Hi Paulo and thanks for your question.
And, as well as answering your question,
I'm going to answer a similar one at the same time from Renato.
And the use of preposition is a topic that worries many of my students
I know because I'm often asked for advice about which preposition to use where,
especially around the time that assignments are due in!
I do think that prepositions are one of the most difficult areas of English to master,
because there are so many prepositions and so many different combinations of verb and preposition that have to be memorised individually.
But let's begin with your first example Paulo, when we say Happy Birthday.
And when we use Happy Birthday we're using a set expression or a greeting like Happy Christmas, Happy New Year or Congratulations,
and if you wanted to follow this expression with a pronoun,
you would have to use the preposition to with it - Happy Christmas to you! Happy Birthday to you!
although it might be more common simply to use the greeting without a pronoun in speech.
In your second example Paulo, for is followed by a pronoun, you, and functions as a preposition showing the intended recipient:
The parcel is for Jenny. The flowers are for mother. Here is a gift for you.
So now to answer Renato's question.
In the examples you give, Renato, I'd say that the first example, with the infinitive verb, is the better choice.
You've only given me part of the sentence – an efficient method to solve my problem
but the phrase suggests an outcome or the solution to a problem.
Let's put this into a complete sentence by adding a verb:
I discovered an efficient method to solve my problem.
In this case, the second verb always appears in the infinitive form because these are all examples of the infinitive showing purpose:
I watched television all day to relieve my boredom.
I made her a chocolate cake to make her feel better.
He went into town to order his new computer.
In each of the examples above, there is an outcome or an intention which is reported by the to clause,
similar to the one in your own example, Renato.
Finally, to look at the last example, if I said Shall we go out to eat lunch?
最后，让我们来看一下最后一个例子，当我说Shall we go out to eat lunch?的时候，
the verb go out would be followed by a verb, wouldn't it?
However, in your own example, Renato – Let's go out for lunch
但是，雷纳托，在你自己的例子中：Let's go out for lunch
the verb is followed by a noun (lunch), so we have to use the preposition for in this expression!
Well Paulo, thanks for your question and I hope that both the answers will be useful to you.