Reverse Bridge Theorem under Constraint Partition

Algorithm 1

The solving procedure for finding CMc of $P_c$.

Procedure RBTH_CNLP $(P_c,x,\bar{\alpha },\bar{\beta })$

    $\alpha \to 0,\beta \to 0$

    repeat

     increase  ${\alpha }_i$ by $\delta$ if $(h_i(x)\ne 0\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{and}\mathrm{\hspace{0.17em}\hspace{0.17em}}{\alpha }_i<{\bar{\alpha }}_i)$ for $i=1,\dots ,m$;

     increase  ${\beta }_j$ by $\delta$ if $(g_j(x)\hspace{0.17em}\hspace{0.17em}\nleqq \hspace{0.17em}\hspace{0.17em}0\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{and}\mathrm{\hspace{0.17em}\hspace{0.17em}}{\beta }_j<{\bar{\beta }}_j)$ for $j=1,\dots ,r$;

     repeat

      perform descent of $L_c(x,\alpha ,\beta )$ with respect to x;

     until a local minimum of $L_c(x,\alpha ,\beta )$ is found;

    until a CMc of $P_c$ is found or (${\alpha }_i>{\bar{\alpha }}_i$ for all $h_i(x)\ne 0$ and ${\beta }_j>{\bar{\beta }}_j$ for all $g_j(x)\hspace{0.17em}\hspace{0.17em}\nleqq \hspace{0.17em}\hspace{0.17em}0$).

    return CMc if found;

end_procedure