st: Maximize a function that contains an integral

[Bob Hammond <robert_hammond@ncsu.edu>]

All,

I want to find the maximum of a function that contains an integral 
without an analytical solution.  The difficulty is that the argument of 
the maximization is the lower bound of the integral.  Consider the 
following trivial example: find the x that maximizes f(x), where f(x) is 
the integral of z (with respect to z) from a lower bound of x to an 
upper bound of 0.  Computing the integral by hand shows that x* = 0 
(because f(x) = -0.5x^2) but I would like to know how to code this 
without invoking the analytical solution.

Note that I cannot simply differentiate f(x) with respect to x and use a 
root finder (e.g., mm_root) to find the optimum because my f'(x) also 
contains an integral.

I've used Stata's integ command to great effect in similar applications 
but do not know how to embed it into my problem of maximizing f(x).  It 
occurred to me that I could use ml instead of Mata but I'm not sure how 
that would be done since the argument of the optimization problem is the 
lower bound of the integration. 

The most promising approach that I've found seems to be "Quadrature on 
sparse grids":

http://sparse-grids.de/#Stata

which contains a Mata function nwspgr() in a zip file.  I've spent some 
time with this function and it's very accurate but I cannot figure out 
how to manipulate the integration bounds such that the lower bound is a 
variable that I can feed in from an optimizer such as Mata's optimize.

I will contact the authors of nwspgr() but I wanted to check a few 
things before doing so:

1. Is there an existing Mata equivalent to integ (which easily allows 
manipulation of the integration bounds)?
2. Relative to using Mata and (possibly) nwspgr(), would it be simpler 
to use integ and ml?  If so, I would appreciate any tips.

Thanks,

Bob
--
------------------------------------------------------------------------
Bob Hammond
Department of Economics
North Carolina State University
Office: (919) 513-2871
Fax: (919) 515-7873
http://www4.ncsu.edu/~rghammon/
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