Minimization and Optimization. Minimize output by optimizing inputs

Hello

Trying to minimize the output by finding the optimium input values. I am unable to construct the minimization part of the code with the constrains I have. Any help would be greatly appreciated.

% Objective:Optimize the lengths to minimze the power using a new variable.

a = 0.1:20;

b = 0.1:20;

d = a + 0.1:20;

c = b + 0.1:20;

e = d + 0.1:20

%Constants

k= 2;

w=1;

v=1.5

%Variables and their constrains

AB = sqrt(a.^2 + b.^2);

BC = sqrt( c.^2 + ((e-d)/2).^2 );

CS = sqrt( c.^2 + ((e-d)/2).^2 );

VAB = sqrt(((((a.*v).^2/(((b.^2).*4))) + (v^2)/2 )));

% VBS = sqrt(((a*v)^2/((4*b*b)) + (v^2)/2 ));

VCS = ((2*c)./(e-d)).*sqrt(AB.^2);

VBC= CS.^2 + BC.^2;

%Actual Power

%P = (AB.*VAB + BC.*VBC + CS.*VCS).*k*w; % Power

%

% Objective

%To search and find the values for a, b, c d and e to minimize power (P)

% Not sure how to write the function for above

% fun = @(x)(x(:,1) + x(:,2) + x(:,3) + x(:,4) + x(:,5)).*k.*w;

%

% [X1, X2, X3, X4, X5] = ndgrid(0:.1:2); % Should I give this condition in a nested loop?

%

% X = [X1(:), X2(:), X3(:), X4(:), X5(:)];

%

% P = fun(X);

% [bestP, idx] = min(P(:))

% best_X = X(idx,:)

Answer 1

[Alan Weiss]

Sounds like you would be best served with the Problem-Based Optimization Workflow. Declare the variables that can change as optimization variables, set the problem objective to P, and call solve.

Alan Weiss

MATLAB mathematical toolbox documentation

[Anand Ra]

@Alan Weiss

I attempted what you suggested. Unsure where I am going wrong.

%defining optimization variables and an optimization problem object.
a1 = optimvar('a');
b1 = optimvar('b');
c1 = optimvar('c');
d1 = optimvar('d');
e1 = optimvar('e');

prob = optimproblem;
k= 2;
w=1;
v=1.5;

%Variables

AB = sqrt(a.^2 + b.^2);

Unrecognized function or variable 'a'.

BC = sqrt( c.^2 + ((e-d)/2).^2 );
CS = sqrt( c.^2 + ((e-d)/2).^2 );


VAB = sqrt(((((a.*v).^2/(((b.^2).*4))) + (v^2)/2 )));
% VBS = sqrt(((a*v)^2/((4*b*b)) + (v^2)/2 ));
VCS = ((2*c)./(e-d)).*sqrt(AB.^2);
VBC= CS.^2 + BC.^2;

%objective function as an expression in the optimization variables.
P =  (AB.*VAB + BC.*VBC + CS.*VCS).*k*w;

%the objective function in prob.
prob.Objective = P;


%constraints
cons 1 = a = 0.1:20;
cons 2 =b = 0.1:20;
cons 3 = d = a + 0.1:20;
cons 4 = c = b + 0.1:20;
cons e = d + 0.1:20;
prob.Constraints.cons1 = cons1;
prob.Constraints.cons2 = cons2;
prob.Constraints.cons3 = cons3;
prob.Constraints.cons4 = cons4;
prob.Constraints.cons5 = cons5;

[Alan Weiss]

I am not sure what you are trying to do, but you need to declare variables as documented.

a1 = optimvar('a'); % Don't do this
% Instead, use
a = optimvar('a');

Furthermore, I don't think that your constraints are really constraints, but instead should be bounds. For example,

cons 1 = a = 0.1:20; % Several mistakes here.
% I think what you mean is
a = optimvar('a','LowerBound',0.1,"UpperBound",20);
% Similarly for the other variables: declare with correct names and with
% bounds

Try it this way and see if you like the results.

Alan Weiss

MATLAB mathematical toolbox documentation

[Anand Ra]

@Alan Weiss

I updated the code accordingly. Not sure what I am missing. Unable to generate the output.

