# Candidate methods for maximizing multivariable constrained nonlinear loglikelihood function

I want to approximate the maximum value of a nonlinear loglikelihood function with 53 strictly positive variables via numerical methods that do not use derivatives. According to literature there are direct methods that avoid taking derivatives (http://dx.doi.org/10.1007/BF02480272); however, I cannot find a clear article on how to implement such a method using the loglikelihood described below. A candidate method I found is the Nelder-Mead algorithm. Moreover, the loglikelihood is concave in at least 2 variables, which might reject some candidate methods. The loglikelihood is derived from a Hawkes process with stochastic intensity values and is as follows

A single observation consists of a stochastic process $$X(t)$$ taking on values in a discrete state space $$\{1,\dots,h\}$$ over the time horizon $$[0,T]$$ with $$n$$ transition times, $$\{1,\dots,t_{n}\}$$ $$$$\begin{split} \log L = &-\int^{T}_{0}\sum^{h-1}_{j=1}q_{j}1_{\{X(u)=j\}}du+\\ &\sum^{n}_{i=1}\frac{\alpha_{1}}{\beta_{1}}(e^{-\beta(T-t_{1,i}}-1)+\\ &\sum^{n}_{i=1}\frac{\alpha_{2}}{\beta_{2}}(e^{-\beta(T-t_{2,i}}-1)+\\ &\sum^{n}_{i=1}\log(\sum^{h-1}_{j,k=1}q_{jk}1_{\{X(t_{i}^{-})=j,X(t_{i})=k\}}+\\ &\alpha_{1}A_{1}(i,\beta_{1})+\alpha_{2}A_{2}(i,\beta_{2}))=f(\bar{Q},\bar{\alpha},\bar{\beta}) \end{split}$$$$ with $$A_{m}(i,\beta_{m})=\sum_{t_{j}, $$A_{1}(i,\beta_{m})=0$$, $$\bar{\alpha}$$, $$\bar{\beta}$$ and $$\bar{Q} = \begin{bmatrix} -q_{1}&q_{12}&\cdots&q_{1h}\\ q_{21}&-q_{2}&\cdots&q_{2h}\\ \vdots&\vdots&\ddots&\vdots\\ q_{h1}&q_{h2}&\cdots&-q_{h}\\ \end{bmatrix}$$

All-in-all my questions are is the Nelder-Mead algorithm appropriate with regards to i) the number of variables, ii) concavity in 2 or more variables and iii) the strictly positive contraint on all variables? Are there other candidate methods? Are there by chance clear articles on these methods?