I am a memeber in an LLC. Our Operating Agreement consists of a strict
non compete clause. It states that if one of members violates this
clause, the partners could seek a restraining order to stop the
violating business actvity, as well as seek damages. My question is:
How does a judge detmermine damages? What is the process the judge
goes through when making this determination? Thank you.

It depends a great deal on the state in which you live and that
state's case law regarding non-compete clauses.  But in general,
damages for breach of contract are determined by the value of the
opportunity that was lost by the plantiff as a result of defendant's
(your) breach.  For example, if you breach the non-compete and as a
result the LLC is deprived of $10,000 in revenue, that's how much the
damages would be.  Damages cannot, however, be speculative and must be
supported by evidence. In other words, the LLC must show that it
actually suffered quantifiable losses as a direct and proximate result
of your breach.

Query: does the operating agreement have a liquidated damages clause? 
Many times when damages are difficult to quantify (like these would
be, I imagine), the parties agree in the contract that in the event of
breach the damages are already determined to be some dollar amount,
called "liquidated damages."  If that's the case, then all the judge
needs to do is find that you did breach the contract by competing, and
then he/she simply awards plaintiff the amount of the liquidated sum
stated in the contract as damages.

(Assuming the clause was otherwise valid, the judge would also have to
find that the amount designated in a liquidated damages clause was
reasonable. That is to say, a liquidated damages clause is supposed to
approximate what actual damages would be, as opposed to being a
punitive provision.)

Yes, and this is why most liquidated damages clauses contain language
that says, in substance, "The parties agree that this liquidated sum
is a fair, fully negotiated approximation of damages to [party A]
occasioned by [party B's] breach, and further agree that such
liquidated sum is imposed in lieu of quantification of actual damages
and is not imposed as a penalty or other forfeiture."