Scaling in optimization problems

Assume a simple example, with an optimization problem:max subject to , with and given. I want to solve it with fmincon (because there will be other constraints as well), but with the derivate with dV/dc will get ever smaller with t.

A similar problem occurs when we have:

, where alpha is close to zero.

Is there a way to scale the problem such that the derivates will be of similar size? Otherwise the solution for large t (or smalle alpha) will be less accurate, right?

(One work around would be to use a value function, and solve for each t, but I don't want to do that).

Answer 1

[Matt J]

the derivate with dV/dc will get ever smaller with t.

It's not clear that that matters because the values of c are also influenced by the constraints and their derivatives.

Regardless, though, most fmincon algorithms are Newton-like, which means they already normalize dV/dc by second derivatives.