Sequential Quadratic Programming Matlab Implementation - matlab

Question

I have a problem with my MATLAB code that solves a nonlinear quadratic problem with SQP algorithm (Sequential quadratic programming) but in the "QP-SUB PROBLEM" section of the code that i have formulated analytically a "num2str"error appears and honestly, i don't know how to fix that and also have to tell you that this method uses KT conditions for a better solution .
In every section of the code i write a comment for better understanding and function with constraints can be found in the code below :
% Maximize f(x1,x2) = x1^4 -2x1^2x2 +x1^2 +x1x2^2 -2x1 +4
%
% h1(x1,x2) = x1^2 + x2^2 -2 = 0
% g1(x1,x2) = 0.25x1^2 +0.75x2^2 -1 <=0
% 0 <= x1 <= 4; 0 <= x2 <= 4
%
%--------------------------------------------------------
% The KT conditions for the QP subproblem
% are applied analytically
% There are two cases for a single inequality constraint
% Case (a) : beta = 0 g < 0
% Case (b) : beta ~= 0, g = 0
% The best solution is used
% --------------------------------------------------------
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% management functions
clear % clear all variable/information in the workspace - use CAUTION
clear global % again use caution - clears global information
clc % position the cursor at the top of the screen
close % closes the figure window
format compact % avoid skipping a line when writing to the command window
warning off %#ok<WNOFF> % don't report any warnings like divide by zero etc.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% DATA ---------------------
%%% starting design
xb(1) = 3; xb(2) = 2;
it = 10; % number of iterations
%%% plot range for delx1 : -3 , +3
dx1L = -3; dx1U = +3;
dx2L = -3; dx2U = +3;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Define functions
syms x1 x2 f g h
syms gradf1 gradf2 gradh1 gradh2 gradg1 gradg2
f = x1^4 - 2*x1*x1*x2 + x1*x1 + x1*x2*x2 - 2*x1 + 4;
h = x1*x1 + x2*x2 - 2;
g = 0.25*x1*x1 +0.75*x2*x2 -1;
%%% the gradient functions
gradf1 = diff(f,x1);
gradf2 = diff(f,x2);
% the hessian
hess = [diff(gradf1,x1), diff(gradf1,x2); diff(gradf2,x1), diff(gradf2,x2)];
% gradient of the constraints
gradh1 = diff(h,x1);
gradh2 = diff(h,x2);
gradg1 = diff(g,x1);
gradg2 = diff(g,x2);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% graphical/symbolic solution for SLP
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
fprintf('***********************')
fprintf('\nSQP - Example 7.1')
fprintf('\n*********************\n')
for i = 1:it
%figure;
f1 = subs(f,{x1,x2},{xb(1),xb(2)});
g1 = subs(g,{x1,x2},{xb(1),xb(2)});
h1 = subs(h,{x1,x2},{xb(1),xb(2)});
%%% Print Information
fprintf('\n******************************')
fprintf('\nIteration : '),disp(i)
fprintf('******************************\n')
fprintf('Linearized about [x1, x2] : '),disp([xb(1) xb(2)])
fprintf('Objective function value f(x1,x2) : '),disp(f1);
fprintf('Equality constraint value value h(x1,x2) : '),disp(h1);
fprintf('Inequality constraint value value g(x1,x2) : '),disp(g1);
%fprintf('\nsolution for [delx1 delx2] : '),disp(sol')
% hold on
% calculate the value of the gradients
% f1 = subs(f,{x1,x2},{xb(1),xb(2)});
% g1 = subs(g,{x1,x2},{xb(1),xb(2)});
% h1 = subs(h,{x1,x2},{xb(1),xb(2)});
fprintf('\n-----------------------')
fprintf('\nQP - SUB PROBLEM')
fprintf('\n---------------------\n')
gf1 = double(subs(gradf1,{x1,x2},{xb(1),xb(2)}));
gf2 = double(subs(gradf2,{x1,x2},{xb(1),xb(2)}));
hess1 = double(subs(hess,{x1,x2},{xb(1),xb(2)}));
gh1 = double(subs(gradh1,{x1,x2},{xb(1),xb(2)}));
gh2 = double(subs(gradh2,{x1,x2},{xb(1),xb(2)}));
gg1 = double(subs(gradg1,{x1,x2},{xb(1),xb(2)}));
gg2 = double(subs(gradg2,{x1,x2},{xb(1),xb(2)}));
% the QP subproblem
syms dx1 dx2 % change in design
fquad = f1 + [gf1 gf2]*[dx1; dx2] + 0.5*[dx1 dx2]*hess1*[dx1 ;dx2];
hlin = h1 + [gh1 gh2]*[dx1; dx2];
glin = g1 + [gg1 gg2]*[dx1; dx2];
Fquadstr = strcat(num2str(f1),' + ',num2str(gf1), ...
