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AarianAarian
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initial commit UOA
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Lines changed: 784 additions & 0 deletions

‎ConjugateG.m‎

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function X_est = ConjugateG(x,iteration )
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A = [3 -1 0 ; -1 3 -1 ; 0 -1 3] ;
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b= [1;2;3] ;
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r = ones(3,iteration) ;
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p = ones(3,iteration) ;
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X = ones(3,iteration) ;
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alpha = ones(1,iteration) ;
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beta = ones(1,iteration) ;
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r(: , 1) = A*x(:,1) -b ;
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p(:,1) = -r(: , 1) ;
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k = 1 ;
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X(:,1) = x ;
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while r(: , k) ~= 0
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alpha(k) = (- r(: , k)' * p(: , k)) / (p(:,k)' * A * p(:,k)) ;
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X (:,k+1) = X(:,k) + alpha(k) * p(:,k) ;
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r (:,k+1) = A*X(:,k+1) -b ;
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beta (k+1) = (r(:,k+1)' * A * p(:,k))/(p(:,k)' * A * p(:,k)) ;
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p(:,k+1) = -r(:,k+1) + beta(k+1) * p(:,k) ;
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k = k+1 ;
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end
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X_est = X(:,end)
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X;
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r
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end
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‎Cuachy.m‎

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function P_c = Cuachy(xx, Delta)
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f = @(x) (x(2)-x(1)^2)^2+(1-x(1))^2 ;
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gf = @(x) [ 2*x(1) - 4*x(1)*(- x(1)^2 + x(2)) - 2 ; - 2*x(1)^2 + 2*x(2)];
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hf = @(x) [ 2+12*x(1)^2-4*x(1) , -4*x(1) ; -4*x(1) , 2 ];
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BB = hf(xx) ;
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gg = gf (xx) ;
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Toe = @(g,B) (g'*B*g <=0) * 1 + (g'*B*g >0) * min( 1 , norm(g,2)^3/(Delta*g'*B*g) ) ;
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P_c = (-Toe(gg,BB) * Delta/norm(gg,2)) * gg;
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min_val_Cauchy_model = f(xx)+ gg'*P_c
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end

