solve non-linear optimization problem

how to maximize this nonlinear function z and find the value of x and y,

z= 0.031 {0.009[exp⁡(x-z)-1]+x/1.14}-(-1+√(1+y^4/x))/y

10<=x<=20 and y>=0

Answer 1

[Matt J]

With fmincon, if you have Optimization Toolbox.

https://www.mathworks.com/help/optim/ug/fmincon.html

[Hayam Wahdan]

The main problem of this optimization problem, the objective function z is calculated in terms of the same value of z

[Matt J]

You "objective" is really an equality constraint:

[Hayam Wahdan]

After this assumption, the problem contains three decision variables x,y and z?

[Matt J]

Yes, that's right.

[Hayam Wahdan]

How fminco function solve this optimization problem

Min  -Z
St
Z-....=0
d>=100
d<=300
v>=0

[Matt J]

Did you already get it to work on your original problem, and if so how is this new version different?

[Hayam Wahdan]

I want to ask, can this function solve this optimization problem subject to non-linear constraints

[Walter Roberson]

Yes, see the nonlcon parameter of fmincon.

[Matt J]

I want to ask, can this function solve this optimization problem subject to non-linear constraints

Yes, see this example in the documentation:

https://www.mathworks.com/help/optim/ug/fmincon.html#busow0u-1

Note that you don't really have to ensure that the initial guess satisfy the constraints, as they do in the example, but it can be beneficial to do so.

[Hayam Wahdan]

Can you help me in writing this MATLAB function Min -Z St z= 0.031 {0.009[exp(x-z)-1]+x/1.14}-(-1+√(1+y^4/x))/y 10<=x<=20 and y>0

[Walter Roberson]

fmincon thinks you can get an arbitrarily large result if you make y arbitrarily large.

format long g
obj = @(xyz) -xyz(3);
A = []; b = []; Aeq = []; beq = [];
lb = [10, realmin, -inf];
ub = [20 1e9 1e9];
nl = @(x,y,z) 0.031*(0.009*(exp(x-z)-1)+x./1.14)-(-1+sqrt(1+y^4./x))./y - z

nl = function_handle with value:
    @(x,y,z)0.031*(0.009*(exp(x-z)-1)+x./1.14)-(-1+sqrt(1+y^4./x))./y-z

nonlcon = @(xyz) deal(nl(xyz(1), xyz(2), xyz(3)), []);
xyz0 = [.1 .2 .3]

xyz0 = 1×3

0.1 0.2 0.3

[best, fval] = fmincon(obj, xyz0, A, b, Aeq, beq, lb, ub, nonlcon)

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in 
feasible directions, to within the value of the optimality tolerance,
and constraints are satisfied to within the value of the constraint tolerance.

best = 1×3

17.9264664551072 190469849.537475 999999999.999999

fval = 
         -999999999.999999

[Walter Roberson]

Please re-check your equations. I suspect that the sqrt() is intended to extend over the /y as well, which is not what you have coded.

[Hayam Wahdan]

Thank you very much for your quick reply,

when i tried this code, gave me error in this line

[best, fval] = fmincon(obj, v0, A, b, Aeq, beq, lb, ub, nonlcon)

[Walter Roberson]

What was the error message?

[Hayam Wahdan]

Error using barrier

Nonlinear constraint function is undefined at initial point. Fmincon cannot continue.

Error in fmincon (line 798)

[X,FVAL,EXITFLAG,OUTPUT,LAMBDA,GRAD,HESSIAN] = barrier(funfcn,X,A,B,Aeq,Beq,l,u,confcn,options.HessFcn, ...

Error in nonlinear (line 11)

[best, fval] = fmincon(obj, vdJ0, A, b, Aeq, beq, lb, ub, nonlcon)

[Walter Roberson]

please post the complete code you used.

[Hayam Wahdan]

format long g
obj = @(xyz) -xyz(3);
A = []; b = []; Aeq = []; beq = [];
lb = [100, realmin, -inf];
ub = [300 1e9 1e9];
%nl = @(x,y,z) 0.031*(0.009*(exp(x-z)-1)+x./1.14)-(-1+sqrt(1+y^4./x))./y - z
w=10^-15*exp(-2.897*10^22*25^-1);
nl = @(x,y,z) 0.055*(w*(exp(5802*(x-19.5*z))-1)+x./1.14)-(-1+sqrt(1+0.5272*y^4./x^2.))/(2.6328 *10^9*y) - z;
nonlcon = @(xyz) deal(nl(xyz(1), xyz(2), xyz(3)), []);
xyz0 = [.1 .2 .3];
[best, fval] = fmincon(obj, xyz0, A, b, Aeq, beq, lb, ub, nonlcon)

[Walter Roberson]

w=10^-15*exp(-2.897*10^22*25^-1);

You badly need to rescale !!

exp(-1158800000000000000000) is about

Effectively your w is 0.

[Hayam Wahdan]

Thank you very much, I will rescale it