Page 50 - Sustainability in the process industry

Pro c ess O p timization 27

method should be based on a clear understanding, and toward this
end it is useful to bear in mind three aspects of optimization
problems:
1. An optimization problem is convex if it minimizes (or
maximizes) a convex objective function and if all the
constraints are convex (Williams, 1999).
2. An optimization problem is linear if its objective function
and all the constraints are linear; note that all linear
optimization problems are also convex (Williams, 1999).
3. A variable that can assume only integer (whole number)
values is an integer variable; integer variables that are
constrained only to values of 0 or 1 are binary variables.
Optimization problems are classified in terms of the following
specific features:
• Objective type: Minimization or maximization problems.
• Presence of constraints: Unconstrained versus constrained.
Most practical tasks involve the formulation of constrained
optimization problems.
• Problem convexity: Convex versus nonconvex.
• Linear and nonlinear problems: This aspect depends on the
nature of the objective function and/or the constraints. Most
process optimization problems are bilinear (i.e., containing
products of two optimization variables), for example, they
feature componentwise mass balances involving products of
mass flow rates and concentration and enthalpy balances
involving products of mass flow rates and enthalpies.
• Absence of integer variables: In such cases the entire problem is
continuous and so linear programming (LPR) or nonlinear
programming (NLP) models are employed.
• Presence of integer variables: In such cases the problem is
referred to as “integer.” Integer problems are further
subdivided into pure integer programming (IP) models,
which involve only integer variables; mixed integer linear
programming (MILP) models, which involve linear
relationships with both integer and continuous variables;
and mixed integer nonlinear programming (MINLP)
models, which involve nonlinear relationships with both
integer and continuous variables.

Further details on these classifications and their properties can
be found in Floudas (1995) and Williams (1999). The most important