# Which optimization algorithm to choose?

#### Marie Laure Delignette Muller, Christophe Dutang

#### 2016-07-01

# 1 Quick overview of main optimization methods

We present very quickly the main optimization methods. Please refer to Numerical Optimization (Nocedal & Wright, 2006) or Numerical Optimization: theoretical and practical aspects (Bonnans, Gilbert, Lemarechal & Sagastizabal, 2006) for a good introduction. We consider the following problem \(\min_x f(x)\) for \(x\in\mathbb{R}^n\).

## 1.1 Derivative-free optimization methods

The Nelder-Mead method is one of the most well known derivative-free methods that use only values of \(f\) to search for the minimum. It consists in building a simplex of \(n+1\) points and moving/shrinking this simplex into the good direction.

1. set initial points \(x_1, \dots, x_{n+1}\).
2. order points such that \(f(x_1)\leq f(x_2)\leq\dots\leq f(x_{n+1})\).
3. compute \(x_o\) as the centroid of \(x_1, \dots, x_{n}\).
4. Reflection:
   - compute the reflected point \(x_r = x_o + \alpha(x_o-x_{n+1})\).
   - if \(f(x_1)\leq f(x_r)<f(x_n)\), then replace \(x_{n+1}\) by \(x_r\), go to step 2.
   - else go step 5.
5. Expansion:
   - if \(f(x_r)<f(x_1)\), then compute the expansion point \(x_e= x_o+\gamma(x_o-x_{n+1})\).
   - if \(f(x_e) <f(x_r)\), then replace \(x_{n+1}\) by \(x_e\), go to step 2.
   - else \(x_{n+1}\) by \(x_r\), go to step 2.
   - else go to step 6.
6. Contraction:
   - compute the contracted point \(x_c = x_o + \beta(x_o-x_{n+1})\).
   - if \(f(x_c)<f(x_{n+1})\), then replace \(x_{n+1}\) by \(x_c\), go to step 2.
   - else go step 7.
7. Reduction:
   - for \(i=2,\dots, n+1\), compute \(x_i = x_1+\sigma(x_i-x_{1})\).

The Nelder-Mead method is available in `optim`. By default, in `optim`, \(\alpha=1\), \(\beta=1/2\), \(\gamma=2\) and \(\sigma=1/2\).

## 1.2 Hessian-free optimization methods

For smooth non-linear function, the following method is generally used: a local method combined with line search work on the scheme \(x_{k+1} =x_k + t_k d_{k}\), where the local method will specify the direction \(d_k\) and the line search will specify the step size \(t_k \in \mathbb{R}\).

### 1.2.1 Computing the direction \(d_k\)

A desirable property for \(d_k\) is that \(d_k\) ensures a descent \(f(x_{k+1}) < f(x_{k})\). Newton methods are such that \(d_k\) minimizes a local quadratic approximation of \(f\) based on a Taylor expansion, that is \(q_f(d) = f(x_k) + g(x_k)^Td +\frac{1}{2} d^T H(x_k) d\) where \(g\) denotes the gradient and \(H\) denotes the Hessian.

The consists in using the exact solution of local minimization problem \(d_k = - H(x_k)^{-1} g(x_k)\).
In practice, other methods are preferred (at least to ensure positive definiteness). The method approximates the Hessian by a matrix \(H_k\) as a function of \(H_{k-1}\), \(x_k\), \(f(x_k)\) and then \(d_k\) solves the system \(H_k d = - g(x_k)\). Some implementation may also directly approximate the inverse of the Hessian \(W_k\) in order to compute \(d_k = -W_k g(x_k)\). Using the Sherman-Morrison-Woodbury formula, we can switch between \(W_k\) and \(H_k\).

To determine \(W_k\), first it must verify the secant equation \(H_k y_k =s_k\) or \(y_k=W_k s_k\) where \(y_k = g_{k+1}-g_k\) and \(s_k=x_{k+1}-x_k\). To define the \(n(n-1)\) terms, we generally impose a symmetry and a minimum distance conditions. We say we have a rank 2 update if \(H_k = H_{k-1} + a u u^T + b v v^T\) and a rank 1 update if $H_k = H_{k-1} + a u u^T $. Rank \(n\) update is justified by the spectral decomposition theorem.

