Algorithm 1

Step 1. Determine some $\left({\widehat{x}}_1\mathrm{,}\cdots \mathrm{,}{\widehat{x}}_m\right)$ satisfying (12)-(14) by using Proposition 1 or Corollary 2. Such a point exists by assumption. Let $z_1=f\left({\widehat{x}}_1\mathrm{,}\cdots \mathrm{,}{\widehat{x}}_m\right)$ . Fix $\delta >0$ as the maximum acceptable error in finding the minimum value of the objective function in $T^{\prime }$ . Set $i=1$ .

Step 2. Set $z_{i+1}=z_i-\delta$ . Apply Proposition 1 or Corollary 2 to $S^{\prime }$ with $z=z_{i+1}$ . If a solution exists, go to Step 3. Otherwise go to Step 4.

Step 3. Set $i=i+1$ and go to Step 2.

Step 4. The optimal objective function value for $T^{\prime }$ lies in the interval $\left[z_{i+1}\mathrm{,}{z}_i\right]$ , where $z_i-z_{i+1}<\delta$ . A solution $\left(x_1^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}\cdots \mathrm{,}{x}_m^{\mathrm{<mo>\; *\; </mo>}}\right)$ to $S^{\prime }$ for $z=z_i$ in Step 2 is either an exact or approximate solution to $T^{\prime }$ . The objective function value $f\left(x_1^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}\cdots \mathrm{,}{x}_m^{\mathrm{<mo>\; *\; </mo>}}\right)$ for $T^{\prime }$ is at most $\delta$ larger than the minimum value.