‎NewtonCubic.m‎

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function F = Newton_backtrack(x1,c,itr_num)
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f = @(x) 100*(x(2)-x(1)^2)^2+(1-x(1))^2 ;
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gf = @ (x) [-2*(1-x(1))-400*x(1)*(x(2)-x(1)^2) ; 200*(x(2)-x(1)^2)] ;
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hf = @ (x) [-400*x(2)+1200*x(1)^2+2 -400*x(1) ; -400*x(1) 200 ] ;
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phi = @ (Alpha,x,p) 100*((Alpha*p(2)+x(2))-(Alpha*p(1)+x(1))^2)^2+(1-(Alpha*p(1)+x(1)))^2;
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gphi = @ (Alpha,x,p) 200*(p(2) - 2*p(1)*(x(1) + Alpha*p(1)))*(x(2) + Alpha*p(2) - (x(1) + Alpha*p(1))^2) + 2*p(1)*(x(1) + Alpha*p(1) - 1);
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%x0 = [0,1]'
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alpha_arr = .9*ones(1,itr_num) ;
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X = ones(2,itr_num) ;
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F = ones(1,itr_num) ;
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%P=[0;0]
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x_nxt = x1 ;
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for k = 1:itr_num
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X(:,k) = x_nxt ;
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P = -gf( X(:,k) ) ;
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ii=2;
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alpha_arr_check = .9*ones(1,10000);
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A_0 = 0.2 ;
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A_n1 = 4 ;
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A_p1 =1 ;
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if gf( X(:,k))' * (((hf(X(:,k)))) *gf( X(:,k) )) >= 0
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%disp('posdef')
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P = -inv((hf(X(:,k)))) *gf( X(:,k) ) ;
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else
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%disp('indef')
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%hf(X(:,k))
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P = -inv((hf(X(:,k)))) *gf( X(:,k) ) ;
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X(:,k) ;
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end
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while ( phi(A_0 , X(:,k) , P) > phi(0, X(:,k) , P) + c* A_0 *gphi(0 , X(:,k) , P) )%eval( subs(phi, [Alpha , x1,x2,p1, p2 ] , [alpha_arr_check(ii) , X(1,k) ,X(2,k) , P(1) , P(2) ])) <= eval( subs(phi, [Alpha , x1,x2,p1, p2 ] , [0 , X(1,k) ,X(2,k) , P(1) , P(2) ])) + c * alpha_arr_check(ii) * eval( subs(gphi, [Alpha , x1,x2,p1, p2 ] , [alpha_arr_check(ii) , X(1,k) ,X(2,k) , P(1) , P(2) ])) )
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%disp('in while')
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%disp(A_0)
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%disp(phi(A_0 , X(:,k) , P))
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%disp(phi(0, X(:,k) , P) + c* A_0 *gphi(0 , X(:,k) , P))
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A_0 = -.5*gphi(0 , X(:,k) , P)*A_n1 ^2/(phi(A_n1 , X(:,k) , P)-phi(0, X(:,k) , P)-gphi(0 , X(:,k) , P)*A_n1);
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%d1 = eval(subs(gphi,Alpha,alpha_arr_check(ii-1)) + subs(gphi,Alpha,alpha_arr_check(ii)) -3* (subs(phi,Alpha,alpha_arr_check(ii-1)) - subs(phi,Alpha,alpha_arr_check(ii)))/(alpha_arr_check(ii-1)-alpha_arr_check(ii)));
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%%subs (d1,)
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%d2 = sign(alpha_arr_check(ii)-alpha_arr_check(ii-1)) *sqrt(d1^2 - eval(subs(gphi,Alpha,alpha_arr_check(ii-1))*subs(gphi,Alpha,alpha_arr_check(ii)))) ;
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%alpha_arr_check(ii+1) = alpha_arr_check(ii) - (alpha_arr_check(ii) - alpha_arr_check(ii-1)) *( (eval(subs(gphi,Alpha,alpha_arr_check(ii))) + d2 - d1 /(eval( subs(gphi,Alpha,alpha_arr_check(ii)) - subs(phi,Alpha,alpha_arr_check(ii-1))) +2*d2 ) )) ;
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%d1
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%d2
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d1 = gphi(A_n1 , X(:,k) , P) + gphi(A_0 , X(:,k) , P) - 3* (phi(A_n1 , X(:,k) , P) - phi(A_0 , X(:,k) , P))/(A_0 - A_n1);
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d2 = sign (A_0 - A_n1) *sqrt(d1^2 - gphi(A_n1 , X(:,k) , P) * gphi(A_0 , X(:,k) , P));
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A_p1 = A_0 - (A_0 - A_n1) *( gphi(A_0 , X(:,k) , P) + d2 - d1) / (A_0 - A_n1 +2*d2);
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%A_p1 = A_0
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%AB = [A_n1^2 ,-A_0^2 ; -A_n1^3,A_0^3] * [phi(A_0 , X(:,k) , P)-phi(0, X(:,k) , P)-gphi(0 , X(:,k) , P)*A_0;
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%phi(A_n1 , X(:,k) , P)-phi(0, X(:,k) , P)-gphi(0 , X(:,k) , P)*A_n1 ] *1/(A_n1^2 * A_0^2*(A_0 - A_n1))
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%aa = AB(1) ;
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%bb = AB(2) ;
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%A_p1 = (-bb + sqrt(bb^2 - 3*aa*gphi(0 , X(:,k) , P)))/(3*aa)
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%
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%ii = ii + 1;
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A_n1 = A_0 ;
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A_0 = A_p1;
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%if abs(A_0 - A_n1) <= 0.001 || A_0 <= 0.01
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% A_0 = .5*A_n1
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%end
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end
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alpha_arr(k) = A_0;
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F(k) = f(X(:,k)) ;
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x_nxt = X(:,k) + alpha_arr(k) * P ;
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P;
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alpha_arr(k) ;
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end
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% cvx_begin
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% variable x(2)
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% minimize(100*(x(2)-x(1)^2)^2+(1-x(1))^2 )
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% subject to
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%
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% cvx_end
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% x
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disp('X')
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X(:,end)
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disp('Alpha')
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alpha_arr(end)
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%figure
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%
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%subplot(1,2,1);
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%plot(1:itr_num , (alpha))
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%
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%
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%title(strcat(' x_0 = ',mat2str(x1)))
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%xlabel('iteration')
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%ylabel('alpha(BackTraking)')
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%
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%
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%subplot(1,2,2);
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%plot(1:itr_num , log(F))
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%%surf(X(1,:) , X(2,:) , F)
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%title(strcat('Newton_backtrack', ' x_0 = ',mat2str(x1) , ' F* = ' ,mat2str(F(end)) , ' X* = ' ,mat2str(X(:,end)) ))
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%xlabel('iteration')
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%ylabel('log(F_k*)')
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return
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end