There are two rank-2 updates which are symmetric and preserve positive definiteness

- DFP minimizes \(\min || H - H_k ||_F\) such that \(H=H^T\): \[ H_{k+1} = \left (I-\frac {y_k s_k^T} {y_k^T s_k} \right ) H_k \left (I-\frac {s_k y_k^T} {y_k^T s_k} \right )+\frac{y_k y_k^T} {y_k^T s_k} \Leftrightarrow W_{k+1} = W_k + \frac{s_k s_k^T}{y_k^{T} s_k} - \frac {W_k y_k y_k^T W_k^T} {y_k^T W_k y_k} . \]
- BFGS minimizes \(\min || W - W_k ||_F\) such that \(W=W^T\): \[ H_{k+1} = H_k - \frac{ H_k y_k y_k^T H_k }{ y_k^T H_k y_k } + \frac{ s_k s_k^T }{ y_k^T s_k } \Leftrightarrow W_{k+1} = \left (I-\frac {y_k s_k^T} {y_k^T s_k} \right )^T W_k \left (I-\frac { y_k s_k^T} {y_k^T s_k} \right )+\frac{s_k s_k^T} {y_k^T s_k} . \]

In `R`, the so-called BFGS scheme is implemented in `optim`.

Another possible method (which is initially arised from quadratic problems) is the nonlinear conjugate gradients. This consists in computing directions \((d_0, \dots, d_k)\) that are conjugate with respect to a matrix close to the true Hessian \(H(x_k)\). Directions are computed iteratively by \(d_k = -g(x_k) + \beta_k d_{k-1}\) for \(k>1\), once initiated by \(d_1 = -g(x_1)\). \(\beta_k\) are updated according a scheme:

- \(\beta_k = \frac{ g_k^T g_k}{g_{k-1}^T g_{k-1} }\): Fletcher-Reeves update,
- \(\beta_k = \frac{ g_k^T (g_k-g_{k-1} )}{g_{k-1}^T g_{k-1}}\): Polak-Ribiere update.

There exists also three-term formula for computing direction \(d_k = -g(x_k) + \beta_k d_{k-1}+\gamma_{k} d_t\) for \(t<k\). A possible scheme is the Beale-Sorenson update defined as \(\beta_k = \frac{ g_k^T (g_k-g_{k-1} )}{d^T_{k-1}(g_{k}- g_{k-1})}\) and \(\gamma_k = \frac{ g_k^T (g_{t+1}-g_{t} )}{d^T_{t}(g_{t+1}- g_{t})}\) if \(k>t+1\) otherwise \(\gamma_k=0\) if \(k=t\). See Yuan (2006) for other well-known schemes such as Hestenses-Stiefel, Dixon or Conjugate-Descent. The three updates (Fletcher-Reeves, Polak-Ribiere, Beale-Sorenson) of the (non-linear) conjugate gradient are available in `optim`.

### 1.2.2 Computing the stepsize \(t_k\)

Let \(\phi_k(t) = f(x_k + t d_k)\) for a given direction/iterate \((d_k, x_k)\). We need to find conditions to find a satisfactory stepsize \(t_k\). In literature, we consider the descent condition: \(\phi_k'(0) < 0\) and the Armijo condition: \(\phi_k(t) \leq \phi_k(0) + t c_1 \phi_k'(0)\) ensures a decrease of \(f\). Nocedal & Wright (2006) presents a backtracking (or geometric) approach satisfying the Armijo condition and minimal condition, i.e. Goldstein and Price condition.

- set \(t_{k,0}\) e.g. 1, \(0 < \alpha < 1\),
- Repeat until Armijo satisfied,
  - \(t_{k,i+1} = \alpha \times t_{k,i}\).
- end Repeat

This backtracking linesearch is available in `optim`.

## 1.3 Benchmark

To simplify the benchmark of optimization methods, we create a `fitbench` function that computes the desired estimation method for all optimization methods. This function is currently not exported in the package.

    fitbench <- function(data, distr, method, grad=NULL, control=list(trace=0, REPORT=1, maxit=1000), lower=-Inf, upper=+Inf, ...)

