , , xy  vicinity of x y (3.1)
,
Further details are given in specialized textbooks on optimization
(see, e.g., Edgar and Himmelblau, 1988; Floudas, 1995; Luenberger
and Ye, 2008). Rigorous definitions can also be found by searching
the Web for “KKT optimality conditions” (aka Karush–Kuhn–Tucker
conditions, an extension of the method of Lagrange multipliers).

3.5.2 Conditions for Global Optimality
Additional conditions are applied to attain global optimality when
using local search algorithms, which in this case require the problem