4 1 Motivating Examples
because its solution is just the given unique feasible decision. The feasibil-
ity region is formally defined through equality and inequality conditions that
are denominated the constraints of the problem. Each one of these conditions
is mathematically expressed through one real-valued function of the decision
variables. This function is equal to zero, greater than or equal to zero, lower
than or equal to zero.
Therefore a mathematical programming problem, representing and opti-
mization decision framework, presents the formal structure below:
minimize objective function
subject to constraints
Depending upon the type of variables and the mathematical nature of the
objective function and the functions used for the constraints, mathematical
programming problems are classified in different manners. If the variables
involved are continuous and both the objective function and the constraints
are linear, the problem is denominated “linear programming problem.” If any
of the variables involved is integer or binary, while the constraints and the
objective function are both linear, the problem is denominated “mixed-integer
linear programming problem.”
Analogously, if the objective function or any constraint is nonlinear and all
variables are continuous, the problem is denominated “nonlinear programming
problem.” If additionally, any variable is integer, the corresponding problem
is denominated “mixed-integer nonlinear programming problem.”
Generally speaking, linear programming problems are routinely solved
even if they involve hundred of thousands of variables and constraints. Nonlin-
ear programming problems are easily solved provided that they meet certain
regularity conditions related to the mathematical concept of convexity, which
is considered throughout the following chapters of this book. Mixed-integer
linear programming problems are routinely solved provided that the number
of integer variables is sufficiently small, typically below one thousand. Mixed-
integer nonlinear programming problems are generally hard to solve and can
be numerically intractable, because: (a) a high number of integer variables,
and (b) the ugly mathematical properties of the functions involved. These
problems require an in-depth specific analysis before a solution procedure is
tried.
From an engineering point of view, operation problems involving engi-
neering systems are normally continuous problems, either linear or nonlin-
ear. However, design and capacity expansion planning problems are generally
mixed-integer linear or nonlinear problems. The reason is that some design or
planning variables are of integer nature while most operation variables are of
continuous nature.
This book considers particular cases of all these optimization problems.
These cases have structural properties that can be advantageously compu-
tationally exploited. These structural properties are briefly illustrated and
described below.