3.2.4
The DA solution depends continuously on its parameters and the data
whereas the LP solution does not. An informal analysis follows.
Consider an LP solution 
obtained with 
. The solution
is a local equilibrium satisfying
In the above, 
or 0, depending on 
and
the configuration of 
. When 
is close to
for some x, a small change 
may flip
over from one state to the other. This is due to the binary
non-linearity of 
. The flip-over leads to a significantly
different solution. This can be expressed as
where 
is a function which is constantly zero in the domain
. The variation 
with respect to 
may not be zero for some 
and 
, which causes
instability. However, the DA solution, denoted 
, is stable
where 
denotes the DA solution. Conclusion on the stability due
to changes in parameter 
can be drawn similarly.
The same is also true with respect to the data. Given 
and
fixed, the solution depends on the data, i.e.
. Assume
a small variation 
in the data d. The solution 
must
change accordingly to reach a new equilibrium 
to satisfy
the Euler equation. However, there always exist possibilities that
may flip over for some x when
is near the 
, resulting in an abrupt change
in the LP solution 
. This can be represented by
That is, the variation 
with respect to 
may
not be zero for some 
and d. However, the DA model is
stable to such changes, i.e.
because of its continuous nature. From the analysis, it can be concluded that the DA better regularizes ill-posed problems than the LP.