# Maximize function with undetermined coefficient

My question is related to this one:

Parametric plot on multi-dimensional domain

I will use a simple example to state my pronblem.

I have a variable f and g, such as

f = x1+x2*Sin[t]+x3*Sin[2t];
g = x1+x2+x3;


x1, x2, x3 will have some values, then I need

F = Maximize[{f,0<t<10},t][]


which means F is the maximum of f as function of t.

Then I will make {x1,0,10,1}, {x2,0,10,1}, {x3,0,10,1} to obtain many Fs and gs, and Plot F vs g.

The problems is when I write the code like this (as suggested in the Parametric plot on multi-dimensional domain):

ListPlot[Transpose[{F, g} /. Thread[{x1, x2, x3,->Transpose@Tuples[{Range[0, 10, 1], Range[0, 10, 1], Range[0, 10, 1]}]]


Or make it simple, just generate a table of F

Table[F,{x1,0,10,1}, {x2,0,10,1}, {x3,0,10,1}]


The problem is it seems that Mathematica will run the Maximize first, then give the x1, x2, x3's value to F. What I want is give values to x1, x2, x3 first, then take the maximum to obtain F.

Is there anyone know how to solve this.

• There are errors in you ListPlot: a comma instead of a }in the Thread part, and Transpose and ListPlot don't have their closing ] brackets anywhere. – corey979 Nov 7 '16 at 19:03

I'd do

f = x1 + x2*Sin[t] + x3*Sin[2 t];
g = x1 + x2 + x3;

tuples = N @ Tuples[Range[0, 10, 1], {3}];

F[i_] := Maximize[{f /. Thread[{x1, x2, x3} -> tuples[[i]]], 0 < t < 10}, t][]
G[i_] := g /. Thread[{x1, x2, x3} -> tuples[[i]]]

data = Table[{G[i], F[i]}, {i, Length @ tuples}];

plot = ListPlot[data, Frame -> True, AspectRatio -> 1, FrameLabel -> {"G", "F"}] • Yes, it works. Thank you. – Qi Zhong Nov 7 '16 at 19:32
• @QiZhong I've noticed that you haven't accepted any answer to none of your questions. If you find some answers useful/helpful, consider accepting them by clicking the gray checkmark. – corey979 Nov 11 '16 at 13:50
• Sorry, I thought the click the upper arrow is enough. Now I have clicked the checkmark. I have a problem. That is it takes about 10 minutes to run my code for 1000+points. The reason I think is the function Maximize. Is there any way to reduce the time for running Maximize, for example reducing the precision of the result? – Qi Zhong Nov 14 '16 at 21:28
• Maximize tries to find a global maximum symbolically, and is not that fast. You may try with NMaximize or related functions, like FindMaximum or so. On the other hand, 10 mins is not that long. – corey979 Nov 14 '16 at 21:39
• Do you know how to control the Precision of G[x]? For example G=1.8364958495, I want G=1.8365 by setting the Precision. – Qi Zhong Nov 15 '16 at 16:52

If I understand correctly, this should work:

f[x1_, x2_, x3_] = x1 + x2*Sin[t] + x3*Sin[2 t];
F[x1_, x2_, x3_] := Maximize[{f[x1, x2, x3], 0 < t < 10}, t][]
Table[F[x1, x2, x3], {x1, 0, 10, 1}, {x2, 0, 10, 1}, {x3, 0, 10, 1}]

• Yes, the table can work in this way. But "Transpose[{F, g} /. Thread[{x1, x2, x3,->Transpose@Tuples..." doesn't work. – Qi Zhong Nov 7 '16 at 18:50