‎Newton_backtrack.m‎

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function F = Newton_backtrack(x1,ro,c,itr_num)
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f = @(x) 100*(x(2)-x(1)^2)^2+(1-x(1))^2 ;
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gf = @ (x) [-2*(1-x(1))-400*x(1)*(x(2)-x(1)^2) ; 200*(x(2)-x(1)^2)] ;
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hf = @ (x) [-400*x(2)+1200*x(1)^2+2 -400*x(1) ; -400*x(1) 200 ] ;
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%x0 = [0,1]'
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alpha = ones(1,itr_num) ;
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X = ones(2,itr_num) ;
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F = ones(1,itr_num) ;
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%P=[0;0]
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x_nxt = x1 ;
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for k = 1:itr_num
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X(:,k) = x_nxt ;
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%disp ('hessian :')
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%hf(X(:,k))
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if gf( X(:,k))' * (((hf(X(:,k)))) *gf( X(:,k) )) >= 0
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%disp('posdef')
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P = -inv((hf(X(:,k)))) *gf( X(:,k) ) ;
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else
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%disp('indef')
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%hf(X(:,k))
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P = -inv((hf(X(:,k)))) *gf( X(:,k) ) ;
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X(:,k) ;
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end
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while ( f( X(:,k) + alpha(k) * P) > f( X(:,k)) + c*alpha(k)* gf( X(:,k))' * P )
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alpha(k) = ro * alpha (k) ;
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end
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F(k) = f(X(:,k)) ;
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x_nxt = X(:,k) + alpha(k) * P ;
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end
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% cvx_begin
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% variable x(2)
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% minimize(100*(x(2)-x(1)^2)^2+(1-x(1))^2 )
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% subject to
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%
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% cvx_end
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% x
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disp('X')
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X(:,end)
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disp('alpha')
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alpha(end)
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figure
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subplot(1,2,1);
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plot(1:itr_num , (alpha))
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title(strcat(' x_0 = ',mat2str(x1)))
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xlabel('iteration')
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ylabel('alpha(BackTraking)')
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subplot(1,2,2);
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plot(1:itr_num , log(F))
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%surf(X(1,:) , X(2,:) , F)
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title(strcat('Newton_backtrack', ' x_0 = ',mat2str(x1) , ' F* = ' ,mat2str(F(end)) , ' X* = ' ,mat2str(X(:,end)) ))
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xlabel('iteration')
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ylabel('log(F_k*)')
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return
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end

