For the computation of the capacity per unit cost

an extension of the basic theorem connect the capacity per unit cost with the capacity-cost function was made

and the single peak of the generalized capacity-cost function per unit cost was proved.Then

an algorithm to locate the optimal point of the generalized capacity-cost function per unit cost based on the single peak of the function was proposed.Meanwhile

it was proved that the computation of the capac-ity-cost function

which is a constrained channel capacity issue

can be expressed as a convex optimization

to which the objective function is self-concordant.The algorithm and its computational complexity were given.