# 2 Numerical illustration with the beta distribution

## 2.1 Log-likelihood function and its gradient for beta distribution

### 2.1.1 Theoretical value

The density of the beta distribution is given by \[ f(x; \delta_1,\delta_2) = \frac{x^{\delta_1-1}(1-x)^{\delta_2-1}}{\beta(\delta_1,\delta_2)}, \] where \(\beta\) denotes the beta function, see the NIST Handbook of mathematical functions http://dlmf.nist.gov/. We recall that \(\beta(a,b)=\Gamma(a)\Gamma(b)/\Gamma(a+b)\). There the log-likelihood for a set of observations \((x_1,\dots,x_n)\) is \[ \log L(\delta_1,\delta_2) = (\delta_1-1)\sum_{i=1}^n\log(x_i)+ (\delta_2-1)\sum_{i=1}^n\log(1-x_i)+ n \log(\beta(\delta_1,\delta_2)) \] The gradient with respect to \(a\) and \(b\) is \[ \nabla \log L(\delta_1,\delta_2) = \left(\begin{matrix} \sum\limits_{i=1}^n\ln(x_i) - n\psi(\delta_1)+n\psi( \delta_1+\delta_2) \\ \sum\limits_{i=1}^n\ln(1-x_i)- n\psi(\delta_2)+n\psi( \delta_1+\delta_2) \end{matrix}\right), \] where \(\psi(x)=\Gamma'(x)/\Gamma(x)\) is the digamma function, see the NIST Handbook of mathematical functions http://dlmf.nist.gov/.

### 2.1.2 `R` implementation

As in the `fitdistrplus` package, we minimize the opposite of the log-likelihood: we implement the opposite of the gradient in `grlnL`. Both the log-likelihood and its gradient are not exported.

    lnL <- function(par, fix.arg, obs, ddistnam)
      fitdistrplus:::loglikelihood(par, fix.arg, obs, ddistnam)
    grlnlbeta <- fitdistrplus:::grlnlbeta

## 2.2 Random generation of a sample

    #(1) beta distribution
    n <- 200
    x <- rbeta(n, 3, 3/4)
    grlnlbeta(c(3, 4), x) #test

    ## [1] -133  317

    hist(x, prob=TRUE)
    lines(density(x), col="red")
    curve(dbeta(x, 3, 3/4), col="green", add=TRUE)
    legend("topleft", lty=1, col=c("red","green"), leg=c("empirical", "theoretical"))

## 2.3 Fit Beta distribution

Define control parameters.

    ctr <- list(trace=0, REPORT=1, maxit=1000)

Call `mledist` with the default optimization function (`optim` implemented in `stats` package) with and without the gradient for the different optimization methods.

    unconstropt <- fitbench(x, "beta", "mle", grad=grlnlbeta, lower=0)

In the case of constrained optimization, `mledist` permits the direct use of `constrOptim` function (still implemented in `stats` package) that allow linear inequality constraints by using a logarithmic barrier.

Use a exp/log transformation of the shape parameters \(\delta_1\) and \(\delta_2\) to ensure that the shape parameters are strictly positive.

    dbeta2 <- function(x, shape1, shape2, log)
      dbeta(x, exp(shape1), exp(shape2), log=log)
    #take the log of the starting values
    startarg <- lapply(fitdistrplus:::start.arg.default(x, "beta"), log)
    #redefine the gradient for the new parametrization
    grbetaexp <- function(par, obs, ...)
        grlnlbeta(exp(par), obs) * exp(par)

    expopt <- fitbench(x, distr="beta2", method="mle", grad=grbetaexp, start=startarg)
    #get back to original parametrization
    expopt[c("fitted shape1", "fitted shape2"), ] <- exp(expopt[c("fitted shape1", "fitted shape2"), ])

Then we extract the values of the fitted parameters, the value of the corresponding log-likelihood and the number of counts to the function to minimize and its gradient (whether it is the theoretical gradient or the numerically approximated one).

## 2.4 Results of the numerical investigation

Results are displayed in the following tables: (1) the original parametrization without specifying the gradient (`-B` stands for bounded version), (2) the original parametrization with the (true) gradient (`-B` stands for bounded version and `-G` for gradient), (3) the log-transformed parametrization without specifying the gradient, (4) the log-transformed parametrization with the (true) gradient (`-G` stands for gradient).

| BFGS | NM | CGFR | CGPR | CGBS | L-BFGS-B | NM-B
--- | --- | --- | --- | --- | --- | --- | ---
fitted shape1 | 2.665 | 2.664 | 2.665 | 2.665 | 2.665 | 2.665 | 2.665
fitted shape2 | 0.731 | 0.731 | 0.731 | 0.731 | 0.731 | 0.731 | 0.731
fitted loglik | 114.165 | 114.165 | 114.165 | 114.165 | 114.165 | 114.165 | 114.165
func. eval. nb. | 23.000 | 47.000 | 225.000 | 271.000 | 137.000 | 11.000 | 47.000
grad. eval. nb. | 5.000 | NA | 55.000 | 73.000 | 37.000 | 11.000 | NA
time (sec) | 0.010 | 0.020 | 0.040 | 0.040 | 0.030 | 0.010 | 0.010

