Reverse Bridge Theorem under Constraint Partition

Algorithm 3

The resolving procedure for finding CMm of $P_m$.

Procedure RBTH_MINLP $(P_m,x,y,\bar{\alpha },\bar{\beta })$

    $\alpha \to 0\text{,}$$\beta \to 0$;

    repeat

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

     repeat

      perform descent of $L_m(x,y,\alpha ,\beta )$ with respect to  $x$ for given  $y$;

     until a local minimum of $L_m(x,y,\alpha ,\beta )$ with respect to  $x$ is found;

     repeat

      perform descent of $L_m(x,y,\alpha ,\beta )$ with respect to  $y$ for given  $x$;

     until a local minimum of $L_m(x,y,\alpha ,\beta )$ with respect to  $y$ is found;

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

    return CMm if found;

end_procedure