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Q:

Convex optimization without constraints

I am doing an optimization problem to find max/min of integral function. It is convex, for instance,
$$
\int_0^1{(a + 2x – 7)dx} \to \min
$$
$$
\int_0^1{4x^2 – 6x + 2}dx \to \max
$$
And I am using SCIP with MIPS solver. But in this problem I have free variables $0\leq x \leq 1$ and I do not have constraints (not even LP). I want to know, can I use SCIP for this kind of problems? Is there any approximation that I can check? Or is there any other software that can be better in this situation?
UPDATE: I want to optimize only integral function in this kind of problem and the problem have to be convex.

A:

Intuitively, it may seem that SCIP is a good choice for this optimization problem. However, there are still some difficulties for you.
First of all, if you do not have a convex objective function, what would you like to use as a convex relaxation? SCIP has a LP relaxation method for convex optimization problems. But does it work well for your optimization problem?
Secondly, if you do have an objective function, but do not have any constraints, what would you like to use as constraints? SCIP has a constraint generation method for SCIP solvers. But does it work well for your optimization problem?
I think the most important problem is the second one. You probably would like to relax some inequalities when you use a convex relaxation method. But, will this method can work well with a convex objective function? To my opinion, this method is not suitable.
You may want to take these problems into consideration. Then, you can try to find a method which is suitable for your problem, for instance, using Lagrangian dual and dual simplex method.

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