'*','dx1',' + ',num2str(gf2),' * ','dx2', ...
' + 0.5*',num2str(hess1(1,1)),' * dx1^2', ...
' +',num2str(hess1(1,2)),' * dx1*dx2', ...
' + 0.5*',num2str(hess1(2,2)),' * dx2^2');
hlinstr = strcat(num2str(h1),' + ',num2str(gh1), ...
'*','dx1',' + ',num2str(gh2),' * ','dx2');
glinstr = strcat(num2str(g1),' + ',num2str(gg1), ...
'*','dx1',' + ',num2str(gg2),' * ','dx2');
fprintf('Quadratic Objective function f(x1,x2): \n'),disp(Fquadstr);
fprintf('Linearized equality h(x1,x2): '),disp(hlinstr);
fprintf('Linearized inequality g(x1,x2): '),disp(glinstr);
fprintf('\n')
% define Lagrangian for the QP problem
syms lamda beta
F = fquad + lamda*hlin + beta*glin;
fprintf('Case a: beta = 0\n');
Fnobeta = fquad + lamda*hlin;
%%% initialize best solution
dx1best = 0;
dx2best = 0;
Fbbest = 0;
%%%%%%%%%%%%%%%%%%%%%%%
%%% solve case (a) %%%
%%%%%%%%%%%%%%%%%%%%%%%
xcasea = solve(diff(Fnobeta,dx1),diff(Fnobeta,dx2),hlin);
sola = [double(xcasea.dx1) double(xcasea.dx2) double(xcasea.lamda)];
dx1a = double(xcasea.dx1);
dx2a = double(xcasea.dx2);
lamdaa = double(xcasea.lamda);
hlina = double(subs(hlin,{dx1,dx2},{dx1a,dx2a}));
glina = double(subs(glin,{dx1,dx2},{dx1a,dx2a}));
Fa = double(subs(Fnobeta,{dx1,dx2,lamda},{dx1a,dx2a,lamdaa}));
%%% results for case (a)
x1a = dx1a + xb(1);
x2a = dx2a + xb(2);
fv = double(subs(f,{x1,x2},{x1a,x2a}));
hv = double(subs(h,{x1,x2},{x1a,x2a}));
gv = double(subs(g,{x1,x2},{x1a,x2a}));
fprintf('Change in design vector: '),disp([dx1a dx2a]);
fprintf('The linearized quality constraint: '),disp(hlina);
fprintf('The linearized inequality constraint: '),disp(glina);
fprintf('New design vector: '),disp([x1a x2a]);
fprintf('The objective function: '),disp(fv);
fprintf('The equality constraint: '),disp(hv);
fprintf('The inequality constraint: '),disp(gv);
if (glina <= 0)
xb(1) = xb(1) + dx1a;
xb(2) = xb(2) + dx2a;
fbest = Fa;
dx1best = dx1a;
dx2best = dx2a;
end
%%%%%%%%%%%%%%%%%%%%%%%
%%% solve case (b) %%%
%%%%%%%%%%%%%%%%%%%%%%%
fprintf('\n Case b: g = 0\n');
xcaseb = solve(diff(F,dx1),diff(F,dx2),hlin,glin);
solb = [double(xcaseb.dx1) double(xcaseb.dx2) double(xcaseb.lamda) double(xcaseb.beta)];
dx1b = double(xcaseb.dx1);
dx2b = double(xcaseb.dx2);
betab = double(xcaseb.beta);
lamdab = double(xcaseb.lamda);
hlinb = double(subs(hlin,{dx1,dx2},{dx1b,dx2b}));
glinb = double(subs(glin,{dx1,dx2},{dx1b,dx2b}));
Fb = double(subs(F,{dx1,dx2,lamda,beta},{dx1b,dx2b,lamdab,betab}));
