Items tagged with maximize

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Let be the number z so that |z+3-2*I| + |z-3-8*I| = 6*sqrt(2). Find min and max of the modulus of z. How can I find min and max of modulus of z with Maple.

Thank for your help!

i have an optimization problem, i want to maximize an expression using assumption, what should i do?


 

restart:with(Optimization):

M1:=Matrix((1,4),[sqrt(p),0,0,sqrt(1-p)]);

M1 := Matrix(1, 4, {(1, 1) = p^(1/2), (1, 2) = 0, (1, 3) = 0, (1, 4) = (1-p)^(1/2)})

(1)

M2:=Matrix((1,4),[cos(theta[1])*cos(theta[2]),exp(I*phi[1])*sin(theta[1])*cos(theta[2]),exp(I*phi[2])*sin(theta[2])*cos(theta[1]),exp(I*(phi[1]+phi[2]))*sin(theta[1])*sin(theta[2])])^+;

M2 := Matrix(4, 1, {(1, 1) = cos(theta[1])*cos(theta[2]), (2, 1) = exp(I*phi[1])*sin(theta[1])*cos(theta[2]), (3, 1) = exp(I*phi[2])*sin(theta[2])*cos(theta[1]), (4, 1) = exp(I*(phi[1]+phi[2]))*sin(theta[1])*sin(theta[2])})

(2)

#Real:=rhs(op(op(2,Re(M1.M2))));

PP:=Re(M1.M2)(1,1);

Re(p^(1/2)*cos(theta[1])*cos(theta[2])+(1-p)^(1/2)*exp(I*(phi[1]+phi[2]))*sin(theta[1])*sin(theta[2]))

(3)

maximize(PP) assuming 0<p ,p<1;

Error, (in assuming) when calling 'maximize'. Received: 'invalid input: `minimize/continuous` expects its 2nd argument, yFP, to be of type {name, list(name)}, but received `theta[1]` = -infinity'

 

 


 

Download optimize.mw

When using maximize on a relatively complicated function (see attached Maple file and PDF), it runs extremely slow. No return after 3 minutes.

My hardward: i7 2.3G, 8G DDR3 MEM, 500G SSD.

Maybe someone is interested to try the Maple code if your workstation is more powerful? :)

MaximizePerformance.mw

MaximizePerformance.pdf

The commands below are from a response by Carl Love to a question posed on February 27, 2016.

Variable t is not mentioned in the plot of f. Does t assume one or more particular values in the construction of the plot?

When t is given a specific value before executing the plot command, the resulting plot appears to be independent of t's value.

What is the logic behind the plot's construction?

f:= cos(2*t/m) + cos(2*(t+5)/m):
plot('maximize'(f), m= 1..10);

Let us consider 

maximize(int(exp(-x^4), x = k .. 3*k), location);

Error, (in maximize) invalid input: iscont expects its 1st argument, f, to be of type algebraic, but received x = k .. 3*k
whereas the expected output is 

[(2*((1/40)*GAMMA(1/4, (1/80)*ln(3))*5^(1/4)*ln(3)^(3/4)-(1/40)*GAMMA(1/4, (81/80)*ln(3))*5^(1/4)*ln(3)^(3/4)))*5^(3/4)*(1/ln(3))^(3/4), [k = (1/10)*10^(3/4)*ln(3)^(1/4)]]

as Mma 11 produces. The following 

RealDomain:-solve(diff(int(exp(-x^4), x = k .. 3*k), k));
  -(1/10)*5^(3/4)*ln(3)^(1/4), (1/10)*5^(3/4)*ln(3)^(1/4)

is not a workaround because of 

int(exp(-x^4), x = (1/10)*5^(3/4)*ln(3)^(1/4) .. (3/10)*5^(3/4)*ln(3)^(1/4));
  FAIL

 

Consider the following expression obtained from the solve command: Note that this uses the two variable arctan function.

