What is "Variational Problem"? It is a very common concept/category method in computer vision society. Such as Level Set, Snake/Gradient Vector Flow (GVF) and Partial Difference Function (PDE) are methods of this category.
Let me try to descibe the definition of variational method as follows.
A function of integral is a particular function called Functional. Solving one problem can be modeled as finding the extreme value (maxima/minimum) of this functional. Similarly with finding the extreme value of a function, finding the extreme of a functional is equal to the derivative of this functional. That is variational problem - Find the derivative of a functional equation.