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Optimization

Optimization. MATLAB Exercises Assoc. Prof. Dr. Pelin GÜNDEŞ. Introduction to MATLAB. Introduction to MATLAB. Introduction to MATLAB. % All text after the % sign is a comment. MATLAB ignores anything to the right of the % sign.

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Optimization

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  1. Optimization MATLAB Exercises Assoc. Prof. Dr. Pelin GÜNDEŞ

  2. Introduction to MATLAB

  3. Introduction to MATLAB

  4. Introduction to MATLAB • % All text after the % sign is a comment. MATLAB ignores anything to the right of the % sign. • ; A semicolon at the end of a line prevents MATLAB from echoing the information you enter on the screen • … A succession of three periods at the end of the line informs MATLAB that code will continue to the next line. You can not split a variable name across two lines. You can not continue a comment on another line.

  5. Introduction to MATLAB • ^c You can stop MATLAB execution and get back the command prompt by typing ^c (Ctrl-C) – by holding down ‘Ctrl’ and ‘c’ together. • help command_name • helpwin Opens a help text window that provides more information on all of the MATLAB resources installed on your system. • helpdesk Provides help using a browser window.

  6. Introduction to MATLAB • = The assignment operator. The variable on the left-hand side of the sign is assigned the value of the right-hand side. • == Within a if construct • MATLAB is case sensitive. An a is different than A. All built-in MATLAB commands are in lower case. • MATLAB does not need a type definition or dimension statement to introduce variables.

  7. Introduction to MATLAB • Variable names start with a letter and contain up to 31 characters (only letters, digits and underscore). • MATLAB uses some built in variable names. Avoid using built in variable names. • Scientific notation is expressed with the letter e, for example, 2.0e-03, 1.07e23, -1.732e+03. • Imaginary numbers use either i or j as a suffix, for example 1i, -3.14j,3e5i

  8. Introduction to MATLAB Arithmetic Operators • + Addition • - Subtraction • * Multiplication • / Division • ^ Power • ‘ Complex conjugate transpose (also array transpose)

  9. Introduction to MATLAB • In the case of arrays, each of these operators can be used with a period prefixed to the operator, for example, (.*) or (.^) or (./). This implies element-by-element operation in MATLAB. • , A comma will cause the information to echo

  10. Exercise >> a=2;b=3;c=4,d=5;e=6, c = 4 e = 6 % why did only c and e echo on the screen?

  11. Exercise >> who % lists all the variables on the screen Your variables are: a b c d e >> a % gives the value stored in a a = 2

  12. Exercise >> A=1.5 % Variable A A = 1.5000 >> a,A % Case matters a = 2 A = 1.5000

  13. Exercise >> one=a;two=b;three=c; >> % assigning values to new variables >> four=d;five=e;six=pi; % value of pi available >> f=7; >> A1=[a b c;d e f]; % A1 is a 2 by 3 matrix % space seperates columns % semi-colon seperates rows >> A1(2,2) % accesses the matrix element on the second raw and second column ans = 6

  14. Exercise >> size(A1) % gives you the size of the matrix (row, columns) ans = 2 3 >> AA1=size(A1) % What should happen here? From previous % statement the size of A1 contains two numbers arranged as a row % matrix. This is assigned to AA1 AA1 = 2 3

  15. Exercise >> size(AA1) % AA1 is a one by two matrix ans = 1 2 >> A1’ % this transposes the matrix A1 ans = 2 5 3 6 4 7

  16. Exercise >> B1=A1’ % the transpose of matrix A1is assigned to B1. B1 is a % three by two matrix B1 = 2 5 3 6 4 7 >> C1=A1*B1 % Matrix multiplication C1 = 29 56 56 110

  17. Exercise >> C2=B1*A1 C2 = 29 36 43 36 45 54 43 54 65 >> C1*C2 % Read the error matrix ??? Error using ==> * Inner matrix dimensions must agree.

  18. Exercise >> D1=[1 2]' % D1 is a column vector D1 = 1 2 >> C1,C3=[C1 D1] % C1 is augmented by an extra column C1 = 29 56 56 110 C3 = 29 56 1 56 110 2

  19. Exercise >> C2 C2 = 29 36 43 36 45 54 43 54 65 >> C3=[C3;C2(3,:)] % The column represents all the columns C3 = 29 56 1 56 110 2 43 54 65

  20. Exercise >> C4=C2*C3 C4 = 4706 7906 2896 5886 9882 3636 7066 11858 4376 >> C5=C2.*C3 % The .* represents the product of each element of % C2 with the corresponding element of C3 C5 = 841 2016 43 2016 4950 108 1849 2916 4225

  21. Exercise >> C6=inverse(C2) ??? Undefined function or variable 'inverse'. % Apparently, inverse is not a command in MATLAB, if command % name is known, it is easy to obtain help >>lookfor inverse % this command will find all files where it comes % across the word “inverse” in the initial comment lines. The % command we need appears to be INV which says inverse of a % matrix. The actual command is in lower case. To find out how to use % it: >> help inv inv(C2) % inverse of C2

