Samenvatting chapter 4 - Numerical Modelling and Design of El. & Mech sys. (E048400A)

Chapter 4
: Constrained
gradient-based optimization
4 11 . Intro + Problem
formulation
~s
optim problems
. are
rarely unconstrained


Build
further the methods from Chapter 3 ?
↳
on n



first introduce
optimality conditions for constrained
opt problem
then
~
focus on 3 main methods
2
{
w
Penalty methods
=
sequential quads progr .
.
(SQP)
·

interior-point methods
in
general : minf(x
t X /Mx dim ) variable vector
2 . .



g(x)20 -

-
ps vecter constraints
ha = eg h : --R
: /R -

" inequality
vector
0 ~
D
equality constraints
↳ both
of (f) .
+ contraints
(g ,
h)
_
↳ con be non-lineair f
ㅇ

↳ should be
2
continuous

constraints ?
-to solve a contrained
problem - Also
requir gradients of all

값 값
1
…




Yn Similar
Ig (Mg
=



: :
-- .




,




재행 ? :
↓ xnx)
(MnXMx) ↳
for inequality constraint
.



G 2) conditions
.


Optimality
↳ Not as
straight forward as those
for unconstrained

start for
~
equality contr .


*
Equality Constraints


~
again start fum order Tayler series
expansion of objective function .




f(x p) f(x)
+ = +
ffxxp
~ Since ** 0 min =>
OfSX 1Tp *
3 0 ④



If unconstrained
only
the
inequality would be If (x *

problem
-
to
satisfy
were 1 =
,
way

-

If problem is constrained ,
function increase & still applies ,
but
p
must be also a



feasible direction
direction 12 order
to
find feasible write
Taylor expansion
~ can
,
we a


constraint
for each
equality
一

,hj(x hj(x) ohjkx)
+
+
p) = +
pj 1
,..., order
mn
first
=




here
dso A feasibility
~
assume
higher order
neglected because
of small step size
T




jThe ne
Assuming is
feasible point-phj(x) 0
for all
=
X a




#‰




→
here Mx =
Mn = 2

↳
feasible space reduced
to
single point
this has to
r means
any feasible di. => no
optim .



freedom !
lis in
nulspace of Jacobian
of the constraints In?




_
Assume rank /i µ
In has full de Constraint
grad linearly indep. (
>
-
.
e . .
Or




>
-



feasible space is sulspace of dimension Mx -



Mp .




for opt . To be
possible nx > un Why ?

~
for one contraint ,
we have thip = o


↳
feasible space corresponds to
tangent plane




- >
for 2 or moe contraints ,
feorible space
reduces to intersection of all tangent hyperplanes ?

~
in 3D -- a line




~
for constrained
optimality > need to
satisfy both If 1 p3 *
, 0 and
Ip(x)p =




-for equality constrained ,
if p
is
feasible ,
-p
must dro be
feasible
=>

ratify
My way to OfIp 70 is
if IfTp =




~ in sum ,
for
*
to be a contrained
optimum ,
we
requir Of *
p
= 0
for all
p such
that

Yu ( t ip
* = o

, other words, vanish
projection of dy function's gradient
~ must
in .
into
feasible space

If(x upon (th(x
Chasx Ohnn' ** )
* *
or
*
) = ) , ) , . . . ,




hers this is illustrated
for case
2 constraints

in 3D




>
-
constrained optimum If * p = o
require If to be I to nullspace of In
motic contains all vectors I to it's nullspace
~ row
space of a


because ofTp
>
-
rowe are
gradients of the constraints


-
objective funct gradient must be a linear comb
of gradients of constraints


Ef ( +
*
) =

λ jchjs + * 1

associated with each constraints
Lagrange multipliers
↳ ,
one


=
first-order optimality condition
for equality contraint are


「
ef ( x *
1 -

yn /x ) λ

h(x) 0
=




Def In constrained optio ,
the
lagrangian function being
a scalar is
defined a




2(x, x) =

f(x) + h5xX
↳
hagrang
Considered to
;an multiplien ora


be
independant
Theorem
gradient of h wrt
king both andit
h EfA reYn *
*
e. =
* *
) λ = 0




第h = hir *
) = 0


with x linear
satisfying independence constraint
qualification
With
lagrangian ,
we
transformed a contrained problem I
design var
, M eg. Contraints


into unconstrained
Adding
by
variable x / MM I
>
new
-



problem
.




desivation
of first-order optimally assumes
gradients of constraints
linearly independent In full rew
>

rank




一