Number of decision variables vs objective space dimension?

Question

Are these two different from each other?

In the context of optimization problems (esp. evolutionary optimization), I've encountered the term decision variables and as its definition and practice suggests, these are the variables we want to find the best value for to find the optimum objective function value.

What confuses me is that sometimes the number of decision variables and the dimension of the problem are treated separately. Aren't they the same? For example, if I have a 2D function f(x1,x2) that I want to optimize, aren't x1 and x2 the decision variables? So, both these two numbers would be 2, wouldn't it?

Are there any problems in which these two are different? Is there any difference in the constraint optimization problems?

Or, If they are always the same, why the difference in the terms?

Answer 1

Based on wikipedia, a mathematical optimization problem can represented as:

• Given a function f: A -> R, from a set A to real numbers R
• Sought a value x0 such f(x0) is smaller than f(x) for all x in A for a minimization.

The function f takes on argument, x0, this is the decision variable. So the space A, the problem space has one dimension. The dimension of problem and the number of decision variable are the same concept. If f would takes two arguments, f(x0, x1), there would be two decision variables.

The dimension of objective space is the number of variables return by the function f. In our case, f map a set of solution A to real number R. The dimension of the objective space is therefore 1.

We could define a multi-objective optimization problem where the function f returns a vector or where we try to optimize multiple function f_k at a time. The problem would then be define as :

• Given a set of function (f1, f2, ..., fk) : A -> R^k, from a set A to real numbers R^k
• Sought a value x0 such (f1(x0), f2(x0), ..., fk(x0)) dominates every (f1(x), f2(x), ..., fk(x)) for all x in A for a minimization.

The problem dimension is 1 and the objective space has k dimensions. The objectives can be combined to a single objective using a weighted sum or can be optimized using a concept of multi-criteria dominance such as the Pareto dominance.