What is this sort of optimisation called?

Question

I am reading a book in mathematical finance. There is something about constrained optimisation. They have specialised it for the financial market, but I am wondering what the general name for this kind of constrained optimisation is.

It is in section 2.5 in this book: https://graduateeconomist.files.wordpress.com/2012/07/published-1997-introduction-to-mathematical-finance-discrete-time-models-by-stanley-r-pliska.pdf, the chapter "short sale restrictions"

I will describe the method without all the finance stuff, do you see where it comes from?

1. We want to maximize a function $E(1+r+(R_1-r)*F_1+(R_2-r)*F_2+..+(R_n-r)*F_n)$, where F is a vector that can be in a set K, and R is a given vector, and r is a constant. So we want to maximize this expression over F, but F has to be in K.
2. We define a support function $\delta(\kappa)=\sup_{\textbf{F} \in K}(-\textbf{F}\cdot\kappa)$ .
3. We denote the set $\tilde{K}=\{\kappa:\delta(\kappa)<\infty\}$
4. For each $\kappa \in \tilde{K}$, we define $R_i^\kappa=R_i+\delta(\kappa)+\kappa_i$, and $r^k=r+\delta(\kappa)$.

5. For each $\kappa \in \tilde{K}$ we then max $E(1+r^{\kappa}+(R_1^{\kappa}-r^{\kappa})*F_1^{\kappa}+(R_2^{\kappa}-r^{\kappa})*F_2^{\kappa}+..+(R_n^{\kappa}-r^{\kappa})*F_n^{\kappa})$=$J(\kappa)$. We max over F, the point here is that we max a similar problem as in 1), but now it is unconstrained.

6. We then minimize $J(\kappa)$, and get $\hat\kappa$, corresponding to $\hat\kappa$ is $F^\hat\kappa$.

7. If now $F^{\hat\kappa}\in K$, and $\delta(\hat\kappa)+F^{\kappa}\cdot\hat\kappa=0$. Then $F^{\hat\kappa}$ solves the problem in 1), and the solution is $J(\hat\kappa)$.

What is this method called in general?, when it is not related to mathematical finance?

Simply put it, we have a optimisation problem, constrained to a set. Instead of optimising directly, we enlarge the problem to a more general one, in this general problem we do unconstrained optimisation, then we minimise all these again, and if the condition in 7 is satisfied we have indeed solved the original constrained one.