| G-BFGS | G-CGFR | G-CGPR | G-CGBS | G-BFGS-B | G-NM-B | G-CGFR-B | G-CGPR-B | G-CGBS-B
--- | --- | --- | --- | --- | --- | --- | --- | --- | ---
fitted shape1 | 2.665 | 2.665 | 2.665 | 2.665 | 2.665 | 2.665 | 2.665 | 2.665 | 2.665
fitted shape2 | 0.731 | 0.731 | 0.731 | 0.731 | 0.731 | 0.731 | 0.731 | 0.731 | 0.731
fitted loglik | 114.165 | 114.165 | 114.165 | 114.165 | 114.165 | 114.165 | 114.165 | 114.165 | 114.165
func. eval. nb. | 20.000 | 251.000 | 225.000 | 138.000 | 25.000 | 47.000 | 263.000 | 188.000 | 176.000
grad. eval. nb. | 5.000 | 71.000 | 69.000 | 43.000 | 5.000 | NA | 69.000 | 59.000 | 47.000
time (sec) | 0.010 | 0.150 | 0.170 | 0.100 | 0.030 | 0.040 | 0.210 | 0.180 | 0.140

| BFGS | NM | CGFR | CGPR | CGBS
--- | --- | --- | --- | --- | ---
fitted shape1 | 2.665 | 2.664 | 2.665 | 2.665 | 2.665
fitted shape2 | 0.731 | 0.731 | 0.731 | 0.731 | 0.731
fitted loglik | 114.165 | 114.165 | 114.165 | 114.165 | 114.165
func. eval. nb. | 18.000 | 41.000 | 136.000 | 116.000 | 113.000
grad. eval. nb. | 5.000 | NA | 27.000 | 29.000 | 29.000
time (sec) | 0.010 | 0.000 | 0.020 | 0.030 | 0.030

| G-BFGS | G-CGFR | G-CGPR | G-CGBS
--- | --- | --- | --- | ---
fitted shape1 | 2.665 | 2.665 | 2.665 | 2.665
fitted shape2 | 0.731 | 0.731 | 0.731 | 0.731
fitted loglik | 114.165 | 114.165 | 114.165 | 114.165
func. eval. nb. | 20.000 | 175.000 | 125.000 | 112.000
grad. eval. nb. | 5.000 | 39.000 | 41.000 | 35.000
time (sec) | 0.020 | 0.100 | 0.100 | 0.100

Using `llsurface`, we plot the log-likehood surface around the true value (green) and the fitted parameters (red).

    llsurface(min.arg=c(0.1, 0.1), max.arg=c(7, 3),
              plot.arg=c("shape1", "shape2"), nlev=25,
              plot.np=50, data=x, distr="beta", back.col = FALSE)
    points(unconstropt[1,"BFGS"], unconstropt[2,"BFGS"], pch="+", col="red")
    points(3, 3/4, pch="x", col="green")

We can simulate bootstrap replicates using the `bootdist` function.

    b1 <- bootdist(fitdist(x, "beta", method="mle", optim.method="BFGS"), niter=100, parallel="snow", ncpus=2)
    summary(b1)

    ## Parametric bootstrap medians and 95% percentile CI
    ##        Median  2.5% 97.5%
    ## shape1   2.73 2.272 3.283
    ## shape2   0.75 0.652 0.888

    plot(b1)
    abline(v=3, h=3/4, col="red", lwd=1.5)

# 3 Numerical illustration with the negative binomial distribution

## 3.1 Log-likelihood function and its gradient for negative binomial distribution

### 3.1.1 Theoretical value

The p.m.f. of the Negative binomial distribution is given by \[ f(x; m,p) = \frac{\Gamma(x+m)}{\Gamma(m)x!} p^m (1-p)^x, \] where \(\Gamma\) denotes the beta function, see the NIST Handbook of mathematical functions http://dlmf.nist.gov/. There exists an alternative representation where \(\mu=m (1-p)/p\) or equivalently \(p=m/(m+\mu)\). Thus, the log-likelihood for a set of observations \((x_1,\dots,x_n)\) is \[ \log L(m,p) = \sum_{i=1}^{n} \log\Gamma(x_i+m) -n\log\Gamma(m) -\sum_{i=1}^{n} \log(x_i!) + mn\log(p) +\sum_{i=1}^{n} {x_i}\log(1-p) \] The gradient with respect to \(m\) and \(p\) is \[ \nabla \log L(m,p) = \left(\begin{matrix} \sum_{i=1}^{n} \psi(x_i+m) -n \psi(m) + n\log(p) \\ mn/p -\sum_{i=1}^{n} {x_i}/(1-p) \end{matrix}\right), \] where \(\psi(x)=\Gamma'(x)/\Gamma(x)\) is the digamma function, see the NIST Handbook of mathematical functions http://dlmf.nist.gov/.