p:=arctan((-cos(theta)^3-(1/2)*cos(theta)^2-(1/2)*cos(theta)*((2*cos(theta)+1)*(2*cos(theta)-3)*(cos(theta)+1)^2)^(1/2)+2*cos(theta)-(1/2)*((2*cos(theta)+1)*(2*cos(theta)-3)*(cos(theta)+1)^2)^(1/2)+3/2)^(1/2), -(1/2)*cos(theta)-1/2-(1/2)*((2*cos(theta)+1)*(2*cos(theta)-3)*(cos(theta)+1)^2)^(1/2)):

#ploting the expression shows a non-zero value at theta = Pi,  however if I convert p to a function using


f:=unapply(p,theta):


# then f(Pi);  gives a value of 0

#On the other hand maximize(p,theta=3*Pi/4..5*Pi/4,location); shows a non-zero value of 4*Pi/5 at theta = Pi,  which agrees with the plot of p, namely, it returns:

-arctan((10-2*5^(1/2))^(1/2)/(5^(1/2)+1))+Pi,

{[{theta = Pi}, -arctan((10-2*5^(1/2))^(1/2)/(5^(1/2)+1))+Pi]}

Is this a bug? Or what?

Thanks, 

Edwin

When I put maximize(cos(t)), everything is fine.

When I put maximize(cos(Pi)), everything is fine.

When I put maximize(cos(t*Pi)), it says invalid limiting point??? What went wrong?

 

Hi all,

I have a function f(x) and want to know at which x-value it attains its supremum.

Tried this, but it doesn't work (or at least hasn't been able to solve the equation in 10+ min):

M := maximize(f, x = 0 .. 1);

solve(f = M, x);

Does anyone know a way to do this?

Thanks,

Paul

a := 1; w0 := 1; w1 := 3; K := 10; h0 := .25; A := 3; alpha := .7; b := .6; c := .2

f := A*x^(b*alpha)*(alpha*A*x^(b*alpha)/(a+w0*h0+(x-h0)*w1))^(alpha/(1-alpha))-(alpha*A*x^(b*alpha)/(a+w0*h0+(x-h0)*w1))^(1/(1-alpha))

16.94206235*x^.42*(x^.42/(.50+3*x))^2.333333333-11.85944364*(x^.42/(.50+3*x))^3.333333333

(1)

NULL

M := maximize(f, x = 0 .. 1)

.8295280335

(2)

solve(f = M, x)

Warning,  computation interrupted

 

``

plot(f, x = 0 .. 1)

 

``

``

``



Download maximum.mw

I've got this huge chunk of code which leads to an optimiazation at the very last line (Bestangles:=minimize(maximize()-minimize))). This minization is taking a very long time (havent solved it yet) and I would very much like to reduce that time. As I've understood maple does optimization by differentiating and then finding all extremes and comparing. Would this mean that since I minimize and optimize within a minimization command, it differentiates a ton of times? And if this is the case, can I somehow do the differentiation beforehand, since it is the same function being differentiate all the time? Or is there some other way I can improve the code? 
Thanks alot!

Heres the full code:

I am not sure how/why, but here is the worksheet.

test.mw

 

The function evalutes fine and can be used for sequence. But it does not seem to be working with plot or Maximize.

V is assumed to between 0 and 1.

Need some help.

 

Thanks,

 

casper

Hi,

  I have the following input

f:=x^2*exp(-1.2*x);
maximize(f, x=0..100,location);

 Maple gives me the location is x=1.25. However, how should I do to obtain this position?  If I write

 

a:=maximize(f, x=0..100,location);

 

  Seems it do not work :(

 

  I may try fsolve at the maximum value, but it seems to be awkward..

 

Thank you very much!

  

Hi there,

I am trying to maximize a function given a set of values to a parameter in the function. The function is an differential equation belonging to a system of two differential equations.

I have a for loop to state different values to the parameter.

