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Jiro comments on The Number Choosing Game: Against the existence of perfect theoretical rationality - Less Wrong Discussion
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I'm kind of defining perfect rationality as the ability to maximise utility (more or less). If there are multiple optimal solutions, then picking any one maximises utility. If there is no optimal solution, then picking none maximises utility. So this is problematic for perfect rationality as defined as utility maximisation, but if you disagree with the definition, we can just taboo "perfect rationality" and talk about utility maximisation instead. In either case, this is something people often assume exists without even realising that they are making an assumption.
This statement is not necessarily true when there is no optimal solution because the solutions are part of an infinite set of solutions. That is, it is not true in exactly the situation described in your problem.
Sorry, that was badly phrased. It should have been: "If there is no optimal solution, then no matter what solution you pick you won't be able to maximise utility"