You define
w(k)=beta(k)*(-k_(k) *w_k(k)-k_(k)*log2(lamda(k)*(1-tau^2))*Pt + 2*k_(k)*log2(sqrt(segma_squared))-v_k(k));
and want to minimize the sum over k of this expression with respect to w_k where you set the constraint as -Inf < w_k(k) <= 0.
Now if you multiply out the expression for w(k), you can write
w(k) = beta(k)*(-k_(k)) w_k(k) + beta(k)*(-k_(k)*log2(lamda(k)*(1-tau^2))*Pt + 2*k_(k)*log2(sqrt(segma_squared))-v_k(k))
= T1 + T2
with
T1 = beta(k)*(-k_(k)) w_k(k)
and
T2 = beta(k)*(-k_(k)*log2(lamda(k)*(1-tau^2))*Pt + 2*k_(k)*log2(sqrt(segma_squared))-v_k(k))
T2 does not depend on w_k(k), so it can be discarded in the optimization for w_k.
So you try to find w_k such that
sum(beta(k)*(-k_(k))*w_k(k))
is minimized where -Inf < w_k <=0.
Now beta and k_ are both >=0 and w_k is constrained to be <= 0.
Thus the sum is minimized (with value 0) if w_k(k) = 0 for all k.
