, Notations
=
[] u
u
=
=Ce
[ur un vz ... un]T
minimisation problem :
*
m (u mo of m(u)
m =
min is the minimum
values of the variables in u that minimisem(M) ut m(r) ut is the minimiser
agmin
: =
of m(u)
m
*
maxm(y) = m(l
augmax
= or u
M
min-ml
*
u =
algmaxm(1) =
ag
m(u)
* *
m =
local minima
Difference between and
global minimum
Global minimum :
point where the function attains its lowest possible value across the entre domain
for a function n(u) ,
global minimum m(u* ) : m(ut) = m(u) FrE/R
Local function min value within
minimum :
point where the attains a
righborhood around the point
m(u* ) = m(u) for all u in a
neighborhood around u
*
Steepest descent
algorithm
-
iteative optimisation
algorithm used to find a local minimum
of a
differentiable function .
-
At each step ,
the
algorithm moves in the direction
of the steepest
negative gradient (the direction in which the function
decreases the most rapidly) .
By taking small steps in this direction ,
the
algorithm gradually "descends" toward a .
nin
Mathematical formulation :
Even a differential function m(r) , the steepest descent
algorithm updates the current
guess refor the min
ur =
u -
xm(uk)
·
U : current
guess for the min at ituation k
· Ch :
step size at ituation k
Ym(rh) u
gradient of function evaluated at
·
: m
+1
uk updated guees for . min
·
:
-
Ym (m) - of steepest ascent
direct -
Xm(m) -> steepest descent
Stepsize &-how far the If is too small
algorithm moves
along the gradient direction . I -
convergence is slow
If X too
large I
night overshoot the minimum and fail to
until the
-
Itrative process >
-
it repeats the process
of updating u
change in the
function value converge
.
or the
gradient becomes
sufficiently small
=
[] u
u
=
=Ce
[ur un vz ... un]T
minimisation problem :
*
m (u mo of m(u)
m =
min is the minimum
values of the variables in u that minimisem(M) ut m(r) ut is the minimiser
agmin
: =
of m(u)
m
*
maxm(y) = m(l
augmax
= or u
M
min-ml
*
u =
algmaxm(1) =
ag
m(u)
* *
m =
local minima
Difference between and
global minimum
Global minimum :
point where the function attains its lowest possible value across the entre domain
for a function n(u) ,
global minimum m(u* ) : m(ut) = m(u) FrE/R
Local function min value within
minimum :
point where the attains a
righborhood around the point
m(u* ) = m(u) for all u in a
neighborhood around u
*
Steepest descent
algorithm
-
iteative optimisation
algorithm used to find a local minimum
of a
differentiable function .
-
At each step ,
the
algorithm moves in the direction
of the steepest
negative gradient (the direction in which the function
decreases the most rapidly) .
By taking small steps in this direction ,
the
algorithm gradually "descends" toward a .
nin
Mathematical formulation :
Even a differential function m(r) , the steepest descent
algorithm updates the current
guess refor the min
ur =
u -
xm(uk)
·
U : current
guess for the min at ituation k
· Ch :
step size at ituation k
Ym(rh) u
gradient of function evaluated at
·
: m
+1
uk updated guees for . min
·
:
-
Ym (m) - of steepest ascent
direct -
Xm(m) -> steepest descent
Stepsize &-how far the If is too small
algorithm moves
along the gradient direction . I -
convergence is slow
If X too
large I
night overshoot the minimum and fail to
until the
-
Itrative process >
-
it repeats the process
of updating u
change in the
function value converge
.
or the
gradient becomes
sufficiently small
