Finite-budget deterministic direct search with positive-spanning polling

Let $U\subseteq {\mathbb{R}}^n$ be open, $f:U\to \mathbb{R}$ be an objective accessible only by function evaluations, and let $x_0\in U$. Fix an ordered finite positive-spanning poll list $D=(d_1,\dots ,d_m)$, initial mesh ${\mathrm{\Delta }}_0>0$, cap ${\mathrm{\Delta }}_{\max }\ge {\mathrm{\Delta }}_0$, forcing function $\rho :(0,{\mathrm{\Delta }}_{\max }]\to (0,\mathrm{\infty })$ with $\rho (\mathrm{\Delta })\mathrm{/}\mathrm{\Delta }\to 0$ as $\mathrm{\Delta }\downarrow 0$, factors $0<{\gamma }_{dec}<1<{\gamma }_{inc}$, and integer poll budget $N\ge 0$. Evaluate $f(x_0)$ first and include it in the best-point record. At poll $k$, inspect $x_k+{\mathrm{\Delta }}_k{d}_i$ in list order, ignoring points outside $U$ without an objective call. Accept the first in-domain point $y$ satisfying $f(y)<f(x_k)-\rho ({\mathrm{\Delta }}_k)$; then set $x_{k+1}=y$ and ${\mathrm{\Delta }}_{k+1}=\min ({\mathrm{\Delta }}_{max},{\gamma }_{inc}{\mathrm{\Delta }}_k)$. If none succeeds, declare an unsuccessful poll, set $x_{k+1}=x_k$, and ${\mathrm{\Delta }}_{k+1}={\gamma }_{dec}{\mathrm{\Delta }}_k$. Return the least-index best point among all evaluated points, current center, mesh, evaluation count, and status budget-exhausted after at most $N$ polls. A finite budget is not a stationarity certificate.