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Unformatted text preview: ions, oftentimes constraints are linear. In that case GCQ is automatically satisﬁed, so you don’t need to check it (exercise). It is known that
the GCQ is the weakest possible condition [1]. 5 Suﬃcient condition The KarushKuhnTucker theorem provides necessary conditions for optimalSecond Order
ity: if the constraint qualiﬁcation holds, then a local solution must satisfy
Sufficient Condition
the KarushKuhnTucker conditions (ﬁrstorder conditions and complementary
Not needed
slackness conditions). Note that the KKT conditions are equivalent to
∇x L(¯, λ, µ) = 0,
x (4) where L(x, λ, µ) is the Lagrangian. (4) is the ﬁrstorder necessary condition of
the unconstrained minimization problem
min L(x, λ, µ). (5) x ∈ RN Below I give a suﬃcient condition for optimality.
Proposition 6. Suppose that x is a solution to the unconstrained minimization
¯
problem (5) for some λ ∈ RI and µ ∈ RJ . If gi (¯) ≤ 0 and λi gi (¯) = 0 for all
x
x
+
i and hj (¯) = 0 for all j , then x is a solution to the constrained minimization
x
¯
problem (1).
Proof. Take any x such that gi (x) ≤ 0 for all i and hj (x) = 0 for all j . Then
J I µj hj (¯)
x λi gi (¯) +
x f (¯) = f (¯) +
x
x j =1 i=1 = L(¯, λ, µ) ≤ L(x, λ, µ)
x
J I µj hj (x) ≤ f (x). λi gi (x) + = f (x) +
i=1 j =1 The ﬁrst line is due to λi gi (¯) = 0 for all i and hj (¯) = 0 for all j . The second
x
x
line is the assumption that x minimizes L(·, λ, µ). The third line is due to λi ≥ 0
¯
and gi (x) ≤ 0 for all i and hj (x) = 0 for all j . 6 Constrained maximization Finally, we brieﬂy discuss maximization. Although maximization is equivalent
to minimization by ﬂipping the sign of the objective function, doing so every
time is awkward. So consider the maximization problem
maximize f (x) subject to gi (x) ≥ 0
hj (x) = 0 (i = 1, . . . , I )
(j = 1, . . . , J ). (6) 2014W Econ 172B Operations Research (B) Alexis Akira Toda (6) is equivalent to the minimization problem
minimize − f (x) subject to − gi (x) ≤ 0
− hj (x) = 0 (i = 1, . . . , I )
(j = 1, . . . , J ). (7) Assuming that x is a local solution and the constraint qualiﬁcation holds, then
¯
the KKT conditions are
I − ∇f (¯) −
x J λi ∇gi (¯) −
x
i=1 µj ∇hj (¯) = 0,
x (8a) j =1 (∀i) λi (−gi (¯)) = 0.
x (8b) But (8) is equivalent to (3). For this reason, it is customary to formulate a
maximization problem as in (6) so that the inequality constraints are always
“greater than or equal to zero”.
As an example, consider a consumer with utility function
u(x) = α log x1 + (1 − α) log x2 ,
where 0...
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This document was uploaded on 02/18/2014 for the course ECON 172b at UCSD.
 Winter '08
 Foster,C

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