how to solve a multi-objective nonlinear optimization problem with constraints ?

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tty 2022-12-18

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回答： Alan Weiss 2022-12-22

采纳的回答： Alan Weiss

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Hello , I have a nonlinear eqt that describes the evolution of my system : dxdt=Ax(t)+Bu(t),with

A=[sqrt(2) 1;2 sqrt(3)]; B=[1 4;3 1]; and u=K1*x+b1. K1 is 2x2 matrix and b1 is 2x1.

I want to find K1 and b1 such that I maximize xdot11 at a point v1 under these constraints :

n1'*B*(K1*v1+b1)+n1'*A*v1<=0;

n4'*B*(K1*v1+b1)+n4'*A*v1<=0;

n1'*xdot11<= 0 ;

K1*v1+b1 <= 2;

K1*v1+b1 >=- 2;

n4'*xdot12<= 0 ;

n1=[0;-1]; n4=[-1;0]

My code is :

clc;

close all;

v1=[80;60];

n1=[0;-1]; n4=[-1;0]; % normal vectors

A=[sqrt(2) 1;2 sqrt(3)];

B=[1 4;3 1];

prob1 = optimproblem;

K1 = optimvar("K1",2,2); % create the optim. variable K1

b1 = optimvar("b1",2,1);

xdot11=((A(1,1)+B(1,1)*K1(1,1)+B(1,2)*K1(2,1))*v1(1))+((A(1,2)+B(1,1)*K1(1,2)+B(1,2)*K1(2,2))*v1(2))+B(1, ...

1)*b1(1)+B(1,2)*b1(2);

xdot21=((A(2,1)+B(2,1)*K1(1,1)+B(2,2)*K1(2,1))*v1(1))+((A(2,2)+B(2,1)*K1(1,2)+B(2,2)*K1(2,2))*v1(2))+B(2, ...

1)*b1(1)+B(2,2)*b1(2);

Obj =-xdot11;

prob1.Objective=Obj

prob1 =

OptimizationProblem with properties: Description: '' ObjectiveSense: 'minimize' Variables: [1×1 struct] containing 2 OptimizationVariables Objective: [1×1 OptimizationExpression] Constraints: [0×0 struct] containing 0 OptimizationConstraints See problem formulation with show.

constr1=n1'*B*(K1*v1+b1)+n1'*A*v1<=0;

constr2=n4'*B*(K1*v1+b1)+n4'*A*v1<=0;

constr3=n1'*xdot11<= 0 ;

constr4=K1*v1+b1 <= 2;

constr5=K1*v1+b1 >=- 2;

constr6=n4'*xdot21<= 0 ;

prob1.Constraints.Constr1=constr1;

prob1.Constraints.Constr2=constr2;

prob1.Constraints.Constr3=constr3;

prob1.Constraints.Constr4=constr4;

prob1.Constraints.Constr5=constr5;

prob1.Constraints.Constr6=constr6;

show(prob1);

OptimizationProblem : Solve for: K1, b1 minimize : -80*K1(1, 1) - 320*K1(2, 1) - 60*K1(1, 2) - 240*K1(2, 2) - b1(1) - 4*b1(2) - 173.1371 subject to Constr1: -240*K1(1, 1) - 80*K1(2, 1) - 180*K1(1, 2) - 60*K1(2, 2) - 3*b1(1) - b1(2) <= 263.923 subject to Constr2: -80*K1(1, 1) - 320*K1(2, 1) - 60*K1(1, 2) - 240*K1(2, 2) - b1(1) - 4*b1(2) <= 173.1371 subject to Constr3: -80*K1(1, 1) - 320*K1(2, 1) - 60*K1(1, 2) - 240*K1(2, 2) - b1(1) - 4*b1(2) <= 173.1371 subject to Constr4: 80*K1(1, 1) + 60*K1(1, 2) + b1(1) <= 2 80*K1(2, 1) + 60*K1(2, 2) + b1(2) <= 2 subject to Constr5: 80*K1(1, 1) + 60*K1(1, 2) + b1(1) >= -2 80*K1(2, 1) + 60*K1(2, 2) + b1(2) >= -2 subject to Constr6: -240*K1(1, 1) - 80*K1(2, 1) - 180*K1(1, 2) - 60*K1(2, 2) - 3*b1(1) - b1(2) <= 263.923

x0.K1=zeros(2);

x0.b1=[1;1];

sol =fsolve(prob1,x0);

Error using fsolve
FSOLVE requires the following inputs to be of data type double: 'X0'.

when I run my code , I get this error :

Unable to perform assignment because dot indexing is not supported for variables of this type.

I tried to solve it with fmincon but it requires to put all of the control variables into one vector x and I didn't know how to define the x0 in this case..

I will be very thankfull for your help.

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tty 2022-12-18

编辑：tty 2022-12-18

在 MATLAB Online 中打开

Thank you for your reply .I tried to solve it with fmincon but i don't know how shoud I define the intial value x0. Sorry if it is a dummy qst but this is my first time to solve a prob with fmincon.