Maple yields the error:

Error, (in Optimization:-NLPSolve) cannot evaluate the solution further right of 0.17757507e-4, probably a singularity

When trying to maximize the function.

Supposed that I was doing something wrong in the loop, if I reproduce the contents of the loop outside, and set a value for the parameter. If I plot the solution of the ordinary differential equation, I can see where the maximum lies.

Having plot it, the Optimizamtion:-Maximize works as expected.

However, omitting the plot has a weird effect: I only get the same result depending on the bounds I set for the Maximization:

de1 := diff(A(t), t) = r*m*(1-g)*A(t)-piecewise(t < 8, r*A(t), t >= 8, (r+k)*A(t));
de2 := diff(G(t), t) = r*m*g*A(t)-l*G(t);

ics := A(0) = 25.0, G(0) = 0.;
num := dsolve({de1, de2, ics}, {A(t), G(t)}, type = numeric, output = listprocedure, parameters = [g]);

num(parameters = [g = .15]);
val := eval(G(t), num);

# odeplot(val, [t, G(t)], t = 0 .. 100);


Maximize(val);
Error, (in Optimization:-NLPSolve) cannot evaluate the solution further right of 0.17757507e-4, probably a singularity

val2 := Maximize(val);

Error, (in Optimization:-NLPSolve) cannot evaluate the solution further right of 0.17757507e-4, probably a singularity

val3 := Maximize(val(t), t = 0 .. 60);

  [10267.824035766165, [t = 8.25727747134303]]

val4 := Maximize(val(t), t = 0 .. 100);

[6.863211343195069e-9, [t = 59.84184367042171]]

 

The right answer is [10267.824035766165, [t = 8.25727747134303]]: Why do I get two different answers even if in that range there is only one relative maximum?

I ignore whether the way I am specifying the arguments for the Maximize function is correct. val is a procedure.

 

What am I missing?

Attached is the worksheet: MaplePrimes_malaria_param_variation_2.mw

 

Thanks,

jon

I want to solve maximize of equation,but the maximize failed to solve it,who can help me.thanks.

c[1] := (1/8)*w*{(1/((x+y+z)^2+1))^(3/2)+(1/((x+y)^2+1))^(3/2)+(1/((x+z)^2+1))^(3/2)+(1/((y+z)^2+1))^(3/2)+(1/(x^2+1))^(3/2)+(1/(y^2+1))^(3/2)+(1/(z^2+1))^(3/2)+1}+(1/8)*{x/((x+y+z)^2+1)+x/((x+y)^2+1)+x/((x+z)^2+1)+x/(x^2+1)}:

c[2] := (1/8)*w*{(1/((x+y+z)^2+1))^(3/2)+(1/((x+y)^2+1))^(3/2)+(1/((x+z)^2+1))^(3/2)+(1/((y+z)^2+1))^(3/2)+(1/(x^2+1))^(3/2)+(1/(y^2+1))^(3/2)+[1/(z^2+1)]^(3/2)+1}+(1/8)*{y/((x+y+z)^2+1)+y/((x+y)^2+1)+y/((y+z)^2+1)+y/(y^2+1)}:

t[1] := diff(c[1], x);

(1/8)*w*{-(3/2)*(1/((x+y+z)^2+1))^(1/2)*(2*x+2*y+2*z)/((x+y+z)^2+1)^2-(3/2)*(1/((x+y)^2+1))^(1/2)*(2*x+2*y)/((x+y)^2+1)^2-(3/2)*(1/((x+z)^2+1))^(1/2)*(2*x+2*z)/((x+z)^2+1)^2-3*(1/(x^2+1))^(1/2)*x/(x^2+1)^2}+(1/8)*{1/((x+y+z)^2+1)-x*(2*x+2*y+2*z)/((x+y+z)^2+1)^2+1/((x+y)^2+1)-x*(2*x+2*y)/((x+y)^2+1)^2+1/((x+z)^2+1)-x*(2*x+2*z)/((x+z)^2+1)^2+1/(x^2+1)-2*x^2/(x^2+1)^2}