### 3.1.2 `R` implementation

As in the `fitdistrplus` package, we minimize the opposite of the log-likelihood: we implement the opposite of the gradient in `grlnL`.

    grlnlNB <- function(x, obs, ...)
    {
      m <- x[1]
      p <- x[2]
      n <- length(obs)
      c(sum(psigamma(obs+m)) - n*psigamma(m) + n*log(p),
        m*n/p - sum(obs)/(1-p))
    }

## 3.2 Random generation of a sample

    #(1) beta distribution
    n <- 200
    trueval <- c("size"=10, "prob"=3/4, "mu"=10/3)
    x <- rnbinom(n, trueval["size"], trueval["prob"])

    hist(x, prob=TRUE, ylim=c(0, .3))
    lines(density(x), col="red")
    points(min(x):max(x), dnbinom(min(x):max(x), trueval["size"], trueval["prob"]), col="green")
    legend("topleft", lty=1, col=c("red","green"), leg=c("empirical", "theoretical"))

## 3.3 Fit a negative binomial distribution

Define control parameters and make the benchmark.

    ctr <- list(trace=0, REPORT=1, maxit=1000)
    unconstropt <- fitbench(x, "nbinom", "mle", grad=grlnlNB, lower=0)
    unconstropt <- rbind(unconstropt, "fitted prob"=unconstropt["fitted mu",] / (1+unconstropt["fitted mu",]))

In the case of constrained optimization, `mledist` permits the direct use of `constrOptim` function (still implemented in `stats` package) that allow linear inequality constraints by using a logarithmic barrier.

Use a exp/log transformation of the shape parameters \(\delta_1\) and \(\delta_2\) to ensure that the shape parameters are strictly positive.

    dnbinom2 <- function(x, size, prob, log)
      dnbinom(x, exp(size), 1/(1+exp(-prob)), log=log)
    #transform starting values
    startarg <- fitdistrplus:::start.arg.default(x, "nbinom")
    startarg$mu <- startarg$size / (startarg$size+startarg$mu)
    startarg <- list(size=log(startarg[[1]]), prob=log(startarg[[2]]/(1-startarg[[2]])))

    #redefine the gradient for the new parametrization
    Trans <- function(x)
      c(exp(x[1]), plogis(x[2]))
    grNBexp <- function(par, obs, ...)
        grlnlNB(Trans(par), obs) * c(exp(par[1]), plogis(x[2])*(1-plogis(x[2])))

    expopt <- fitbench(x, distr="nbinom2", method="mle", grad=grNBexp, start=startarg)
    #get back to original parametrization
    expopt[c("fitted size", "fitted prob"), ] <- apply(expopt[c("fitted size", "fitted prob"), ], 2, Trans)

Then we extract the values of the fitted parameters, the value of the corresponding log-likelihood and the number of counts to the function to minimize and its gradient (whether it is the theoretical gradient or the numerically approximated one).

## 3.4 Results of the numerical investigation

Results are displayed in the following tables: (1) the original parametrization without specifying the gradient (`-B` stands for bounded version), (2) the original parametrization with the (true) gradient (`-B` stands for bounded version and `-G` for gradient), (3) the log-transformed parametrization without specifying the gradient, (4) the log-transformed parametrization with the (true) gradient (`-G` stands for gradient).

