priscillianaw1

2022-09-03

Define the term unconstrained optimization?

### Answer & Explanation

Sanaa Hudson

Unconstrained optimization problems consider the problem of minimizing an objective function that depends on real variables with no restrictions on their values. Mathematically, let $x\in {R}_{n}$ be a real vector with $n\ge 1$ components and
let $f:{R}_{n}\to R$ be a smooth function. Then, the unconstrained optimization problem is
$mi{n}_{x}f\left(x\right)$.
Unconstrained optimization problems arise 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.
Unconstrained optimization problem can be presented as
minx∈${R}_{n}f\left(x\right)$
where $f\in {R}_{n}$ is a smooth function.
In fact, the above eq is unconstrained minimization problem. But, it is well known that the unconstrained minimization problem is equivalent to an unconstrained maximization problem, i.e.
minf(x)=−max(−f(x))
as well as
maxf(x)=−min(−f(x)).

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