Nelder-Mead is a derivative-free simplex method.

Definition and Syntax

bool nm(arma::vec& init_out_vals, std::function<double (const arma::vec& vals_inp, arma::vec* grad_out, void* opt_data)> opt_objfn, void* opt_data);
bool nm(arma::vec& init_out_vals, std::function<double (const arma::vec& vals_inp, arma::vec* grad_out, void* opt_data)> opt_objfn, void* opt_data, algo_settings_t& settings);


Function arguments:

• init_out_vals a column vector of initial values; will be replaced by the solution values.
• opt_objfn the function to be minimized, taking three arguments:
• vals_inp a vector of inputs;
• grad_out an empty vector, as Nelder-Mead does not require the gradient to be known/exist; and
• opt_data additional parameters passed to the function.
• opt_data additional parameters passed to the function.
• settings parameters controlling the optimization routine; see below.

Optimization control parameters:

• double err_tol the value controlling how small $\| \nabla f \|$ should be before 'convergence' is declared.
• int iter_max the maximum number of iterations/updates before the algorithm exits.
• bool vals_bound whether the search space is bounded. If true, then
• arma::vec lower_bounds this defines the lower bounds.
• arma::vec upper_bounds this defines the upper bounds.

• double nm_par_alpha reflection parameter.
• double nm_par_gamma expansion parameter.
• double nm_par_beta contraction parameter.
• double nm_par_delta shrikage parameter.

Details

Let $x^{(i)}$ denote the simplex values at stage $i$ of the algorithm. The Nelder-Mead updating rule is as follows.

• Step 1. Sort the vertices in order of function values, from smallest to largest:
• $$f(x^{(i)}(1,:)) \leq f(x^{(i)}(2,:)) \leq \cdots \leq f(x^{(i)}(n+1,:))$$
• Step 2. Calculate the centroid value up to the $n$th vertex: $$\bar{x} = \frac{1}{n} \sum_{j=1}^n x^{(i)}(j,:)$$ and compute the reflection point: $$x^r = \bar{x} + \alpha (\bar{x} - x^{(i)}(n+1,:))$$ where $\alpha$ is set by nm_par_alpha.

If $f(x^r) \geq f(x^{(i)}(1,:))$ and $f(x^r) < f(x^{(i)}(n,:))$, then $$x^{(i+1)}(n+1,:) = x^r, \ \ \textbf{ and go to Step 1.}$$ Otherwise continue to Step 3.
• Step 3. If $f(x^r) \geq f(x^{(i)}(1,:))$ then go to Step 4, otherwise compute the expansion point: $$x^e = \bar{x} + \gamma (x^r - \bar{x})$$ where where $\gamma$ is set by nm_par_gamma.

Set $$x^{(i+1)}(n+1,:) = \begin{cases} x^e & \text{ if } f(x^e) < f(x^r) \\ x^r & \text{ else } \end{cases}$$ and go to Step 1.
• Steps 4 & 5. If $f(x^r) < f(x^{(i)}(n,:))$, then compute the outside or inside contraction: $$x^{c} = \begin{cases} \bar{x} + \beta(x^r - \bar{x}) & \text{ if } f(x^r) < f(x^{(i)}(n+1,:)) \\ \bar{x} - \beta(x^r - \bar{x}) & \text{ else} \end{cases}$$ where $\beta$ is set by nm_par_beta.

If $f(x^c) < f(x^{(i)}(n+1,:))$, then $$x^{(i+1)}(n+1,:) = x^c, \ \ \textbf{ and go to Step 1.}$$ Otherwise go to Step 6.
• Step 6. Shrink the simplex toward $x^{(i)}(1,:)$: $$x^{(i+1)}(j,:) = x^{(i)}(1,:) + \delta (x^{(i)}(j,:) - x^{(i)}(1,:)), \ \ j = 2, \ldots, n+1$$ where $\delta$ is set by nm_par_delta. Go to Step 1.