On a class of optimization problems with no “effectively computable” solution

It is well known that large random structures may have nonrandom macroscopic properties. We give an example of nonrandom properties for a class of large optimization problems related to the computational problem $MAX\,FLS^=$ of calculating the maximum number of consistent equations in a given overdetermined system of linear equations.
For this class we establish the following. There is no “efficiently computable” optimal strategy. When the size of a random instance of the optimization problem goes to infinity, the probability that the uniform mixed strategy is $\varepsilon$-optimal goes to one. Moreover, there is no “efficiently computable” strategy that is substantially better for each instance of the optimization problem.