Let $\mathbb{X}$ be a Banach space equipped with some norm $||\cdot||_\mathbb{X}$ and $F:\mathbb{X}\to\mathbb{R}$ be some linear functional. Suppose we are given a set $A\subseteq\mathbb{X}$ which is convex and uniformly bounded i.e. $\sup_{f\in A}||f||_{\mathbb{X}}<+\infty$. Consider the following optimization problem $$\inf_{f\in A} F(f):=\mu$$ My question is: If we know that the infimum of $F$ on $A$ is finite but not attainable, how do we estimate this infimum value? I know that in such a case it is desirable to construct a sequence $(f_k)\subseteq A$ such that $\lim_kF(f_k)=\mu$ i.e. construct a minimizing sequence. If we are lucky enough that $F(f_k)$ are easily computable then we just look at the limit of a sequence of real numbers $(F(f_k))$. However are there explicit iterative methods for constructing such sequences? What relevant research is done in this direction?

**Some context**: This question is related to this specific situation. Let $\mathbb{X}=L^1(\mathbb{R}_+,\mu)$ be the space of integrable functions with respect to a given measure $\mu$ and $F:L^1(\mathbb{R}_+,\mu)\to\mathbb{R}$ a linear functional defined as
$$L(f):=\int_{\mathbb{R}_+}f(t)\,d\mu(t)$$
Let $A:=\{f\in L^1(\mathbb{R}_+,\mu), f\geqslant 0:||Tf||_{\infty}\leqslant 1\}$ where $T:L^1(\mathbb{R}_+,\mu)\to\text{ran}(T)$ is an integral operator defined as
$$(Tf)(t):=\int^1_0f(tx)\,d\beta(x)$$
with $\beta(x)$ some given smooth function on $[0,1]$.