Variational principles
Standard. Useful because nearly every bound on $\lambda_{\min}^+$ is a Rayleigh-quotient argument.
Rayleigh quotient
For symmetric $A$ and nonzero $x$ define $R_A(x)=\langle Ax,x\rangle/\langle x,x\rangle$. If $A\succeq 0$, then $R_A(x)\ge 0$.
For PSD $A$, let $(\ker A)^\perp$ be the orthogonal complement of the kernel. Then $$\lambda_{\min}^+(A)=\min_{x\in(\ker A)^\perp,\ x\ne 0} R_A(x).$$
Courant–Fischer
Let $\lambda_1\le\cdots\le\lambda_n$ be eigenvalues of symmetric $A$. The min–max principle states $$\lambda_k=\min_{\dim S=k}\ \max_{x\in S\setminus\{0\}} R_A(x).$$
A practical corollary (test subspaces)
To upper bound $\lambda_{\min}^+(A)$ you may choose a subspace $S\subseteq(\ker A)^\perp$ of small dimension and compute $\max_{x\in S,\|x\|=1}\langle Ax,x\rangle$. If $S$ has codimension $1$ inside $(\ker A)^\perp$, this often becomes sharp.
If $H\subseteq (\ker A)^\perp$ has codimension $1$, then $$\lambda_{\min}^+(A)\le \max_{x\in H,\ \|x\|=1}\langle Ax,x\rangle.$$
A full proof appears in codim1-courant-fischer.