(1)

t[2] := diff(c[2], y);

(1/8)*w*{-(3/2)*(1/((x+y+z)^2+1))^(1/2)*(2*x+2*y+2*z)/((x+y+z)^2+1)^2-(3/2)*(1/((x+y)^2+1))^(1/2)*(2*x+2*y)/((x+y)^2+1)^2-(3/2)*(1/((y+z)^2+1))^(1/2)*(2*y+2*z)/((y+z)^2+1)^2-3*(1/(y^2+1))^(1/2)*y/(y^2+1)^2}+(1/8)*{1/((x+y+z)^2+1)-y*(2*x+2*y+2*z)/((x+y+z)^2+1)^2+1/((x+y)^2+1)-y*(2*x+2*y)/((x+y)^2+1)^2+1/((y+z)^2+1)-y*(2*y+2*z)/((y+z)^2+1)^2+1/(y^2+1)-2*y^2/(y^2+1)^2}

(2)

eliminate({t[1], t[2]}, w);

[{w = -{1/((x+y+z)^2+1)-y*(2*x+2*y+2*z)/((x+y+z)^2+1)^2+1/((x+y)^2+1)-y*(2*x+2*y)/((x+y)^2+1)^2+1/((y+z)^2+1)-y*(2*y+2*z)/((y+z)^2+1)^2+1/(y^2+1)-2*y^2/(y^2+1)^2}/{-(3/2)*(1/(x^2+2*x*y+2*x*z+y^2+2*y*z+z^2+1))^(1/2)*(2*x+2*y+2*z)/((x+y+z)^2+1)^2-(3/2)*(1/(x^2+2*x*y+y^2+1))^(1/2)*(2*x+2*y)/((x+y)^2+1)^2-(3/2)*(1/(y^2+2*y*z+z^2+1))^(1/2)*(2*y+2*z)/((y+z)^2+1)^2-3*(1/(y^2+1))^(1/2)*y/(y^2+1)^2}}, {{1/((x+y+z)^2+1)-x*(2*x+2*y+2*z)/((x+y+z)^2+1)^2+1/((x+y)^2+1)-x*(2*x+2*y)/((x+y)^2+1)^2+1/((x+z)^2+1)-x*(2*x+2*z)/((x+z)^2+1)^2+1/(x^2+1)-2*x^2/(x^2+1)^2}*{-(3/2)*(1/(x^2+2*x*y+2*x*z+y^2+2*y*z+z^2+1))^(1/2)*(2*x+2*y+2*z)/((x+y+z)^2+1)^2-(3/2)*(1/(x^2+2*x*y+y^2+1))^(1/2)*(2*x+2*y)/((x+y)^2+1)^2-(3/2)*(1/(y^2+2*y*z+z^2+1))^(1/2)*(2*y+2*z)/((y+z)^2+1)^2-3*(1/(y^2+1))^(1/2)*y/(y^2+1)^2}-{1/((x+y+z)^2+1)-y*(2*x+2*y+2*z)/((x+y+z)^2+1)^2+1/((x+y)^2+1)-y*(2*x+2*y)/((x+y)^2+1)^2+1/((y+z)^2+1)-y*(2*y+2*z)/((y+z)^2+1)^2+1/(y^2+1)-2*y^2/(y^2+1)^2}*{-(3/2)*(1/(x^2+2*x*y+2*x*z+y^2+2*y*z+z^2+1))^(1/2)*(2*x+2*y+2*z)/((x+y+z)^2+1)^2-(3/2)*(1/(x^2+2*x*y+y^2+1))^(1/2)*(2*x+2*y)/((x+y)^2+1)^2-(3/2)*(1/(x^2+2*x*z+z^2+1))^(1/2)*(2*x+2*z)/((x+z)^2+1)^2-3*(1/(x^2+1))^(1/2)*x/(x^2+1)^2}}]

(3)

w = -(1/((x+y+z)^2+1)-y*(2*x+2*y+2*z)/((x+y+z)^2+1)^2+1/((x+y)^2+1)-y*(2*x+2*y)/((x+y)^2+1)^2+1/((y+z)^2+1)-y*(2*y+2*z)/((y+z)^2+1)^2+1/(y^2+1)-2*y^2/(y^2+1)^2)/(-(3/2)*sqrt(1/(x^2+2*x*y+2*x*z+y^2+2*y*z+z^2+1))*(2*x+2*y+2*z)/((x+y+z)^2+1)^2-(3/2)*sqrt(1/(x^2+2*x*y+y^2+1))*(2*x+2*y)/((x+y)^2+1)^2-(3/2)*sqrt(1/(y^2+2*y*z+z^2+1))*(2*y+2*z)/((y+z)^2+1)^2-3*sqrt(1/(y^2+1))*y/(y^2+1)^2);

w = -(1/((x+y+z)^2+1)-y*(2*x+2*y+2*z)/((x+y+z)^2+1)^2+1/((x+y)^2+1)-y*(2*x+2*y)/((x+y)^2+1)^2+1/((y+z)^2+1)-y*(2*y+2*z)/((y+z)^2+1)^2+1/(y^2+1)-2*y^2/(y^2+1)^2)/(-(3/2)*(1/(x^2+2*x*y+2*x*z+y^2+2*y*z+z^2+1))^(1/2)*(2*x+2*y+2*z)/((x+y+z)^2+1)^2-(3/2)*(1/(x^2+2*x*y+y^2+1))^(1/2)*(2*x+2*y)/((x+y)^2+1)^2-(3/2)*(1/(y^2+2*y*z+z^2+1))^(1/2)*(2*y+2*z)/((y+z)^2+1)^2-3*(1/(y^2+1))^(1/2)*y/(y^2+1)^2)

(4)

sub(w = -(1/((x+y+z)^2+1)-y*(2*x+2*y+2*z)/((x+y+z)^2+1)^2+1/((x+y)^2+1)-y*(2*x+2*y)/((x+y)^2+1)^2+1/((y+z)^2+1)-y*(2*y+2*z)/((y+z)^2+1)^2+1/(y^2+1)-2*y^2/(y^2+1)^2)/(-(3/2)*(1/(x^2+2*x*y+2*x*z+y^2+2*y*z+z^2+1))^(1/2)*(2*x+2*y+2*z)/((x+y+z)^2+1)^2-(3/2)*(1/(x^2+2*x*y+y^2+1))^(1/2)*(2*x+2*y)/((x+y)^2+1)^2-(3/2)*(1/(y^2+2*y*z+z^2+1))^(1/2)*(2*y+2*z)/((y+z)^2+1)^2-3*(1/(y^2+1))^(1/2)*y/(y^2+1)^2), c[1]);

sub(w = -(1/((x+y+z)^2+1)-y*(2*x+2*y+2*z)/((x+y+z)^2+1)^2+1/((x+y)^2+1)-y*(2*x+2*y)/((x+y)^2+1)^2+1/((y+z)^2+1)-y*(2*y+2*z)/((y+z)^2+1)^2+1/(y^2+1)-2*y^2/(y^2+1)^2)/(-(3/2)*(1/(x^2+2*x*y+2*x*z+y^2+2*y*z+z^2+1))^(1/2)*(2*x+2*y+2*z)/((x+y+z)^2+1)^2-(3/2)*(1/(x^2+2*x*y+y^2+1))^(1/2)*(2*x+2*y)/((x+y)^2+1)^2-(3/2)*(1/(y^2+2*y*z+z^2+1))^(1/2)*(2*y+2*z)/((y+z)^2+1)^2-3*(1/(y^2+1))^(1/2)*y/(y^2+1)^2), (1/8)*w*{(1/((x+y+z)^2+1))^(3/2)+(1/((x+y)^2+1))^(3/2)+(1/((x+z)^2+1))^(3/2)+(1/((y+z)^2+1))^(3/2)+(1/(x^2+1))^(3/2)+(1/(y^2+1))^(3/2)+(1/(z^2+1))^(3/2)+1}+(1/8)*{x/((x+y+z)^2+1)+x/((x+y)^2+1)+x/((x+z)^2+1)+x/(x^2+1)})

