Reverse Bridge Theorem under Constraint Partition

Algorithm 2

The solving procedure for finding CMd of $P_d$.

Procedure RBTH_DNLP $(P_d,y,\bar{\alpha },\bar{\beta })$

    $\alpha \to 0,\beta \to 0$;

    repeat

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

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

     repeat

      perform descent of $L_d(y,\alpha ,\beta )$ with respect to $y$;

     until a local minimum of $L_d(y,\alpha ,\beta )$ is found;

    until a CMd of $P_d$ is found or (${\alpha }_i>{\bar{\alpha }}_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(y)\ne 0$$\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{and}\mathrm{\hspace{0.17em}\hspace{0.17em}}{\beta }_j>{\bar{\beta }}_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(y)\nleqq 0$).

    return CMd if found;

end_procedure