MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Civil and Environmental Engineering
1.731 Water Resource Systems
Lecture 5 & 6, Optimality Conditions, Sept. 21 & 26, 2006
How do we know when a particular candidate solution x* is a local maximum?
Necessary (Kuhn-Tucker) conditions for a candidate solution x* to be a local maximum are:
1. Feasibility
2. Stationarity
3. Inequality Lagrange multipliers
4. Curvature
Preliminaries:
x* is a local maximum if F(x*) ≥ F(x) for all feasible x near x*
m*A active constraints at x*:
gi(x* )= 0
i ∈ C (x*) = active set
m*I inactive constraints at x*:
gi(x* ) < 0
i ∉ C (x*)
m*A + m*I = m
Form an m*A by n matrix with rows the gradient vectors ∂g i ( x*) / ∂x j of the m*A constraint
functions active at x* . If Rank [ ∂g i ( x*) / ∂x j ] = ρ *A < m*A the problem is degenerate.
Otherwise ρ *A = m*A and the problem is non-degenerate.
The set of m*A constraints active at x* define an n − ρ *A ≥ 0 dimensional constraint surface in
the n dimensional decision space.
Any ρ *A linearly independent gradient vectors form a basis for a ρ *A dimensional gradient
space. Any n − ρ *A tangent vectors pi (i = 1,…, n − ρ *A ) normal to all the gradient vectors form
a basis for an n − ρ *A dimensional tangent space.
Orthogonality condition satisfied by any vector pi in tangent space:
∂g j ( x*)
∂g j ( x*)
→ pi
=0
pi ⊥
i ∈ C (x*)
∂xi
∂xi
The tangent space can be viewed as a plane that intersects the n − ρ *A constraint surface at x* .
This plane approximates the constraint surface for x sufficiently close to x* .
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Constraint gradient space
∂gi ( x)
∂x j
Dim = ρ *
Constraint tangent space
pi ⊥
∂g i ( x)
∂x j
Dim = n − ρ *
x
Intersection of active
constraint surfaces
Statement of Necessary Conditions for a Local Maximum:
1. Feasibility
x* must lie in the feasible region F:
gi(x* ) = 0
i = 1,…r
gi(x* ) ≤ 0
i = r+1, …, m
2. Stationarity
Objective function gradient at x* must lie in the constraint gradient space (i.e. it has no
projection onto the constraint tangent plane).
For non-degenerate problems this implies:
∂g i ( x*)
∂F ( x*)
= λi
∂xj
∂xj
i ∈ C (x*)
The λi are Lagrange multipliers for the active constraints at x*.
If x* is a local maximum this system of n linear equations in the ρ *A = m*A unknown λi‘s
must have a solution (i.e. it must be consistent).
For degenerate problems include only ρ *A < m*A linearly independent constraints and set
λi = 0 for the remaining redundant constraints
Adopt convention that λi = 0 for inactive constraints as well as redundant constraints
so the stationarity condition can include all constraints:
∂g i ( x*)
∂F ( x*)
= λi
∂x j
∂xj
λi g i ( x*) = 0 for each i ∈ C (x*) (no sum over i)
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Define Lagrangian function to be:
L( x*, λ ) = F ( x*) − λi g i ( x*)
Then stationarity condition requires:
∂L( x*, λ )
=0
∂x j
3. Inequality Lagrange multipliers
If x* is a local maximum then the Lagrange multipliers for all inequality constraints
active at x* must be non-negative: λi ≥ 0, i ∈ C (x*).
4. Curvature
Projection of Lagrangian onto the constraint tangent space must have a negative semidefinite Hessian.
Projection operator is an n by n − ρ *A matrix Zik with columns composed of the n − ρ *A
constraint tangent space basis vectors. These basis vectors are linearly independent
solutions pi of:
∂g j ( x*)
∂g ( x*)
pi ⊥ i
→ pi
=0
∂x j
∂xi
⎡
Z ik = ⎢ p1i K
⎣
n − ρ *A ⎤
pi
⎥
⎦
:
Hessian of the projected Lagrangian is Wkl:
Wkl =
∂ 2 L( x*, λ )
Z ik Z lj = 0
∂x j ∂xi
If x* is a local maximum, Wkl must be negative semidefinite → Wkl ≤ 0.
