Algorithm 1 GPSPositiveBasis2N algorithm

Require:    Objective function $F_{\mathrm{Obj}}$    Initial point ${\mathbf{x}}_0\in {\mathbb{R}}^n$    Lower bound $\mathbf{LoB}$, upper bound $\mathbf{UpB}$    Initial step size ${\Delta }_0>0$, tolerance $\tau >0$    Maximum iterations $K_{max}$ Ensure: Optimal solution ${\mathbf{x}}^{*}$, optimal value $f^{*}$  1:${\mathbf{x}}_{\mathrm{best}}\leftarrow {\mathbf{x}}_0$  2:$f_{\mathrm{best}}\leftarrow F_{\mathrm{Obj}}\left({\mathbf{x}}_{\mathrm{best}}\right)$  3:$\Delta \leftarrow {\Delta }_0$  4:Generate positive basis $D={\left[\begin{array}{cc}{I}_n& -I_n\end{array}\right]}^T$  5:for $k=1\to K_{max}$ do  6:      $\mathrm{improved}\leftarrow \mathrm{false}$  7:      for each direction ${\mathbf{d}}_i\in D$ do  8:            ${\mathbf{x}}_{\mathrm{cand}}\leftarrow {\mathbf{x}}_{\mathrm{best}}+\Delta {\mathbf{d}}_i$  9:            ${\mathbf{x}}_{\mathrm{cand}}\leftarrow clip({\mathbf{x}}_{\mathrm{cand}},\mathbf{LoB},\mathbf{UpB})$ 10:           $f_{\mathrm{cand}}\leftarrow F_{\mathrm{Obj}}\left({\mathbf{x}}_{\mathrm{cand}}\right)$ 11:           if $f_{\mathrm{cand}}<f_{\mathrm{best}}$ then 12:               ${\mathbf{x}}_{\mathrm{best}}\leftarrow {\mathbf{x}}_{\mathrm{cand}}$ 13:               $f_{\mathrm{best}}\leftarrow f_{\mathrm{cand}}$ 14:               $\mathrm{improved}\leftarrow \mathrm{true}$ 15:          end if 16:      end for 17:      if improved then 18:            $\Delta \leftarrow 1.5\Delta$               ▷ Step expansion 19:      else 20:            $\Delta \leftarrow 0.5\Delta$             ▷ Step contraction 21:      end if 22:      if $\Delta <\tau$ then 23:            break 24:      end if 25:end for 26:return ${\mathbf{x}}_{\mathrm{best}},f_{\mathrm{best}}$