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Perhaps a FindMinimum issue...
 To: mathgroup at smc.vnet.net
 Subject: [mg112673] Perhaps a FindMinimum issue...
 From: mpolko lokta <mpolkolokta at gmail.com>
 Date: Sun, 26 Sep 2010 02:43:15 0400 (EDT)
I have been trying to:
min 1/2*Transpose[w].V.w,
subject to the constraints
Total[w] == 1
and
Thread[w >= conformable vector of zeros]
for a number of different V matrices using FindMinimum.
One common feature of all the optimization problems I was trying to
solve is that V was always singular. The dimensions of the V matrix
ranged from 50  250 items.
The problem I am facing is that FindMinimum returns consistently
suboptimal results.
A hint about that was that some of the lagrange multipliers
corresponding to the nonnegativity constraints I calculated using the
proposed solution from FindMinimum and the first order conditions of
the Lagrangean of the optimization problem were actually negative
(they should have been positive or zero). Some of the negative
multipliers could be chopped (using Chop[] with default options) to
zero but others still persisted.
To make sure that I was actually getting suboptimal results that were
not due to rounding errors etc. I used a number of oher software that
provided routines for solving constrained quadratic programming
problems (both proprietary and open source) to solve a small subset of
the original problems Mathematica was supposedly providing me with
suboptimal results.
Unfortunately, I verified that FindMinimum was actually producing
suboptimal results!
After that comparison I tried using Minimize on a small subset of the
initial problems and I got results that were "closer" to the "right"
results provided by the other QP solvers, although Minimize's
solutions were still less than optimum. Another drawback of using
Minimize was the time it needed to provide an answer.
Does anybody know if the behavior of FindMinimum is due to the
singularity of the V matrix and has anybody any suggestion on how to
speed up the time it takes for Minimize to converge?
Thanks!
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