X - UWSpace - University of Waterloo

1T 17 t-IT T
nlk*'T )' and &'-' = (0, ,a1 ... * 0 1 ) corne from (k-1) previous dual solutions of SPI;
k- 1
pk-"' is a (k-1)-dimensional row vector variable, i.e. #*'= (pib', p?-', ... , pk-~ );
is an
unrestricted scalar variable; and ek-' is again the (k-1)-dimensional row vector with al1
-
components equal to 1. Because SD: is a restriction of D and the restriction can be loosened
at each itention. by the inclusion of another nl in t?'"' and oi in R~'", it follows that z.5 2:
- k-I<
-2 - . .. 6 z2'.
When k=l. there are no variables & or $,. and there are no cuts. nor are there )c or C(
variables. because the algorithm begins by solving both SPI and SPr simultaneously, so
ihere are no solutions yet avdable from both subproblems.
After both subproblerns are solved concurrent1 y. SPI sen& dual information (called
proposals) to SP2, and receives prima1 information (proposais) from SP2, while SP2 sends
primal information to SPI, and receives dud proposais from SPI. This communication scheme
continues until the gap between their objective function values reduces to within a prescribed
tolennce. Figure 3.1 shows this communication scheme.