Write down the KKT conditions of (19.1) and (19.59), and establish a one-to-one correspondence…

(a) Write down the KKT conditions of (19.1) and (19.59), and establish a one-to-one correspondence between KKT points of these problems (despite the different numbers of variables and multipliers).

(b) The multipliers z correspond to the equality constraints (19.1c) and should therefore be unsigned. Nonetheless, argue that (19.2) with p. = 0 together with (19.3) can be seen as the KKT conditions of problem (19.1). Moreover, argue that the multipliers z in (19.2) can be seen as the multipliers of the inequalities c, in (19.59).

(c) Suppose z is feasible for (19.59). Show that LICQ holds at i for (19.59) if and only if LICQ holds at (2, §) for (19.1), with s = c,(2).

(d) Repeat part (c) assuming that the M FCQ condition holds (see Definition 12.6) instead of LICQ.
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