(5)

subs(w = -(1/((x+y+z)^2+1)-y*(2*x+2*y+2*z)/((x+y+z)^2+1)^2+1/((x+y)^2+1)-y*(2*x+2*y)/((x+y)^2+1)^2+1/((y+z)^2+1)-y*(2*y+2*z)/((y+z)^2+1)^2+1/(y^2+1)-2*y^2/(y^2+1)^2)/(-(3/2)*(1/(x^2+2*x*y+2*x*z+y^2+2*y*z+z^2+1))^(1/2)*(2*x+2*y+2*z)/((x+y+z)^2+1)^2-(3/2)*(1/(x^2+2*x*y+y^2+1))^(1/2)*(2*x+2*y)/((x+y)^2+1)^2-(3/2)*(1/(y^2+2*y*z+z^2+1))^(1/2)*(2*y+2*z)/((y+z)^2+1)^2-3*(1/(y^2+1))^(1/2)*y/(y^2+1)^2), c[2]);

-(1/8)*(1/((x+y+z)^2+1)-y*(2*x+2*y+2*z)/((x+y+z)^2+1)^2+1/((x+y)^2+1)-y*(2*x+2*y)/((x+y)^2+1)^2+1/((y+z)^2+1)-y*(2*y+2*z)/((y+z)^2+1)^2+1/(y^2+1)-2*y^2/(y^2+1)^2)*{(1/((x+y+z)^2+1))^(3/2)+(1/((x+y)^2+1))^(3/2)+(1/((x+z)^2+1))^(3/2)+(1/((y+z)^2+1))^(3/2)+(1/(x^2+1))^(3/2)+(1/(y^2+1))^(3/2)+[1/(z^2+1)]^(3/2)+1}/(-(3/2)*(1/(x^2+2*x*y+2*x*z+y^2+2*y*z+z^2+1))^(1/2)*(2*x+2*y+2*z)/((x+y+z)^2+1)^2-(3/2)*(1/(x^2+2*x*y+y^2+1))^(1/2)*(2*x+2*y)/((x+y)^2+1)^2-(3/2)*(1/(y^2+2*y*z+z^2+1))^(1/2)*(2*y+2*z)/((y+z)^2+1)^2-3*(1/(y^2+1))^(1/2)*y/(y^2+1)^2)+(1/8)*{y/((x+y+z)^2+1)+y/((x+y)^2+1)+y/((y+z)^2+1)+y/(y^2+1)}

(6)

"#"Iwant to maximize the equation (5)and (6),under the conditon of x,y,z are negative or positive at the same time.

 

NULL

 

Download maximize.mw

I want to find the greatest value of this expression 

f:=(x,y,z)->sqrt((x+1)*(y^2+2)*(z^3+3))+sqrt((y+1)*(z^2+2)*(x^3+3))+sqrt((z+1)*(x^2+2)*(y^3+3));

with x>0, y>0 , z>0,x+y+z=3.

I tried

restart:

 f:=(x,y,z)->sqrt((x+1)*(y^2+2)*(z^3+3))+sqrt((y+1)*(z^2+2)*(x^3+3))+sqrt((z+1)*(x^2+2)*(y^3+3));

DirectSearch[GlobalOptima](f(x,y,z), {x>0, y>0 , z>0,x+y+z=3},maximize);

I got the output

[HFloat(infinity), [x = .591166078050740e52, y = .183647204560715e52, z = .786638021216969e52], 1249]

 

 

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