Optimal Performance- Based Seismic Design - Structural Seismic Design Optimization and Earthquake Engineering - page 186

Figure 4. Design cycles: a) traditional, b)optimal

tions. The solution of constrained optimization
problem places within feasible space. The feasible
space for problems with equality constraints is
the boundary constructed by the intersection of
all equality constraints. In the engineering design
problems, we rarely confront to problems with
equality constraints. Considerable research effort
has been paid to propose some solution algorithms
that efficiently solve the problem. Depending on
the order of derivatives that optimization algo-
rithms use for solution, Haftka (1992) divides
them into three main groups. They are: zero order,
first order and second order algorithms.
Zero order algorithms do not use any derivative
of objective functions or constraints. There are a
few zero order classical algorithms, but there are
many hierarchical and probabilistic algorithms
such as Genetic Algorithms, Particle Swarm Op-
timization, Ant Colony, etc. Most of zero order
algorithms are used for unconstrained optimization
problems. Since almost all engineering design
optimization problems are constrained problems,
to use a zero order algorithm for their solution,
the constrained optimization problems have to be
converted to unconstrained optimization problems
via one of Penalty Function methods. A topic by
Arora (2004) will assist in this regard. The first
order algorithms use the first order derivatives of
constraints. This group of algorithms comprise

a large part of body of classical optimization
and are most commonly used. The second order
optimization algorithms that use the second or-
der derivatives of constraints are fairly efficient
algorithms, but because of calculation of second
order derivatives, they are not proper algorithms
for problems with numerous design variables.
Many attempts have been made to formulate
the optimization problem in such a way that it
is easily solved. Optimization problems may be
categorized into linear and nonlinear. If the con-
straints and objective function of a problem are
linear, the problem is called a linear programming
problem and may be solve by Simplex method.
Most of structural design optimization problems
are naturally nonlinear problems because of the
nonlinear nature of indeterminate structures. For
determinate structures, the fully stressed design
(FSD) optimization or similar algorithms (such as
simultaneous failure mode (SFM), etc.) which are
a member to member optimization strategy may
be used. In most of engineering problems, the
design constraints cannot be defined in explicit
form in terms of design variables. Accordingly,
the Taylor series expansion is used to express the
constraints in explicit form as follows: