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{equation} \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} \end{equation} with $A_{m}(i,\beta_{m})=\sum_{t_{j}<t_{i}}e^{\beta_{m}(t_{i}-t_{i})}$, $A_{1}(i,\beta_{m})=0$, $\bar{\alpha}$, $\bar{\beta}$ 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?


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