%defining optimization variables and an optimization problem object.

a = optimvar('a','LowerBound',0.1,"UpperBound",20);

b = optimvar('b','LowerBound',0.1,"UpperBound",20);

c = optimvar('c','LowerBound',0.1,"UpperBound",20);

d = optimvar('d','LowerBound',0.1,"UpperBound",20);

e = optimvar('e','LowerBound',0.1,"UpperBound",20);

prob = optimproblem;

k= 2;

w=1;

v=1.5;

%the objective function in prob.

prob.Objective = P;

%constraints

% cons1 = e >= (a+d);

% cons2 = d >=a ;

% cons3 = b <=c ;

% cons4 = (e-d) >= (d-a) ;

% cons5 = (c-b) <= b;

cons1 = e - a- d >= 0.1;

cons2 = d - a >= 0.1 ;

cons3 = c-b >= 0.1;

cons4 = b - a >= 0.1 ;

cons5 = (e-c) >=0.1;

prob.Constraints.cons1 = cons1;

prob.Constraints.cons2 = cons2;

prob.Constraints.cons3 = cons3;

prob.Constraints.cons4 = cons4;

prob.Constraints.cons5 = cons5;

x0.a = 4;

x0.b = 6;

x0.c = 8;

x0.d = 7;

x0.e = 12;

%new variables

AB = sqrt(a.^2 + b.^2);

BC = sqrt( c.^2 + ((e-d)/2).^2 );

CS = sqrt( c.^2 + ((e-d)/2).^2 );

VAB = sqrt(((((a.*v).^2/(((b.^2).*4))) + (v^2)/2 )));

% VBS = sqrt(((a*v)^2/((4*b*b)) + (v^2)/2 ));

VCS = ((2*c)./(e-d)).*sqrt(AB.^2);

VBC= CS.^2 + BC.^2;

%objective function as an expression in the optimization variables.

P = (AB.*VAB + BC.*VBC + CS.*VCS).*k*w;

[Alan Weiss]

Your code defines the objective before defining P. Put the line defining the problem objective after P is completely defined. Then call solve.

Alan Weiss

MATLAB mathematical toolbox documentation

[Anand Ra]

@Alan Weiss, Thank you Sir!

I think I generated the output. However the optimized results vary with the initial guess. Is there a way to fix it?

%defining optimization variables and an optimization problem object.

a = optimvar('a','LowerBound',0.1,"UpperBound",20);
b = optimvar('b','LowerBound',0.1,"UpperBound",20);
c = optimvar('c','LowerBound',0.1,"UpperBound",20);
d = optimvar('d','LowerBound',0.1,"UpperBound",20);
e = optimvar('e','LowerBound',0.1,"UpperBound",20);

prob = optimproblem;
k= 2;
w=1;
v=1.5;

%new variables

AB = sqrt(a.^2 + b.^2);
BC = sqrt( c.^2 + ((e-d)/2).^2 );
CS = sqrt( c.^2 + ((e-d)/2).^2 );


VAB = sqrt(((((a.*v).^2/(((b.^2).*4))) + (v^2)/2 )));
% VBS = sqrt(((a*v)^2/((4*b*b)) + (v^2)/2 ));
VCS = ((2*c)./(e-d)).*sqrt(AB.^2);
VBC= CS.^2 + BC.^2;

%objective function as an expression in the optimization variables.
P =  (AB.*VAB + BC.*VBC + CS.*VCS).*k*w;

%the objective function in prob.
prob.Objective = P;



%constraints



% cons1 = e >= (a+d);
% cons2 = d >=a ;
% cons3 = b <=c ;
% cons4 = (e-d) >= (d-a) ;
% cons5 = (c-b) <= b;

cons1 = e - a- d >= 0.1;
cons2 = d - a >= 0.1 ;
cons3 = c-b >= 0.1;
cons4 = b - a >= 0.1 ;
cons5 = (e-c) >=0.1;

prob.Constraints.cons1 = cons1;
prob.Constraints.cons2 = cons2;
prob.Constraints.cons3 = cons3;
prob.Constraints.cons4 = cons4;
prob.Constraints.cons5 = cons5;



x0.a = 4;
x0.b = 6;
x0.c = 8;
x0.d = 7;
x0.e = 12;


sol = solve(prob,x0)

[Alan Weiss]

If you look at the objective function value

[sol,fval] = solve(prob,x0)

you find that the different solutions all have the same objective function value. So there are many solutions. Is this unexpected? This is not something that I can tell you how to fix. it is for you to determine whether the problem is formulated correctly.

Alan Weiss

MATLAB mathematical toolbox documentation

[Anand Ra]

Got it! Thank you @Alan Weiss