Unconstrained Optimization minimize or maximize an objective function that depends on variables or set or variables with no restriction on their values. Mathematically, the value of $x^*$ such that the function $f(x^*)$ takes the smallest value is given by

$x^*=min_x\, f(x)$

$x$ may be scalar in $\mathbf{R}^1$ or vector in $\mathbf{R}^n$ while $f(x)$  is always a scalar in $\mathbf{R}^1$. Unconstrained optimization has many applications like finance, nonlinear equations, similarity transformations, etc. Unconstrained optimization problems arises directly in some applications but they also arise indirectly from reformulations of constrained optimization problems. Often it is practical to replace the constraints of an optimization problem with penalized terms in the objective function and to solve the problem as an unconstrained problem. Many constrained optimaization problems and algorithims involves solving unconstrained optimization as a subproblem.