# What are the reasons behind using constraints in convex optimization?

We use the following notation to describe a minimum convex optimization problem:

               minimize     f0(x)
subject to   fi(x) ≤ 0,     i = 1, . . . ,m
hi(x) = 0,     i = 1, . . . , p

to describe the problem of finding an x that minimizes f0(x) among all x
that satisfy the conditions fi(x) ≤ 0, i = 1, . . . ,m, and hi(x) = 0, i =
1, . . . , p.


What is the reason for using fi(x)<=0 and hi(x)=0.

How do someone arrived at only these two constraints?

• I don't really understand what you're asking. The answer to your title question is "Because the minimum without any constraints is always just to set all the values to zero, and because the constraints model the system that's being optimized." – David Richerby Mar 1 '18 at 17:15
• I think that maybe your question is why this 'standard form' is general enough and why we don't consider things like $h_i(x) \leq 35$ or something similar. If this is correct, please edit your question to clarify. – Discrete lizard Mar 1 '18 at 17:56