Re: Re: Problem with NMaximize

• To: mathgroup at smc.vnet.net
• Subject: [mg90098] Re: [mg90037] Re: Problem with NMaximize
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Sat, 28 Jun 2008 05:55:38 -0400 (EDT)
• References: <g3vkvi\$kio\$1@smc.vnet.net> <4863CA88.5040100@gmail.com> <200806271014.GAA24089@smc.vnet.net>

```ramiro wrote:
> Thank you very much for your responses, Bobby and Jean-Marc.   I did try
> changing the parametrization (thanks Bobby!) and I got an answer.
>
> However, let me try again with a hopefully more clear example, as I
> don't understand why this doesn't work, in particular, I don't
> understand why is it evaluating negative numbers if I am putting as a
> constraint that it shouldn't?
>
>
> Sample likelihood,
>
> In[196]:=
> logLikelihood1[t1_?NumberQ, t2_?NumberQ, t3_?NumberQ, t4_?NumberQ,
>    t5_?NumberQ, t6_?NumberQ, t7_?NumberQ, t8_?NumberQ, t9_?NumberQ,
>    t10_?NumberQ, t11_?NumberQ, t12_?NumberQ] :=
>   Log[0.00001712341312000713` t1 + 5.648647024829866`*^-6 t10 +
>     8.90597537441381`*^-6 t11 + 1.9727284288726167`*^-6 t12 +
>     4.102725237468317`*^-6 t2 + 3.7864902508468615`*^-6 t3 +
>     9.215772325326653`*^-7 t4 + 0.000057161484895917856` t5 +
>     0.000012892953334779516` t6 + 8.646320198720343`*^-6 t7 +
>     5.877910295858781`*^-6 t8 + 1.6837835562631724`*^-6 t9];
> [...]
> In[198]:= NMaximize[{logLikelihood1[t1, t2, t3, t4, t5, t6, t7, t8,
>    t9, t10, t11, t12], t1 > 0, t2 > 0, t3 > 0, t4 > 0, t5 > 0, t6 > 0,
>    t7 > 0, t8 > 0, t9 > 0, t10 > 0, t11 > 0, t12 > 0,
>   t1 + t10 + t11 + t12 + t2 + t3 + t4 + t5 + t6 + t7 + t8 + t9 ==
>    1}, {t1, t2, t3, t4, t5, t6, t7, t8, t9, t10, t11, t12}]
>
> During evaluation of In[198]:= NMaximize::nrnum: The function value \
> 15.7729-3.14159 I is not a real number at \
> {t1,t10,t11,t12,t2,t3,t4,t5,t6,t7,<<2>>} = {-0.490249,<<9>>,<<2>>}. \
>  >>
>
> Out[198]= {-13.2945, {t1 -> 2.10942*10^-15, t10 -> -2.22045*10^-16,
>   t11 -> -2.22045*10^-16, t12 -> -2.22045*10^-16,
>   t2 -> -1.11022*10^-16, t3 -> -2.22045*10^-16, t4 -> -2.22045*10^-16,
>    t5 -> -6.93889*10^-18, t6 -> -2.22045*10^-16,
>   t7 -> -1.11022*10^-16, t8 -> -4.44089*10^-16, t9 -> 1.}}
> [...]

Handling of constraints can be slightly tricky; sometimes the code will
allow a small amount of slop so as not to chase it's metaphorical tail
trying to satisfy them. Some tactics you might try include forcing a log
of a negative to evaluate to something that is clearly undesirable, and
maybe further constraining so that all variables are lessequal to 1.
Other things you could try include adding penalties based on UnitStep of
negative variables, and maybe play with Precision/AccuracyGoal (I found
that these hurt more than they helped).

Here is code to do some of this.

myLog[t_?NumberQ /; t<=0] := -10^6
myLog[t_] := Log[t]

vec = {0.00001712341312000713,4.102725237468317*^-6,3.7864902508468615*^-6,
9.215772325326653*^-7,0.000057161484895917856,0.000012892953334779516,
8.646320198720343*^-6,5.877910295858781*^-6,1.6837835562631724*^-6,
5.648647024829866*^-6,8.90597537441381*^-6,1.9727284288726167*^-6};

vars = Array[t,Length[vec]];

logLikelihood1[vars:{_?NumberQ..}] := myLog[vec.vars]

In[49]:= InputForm[NMaximize[{logLikelihood1[vars],
Flatten[{Map[0<=#<=1&,vars], Total[vars]==1}]}, vars]]

Out[49]//InputForm=
{-9.76963022735483, {t[1] -> 0., t[2] -> 0., t[3] -> 0., t[4] -> 0.,
t[5] -> 1., t[6] -> 0., t[7] -> 0., t[8] -> 0., t[9] -> 0.,
t[10] -> 0., t[11] -> 0., t[12] -> 0.}}

Daniel Lichtblau
Wolfram Research

```

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