Let $\alpha>0$, $c\in\mathbb R$, $t\in\mathbb N$ with $t\ge2$, $b_0,\ldots,b_{t-2}\ge0$ and $G_{s,\:j}\in\mathbb R$ for $s\in\{0,\ldots,t-2\}$ and $j\in\{0,\ldots,t-1\}$. Consider the function $$f(b):=\alpha\left|b_0G_{0,\:0}+\sum_{j=1}^{t-2}\frac{b_{j-1}G_{j-1,\:j}+b_jG_{j,\:j}}2+b_{t-2}G_{t-2,\:t-1}-c\right|^2+\sum_{j=1}^{t-2}\left|b_{j-1}G_{j-1,\:j}-b_jG_{j,\:j}\right|^2.$$
Assume that we want to minimize $f$ with respect to $b$. What kind of optimization problem is this and how can we solve it numerically?
You want to minimize a quadratic function of nonnegative variable $b\in\mathbb{R}^{t+1}$. This is a quadratic programming (QP) problem, and you can use a QP solver.