Page 135 - Academic Press Encyclopedia of Physical Science and Technology 3rd Chemical Engineering

P1: ZCK/FLS P2: FYK/FLS QC: FYK Revised Pages
Encyclopedia of Physical Science and Technology EN002G-100 May 19, 2001 18:49

Chemical Process Design, Simulation, Optimization, and Operation 757

FIGURE 3 Determination of optimum insulation thickness.

C. Unconstrained Optimization characterized by their stationary points. Stationary points
of a function are the values of the dependent variable (x)
Unconstrained optimization deals with situations where
where the first derivative of the function (termed the gra-
the constraints can be eliminated from the problem by
dient) is zero.
substitution directly into the objective function. Many
optimization techniques rely on the solution of uncon- ∂ f
strained subproblems. The concepts of convexity and con-
∂x 1

cavity will be introduced in this subsection, as well as
∂ f
discussing unimodal versus multimodal functions, single-

variable optimization techniques, and examining multi- ∇ f (x) = ∂x 2 (14)

.
variable techniques. . .
Multimodal functions are functions that have multi- ∂ f
ple minima/maxima over the independent variable range.
∂x n
Since most optimization techniques in use today search for
local optima (minima or maxima), one can see that these Stationary points can be a (1) local maximum, (2) local
techniques could easily fail to find the true optimum, also minimum,or(3)saddlepoint.Theexistenceofastationary
known as the global extrema. Multimodal functions are point is a necessary condition for an optimum.