 Solution

The increasing baseline order is $1/5,1/4,1/2,3/4$. A water-filling algorithm with three active coordinates givesTherefore, in the original coordinate order,The first three shifted coordinates all equal $13/20$, the fourth remains $3/4$, and the allocations sum to one. For an explicit KKT certificate take $\lambda =20/13$, ${\nu }_1={\nu }_2={\nu }_3=0$, and ${\nu }_4=20/13-4/3=8/39$. The strict convexity established above makes this optimum unique.