‎Report1_96131054.pdf‎

525 KB
Binary file not shown.

‎TR.m‎

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function X_star = TR (xx,eta, Delta_hat,Delta0,itration)
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f = @(x) (x(2)-x(1)^2)^2+(1-x(1))^2 ;
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gf = @(x) [ 2*x(1) - 4*x(1)*(- x(1)^2 + x(2)) - 2 ; - 2*x(1)^2 + 2*x(2)];
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hf = @(x) [ 2+12*x(1)^2-4*x(1) , -4*x(1) ; -4*x(1) , 2 ];
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g= gf(xx);
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B = hf(xx) ;
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mdl = @ (p) f(xx) + g'*p + p'* B *p;
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%gf(xx) ;
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%hf(xx) ;
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%P_dog = dogleg ( xx, 10)
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X = ones(2,itration) ;
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P_dog = ones(2,itration) ;
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Delta = ones(1,itration) ;
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Delta(1) = Delta0 ;
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X(:,1) = xx;
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for k = 1:itration
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P_dog(:,k) = dogleg ( X(:,k), Delta(k)) ;
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ro = (f(X(:,k)) - f(X(:,k)+P_dog(:,k)) ) / (mdl([0 0]') - mdl(P_dog(:,k)) );
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if ( ro <.25 )
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Delta(k+1) = Delta(k)/4 ;
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else
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if (ro>.25 & norm(P_dog(:,k),2)==Delta(k))
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Delta(k+1) = min(2*Delta(k) , Delta_hat) ;
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else
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Delta(k+1) = Delta(k) ;
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end
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end
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if(ro > eta)
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X(:,k+1) = X(:,k) +P_dog(:,k) ;
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else
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X(:,k+1) = X(:,k) ;
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end
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end
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X_star = X(:,end);
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X ;
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end

‎dogleg.m‎

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function P_star = dogleg(xx, Delta)
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f = @(x) (x(2)-x(1)^2)^2+(1-x(1))^2 ;
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gf = @(x) [ 2*x(1) - 4*x(1)*(- x(1)^2 + x(2)) - 2 ; - 2*x(1)^2 + 2*x(2)];
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hf = @(x) [ 2+12*x(1)^2-4*x(1) , -4*x(1) ; -4*x(1) , 2 ];
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B = hf(xx) ;
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g = gf (xx) ;
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P_B = -inv(B) *g;
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P_U = - ((g'*g) /(g'*B*g) ) * g;
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f = @(x) (x(2)-x(1)^2)^2+(1-x(1))^2 ;
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P_hat = @(t) ((0<=t)&(t<=1)) * t' *P_U + ((1<t)&(t<=2)) * (P_U + (t'-1) *(P_B - P_U));
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mdl = @ (p) f(xx) + g'*p + p'* B *p;
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syms t x
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%eqn = norm(P_U + (t-1)*(P_B-P_U),2) == Delta
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eqn = (P_U(1) + (t-1)*(P_B(1)-P_U(1))) ^2 + (P_U(2) + (t-1)*(P_B(2)-P_U(2))) ^2 == Delta^2;
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tt = eval(solve(eqn,t));
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%P_star = 0 ;
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if(norm(P_B,2) <= Delta || tt(2) >=2)
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%disp('xx+P_B')
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xx+P_B;
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v_star = f(xx+P_B) ;
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P_star = P_B ;
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else
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if (imag(tt(2)) ~= 0)
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tt(2) = 0 ;
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end
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if ( 0 <=tt(2) & tt(2) < 1 )
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%disp ('0< < 1')
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P_star = tt(2) * P_U ;
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xx+P_star ;
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else
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if (1 <=tt(2) & tt(2) < 2)
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P_star = (P_U + (tt(2)-1) *(P_B - P_U));
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%disp ('1< < 2')
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xx+P_star ;
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end
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end
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%end
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end
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%disp('out of trust')
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%T = 0.1:.1:2;
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%sz = size(T)
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%lst = [] ;
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%my_p = 0;
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%toe = 0.1;
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%min_val_dog_model = 1000 ;
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%for kk = 0:.1:2
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% p_t = P_hat( kk )
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% %kk
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% %disp('norm p_t ')
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% %norm(p_t,2)
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% if(norm(p_t,2) <= Delta)
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% disp('||p_t|| <= Delta')
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% if(min_val_dog_model > mdl(p_t) )
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% min_val_dog_model = mdl(p_t);
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% my_p = p_t;
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% toe = kk ;
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% end
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% %mdl_out = mdl(p_t) ;
67+
% %lst = [lst,mdl_out];
68+
% end
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%end
70+
%[m,i] = min ( mdl_out ) ;
71+
%P_star = P_hat( T(i) ) ;
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%P_star = my_p ;
73+
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%disp ('min_val_dog_model : ')
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%T(i)
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%m
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%min_val_dog_model
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%toe
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%end
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end

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