sci.op-research

Hi all,

I just had a few questions in using KKT conditions to solve
constrained optimization problems. There are not many books give
examples about using the conditions to solve non-trivial problems.
Here are my questions:

max f(x)

such that x1>=a, x2>=b, x3>=c,
and g(x)>=0

And f(x) and g(x) involve other parameters p and q.

Let's say I already know that x2>=b never becomes binding, and x3>=c
never becomes binding, and g(x) >=0 is always binding.

I define the Laplacian to be:

V(x, a, b, c)= f(x) - l1*(-x1+a)-l2*(-x2+b)-l3*(-x3+c)-l4*(-g(x))

where we know that l2=0, l3=0, and l4>0, and g(x)=0.

Now I have one lambda remaining, which is l1.

I divide into two cases:

case 1: If l1=0, then x1>a, I solve for x1, x2, x3. And these involve
the additional parameters such as p and q.

case 2: If l1>0, then x1=a, I solve for x2 and x3.

My questions are:

(1) In case 1: here it seems that the value of "a" doesn't matter. For
any value of "a", I get the same expressions for x1, x2, x3. "a"
probably even doesn't appear in the expressions;

(2) In case 1: What if the solved expression of x1, which depends on
the parameters, gives x1<=a, for certain range of the parameters?

(3) After discussing the two cases, how do I combine them together?

I think I should pick one of these two that gives the largest
objective function value.

Do I plug in these two solutions into the objective function and then
compare?

Or, in this case, since we already know that x2, and x3 should be not
on the boundary, we just need to check the solution in case 1:
depending on the parameters, if the solution in case 1 is indeed with
x1>a, then that's the optimal solution; if the solution in case 1 is
indeed with x1<=a, then we should switch to the second case with x1=a.
Are these two cases complete?

(4) It seems that for all constrained optimization problems, either
the optimal solution is in the interior, or it is one of the corner/
extreme/boundary points. Am I correct? In general, shall I just
compare the objective function values for all these "candidate"
interior points and the boundary points? I understand that since the
first order conditions are really necessary conditions, so there could
be multiple candidate interior points and multiple candidate boundary
points. And then comparing these objective functions to choose the
largest one is very tedious, any better approaches?