13.1
Nonlinear Programming Problems
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and
gi (x1 , x2 , . . . , xn ) =
n
X
ai j x j
(i = 1, 2, . . . , m).
j=1
Note that nonnegativity restrictions on variables can be included simply by appending the additional constraints:
gm+i (x1 , x2 , . . . , xn ) = −xi ≤ 0
(i = 1, 2, . . . , n).
Sometimes these constraints will be treated explicitly, just like any other problem constraints. At other times,
it will be convenient to consider them implicitly in the same way that nonnegativity constraints are handled
implicitly in the simplex method.
For notational convenience, we usually let x denote the vector of n decision variables x1 , x2 , . . . , xn —
that is, x = (x1 , x2 , . . . , xn ) — and write the problem more concisely as
Maximize f (x),
subject to:
gi (x) ≤ bi
(i = 1, 2, . . . , m).
As in linear programming, we are not restricted to this formulation. To minimize f (x), we can of course
maximize − f (x). Equality constraints h(x) = b can be written as two inequality constraints h(x) ≤ b and
−h(x) ≤ −b. In addition, if we introduce a slack variable, each inequality constraint is transformed to an
equality constraint. Thus sometimes we will consider an alternative equality form:
Maximize f (x),
subject to:
h i (x) = bi
xj ≥ 0
(i = 1, 2, . . . , m)
( j = 1, 2, . . . , n).
Usually the problem context suggests either an equality or inequality formulation (or a formulation with both
types of constraints), and we will not wish to force the problem into either form.
The following three simplified examples illustrate how nonlinear programs can arise in practice.
Portfolio Selection An investor has $5000 and two potential investments. Let x j for j = 1 and j = 2
denote his allocation to investment j in thousands of dollars. From historical data, investments 1 and 2 have
an expected annual return of 20 and 16 percent, respectively. Also, the total risk involved with investments 1
and 2, as measured by the variance of total return, is given by 2x12 + x22 + (x1 + x2 )2 , so that risk increases with
total investment and with the amount of each individual investment. The investor would like to maximize his
expected return and at the same time minimize his risk. Clearly, both of these objectives cannot, in general, be
satisfied simultaneously. There are several possible approaches. For example, he can minimize risk subject
to a constraint imposing a lower bound on expected return. Alternatively, expected return and risk can be
combined in an objective function, to give the model:
Maximize f (x) = 20x1 + 16x2 − θ [2x12 + x22 + (x1 + x2 )2 ],
subject to:
g1 (x) = x1 + x2 ≤ 5,
x1 ≥ 0, x2 ≥ 0,
(that is, g2 (x) = −x1 ,
g3 (x) = −x2 ).
The nonnegative constant θ reflects his tradeoff between risk and return. If θ = 0, the model is a linear
program, and he will invest completely in the investment with greatest expected return. For very large θ , the
objective contribution due to expected return becomes negligible and he is essentially minimizing his risk.
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