where the definitions for , , and are the same as in the previous case, is the number of new facilities, and are the weights of the new-new connections. In a AMPL model, emfl.mod in [16], is defined as constant and is decided by the initial guess. The way of defining and does not seem realistic; however, they are still reasonable if one has a good a initial guess. We now convert this problem to SOCP format. First, we introduce , and . Similarly, we need some additional variables to replace , and . By rewriting and appropriately, we obtain the SOCP formulation.

**Example 3.** To consider a simplified problem, we take an AMPL
model, emfl.mod, from [16]. In this model, the coordinates of
the existing facilities are generated randomly, the weight
associated with the new-new connections is fixed at , and the
weight associated with the old-new connection is fixed by the
distance between the old facilities and the initial guesses. We
use the following four cases, (i) (ii)
(iii)
and (iv)