Geology Reference
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QP sub-problems should be repeated. These strat-
egies are called Sequential Linear Programming
(SLP) and Sequential Quadratic Programming
(SQP), respectively. Although SLP and SQP can
often capture the optimum solution, they are not
so much efficient algorithms. Considerable effort
has been paid to find efficient and robust optimiza-
tion algorithms. Among successful optimization
algorithms are Dual method and Conlin algorithm
by Fleury (1983 and 1989) and DOT algorithm
by Vanderplatts (1997). The degree of success
for an optimization algorithm somehow depends
on the nature of the optimization problem. An
algorithm may work very well for a problem and
not well for another. There is not a firm conclu-
sion about the efficiency and robustness of any
especial optimization algorithm. An optimization
algorithm may solve one optimization problem
efficiently, and not solve another. For example if
an optimization problem is expressed in terms of
reciprocals of design variables, it may be solved
in an excellent way, but if it is defined in terms
of direct design variables it may not be solved
properly. This may happen vice versa.
As a conclusion, the choice of the optimization
algorithm for a design optimization problem does
not have a certain rule. However, some algorithms
have shown that in many applications are often
more efficient than the others. Optimality Criteria
(OC) methods are believed to be more efficient
than other classical optimization algorithms. These
methods do not directly consider the objective
function in the solution process. They seek the
optimality of the solution in satisfaction of some
predefined criteria. The FSD and SFM methods
are among this group of optimization algorithms.
Some research works focused their attempt to
combine the principals of OC methods with those
of classical optimization algorithms. The Kuhn-
Tucker-based OC method is an example of this
kind. In this type of OC algorithm, the necessary
conditions for optimality of a design point that
are called Kuhn-Tucker conditions, are chosen
as a basis for the establishment of recursive for-
1
2
F x
( )
=
F x
(
0
)
+∇
F x
(
0
)
T
∆
x
+
∆
x
T
∇
2
F x
(
0
)
∆
x
+
...
(16)
0
where,
x
is the vector of deign variables;
∇
F
(
)
is the gradient of the function F at the current
point and
∇
2
F
(
is the matrix of second order
derivatives, named Hessian matrix of the function.
Although the design constraint may be implicit
function of design variables, its derivatives can
be numerically obtained by sensitivity analysis.
The calculation of Hessian of a function, particu-
larly if it is a function of numerous design variables
is relatively difficult and computationally time
consuming. Some algorithms such as DFP and
BFGS use the first order derivatives to gradually
construct the Hessian matrix or its inverse from
its first order derivatives.
If a function is approximated with its first two
terms of Taylor series, it is called linearized form
of the function. Keeping the first three terms of
Taylor series approximates the function with a
quadratic form. Except for problems with a few
design variables, the quadratic term is rarely used.
Expressing the constraints and objective function
in their linear form, makes it possible to define
a Linear Programming (LP) sub-problem and
solve it by Simplex algorithm. Objective func-
tion is usually given explicitly in terms of design
variables. If the objective function is written in
quadratic form and the constraints expressed in
linear form, a Quadratic Programming (QP) prob-
lem is established. The solution of an optimization
problem can be sought by converting it to LP or
QP sub-problem and solving it. The QP and LP
problems have a straight-forward solution scheme
and most of mathematical programming software
have these solution options. The outcome of LP
or QP sub-problem will not necessarily insure the
feasibility and optimality of the solution because
the constraints of these problems are some ap-
proximation of actual design problem. Therefore,
the process of establishing and solution of LP or
0
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