# How to use simplex method for initial estimates of parameters in Nelson-Siegel-Svensson

I came across a BIS note about the estimation of the Nelson-Siegel-Svensson method. Currently, I'm trying to implement this. However, one step is not fully clear to me. Let me outline the steps of the algorithm to make the question self-contained. Although the note is very short (essentially 2 pages) and up to my question easy accessible.

The proposed algorithm works essentially the following way. For given $$N$$ bonds with prices $$P=(P_1,\dots,P_N)$$ and corresponding cash flow matrix $$C\in\mathbb{R}^{N\times L}$$ (so columns are maturities):

1. Calculate spot rates $$r_l$$ for all maturities, $$r=(r_1,\dots,r_L)$$
2. Calculate discount factors $$d_l$$, $$d=(d_1,\dots,d_L)$$
3. Given a cash flow matrix of all bonds, calculate the theoretical prices as $$\hat{P}=Cd$$
4. Using Newton-Raphson we calculate the estimated yield-to-maturity, $$\hat{ytm}=(\hat{ytm}_1,\dots,\hat{ytm}_N)$$

Now, for the actual objective there is the step which isn't fully clear to me. I quote directly the paper:

5. Computing the function $$\sum_{i=1}^N (ytm_i-\hat{ytm}_i(b_t,P,C))^2$$ (sum of squared yield errors) using first the Simplex algorithm and then the BHHH algorithm in order to determine a new $$b_{t+1}$$

Where $$b_t$$ is the current set of parameters to be estimated. Before stating this they add the following sentence

The optimisation is performed using a numerical non-linear optimisation to maximise a log-likelihood function subject to the constraint on the parameter $$\beta_1$$ (= overnight rate – $$\beta_0$$). First, we use the Simplex algorithm to compute starting values and then the Berndt, Hall, Hall and Hausmann (BHHH) algorithm to estimate the final parameters.

Question I understand that they somehow want to use the simplex method in 5. for getting better starting values and then run a ordinary non-linear optimisation using BHHH. However, the first part using the Simplex (linear programming) is unclear to me. Maybe someone with more expertise in optimisation can shed some light on this.