Thank You Mr. Fung for your reply.

You have stated that problems may occur while applying gradient based methods in optimizing objective functions that involve absolute values. So, is it possible that this problem can be avoided if I use stochastic search algorithms such as PSO or GA ??

Dear Biswajit,

In mathematics, the absolute value or modulus |x| of areal number x is the non-negative value of x without regard to its sign. Namely, |x| = x for a positive x, |x| = −x for anegative x (in which case −x is positive), and |0| = 0. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero.

Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers, thequaternions, ordered rings, fields and vector spaces. The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts.

The absolute value is a particular instance of a norm. Or perhaps, you can think of norms as functions V→RV→R where VV is a vector space over a field FF, and "absolute values" are "norms on the base field".

The absolute value is a function ||:R→[0,∞)||:R→[0,∞); given any real number rr, you get a nonnegative real number that we write |r||r|, and which satisfies the following properties:

1.       |x|≥0|x|≥0 for all x∈Rx∈R; |x|=0|x|=0 if and only if x=0x=0.

2.       |rx|=|r||x||rx|=|r||x| for all r,xr,x.

3.       |x+y|≤|x|+|y||x+y|≤|x|+|y|.

It's a real valued function of real variable, because it takes a real number as an input, and gives a (nonnegative) real number as an output. It's just that instead of calling the function, say, ff, and writing the output as f(x)f(x), we call the function "||||" and write the output as |x||x|.

in my opinion it is quite difficult to compare these two concepts in terms what the best method. Here are links and attached files in subject.

-How are norms different from absolute values? - Mathematics Stack ...

math.stackexchange.com/.../how-are-norms-different-...

- Vector and matrix norms - MATLAB norm - MathWorks

www.mathworks.com › ... › Linear Algebra

Best regaerds

Thank You Mr. Lafifi for your reply.

You have provided a good explanation on the definition of l2 norm and absolute values.

However, can you please point me towards some research where it is established that l2 norms hold sway over l1 norms in terms of  optimization using stochastic methods such as PSO or GA ??  

I totally agree with wai Keung FUNG;

Further what Wai-Keung says above, note that l1 and l2 norm behave differently in the presence of outliers. l2 penalizes any outliers more severely than l1 norm (due to squaring operation). If you don't want this to happen (i.e., don't want to 'chase outliers'), you might want to use l1 norm. See slides 15-18 from the notes  at https://www.dropbox.com/s/ua3xh98v388iofr/natcor_lecture2._slides.pdf?dl=0 .

Thank You for your reply Mr. Paresh Date...

So, you are trying to mean that l2 norm can be more useful with hard constraints while l1 norm can be more useful with soft constraints ??