x1b = dx1b + xb(1);
x2b = dx2b + xb(2);
fv = double(subs(f,{x1,x2},{x1b,x2b}));
hv = double(subs(h,{x1,x2},{x1b,x2b}));
gv = double(subs(g,{x1,x2},{x1b,x2b}));
fprintf('Change in design vector: '),disp([dx1b dx2b]);
fprintf('The linearized quality constraint: '),disp(hlinb);
fprintf('The linearized inequality constraint: '),disp(glinb);
fprintf('New design vector: '),disp([x1b x2b]);
fprintf('The objective function: '),disp(fv);
fprintf('The equality constraint: '),disp(hv);
fprintf('The inequality constraint: '),disp(gv);
fprintf('Multiplier beta: '),disp(betab);
fprintf('Multiplier lamda: '),disp(lamdab);
if (betab > 0) & (Fb <= fbest)
xb(1) = x1b;
xb(2) = x2b;
dx1best = dx1b;
dx2best = dx2b;
end
%%% stopping criteria
if ([dx1best dx2best]*[dx1best dx2best]') <= 1.0e-08
fprintf('\n&&&&&&&&&&&&&&&&&&&&&&&&&&&&&')
fprintf('\nStopped: Design Not Changing')
fprintf('\n&&&&&&&&&&&&&&&&&&&&&&&&&&&&&\n\n')
break;
elseif i == it
fprintf('\n&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&')
fprintf('\nStpped: Number of iterations at maximum')
fprintf('\n&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&\n\n')
break;
end
end

Answer 1

f1, g1, h1 are still syms variable type.
Change them to numeric type using the function double() before applying
the function num2str()
You have already applied double() to the variables
gf1, gf2, gh1, gh2,gg1, gg2 and hess1
right above, no need to touch them
Here is the section you should replace
Fquadstr = strcat(num2str(f1),' + ',num2str(gf1), ...
'*','dx1',' + ',num2str(gf2),' * ','dx2', ...
' + 0.5*',num2str(hess1(1,1)),' * dx1^2', ...
' +',num2str(hess1(1,2)),' * dx1*dx2', ...
' + 0.5*',num2str(hess1(2,2)),' * dx2^2');
hlinstr = strcat(num2str(h1),' + ',num2str(gh1), ...
'*','dx1',' + ',num2str(gh2),' * ','dx2');
glinstr = strcat(num2str(g1),' + ',num2str(gg1), ...
'*','dx1',' + ',num2str(gg2),' * ','dx2');
by this
% apply double() to f1
Fquadstr = strcat(num2str(double(f1)),' + ',num2str(gf1), ...
'*','dx1',' + ',num2str(gf2),' * ','dx2', ...
' + 0.5*',num2str(hess1(1,1)),' * dx1^2', ...
' +',num2str(hess1(1,2)),' * dx1*dx2', ...
' + 0.5*',num2str(hess1(2,2)),' * dx2^2');
% apply double() to h1
hlinstr = strcat(num2str(double(h1)),' + ',num2str(gh1), ...
'*','dx1',' + ',num2str(gh2),' * ','dx2');
% apply double() to g1
glinstr = strcat(num2str(double(g1)),' + ',num2str(gg1), ...
'*','dx1',' + ',num2str(gg2),' * ','dx2');