function fonc(u1)
K1=[u1(1),u1(2);u1(3),u1(4)];
b1=[u1(5);u1(6)];
v1=[80;60];
A=[sqrt(2) 1;2 sqrt(3)];
B=[12.5 4;5 10];
xdot11=-(((A(1,1)+B(1,1)*u1(1)+B(1,2)*u1(3))*v1(1))+((A(1,2)+B(1,1)*u1(2)+B(1,2)*u1(4))*v1(2))+B(1, ...
  1)*u1(5)+B(1,2)*u1(6));
end
function[c1,c2,c3,c4,c5,c6,ceq]=carreconst(u1)
v1=[80;60]; 
A=[sqrt(2) 1;2 sqrt(3)];
B=[1 4;3 1];
K1=[u1(1),u1(2);u1(3),u1(4)];
b1=[u1(5);u1(6)]';
xdot11=-(((A(1,1)+B(1,1)*u1(1)+B(1,2)*u1(3))*v1(1))+((A(1,2)+B(1,1)*u1(2)+B(1,2)*u1(4))*v1(2))+B(1, ...
  1)*u1(5)+B(1,2)*u1(6));
xdot21=((A(2,1)+B(2,1)*u1(1)+B(2,2)*u1(3))*v1(1))+((A(2,2)+B(2,1)*u1(2)+B(2,2)*u1(4))*v1(2))+B(2, ...
    1)*u1(5)+B(2,2)*u1(6);
n1=[0;-1]; n4=[-1;0];
c1=n1'*B*(K1*v1+b1)+n1'*A*v1;
c2=n4'*B*(K1*v1+b1)+n4'*A*v1;
c3=n1'*xdot11;
c4=n4'*xdot21 ;
c5=K1*v1+b1-5;
c6=-K1*v1-b1+5;
ceq=[];
end
clc;
close all;
v1=[80;60];
n1=[0;-1]; n4=[-1;0]; % normal vectors
A=[sqrt(2) 1;2 sqrt(3)];
B=[1 4;3 1];
nonlcon=@carreconst;
fun=@(u1)(-(((A(1,1)+B(1,1)*u1(1)+B(1,2)*u1(3))*v1(1))+((A(1,2)+B(1,1)*u1(2)+B(1,2)*u1(4))*v1(2))+B(1, ...
  1)*u1(5)+B(1,2)*u1(6)));
%x0=...? 
sol=fmincon(fun,x0,[],[],[],[],[],[],nonlcon)

Torsten 2022-12-18

编辑：Torsten 2022-12-18

在 MATLAB Online 中打开

The expressions

n1'*B*(K1*v1+b1)+n1'*A*v1
n4'*B*(K1*v1+b1)+n4'*A*v1
n1'*xdot11
n4'*xdot21 
K1*v1+b1-5
-K1*v1-b1+5

you want to constrain are 1x2 vectors and 2x2 matrices. Is this really what you want ?

I still don't see that you solve your differential equation somewhere.

clc;
close all;
v1=[80;60];
n1=[0;-1]; n4=[-1;0]; % normal vectors
A=[sqrt(2) 1;2 sqrt(3)];
B=[1 4;3 1];
nonlcon=@carreconst;
fun=@(u1)(-(((A(1,1)+B(1,1)*u1(1)+B(1,2)*u1(3))*v1(1))+((A(1,2)+B(1,1)*u1(2)+B(1,2)*u1(4))*v1(2))+B(1, ...
  1)*u1(5)+B(1,2)*u1(6)));
x0=ones(6,1); 
sol=fmincon(fun,x0,[],[],[],[],[],[],nonlcon)
function fonc(u1)
K1=[u1(1),u1(2);u1(3),u1(4)];
b1=[u1(5);u1(6)];
v1=[80;60];
A=[sqrt(2) 1;2 sqrt(3)];
B=[12.5 4;5 10];
xdot11=-(((A(1,1)+B(1,1)*u1(1)+B(1,2)*u1(3))*v1(1))+((A(1,2)+B(1,1)*u1(2)+B(1,2)*u1(4))*v1(2))+B(1, ...
  1)*u1(5)+B(1,2)*u1(6));
end
function[c,ceq]=carreconst(u1)
v1=[80;60]; 
A=[sqrt(2) 1;2 sqrt(3)];
B=[1 4;3 1];
K1=[u1(1),u1(2);u1(3),u1(4)];
b1=[u1(5);u1(6)]';
xdot11=-(((A(1,1)+B(1,1)*u1(1)+B(1,2)*u1(3))*v1(1))+((A(1,2)+B(1,1)*u1(2)+B(1,2)*u1(4))*v1(2))+B(1, ...
  1)*u1(5)+B(1,2)*u1(6));
xdot21=((A(2,1)+B(2,1)*u1(1)+B(2,2)*u1(3))*v1(1))+((A(2,2)+B(2,1)*u1(2)+B(2,2)*u1(4))*v1(2))+B(2, ...
    1)*u1(5)+B(2,2)*u1(6);
n1=[0;-1]; n4=[-1;0];
c(1)=n1'*B*(K1*v1+b1)+n1'*A*v1;
c(2)=n4'*B*(K1*v1+b1)+n4'*A*v1;
c(3)=n1'*xdot11;
c(4)=n4'*xdot21 ;
c(5)=K1*v1+b1-5;
c(6)=-K1*v1-b1+5;
ceq=[];
end

Torsten 2022-12-18

I must admit that I still don't understand

I want to find K1 and b1 such that I maximize xdot11 at a point v1 under these constraints :

What is v1 ? How does it correspond to something you get from or input into your differential equations ?

tty 2022-12-19

编辑：tty 2022-12-20

my system should evolves inside a rectangle region in my state space and leave the rectangle only from the right edge (line between v2 and v3). this rectangle is specified with four vertices v1,v2,v3 and v4.

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Answer 1

Alan Weiss 2022-12-22

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链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1880642-how-to-solve-a-multi-objective-nonlinear-optimization-problem-with-constraints#answer_1134477

For an example that optimizes the solution of an ODE using optimization variables, see Fit ODE Parameters using Optimization Variables. That example shows calling ode45 to solve the ODE, and has an optimization problem built around this ODE solver.

For an entirely different approach, see Discretized Optimal Trajectory, Problem-Based. I don't know if this approach is relevant to your problem.

Good luck,

Alan Weiss

MATLAB mathematical toolbox documentation

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