  22. Exercise >> for i=1:20 f(i)=i^2; end % The for loop is terminated with “end” >> plot(sin(0.01*f)',cos(0.03*f)) >> xlabel('sin(0.01f)') >> ylabel('cos(0.03*f)') >> legend('Example') >> title('A Plot Example') >> grid >> exit % finished with MATLAB

  23. Graphical optimization Minimize f(x1,x2)= (x1-3)2 + (x2-2)2 subject to: h1 (x1,x2): 2x1 + x2 =8 h2(x1,x2): (x1-1)2 + (x2-4)2 =4 g1(x1,x2): x1 + x2 ≤ 7 g2(x1,x2): x1 – 0.25 x22≤ 0 0 ≤ x1≤ 10; 0 ≤ x2≤ 10;

  24. Example 1 % Example 1 (modified graphics)% % % graphical solution using matlab (two design variables) % the following script should allow the graphical solution % to example % % Minimize f(x1,x2) = (x1-3)**2 + (x2-2)**2 % % h1(x1,x2) = 2x1 + x2 = 8 % h2(x1,x2) = (x1-1)^2 + (x2-4)^2 = 4 % g1(x1,x2) : x1 + x2 <= 7 % g1(x1,x2) : x1 - 0.25x2^2 <= 0.0 % % 0 <= x1 <= 10 ; 0 <= x2 <= 10 % % %

  25. Example 1 % % % WARNING : The hash marks for the inequality constraints must % be determined and drawn outside of the plot % generated by matlab % %---------------------------------------------------------------- x1=0:0.1:10; % the semi-colon at the end prevents the echo x2=0:0.1:10; % these are also the side constraints % x1 and x2 are vectors filled with numbers starting % at 0 and ending at 10.0 with values at intervals of 0.1 [X1 X2] = meshgrid(x1,x2); % generates matrices X1 and X2 correspondin % vectors x1 and x2

  26. Example 1 cont’d f1 = obj_ex1(X1,X2);% the objecive function is evaluated over the entire mesh ineq1 = inecon1(X1,X2);% the inequality g1 is evaluated over the mesh ineq2 = inecon2(X1,X2);% the inequality g2 is evaluated over the mesh eq1 = eqcon1(X1,X2);% the equality 1 is evaluated over the mesh eq2 = eqcon2(X1,X2);% the equality 2 is evaluated over the mesh [C1,h1] = contour(x1,x2,ineq1,[7,7],'r-'); clabel(C1,h1); set(h1,'LineWidth',2) % ineq1 is plotted [at the contour value of 8] hold on % allows multiple plots k1 = gtext('g1'); set(k1,'FontName','Times','FontWeight','bold','FontSize',14,'Color','red') % will place the string 'g1' on the lot where mouse is clicked

  27. Example 1 cont’d [C2,h2] = contour(x1,x2,ineq2,[0,0],'r--'); clabel(C2,h2); set(h2,'LineWidth',2) k2 = gtext('g2'); set(k2,'FontName','Times','FontWeight','bold','FontSize',14,'Color','red') [C3,h3] = contour(x1,x2,eq1,[8,8],'b-'); clabel(C3,h3); set(h3,'LineWidth',2) k3 = gtext('h1'); set(k3,'FontName','Times','FontWeight','bold','FontSize',14,'Color','blue') % will place the string 'g1' on the lot where mouse is clicked [C4,h4] = contour(x1,x2,eq2,[4,4],'b--'); clabel(C4,h4); set(h4,'LineWidth',2) k4 = gtext('h2'); set(k4,'FontName','Times','FontWeight','bold','FontSize',14,'Color','blue')

  28. Example 1 cont’d [C,h] = contour(x1,x2,f1,'g'); clabel(C,h); set(h,'LineWidth',1) % the equality and inequality constraints are not written with 0 on the right hand side. If you do write % them that wayyou would have to include [0,0] in the contour commands xlabel(' x_1 values','FontName','times','FontSize',12,'FontWeight','bold'); % label for x-axes ylabel(' x_2 values','FontName','times','FontSize',12,'FontWeight','bold'); set(gca,'xtick',[0 2 4 6 8 10]) set(gca,'ytick',[0 2.5 5.0 7.5 10]) k5 = gtext({'Chapter 2: Example 1','pretty graphical display'}) set(k5,'FontName','Times','FontSize',12,'FontWeight','bold') clear C C1 C2 C3 C4 h h1 h2 h3 h4 k1 k2 k3 k4 k5 grid hold off

  29. Example 1 cont’d Objective function function retval = obj_ex1(X1,X2) retval = (X1 - 3).*(X1 - 3) +(X2 - 2).*(X2 - 2); The first inequality function retval = inecon1(X1, X2) retval = X1 + X2; The second inequality function retval = inecon2(X1,X2) retval = X1 - 0.25*X2.^2;

  30. Example 1 cont’d The first equality function retval = eqcon1(X1,X2) retval = 2.0*X1 + X2; The second equality function retval = eqcon2(X1,X2) retval = (X1 - 1).*(X1 - 1) + (X2 - 4).*(X2 - 4);

  31. Example 2- Graphical solution

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