% Boyd & Vandenberghe "Convex Optimization"
% Joëlle Skaf - 04/24/08
%
% The random variable y is nonnegative and integer valued with a Poisson
% distribution with mean mu > 0. In a simple statistical model, the mean mu
% is modeled as an affine function of a vector u: mu = a'*u + b.
% We are given a number of observations which consist of pairs (u_i,y_i),
% i = 1,..., m, where y_i is the observed value of y for which the value of
% the explanatory variable is u_i. We find a maximum likelihood estimate of
% the model parameters a and b from these data by solving the problem
%           maximize    sum_{i=1}^m (y_i*log(a'*u_i + b) - (a'*u_i + b))
% where the variables are a and b.

% Input data
rand('state',0);
n = 10;
m = 100;
atrue = rand(n,1);
btrue = rand;

u = rand(n,m);
mu = atrue'*u + btrue;

% Generate random variables y from a Poisson distribution
% (The distribution is actually truncated at 10*max(mu) for simplicity)
L  = exp(-mu);
ns = ceil(max(10*mu));
y  = sum(cumprod(rand(ns,m))>=L(ones(ns,1),:));

% Maximum likelihood estimate of model parameters
cvx_begin
    variables a(n) b(1)
    maximize sum(y.*log(a'*u+b) - (a'*u+b))
cvx_end

 
Successive approximation method to be employed.
   For improved efficiency, SDPT3 is solving the dual problem.
   SDPT3 will be called several times to refine the solution.
   Original size: 276 variables, 103 equality constraints
   92 exponentials add 736 variables, 460 equality constraints
-----------------------------------------------------------------
 Cones  |             Errors              |
Mov/Act | Centering  Exp cone   Poly cone | Status
--------+---------------------------------+---------
 92/ 92 | 6.764e-01  3.304e-02  0.000e+00 | Solved
 90/ 92 | 5.602e-02  2.250e-04  0.000e+00 | Solved
 55/ 88 | 1.386e-03  1.261e-07  0.000e+00 | Solved
  0/ 34 | 1.389e-04  1.143e-09  0.000e+00 | Solved
-----------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +102.57