CONSTRAINT QUALIFICATIONS FOR NONLINEAR PROGRAMMING
RODRIGO G. EUST
´
AQUIO
§
, ELIZABETH W. KARAS
¶
,
AND
ADEMIR A. RIBEIRO
¶
Abstract.
This paper deals with optimality conditions to solve nonlinear programming problems. The classical
KarushKuhnTucker (KKT) optimality conditions are demonstrated through a cone approach, using the well known
Farkas’ Lemma. These conditions are valid at a minimizer of a nonlinear programming problem if a constraint
qualiﬁcation is satisﬁed. First we prove the KKT theorem supposing the equality between the polar of the tangent
cone and the polar of the ﬁrst order feasible variations cone. Although this condition is the weakest assumption, it
is extremely diﬃcult to be veriﬁed. Therefore, other constraints qualiﬁcations, which are easier to be veriﬁed, are
studied, as: Slater’s, linear independence of gradients, MangasarianFromovitz’s and quasiregularity. The relations
among them are discussed.
Key words.
Optimality conditions, KarushKuhnTucker, constraint qualiﬁcations.
1. Introduction.
We shall study the nonlinear programming problem
(
P
)
minimize
f
(
x
)
subject to
h
(
x
) = 0
g
(
x
)
≤
0
,
where the functions
f
: IR
n
→
IR,
g
: IR
n
→
IR
p
and
h
: IR
n
→
IR
m
are continuously diﬀerentiable.
The feasible set is Ω =
{
x
∈
IR
n

h
(
x
) = 0
,
g
(
x
)
≤
0
}
.
Given
x
*
∈
Ω, the classical
KarushKuhnTucker
(KKT) conditions say that there exist
La
grangian
multipliers
λ
*
∈
IR
m
and
μ
*
∈
IR
p
such that:
∇
f
(
x
*
) =
m
X
i
=1
λ
*
i
∇
h
i
(
x
*
) +
p
X
j
=1
μ
*
j
∇
g
j
(
x
*
)
,
μ
*
j
≥
0
,
j
= 1
, . . . , p,
μ
*
j
g
j
(
x
*
) = 0
,
j
= 1
, . . . , p.
In nonlinear programming we would like that KKT are necessary conditions for a given point to
be a solution to the problem. When the problem is unconstrained (Ω = IR
n
), the KKT conditions
reduce to
∇
f
(
x
*
) = 0 which is a necessary optimality condition. However, this not always happens,
as shown in the following example.
Example 1.
Consider the problem
(
P
)
with
f
: IR
2
→
IR
and
g
: IR
2
→
IR
2
deﬁned by
f
(
x
) =
x
1
and
g
(
x
) = (
x
2

(1

x
1
)
3
,

x
2
)
T
. Note that
x
*
= (1
,
0)
T
is a minimizer of the problem
but the KKT conditions do not hold.
In this paper we shall discuss assumptions on the constraints in order to ensure that the
KKT conditions hold at a minimizer. Such an assumption is called
constraint qualiﬁcation (CQ)
.
Formally, we say that the constraints
h
(
x
) = 0 and
g
(
x
)
≤
0 satisfy a constraint qualiﬁcation at
x
*
∈
Ω when, given any diﬀerentiable function
f
minimized at
x
*
with respect to Ω, the KKT
conditions are valid.
Several authors have obtained diﬀerent constraint qualiﬁcations. In this work, we will discuss
many of them as well as some relations between them. A special interest is devoted to show the
weakest such qualiﬁcation. In this context, the concept of cones and their polars will be useful.
§
Master Program in Numerical Methods in Engineering, Federal University of Paran´a, Cx. Postal 19081, 81531