       Re: NMINIMIZATION WITH COMPLEX CONJUGATE

• To: mathgroup at smc.vnet.net
• Subject: [mg112871] Re: NMINIMIZATION WITH COMPLEX CONJUGATE
• From: pratip <pratip.chakraborty at gmail.com>
• Date: Mon, 4 Oct 2010 06:06:37 -0400 (EDT)
• References: <i86uvd\$hp3\$1@smc.vnet.net>

```Dear Tarun,

Here is the list of parameters that you mentioned.

p = 0.05;
q = 1
n = 5;

All the variables are separated in real and complex part. Here

x[i]=CompNum[i]=a[i]+I b[i]

and the conjugates are coded like this

(x^*)[i]=TCompNum[i]=a[i]-I b[i]

CompNum[i_] := a[i] + I b[i];
TCompNum[i_] := a[i] - I b[i];

Here is your function to be minimized.

f = Sum[Sqrt[i + 1] p CompNum[i] TCompNum[i + 1] +
p CompNum[i + 1] TCompNum[i] + (q*i + i (i - 1)) CompNum[i]
TCompNum[
i], {i, 0, n}];

Here is the equality constraint

c = \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 0\), \(n\)]
\*SuperscriptBox[\(Abs\ [CompNum[i]]\), \(2\)]\) == 1; c //

You mentioned that x=0. Hence I set the last variable to be equal
to zero be setting the real part and imaginary part to be equal to
zero.

dp = {a[n + 1] -> 0, b[n + 1] -> 0};

Here are the variables to be optimized.

var = Table[CompNum[i], {i, 0, n}] /. a_ + I b_ -> {a, b} // Flatten;

prob = Join[ComplexExpand[f] /. a_ + I b_ -> {a^2 + b^2} /. dp, {c}];

{val, res} =
FindMinimum[prob, var, AccuracyGoal -> 15, MaxIterations -> 10000]

{1.6269*10^-9, {a -> 0.707063, b -> 0.70707, a ->
-0.00465305,
b -> -0.00466767, a -> 0.00500541, b -> 0.0050008,
a -> 0.0000544725, b -> 0.0000995694, a -> -0.00175304,
b -> -0.00175306, a -> -0.00268813, b -> -0.00268786}}

You can check how good the constraints are satisfied. You can see the
left hand side of the constraint is equal to 1 if we insert the
optimization result from above.

c /. al_ == bl_ -> (al) /. res

1.

However the problem is very simple just you have to code it properly
so that Mathematica can solve it.

Pratip

```

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