# A heuristic enhancement of optimisation algorithm

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The majority of the world’s problem deal with directly or indirectly some kind of optimisation. Instance of optimisation of resources or utility function can be seen our daily life. Here I am talking about standard optimisation problem in statistics, maximum likelihood estimate. This blog is a small episode of my recent work. I used R for this optimisation exercise.

There are packages available for optimisation in R. The mostly used are: optim() and optimize() are in stats package. A dedicated function mle() available in stats4 package. This is very useful most of the mean part modeling(eg: glm). What we need for this optimisation is to prepare function for the (log)likelihood and gradient (or hessian). We can also specify the algorithm(methods) as any of the following: “Nelder-Mead”, “BFGS”, “CG”, “L-BFGS-B”, “SANN”, and “Brent”. This opimisation output give the necessary information for the model estimation.

The case I was doing is an extension of bivariate garch model. It includes more than 20 parameters. let me focus pure garch model, garch(1,1), as a prototype to explain the algorithm. Here the model equation is in the variance part. Also the parameter has constraint to ensure the positive variance. The model equation is given below.

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