Chapter 12
Optimization
In design, construction and maintenance of any engineering system, engineers have to take many
technological (and managerial) decisions at several stages. The goal is to either minimize the effort
required or maximize the desired benefit. Both goals are required to be expressed as a function of
certain decision variables, optimizing over which would yield better (if not the best) result. Some
practical instances of use of optimization are: (a) minimizing material volume (and/or stiffness) when
constructing structures like over-bridges, (b) optimizing to determine the material connectivity or
topology in such structures, (c) optimizing the shape of an automobile body to minimize aerodynamic
drag, (d) optimizing the bumper for crashworthiness, and many more.
Numerical implementation of optimization is usually an iterative procedure wherein at every step
the design variables are updated when a better goal value is achieved. An optimization algorithm can
either be intuitive, like in the optimality criteria method, or can be a result of a rigorous derivation
from the zeroth, first or second order approximations of the objective function (or goal) with respect
to the design variables, for instance in the mathematical programming schemes. This chapter aims to
brief the reader on some existing methods in optimization. Such methods can be classified in numerous
ways depending on the number of variables, constraints, their nature (linear or nonlinear), and the
nature of solution (generic or problem specific). We brief some generic methods on single-variable
and multi-variable optimization.
12.1 Classical Optimization
The necessary condition for optimality for a function f(x) in single variable x is well established in
that equating the first derivative f′(x) = df(x)/dx to zero yields the locations of zero slope or the
optima. It can be intuitively observed in Figure 12.1 that such locations correspond to sites wherein
the function changes its trend of monotonic increase (at maxima) or decrease (at minima). Further,
the sign of the second derivative f′′(x) conveys, as a sufficiency condition, the nature of the optima
at the location of zero slope. We would expect at a maximum that the slope would start decreasing
asx increases and vice versa at a minimum, that is, the rate of change of slope dfx
dx
2
2() < 0 at amaximum and > 0 at a minimum.
12.2 Single Variable Optimization
Determining the locations of optima for a function g(x) in single variable amounts to finding the roots