| BFGS | NM | CGFR | CGPR | CGBS | L-BFGS-B | NM-B
--- | --- | --- | --- | --- | --- | --- | ---
fitted size | 40.539 | 44.074 | 40.610 | 41.833 | 42.984 | 43.970 | 43.771
fitted mu | 3.425 | 3.425 | 3.425 | 3.425 | 3.425 | 3.425 | 3.425
fitted loglik | -403.894 | -403.892 | -403.894 | -403.893 | -403.892 | -403.892 | -403.892
func. eval. nb. | 2.000 | 39.000 | 2999.000 | 2674.000 | 2583.000 | 6.000 | 0.000
grad. eval. nb. | 1.000 | NA | 1001.000 | 1001.000 | 1001.000 | 6.000 | NA
time (sec) | 0.000 | 0.000 | 0.530 | 0.500 | 0.500 | 0.000 | 0.000
fitted prob | 0.774 | 0.774 | 0.774 | 0.774 | 0.774 | 0.774 | 0.774

| G-BFGS | G-CGFR | G-CGPR | G-CGBS | G-BFGS-B | G-NM-B | G-CGFR-B | G-CGPR-B | G-CGBS-B
--- | --- | --- | --- | --- | --- | --- | --- | --- | ---
fitted size | 40.538 | 40.538 | 40.538 | 40.538 | 40.538 | 43.771 | 40.538 | 40.538 | 40.538
fitted mu | 3.425 | 3.425 | 3.425 | 3.425 | 3.425 | 3.425 | 3.425 | 3.425 | 3.425
fitted loglik | -403.894 | -403.894 | -403.894 | -403.894 | -403.894 | -403.892 | -403.894 | -403.894 | -403.894
func. eval. nb. | 26.000 | 233.000 | 292.000 | 233.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000
grad. eval. nb. | 1.000 | 15.000 | 19.000 | 15.000 | NA | NA | NA | NA | NA
time (sec) | 0.000 | 0.010 | 0.020 | 0.040 | 0.020 | 0.010 | 0.020 | 0.030 | 0.010
fitted prob | 0.774 | 0.774 | 0.774 | 0.774 | 0.774 | 0.774 | 0.774 | 0.774 | 0.774

| BFGS | NM | CGFR | CGPR | CGBS
--- | --- | --- | --- | --- | ---
fitted size | 40.541 | 43.901 | 42.154 | 43.966 | 43.971
fitted prob | 0.922 | 0.928 | 0.925 | 0.928 | 0.928
fitted loglik | -403.894 | -403.892 | -403.892 | -403.892 | -403.892
func. eval. nb. | 6.000 | 41.000 | 4001.000 | 594.000 | 319.000
grad. eval. nb. | 1.000 | NA | 1001.000 | 151.000 | 65.000
time (sec) | 0.000 | 0.010 | 0.600 | 0.090 | 0.040

| G-BFGS | G-CGFR | G-CGPR | G-CGBS
--- | --- | --- | --- | ---
fitted size | 40.538 | 40.538 | 40.538 | 40.538
fitted prob | 0.922 | 0.922 | 0.922 | 0.922
fitted loglik | -403.894 | -403.894 | -403.894 | -403.894
func. eval. nb. | 18.000 | 112.000 | 67.000 | 105.000
grad. eval. nb. | 1.000 | 9.000 | 5.000 | 9.000
time (sec) | 0.020 | 0.020 | 0.000 | 0.000

Using `llsurface`, we plot the log-likehood surface around the true value (green) and the fitted parameters (red).

    llsurface(min.arg=c(5, 0.3), max.arg=c(15, 1),
              plot.arg=c("size", "prob"), nlev=25,
              plot.np=50, data=x, distr="nbinom", back.col = FALSE)
    points(unconstropt["fitted size","BFGS"], unconstropt["fitted prob","BFGS"], pch="+", col="red")
    points(trueval["size"], trueval["prob"], pch="x", col="green")

We can simulate bootstrap replicates using the `bootdist` function.

    b1 <- bootdist(fitdist(x, "nbinom", method="mle", optim.method="BFGS"), niter=100, parallel="snow", ncpus=2)
    summary(b1)

    ## Parametric bootstrap medians and 95% percentile CI
    ##      Median 2.5%  97.5%
    ## size  40.54 9.89 101.64
    ## mu     3.44 3.19   3.74
    ##
    ## The estimation method converged only for 77 among 100 iterations

    plot(b1)
    abline(v=trueval["size"], h=trueval["mu"], col="red", lwd=1.5)

# 4 Conclusion

Based on the two previous examples, we observe that all methods converge to the same point. This is rassuring.
However, the number of function evaluations (and the gradient evaluations) is very different from a method to another. Furthermore, specifying the true gradient of the log-likelihood does not help at all the fitting procedure and generally slows down the convergence. Generally, the best method is the standard BFGS method or the BFGS method with the exponential transformation of the parameters. Since the exponential function is differentiable, the asymptotic properties are still preserved (by the Delta method) but for finite-sample this may produce a small bias.