SLIDE 1
Example
) 1 ( 1 s.t. max
2 3 1 2 2 2 1 1
≤ − − ≤ + x x x x x
1
2 2 2 1
≤ +x x
) 1 (
2 3 1
≤ − − x x
Nonlinear Inequality Constraints Example Example max x 1 + 2 - - PowerPoint PPT Presentation
Nonlinear Inequality Constraints Example Example max x 1 + 2 - - PowerPoint PPT Presentation
Nonlinear Inequality Constraints Example Example max x 1 + 2 2 s.t. x x 1 1 2 3 ( 1 ) 0 x x 1 2 + x 2 2 3 ( x 1 ) x 0 x 1 1 2 1 2 Convert to standard form min x 1
SLIDE 2
SLIDE 3
) 1 ( 1 s.t. min
2 3 1 2 2 2 1 1
≥ + − − ≥ + − − − x x x x x
[ ]
⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = = , 2 1 * , 1 * λ x
Primal feasibility satisfied Both constraints active
) ) 1 ( ( ) 1 ( ) , (
2 3 1 2 2 2 2 1 1 1
x x x x x x L + − − − − − − − − = λ λ λ
Dual Feasibility:
Convert to standard form
SLIDE 4
) (x f ∇
⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡− = ∇ 1 ) 1 ( 3 2 2 1 ) , (
2 1 2 2 1 1
x x x x L
x
λ λ λ
2 1 1 2 1 1 1 2 2 2 2 2 1 2 3 1 2 1 2
1 2 3 ( 1) 2 1 ( 1) , L x x x L x x x x x x λ λ λ λ λ λ ∂ = − + + − = ∂ ∂ = + − = ∂ − − + = − − + = ≥
[ ]
⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = = , 2 1 * , 1 * λ x
) )( ( 2 *) *, ( ) 1 1 )( )( 3 ( ) 1 )( ( 2 1 *) *, (
2 1 2 2 2 1 1
= − + = ∂ ∂ = − + + − = ∂ ∂ x x L x x L λ λ
* ≥ λ
Check FONC
Solution solves KKT cond Assuming two constraints active
SLIDE 5
Complementarity:
) ( ) ( ) (
2 1
= = = x g x g x gi
i
λ
thus complementarity holds For inactive constraints,
=
i
λ
Complementarity
SLIDE 6
Let’s check necessary conditions Is x* regular, i.e. are the gradients of the active constraints linearly independent?
⎥ ⎦ ⎤ ⎢ ⎣ ⎡− = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∇ ∇ 1 2 ) ( ) (
2 1 T T
x g x g
So x* is regular, and x* is a KKT point, so FONC are satisfied. The null space is empty so SONC:
Z x L Z
xx T
) , (
2
λ ∇
p.s.d. is vacuously satisfied.
More FONC, SONC
SLIDE 7
[ ] [ ]
2 2 2
2 1
− = − − =
+
x x A
Only first constraint is active And non-degenerate (positive multiplier)
⎥ ⎦ ⎤ ⎢ ⎣ ⎡ =
+
1 Z
⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡− = ∇ 1 ) 1 ( 3 2 2 1 ) , (
2 1 2 2 1 1
x x x x L
x
λ λ λ
⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − + = ∇ 1 1 1 ) 1 ( 6 2 ) , (
1 2 1 2
x x L
xx
λ λ λ
A+,Z+,etc.
SLIDE 8
Just showed that x*, λ* is a KKT point. Let’s check SOSC:
+ + ∇
Z x L Z
xx T
) , (
2
λ
is positive definite The general Jacobian of this problem for the active constraints:
⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − − − = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∇ ∇ 1 ) 1 ( 3 2 2 ) ( ) (
2 1 2 1 2 1
x x x x g x g
T T
We only need non-degenerate active constraints
[ ]
2
1 ( , ) 1 1 1 1
T xx
Z L x Z λ
+ +
⎡ ⎤ ⎡ ⎤ ∇ = = > ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦
so SOSC satisfied; x* is a strict local minimizer