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NONLINEAR PROGRAMMING
ECTURE 17: AUGMENTED LAGRANGIAN METHODS
LECTURE OUTLINE
• Multiplier Methods
• Consider the equality constrained problem
minimize f (x)
subject to h(x) = 0,
where f :
n
→ and h :
n → m
are continuously
differentiable.
• The (1st order) multiplier method finds
c k ‖h(x)‖ 2 2 k = arg min x∈n†
L
ck† (x, λ k ) ≡ f (x) + λ k ′^ x h(x) +
and updates λ
k
using
λ k+ = λ k
CONVEX EXAMPLE
• Problem: minx 1 =1(1/2)(x
2 2
) with optimal so-
1
lution x
∗
= (1, 0) and Lagr. multiplier λ
∗
• We have
c k − λ k x k = arg min L ck† (x, λ k ) = ck^ + 1
x∈n† c k − λ k λ k+ = λ k
- c k − 1 c k
- 1 λ k − λ ∗ λ k+ − λ ∗ = c k
- 1
• We see that:
− λ k → λ ∗
= − 1 and x
k → x ∗
= (1, 0) for ev-
ery nondecreasing sequence {c
k
}. It is NOT
necessary to increase c
k
to ∞.
− The convergence rate becomes faster as c
k
becomes larger; in fact |λ
k −λ ∗
| converges
superlinearly if c
k
THE PRIMAL FUNCTIONAL
• Let (x
∗ , λ ∗
) be a regular local min-Lagr. pair sat-
isfying the 2nd order suff. conditions are satisfied.
• The primal functional
p(u) = min f (x), h(x)=u
defined for u in an open sphere centered at u = 0,
and we have
p(0) = f (x ∗ ), ∇p(0) = −λ ∗ , (^0) u (u + 1)^2 1 2 p(u) = p(0) = f(x*) = 1 2
- 0 u (u + 1)^2 1 2 p(u) = - p(0) = f(x*) = - 1 2
(a) (b) p(u) = min 1 2 2 2 (x 1 + x 2 ), p(u) = min x 1 −1=u 1 2 2 2 (−x 1 + x 2 ) x 1 −1=u
AUGM. LAGRANGIAN MINIMIZATION
• Break down the minimization of Lc(·, λ):
c
f (x) + λ
′
h(x) + �h(x)�
2
min
x
Lc(x, λ) = min
u
min
h(x)=u
p(u) + λ
′
u +
c
�u�
2
= min ,
u
where the minimization above is understood to
be local in a neighborhood of u = 0.
• Interpretation of this minimization:
Penalized Primal Function p(0) = f(x) p(u) Slope = - λ min L (^) c (x,λ) x Slope = - λ u(λ,c) 0
- λ'u(λ,c) Primal Function p(u) + 2 ||u||^2 u c
• If c is suf. large, p(u) + λ
′ u + c 2 ‖u‖ 2
is convex in
a neighborhood of 0. Also, for λ ≈ λ
∗
and large c,
the value minx Lc(x, λ) ≈ p(0) = f (x
∗
COMPUTATIONAL ASPECTS
• Key issue is how to select {c
k
− c k
should eventually become larger than the
“threshold” of the given problem.
− c 0
should not be so large as to cause ill-
conditioning at the 1st minimization.
− c k
should not be increased so fast that too
much ill-conditioning is forced upon the un-
constrained minimization too early.
− c k
should not be increased so slowly that
the multiplier iteration has poor convergence
rate.
• A good practical scheme is to choose a mod-
erate value c
0
, and use c
k+ = βc k
, where β is a
scalar with β > 1 (typically β ∈ [5, 10] if a Newton-
like method is used).
• In practice the minimization of L
ck† (x, λ k
) is typ-
ically inexact (usually exact asymptotically). In
some variants of the method, only one Newton
step per minimization is used (with safeguards).
DUALITY FRAMEWORK
• Consider the problem
c
minimize f (x) + ‖h(x)‖
2 2
subject to ‖x − x
∗ ‖ < �, h(x) = 0,
where � is small enough for a local analysis to
hold based on the implicit function theorem, and c
is large enough for the minimum to exist.
• Consider the dual function and its gradient
qc(λ) = min Lc(x, λ) = Lc x(λ, c), λ ‖x−x∗^ ‖<� ∇qc(λ) = ∇λx(λ, c)∇xLc x(λ, c), λ + h x(λ, c) = h x(λ, c).
We have ∇qc(λ
∗ ) = h(x ∗
) = 0 and ∇
2 qc(λ ∗
• The multiplier method is a steepest ascent iter-
ation for maximizing qck†
λ k+ = λ k
