4.5.
MISCELLANEOUS COMMENTS 47
4.5.1 Some mathematical tricks.
It is often the case in practical decision problems that the objective is not well-defined. There may be a number of plausible objective functions. In our LP framework this situation can be formulated
as follows. The constraints are given as usual by Ax ≤ b, x ≥ 0. However, there are, say, k
objective functions c
1 ′
x, . . . , c
k ′
x. It is reasonable then to define a single objective function f
x by f x = minimum {c
1 ′
x, c
2 ′
x, . . . , c
k ′
x}, so that we have the decision problem, Maximize
f x
subject to Ax ≤ b, x ≥ 0 .
4.45 This is not a LP since
f is not linear. However, the following exercise shows how to transform
4.45 into an equivalent LP.
Exercise 1:
Show that 4.45 is equivalent to 4.46 below, in the sense that x
∗
is optimal for 4.45 iff
x
∗
, y
∗
= x
∗
, f x
∗
is optimal for 4.46. Maximize
y subject to
Ax ≤ b, x ≤ 0 y ≤ c
i ′
x , 1 ≤ i ≤ k . 4.46
Exercise 1 will also indicate how to do Exercise 2. Exercise 2:
Obtain an equivalent LP for 4.47: Maximize
n
X
j=1
c
i
x
i
subject to Ax ≤ b, x ≤ 0 ,
4.47 where
c
i
: R → R are concave, piecewise-linear functions of the kind shown in Figure 4.3. The above-given assumption of the concavity of the
c
i
is crucial. In the next exercise, the inter- pretation of “equivalent” is purposely left ambiguous.
Exercise 3: Construct an example of the kind 4.47, where the
c
i
are piecewise linear but not concave, and such that there is no equivalent LP.
It turns out however, that even if the c
i
are not concave, an elementary modification of the Simplex algorithm can be given to obtain a “local” optimal decision. See Miller [1963].
4.5.2 Scope of linear programming.
LP is today the single most important optimization technique. This is because many decision prob- lems can be adequately formulated as LPs, and, given the capabilities of modern computers, the
Simplex method together with its variants is an extremely powerful technique for solving LPs in- volving thousands of variables. To obtain a feeling for the scope of LP we refer the reader to the
book by one of the originators of LP Dantzig [1963].
48
CHAPTER 4. LINEAR PROGRAMMING
. .
.
c
i
x
i
x
i
Figure 4.3: A function of the form used in Exercise 2.
Chapter 5
OPTIMIZATION OVER SETS DEFINED BY INEQUALITY
CONSTRAINTS: NONLINEAR PROGRAMMING
In many decision-making situations the assumption of linearity of the constraint inequalities in LP is quite restrictive. The linearity of the objective function is not restrictive as shown in the first
exercise below. In Section 1 we present the general nonlinear programming problem NP and prove the Kuhn-Tucker theorem. Section 2 deals with Duality theory for the case where appropriate
convexity conditions are satisfied. Two applications are given. Section 3 is devoted to the important special case of quadratic programming. The last section is devoted to computational considerations.
5.1 Qualitative Theory of Nonlinear Programming
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