Its pretty hard to give any problem which you absolutely have to solve using only one particular technique. Nevertheless, there are some problems which require more than just being able to add positive numbers together in order to solve.

Consider, for example, the problem of moving an electric car with regenerative braking along a grid of very hilly roads, using minimal charge from the battery. When you go down a hill, sure, you can add energy to your battery by hitting the brakes. But when you go uphill again, you may have to drain your battery of charge.

Indeed, if you can find a course which is largely downhill, you may very well end uo with a "negative amount of energy charged" to the battery :-)