Role of numerical optimization and linear and nonlinear programming in PDE analysis

This is a quite naive question, but, I'm very new to the field of numerical analysis and I couldn't find a satisfactory answer to it.

• What are numerical optimization and linear and nonlinear programming and what is their role (applications) in PDE analysis?

• What are references that shed light on the relationship between numerical optimization and linear and nonlinear programming and PDE analysis?

$$\cases{f_1(x) = 0\\f_2(x)=0\\\hspace{1.05cm}\vdots\\f_n(x)=0}$$ Which in turn we can try to achieve by doing
$$x_o = \min_x\left\{\sum_{k=1}^n\|f_k(x)\|\right\}$$
Which works because of the non-negativity property of norms the least possible value would be $0$ occurring if and only if all equations are simultaneously fulfilled.