Here is the problem setup
\begin{equation} \begin{array}{c} \text{min} \hspace{4mm} \mathbf{b}^{T}_{}\mathbf{A}^{}_{}\mathbf{b}^{}_{} \\ s.t \hspace{5mm} \mathbf{b} \in \mathbb{R}^{N} \\ \hspace{9mm}b_0=1\\ \hspace{62mm}b_i=b_{i-2}b_{i-1} \hspace{5mm} \forall \hspace{5mm} 2<i\leq N-1 \end{array} \end{equation}
where $\mathbf{A}$ is a $N\times N$ positive semi-definite matrix
To handle $b_0=1$ constrain, I know I can introduce Lagrange multiplier in objective function
\begin{equation} \begin{array}{c} \text{min} \hspace{4mm} \mathbf{b}^{T}_{}\mathbf{A}^{}_{}\mathbf{b}^{}_{} + \lambda(\mathbf{b}^{T}_{}\mathbf{u}^{}_{}-1) \end{array} \end{equation}
where $\mathbf{u}_{}^{}$ is a $N\times 1$ column vector and $u_0=1$ and $u_i=0 \forall 1<i\leq N-1$.
But I dont know how to handle second set of constraints?
EDIT: Example
Let $N=4$. Therefore I am looking for $\mathbf{b}=[b_0 \quad b_1 \quad b_2 \quad b_3]^T = [1 \quad b_1 \quad b_2 \quad b_1b_2]^T$. How do I include $b_3=b_1b_2$ constraint in objective function
From that you have described, the second constraint in your problem happens to be a bilinear constraint, thereby making the problem non-convex. Solvers such as Baron (http://archimedes.cheme.cmu.edu/?q=baron) can be used to solve such problems.