Optimizacion

PART

II
O PTIMIZATION
THEORY
A ND M ETHODS

P

art I I describes modern techniques of optimization and translates these
concepts into computational methods and algorithms. A vast literature on
optimization techniques exists, hence we have focused solely on methods which
have been proved effective for a wide range of problems. Optimization methods
have matured sufficiently during thepast twenty years so t hat fast a nd reliable
methods are available to solve each important class of problem.
Six chapters make up P art I I of this book, covering the following areas:
1.
2.
3.
4.
5.
6.

M athematical concepts (Chapter 4)
One-dimensional search (Chapter 5)
U nconstrained multivariable optimization (Chapter 6)
Linear programming (Chapter 7)
N onlinear programming(Chapter 8)
O ptimization involving staged processes and discrete variables (Chapter 9)

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O PTIMIZATION THEORY AND M ETHODS

The topics are grouped so t hat unconstrained methods are presented first, followed by constrained methods. The last chapter in P art I I deals with discontinuous (integer) variables, a common category of problem in chemical engineering
but one quitedifficult to solve without great effort.
As optimization methods as well as computer hardware have been improved over the past two decades, the degree of difficulty of the problems t hat
can be solved has expanded significantly. Continued improvements in optimization algorithms and computer hardware and software should enable optimization of large-scale nonlinear problems involving thousands ofvariables, b oth
continuous and integer, some of which may be stochastic in nature.

C HAPTER

4
BASIC
C ONCEPTS'
O F O PTIMIZATION

4.1 Continuity of Functions
4.2 Unimodal Versus Multimodal Functions
4.3 Convex and Concave Functions
4.4 Convex Region
4.5 Necessary and Sufficient Conditions for an Extremum of an Unconstrained
Function
4.6 Interpretation of the Objective Function inTerms of Its Quadratic Approximation
References
Problems

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127
129
134
138
145
151
152

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BASIC CONCEPTS O F O PTIMIZATION

I n o rder to understand the strategy of optimization procedures, certain
basic concepts must be described. I n this chapter we will examine the properties
of objective functions and constraints to establish a basis for analyzing optimizationproblems. Those features which are desirable (and also undesirable) in the
formulation of an optimization problem are identified. Both qualitative and
quantitative characteristics of functions will be described. I n addition, we will
present the necessary a nd sufficient conditions to guarantee t hat a supposed
extremum is indeed a minimum o r a maximum.

4.1

CONTINUITY O F F UNCTIONS

I ncarrying out analytical o r numerical optimization you will find it preferable
and more convenient to work with continuous functions of one o r more variables than with functions containing discontinuities. Functions having continuous derivatives are also preferred. W hat does continuity mean? Examine
Fig. 4.1. I n case A , the function is clearly discontinuous. Is case B also discontinuous?
Wedefine the property of continuity as follows. A function of a single
variable x is continuous a t a p oint X o if
(a)
(b)

f (x o) exists

lim f (x) exists
X --+XO

(c)

lim f (x) = f (x o)
X -+XO

I f f (x) is continuous a t every point in region R, t hen f (x) is said to be continuous throughout R. F or case B in Fig. 4.1, the function of x h~s a " kink" in it,
but f (x) doessatisfy the property of continuity. However, f '(x) == d f(x)/dx does
not. Therefore, the function in case B is continuous b ut n ot continuously differentIable.

! (X l )

~~
Xl

Case A
Figure 4.1

! (x 2 )

/\
X2

Case B

Functions with discontinuities in the function a nd/or derivatives.

4.1

E XAMPLE 4.1

C ONTINUITY O F F UNCTIONS

125

ANALYSIS O F F UNCTIONS F...