In analyses dealing with both constrained and unconstrained budgets, the goal is usually optimization. That is, researchers hope to maximize or minimize a certain function, usually called the objective function. While the goal of these analyses is the same, the implications of the goals are not; the optimized function for an unconstrained budget is ideal, whereas an optimized function for a constrained budget is pragmatic.
The solution for either type of optimization function will be presented as both an optimized objective function value and a set of decision variables, which are the variables that the researchers have control over. However, it is often possible to know whether the problem was one of constrained budget or unconstrained budget simply by looking at the solution: solutions with unconstrained budgets tend to be high in number – often of infinite number. This is because unconstrained budgets do not limit the resource or fund allocations, thereby allowing large sets of possibilities.
While the name “unconstrained budget” implies there being no constraints on the budget, optimization problems with unconstrained budgets tend to have hidden constraints. In short, the unconstrained budget problems tend to constrain themselves merely by the form of function. For example, if the optimization function is a two-dimension function that appears as a conic section, it is likely that there is only one maximum solution, despite there being an unconstrained budget. This fact is entirely due to the shape of the optimization function, and the solution should be the same regardless of the type of budget. That is, some unconstrained budget analyses are in practice the same as constrained budget analyses.
The primary difference mathematically and statistically between these two budget types is that they require different methods to arrive at optimal solutions. Unconstrained budgets lead to calculus-type optimization problems, in which the researcher must use derivatives to find the possible optimal values. On the other hand, the inclusion of constraints on the budget changes such a problem to a linear programming function, which requires the use of algorithms to arrive at a solution.