Show that, for linear constraints, if at some point in the reduced gradient method z is zero, that..

Show that, for linear constraints, if at
some point in the reduced gradient method z is zero, that point satisfies the
Karush-Kuhn–Tucker first-order necessary conditions for a constrained minimum.

Consider the minimize f (x) subject to Ax
= b, x 0, where A is m × n. Assume f ∈ C1, that the feasible set
is bounded, and that the nondegeneracy assumption holds. Suppose a “modified”
reduced gradient algorithm is defined following the procedure in Sect. 12.5 but
with two modifications: (1) the basic variables are, at the beginning of an
iteration, always taken as the m largest variables (ties are broken arbitrarily);
(2) the formula for z is replaced by

Establish the global
convergence of this algorithm. 20. Find the exact solution to the presented in
S