I have a co-worker who refuses to take advice on a solution to a problem. I think he will eventually get there however the solution will be far more complex than it needs to be.
He is the lone warrior type, and get's passive aggressive when you try to give him advice.
Do I just let him use the more complex solution which will take more time? I don't want to hurt his pride.
You may hurt his pride, but keep in mind that you will be doing him a favor in the long run. He may choose to ignore your advice (simply to protect his pride), but perhaps in the future he will put your advice to work when it's on his own terms and not yours.
Also, rather than just looking out for your co-worker's feelings, perhaps it will help to remember that you also owe it to your employer to not only do your best, most efficient work, but to help others do the same.
Most importantly, don't let this person drag you down and make you unhappy.
People like that (and many others) only learn through mistakes. Give him a task that he is sure to screw up, and blow his whole leg off. Next time, he might think twice and ask for some advice.
If you can't handle the confrontation / pride hurting, you could try just showing interest in his work, in how he solves problems, etc — ask questions rather than offering advice. Perhaps also ask questions about he would solve things that you are doing. You may find that you prompt him to:
You have to find a way to communicate the information without saying "you're doing it wrong".
One possibility might be to pair-program with him on the problem. Then you can change it from "you're doing it wrong, you should do it like this" to "we're doing it wrong, we should do it like this", or even better, "we might be doing it wrong - could we try doing it like this?". That's much easier for him to take, because you're putting both of you on the same side.
Of course, if he's a lone wolf who won't take advice, then he won't want to pair program. You could try phrasing it as "can i pair with you to understand your solution?", which he might be more likely to accept, but then he'll start off being resistant to any ideas you contribute, because what you've asked for is a one-way conversation.
An alternative might be to apply the Socratic method - ask questions that lead him to work out for himself that you're right. So rather than "that won't work because the temporary files will be deleted at boot", you start with "where are you keeping the data?", "does anything special happen to those files at any time?", "how about at boot?", etc. The advantage of this is that the only facts or assertions introduced into the conversation come from him, prompted by your questions, so if they lead to the conclusion that he's wrong, he can't resist it, because he's got there himself. If he's defensive about his ideas, then it's natural to lead him into this discussion, because he'll want to defend himself by explaining. The problem, of course, is that the flaw in his thinking might not be in his logic, but in his understanding of the facts. For instance, if he doesn't think temporary files get deleted at boot, this approach fails. But then, at least you can drill down to the specific points where he's wrong, and it should be easier to apply a facthammer to those than to the problem as a whole.
A nice feature of the Socratic method is that if it turns out that you're wrong, it will also reveal that!
But mostly, if he gets things wrong and won't listen to advice from colleagues, he needs to be fired. Make sure your boss is aware of this problem.
Are you working in IKEA?
Either you are his 'co-worker' or you have a management function with regards to his position. When you say 'do I let him', I am thinking that you are not in fact a co-worker.
If you are a co-worker tough luck. It sounds like you'll just have to suck it up and if he is as bad as you think, he'll swing enough rope for himself eventually.
If however you have a management role I would encourage you to make your thoughts clear to him, in a non-confrontational manner and in a non-situational context. What I mean is that you review his performance and mentor him in the ways that you think he could have done it better, post project/code cycle. A good dose of the truth as you see it, with good constructive criticism will go a long way to developing both his esteem and career as well as your own.
If you are working as a team and your team has discussed (and agreed) that you should commit to and solve problems together, then I think the one approach is to simply state something like
"How would you feel about working together with me on this task?"
Listen to his response. If he is clearly against it, just drop and let him fail. He will learn from this, eventually.