Probability of Attaining the/a Global Optimum Suppose for a given optimization problem, $A$, the size of the search space $S$ is |$S$|.
If the fitness landscape defined by $A$ is unimodal, then there is clearly only one global optimum. Given that $G$ is the event &quot;reach the global optimum&quot;, can we then infer that $P(G) = \frac{1}{|S|}$, where $P(G)$ denotes the probability of reaching the/a global optimum?
For $N$ ($N &lt; |S|$) such global optima (in the case of a multimodal landscape), the probability increases to $P(G) = \frac{N}{|S|}$
Is this a valid argument or can we not conclude anything regarding the probability of attaining the/a global optimum on the basis of the number of optima and the size of the search space?
I realize that often one cannot know in advance the number of global optima, but say, if the problem was to maximize a simple continuous function over some domain, then the function of interest could easily be plotted and the number of (local) optima examined.
No, you can't infer that.  You seem to be assuming that the optimization process tries one element of the search space, chosen uniformly at random, and hopes that it got lucky.  But that's not how real optimization algorithms work.  They try multiple solutions, and they're not chosen uniformly at random; they're typically chosen in some way that we hope is more likely to hit a good solution.
In general, you cannot conclude anything about the probability of hitting the optimum after some amount of optimization, without further information.  The probability might be much larger than $N/|S|$ (e.g., if the optimization problem is easy or the optimization algorithm is effective on this kind of problem).
In the kinds of optimization problems we typically use a computer to solve, often you can't just &quot;plot it out&quot; as often we're working with a multidimensional search space.