Reverse Bridge Theorem under Constraint Partition

Algorithm 4

The partitioning and resolving procedure for finding CMm of $P_t$.

Procedure RBTH_partition_resolve_mixed $(P_t,z,\overline{\alpha },\bar{\beta },\bar{\gamma },\bar{\eta })$

  $\gamma \to 0$,$\mathrm{\hspace{0.17em}\hspace{0.17em}}\eta \to 0$;

  repeat

    increase $\hspace{0.17em}\hspace{0.17em}{\gamma }_i$ by $\delta$ if $(H_i(z)\ne 0\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{and}\mathrm{\hspace{0.17em}\hspace{0.17em}}{\gamma }_i<{\bar{\gamma }}_i)\mathrm{\hspace{0.17em}\hspace{0.17em}}$for $i=1,\dots ,p$;

    increase $\hspace{0.17em}\hspace{0.17em}{\eta }_j$ by $\delta$ if $(G_j(z)\hspace{0.17em}\nleqq \hspace{0.17em}0\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{and}\mathrm{\hspace{0.17em}\hspace{0.17em}}{\eta }_j<{\bar{\eta }}_j)$ for $j=1,\dots ,q$;

    for  $t=0$ to  N

     call RBTH_MINLP $(P_t^{(t)},z,\overline{\alpha },\bar{\beta })$ to solve $P_t^{(t)}$.

    end for;

  until a CMm of $P_t$ is found or (${\gamma }_i>{\bar{\gamma }}_i\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{for}\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{all}\mathrm{\hspace{0.17em}\hspace{0.17em}}{H}_i(z)\ne 0\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{and}\mathrm{\hspace{0.17em}\hspace{0.17em}}{\eta }_j>{\bar{\eta }}_j\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{for}\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{all}\mathrm{\hspace{0.17em}\hspace{0.17em}}{G}_j(z)\hspace{0.17em}\nleqq \hspace{0.17em}0$).

  return CMm of $P_t$ if found;

end_procedure