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Example:
Consider an two-dimensional example with 2 inequality constraints and 3 candidate solutions
*
= (1,2)
x*A = (0,2) , x*B = (0,1) ., xC
2 2
F(x)= -x - x
1 2
2
g1(x) = x1 − x2 + 1 ≤ 0
g2(x) = x2 - 2 ≤ 0
x2
*
x A = (0,2)
g2(x)≤0
g1(x)≤0
*
xC = (1,2)
F
*
x B = (0,1)
-1
-2
-1
Gradients are:
∂F
∂F
= −2 x1
= −2 x 2
∂x1
∂x 2
∂g1
∂g1
= 2 x1
= −1
∂x1
∂x 2
∂g 2
=0
∂x1
0
-4
1
2
x1
Lagrangian and its Hessian are:
L( x, λ ) = − x12 − x 22 − λ 1[ x12 − x 2 + 1] + λ 2 [ x 2 − 2]
∂L( x, λ ) ⎡− 2 − 2λ 1
=⎢
0
∂xi ∂x j
⎣
∂g 2
=1
∂x 2
0 ⎤
− 2⎥⎦
Evaluate gradients at candidate solutions
x*A = (0,2)
x*B = (0,1)
*
xC
= (1,2)
∂F ( x*) / ∂x j
∂g1 ( x*) / ∂x j
⎡0⎤
⎢ − 4⎥
⎣ ⎦
⎡0⎤
⎢ − 2⎥
⎣ ⎦
⎡ − 2⎤
⎢ − 4⎥
⎣ ⎦
⎡0⎤
⎢−1⎥
⎣ ⎦
⎡0⎤
⎢−1⎥
⎣ ⎦
⎡2⎤
⎢−1⎥
⎣ ⎦
∂g 2 ( x*) / ∂x j m *A
⎡0⎤
1
⎢1⎥
⎣ ⎦
⎡0⎤
1
⎢1⎥
⎣ ⎦
⎡0⎤
⎢1⎥
⎣ ⎦
Check necessary conditions for a local maximum at each x*:
4
2
ρ *A n − ρ *A
1
1
1
1
2
0
Z ij
⎡a ⎤
⎢0⎥
⎣ ⎦
⎡a ⎤
⎢0⎥
⎣ ⎦
None
1. Feasibility:
All 3 candidate solutions are feasible.
2. Stationarity
Consider for all active constraints:
Consistent ? λ1
λ2
Wkl
x *A = (0,2)
Yes
0
-4
-2a < 0
x *B = (0,1)
Yes
+2
0
-6a < 0
*
xC
= (1,2)
Yes
-1
-5
None
2
2
3. Inequality Lagrange Multipliers
Only x*B has non-negative Lagrange multipliers for active inequalities.
4. Curvature:
*
Both x*A and x*B satisfy the curvature condition. This condition does not apply to xC
.
Example Summary:
Only x*B is a local maximum.since it is the only solution that satisfies all 4 conditions. For this
problem x*B is also a global maximum (why?)
Quick Outline of Derivation:
Derivation of necessary conditions is based on Taylor series approximations of gi(x) and F(x):
An infinitesimal feasible arc from x* to x lies wholly inside the feasible region.
Let θ be distance along this arc from x* to x.
x* = x(0)
x = x(θ)
The vector tangent to this arc at x(0) is ∂x j (0) / ∂θ .
Infinitesimal arcs originating at x(0) are feasible [i.e. g i [ x(θ )] = 0 ] if the corresponding
∂x j (0) / ∂θ lies in the constraint tangent space. To see this use a Taylor series expansion of
g i [ x(θ )] :
g i [ x(θ )] = g i [ x(0)] +
∂g i [ x(0)] ∂x j (0)
θ +K = 0
∂x j
∂θ
i ∈ C (x*)
The first term on the right is zero because constraint i is active at x* = x(0).
The second term on the right is zero since ∂x j (0) / ∂θ is orthogonal to all the active constraint
vectors if it lies in constraint tangent space.
The Taylor series expansion of F[x(θ)] along an infinitesimal arc is:
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∂F [ x(0)]
∂ 2 F [ x(0)] 2
F [ x(θ )] = F [ x(0)] +
θ+
θ +K =
∂θ
∂θ 2
F [ x(θ )] = F [ x*] +
∂F [ x*] ∂xi (0)
∂ 2 F [ x*] ∂xi (0) ∂x j (0) 2
θ+
θ +K
∂xi
∂θ
∂xi ∂x j ∂θ
∂θ
If x* is a local maximum F(x) = F[x(θ)] must be ≤ F[x(0)] = F(x*) for all values of θ along the
arc. This implies:
∂F [ x(0)] ∂F [ x*] ∂x j (0)
1).
=
=0
∂θ
∂x j
∂θ
2).
∂ 2 F [ x(0)]
∂ 2 F [ x*] ∂xi (0) ∂x j (0)
=
≤0
∂xi ∂x j ∂θ
∂θ
∂θ 2
The stationarity condition follows from 1). and the curvature condition follows from 2), if the
requirement that ∂xi (0) / ∂θ lies in the constraint tangent space and the definition of the
Lagrangian are invoked.
The stationarity condition takes care of feasible arcs that lie in the tangent space, which are the
only directions that are feasible for equality constraints.
If the constraint is an inequality the feasible arc may also point into the feasible region, away
from the tangent space. Directions into the feasible region are defined by::
∂x j (0) ∂g i [ x(0)]
3).
<0
for i an inequality constraint ∈ C (x*)
∂θ
∂x j
The objective function cannot increase along this feasible arc if x* is a local maximum. So:
∂F [ x(0)] ∂x j (0)
4).
≤0
for i an inequality constraint ∈ C (x*)
∂x j
∂θ
The inequality Lagrange multiplier condition follows from 3) and 4).
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