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Use reproducible random starts and retain the best successful result:

The estimate is $x_{\min }\simeq -0.309213$, $f_{\min }\simeq 0.0192409$. This random-restart optimization improves coverage of the different attraction regions, but their number alone does not guarantee global optimality.

Here an analytic check is possible. For $|x|\ge \pi /5$, the envelope gives $f(x)\ge 1-e^{-{\pi }^2/125}\simeq 0.0759$, above the candidate. On $[0,\pi /5]$, $f\ge 1$. In the remaining interval $[-\pi /5,0]$, the derivative condition is equivalent, with $z=-x$, to $\cot (5z)=0.08z$. The left side decreases strictly from $+\infty$ to $-\infty$ while the right side increases, giving one minimum. This certifies the selected well as global rather than relying only on the restart experiment.

Random-restart optimization repeatedly runs a local optimization algorithm from independently sampled starting points and retains the best result. It exploits multiple basins of attraction without requiring the local solver itself to escape a local extremum.

If each restart has probability $p$ of entering a desired basin, independent restarts miss it with probability $(1-p{)}^N$ after $N$ attempts. This formula quantifies the benefit only for the specified starting distribution and basin; neither $p>0$ nor a finite run certifies a global extremum. Recording the starting distribution and stopping rule makes computational comparisons reproducible.