Question

parameter values (2 , 2 , 2 ,4 )

Minimize f (X , X)-X,-AX2+ bx 12+ Cx x2+ dx , 2 2 Use any programing language of your preference .Use constant lambda . Run optimization for three different starting points . Draw chart Optimality gap vs iterations . Minimize f (X X2)-X-AX2 + bx 2+ cx x 2+ . Use any programing language of your preference .Use constant lambda . Run optimization for three different starting points . Draw chart Optimality gap vs iterations . You can use Wolframalpha to explore function Vary values of lambda Draw chart lambda vs number of iterations . .

Answer 1

SMatlab code for Gradient Descent method clear all close all functions for which minimum have to find fprintf( Displaying the function f(x,y)In' disp(f) sset lambda value set starting value [xx-val, error-steepest-descent ( f , xx, 1 ) ; minimum value using steepest descent for initial guess [2,2] fprintf( "\nminimum value using steepest descent for initial guess [%, set lambda value 8set starting value [xx_val,error]-steepest_descent (f,xx,1); eminimum value using steepest descent for initial guess [2,2 tprint f ("Anmínimum value using steepest descent for initial guess f is.\n',xx(1), xx (2)) fprint f ('K1-%f; x2 % f\n',xx-val (1),xx-val ( 2 )) [%, set lambda value xx-12; 2]: [xx val,error1-steepest_descent(f, x,l); %set starting value minimum value using steepest descent for initial guess [2,2] fprintf ( nminimum value using steepest descent for initial guess 삐 is.\n",xx(1),xx ( 2 )) fprint f ("Kl=%f; x2-% f\n\n'.xx-va l (1),xx-va 1(2)) [f Loop for different lambda value and iteration count 11-linspace(0.05,0.3,20); for i= 1:length(11) %set lambda value set starting value [xxval,error]-steepestdescent(f, xx, 1); it cnt(i)-length(error); fprint f ( '\tFor 1-82.4f convergrnce in iteration count is %d n',1,it_cnt (i)) 용음음응용음음응용음음응용음음응용음음응용음음용용음음용용음음용용음음용용음응용욤음응용욤음음各욤음응용욤음응용욤음응용 %function for steepest descent method

function [xx val,error 1-steepest_descent (f, xx,l) % f(x,y) is the function for finding minima % x is the initial guess % is lambda value ginitialize iteration counter ginitialize error n-0; eps-1; syms X y finding gradient f_x(x, y)-diff(f,x) f_y(x.y)-diff(f,Y)i Computation loop while eps>le-10&& n<500 grad f-double ( [f-x ( xx ( 1 ) , xx ( 2 ) ) ; f-y (xx( 1 ) ,xx(2)))); yy double(xx-l1*gradf) n=n+1 ; eps-norm (xx-yy) error ( n ) =eps ; ххауу; %grad f(x) iterate %counter + 1 update x end xx_val-xx; end 용욤욤응各욤욤응응욤욤各各욤욤各各욤욤各 End of Code 욤욤各各음음各各음음各各음음各욤음응용 Displaying the function f(x,y) (x,y)x-2*y+2*x 2+2*x*y*2+4*y 2 minimum value using steepest descent for initial guess [2.000000,1.0000001 is x1=-0.292893; x2=0 . 292893 minimum value using steepest descent for initial guess 1.000000,2.000000] is x1=-0.292893; x2-0.292893 minimum value using steepest descent for initial guess [ 2.000000,2.000000] is x1=-0.292893; x2=0 . 292893 For 1-0.0500 convergrnce in iteration count is 111 For 1-0.0632 convergrnce in iteration count is 87. For 1-0.0763 convergrnce in iteration count is 70. For 1-0.0895 convergrnce in iteration count is 58. For 1-0.1026 convergrnce in iteration count is 47. For 1-0.1158 convergrnce in iteration count is 42. For 1-0.1289 convergrnce in iteration count is 39. For 1-0.1421 convergrnce in iteration count is 35 For 1-0.1553 convergrnce in iteration count is 32 For 1-0.1684 convergrnce in iteration count is 29 For 1-0.1816 convergrnce in iteration count is 28.

For 1-0.1947 convergrnce in iteration count is 14 For I-0.2079 convergrnce in iteration count is 13. For 1-0.2211 convergrnce in iteration count is 13 For 1-0.2342 convergrnce in iteration count is 12. For 1-0.2474 convergrnce in iteration count is 12 For 1-0.2605 convergrnce in iteration count is 12 For 1 0.2737 convergrnce in iteration count is 12 For 1-0.2868 convergrnce in iteration count is 12. For 1-0.3000 convergrnce in iteration count is 11 Published with MATLAB® R2018a

%%Matlab code for Gradient Descent method
clear all
close all

%functions for which minimum have to find
f=@(x,y) x-2*y+2*x^2+2*x*y^2+4*y^2;

fprintf('Displaying the function f(x,y)\n')
disp(f)

l=0.1;        %set lambda value
xx=[2;1];      %set starting value
[xx_val,error]=steepest_descent(f,xx,l);

%minimum value using steepest descent for initial guess [2,2]
fprintf('\nminimum value using steepest descent for initial guess [%f,%f] is.\n',xx(1),xx(2))
fprintf('x1=%f; x2=%f\n',xx_val(1),xx_val(2))

l=0.1;        %set lambda value
xx=[1;2];      %set starting value
[xx_val,error]=steepest_descent(f,xx,l);

%minimum value using steepest descent for initial guess [2,2]
fprintf('\nminimum value using steepest descent for initial guess [%f,%f] is.\n',xx(1),xx(2))
fprintf('x1=%f; x2=%f\n',xx_val(1),xx_val(2))

l=0.1;        %set lambda value
xx=[2;2];      %set starting value
[xx_val,error]=steepest_descent(f,xx,l);

%minimum value using steepest descent for initial guess [2,2]
fprintf('\nminimum value using steepest descent for initial guess [%f,%f] is.\n',xx(1),xx(2))
fprintf('x1=%f; x2=%f\n\n',xx_val(1),xx_val(2))

%Loop for different lambda value and iteration count
ll=linspace(0.05,0.3,20);
for i=1:length(ll)
  
    l=ll(i);        %set lambda value
    xx=[2;2];      %set starting value
    [xx_val,error]=steepest_descent(f,xx,l);
    it_cnt(i)=length(error);
    fprintf('\tFor l=%2.4f convergrnce in iteration count is %d.\n',l,it_cnt(i))
  
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%function for steepest descent method
function [xx_val,error]=steepest_descent(f,xx,l)
    % f(x,y) is the function for finding minima
    % xx is the initial guess
    % l is lambda value

    n=0;            %initialize iteration counter
    eps=1;          %initialize error
    syms x y
    %finding gradient
    f_x(x,y)=diff(f,x);
    f_y(x,y)=diff(f,y);

    %Computation loop
    while eps>1e-10 && n<500
        gradf=double([f_x(xx(1),xx(2));f_y(xx(1),xx(2))]); %gradf(x)
        yy=double(xx-l*gradf);                             %iterate
        n=n+1;
        eps=norm(xx-yy);                                    %counter+1
        error(n)=eps;
        xx=yy;                                             %update x

    end
    xx_val=xx;
end

%%%%%%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%%%%%