Question

can someone help me with this optimization problem

1. [14 points Write the following two functions in Matlab: (a) function B - hessmod1 (H) This function takes in the Hessian matrix H and returns a positive definite matrix B by the following procedure Compute the spectral decomposition of H using the eig command in Matlab . Replace any negative eigenvalues with the positive quantity δ-0.25. Return B- SD ST, where S is the matrix whose columns are eigenvectors of H, and D is the diagonal matrix where negative eigenvalues have been replaced by δ. (b) function B -hessmod2 (H) This function takes in the Hessian matrix H and returns a positive definite matrix B by the following prex While B is not positive definite (use the chol command to check). Set τ-max(3,29 Set B-H+TI (c) Let H What do your functions hessmod1 and hessmod2 return with as input?

Answer 1

(a) The function M-file hessmod1.m is given below

function B = hessmod1(H)
%HESSMOD1 Takes in the Hessian matrix H and returns a positive definite
% matrix by replacing negative eigen values with delta = 0.25 in the
% spectral decomposition of H

% Input : H - Hessian matrix,
% Output: B - Positive definite matrix

% Spectral decomposition of H where D is the diagonal matrix containing the
% eigen values of H and S is the matrix with eigen vectors as its columns.
[S,D] = eig(H);

% Check for negative eigen values and replace them with delta.
delta = 0.25;
% Check is necessary only for the elements of the diagonal.Extract the
% diagonal and store it in a vector.
V = diag(D);
% Finds the positions where negative value occurs and replaces the value at
% that position with delta. If no negative values occur no changes occur.
V(find(V<0)) = delta;
% Create new diagonal matrix D*.
D = diag(V);
% Calculate B = SD*S'
B = S*D*S';

end

function B hessmod1 (H) %HESSMODĪ Takes in the Hessian matrix H and returns a positive definite matrix by replacing negative eigen values with delta 0.25 in the spectral decomposition of H Input :H -Hessian matrix, Output:B Positive definite matrix % spectral decomposition of H where D is the diagonal matrix containing the % eigen values of H and S is the matrix with eigen vectors as its columns. ,D] eig (H) % Check for negative eigen values and replace them with delta. delta = 0.25; % check is necessary only for the elements of the diagonal . Extract the % diagonal and store it in a vector. V = diag (D); % Finds the positions where negative value occurs and replaces the value at % that position with delta. If no negative values occur no changes occur V(find (V<0)) delta; % Create new diagonal matrix D*. D diag (V) % Calculate B SD*S' end

(b) The function M-file hessmod2.m is given below

function B = hessmod2(H)
%HESSMOD2 Given a Hessian matrix H returns a positive definite matrix B by
% performing the iterations B = H+tauI until B is positive definite
% Here tau is a parameter defined as tau = max(beta,2*tau), for
% initial values of beta = 0.25, tau = 0
%
% Input : H - Hessian matrix
% Output: B - Positive definite matrix.

% Set initial values for beta, tau, and B
beta = 0.25; tau = 0; B = H;
% p value is positive if the matrix is not positive definite
[~,p] = chol(B);
% Creating the identity matrix
I = eye(size(H));
while p~=0
tau = max(beta,2*tau);
B = H+tau*I;
[~,p] = chol(B);
end

end

function Bhessmod2 (H) %HESSMOD2 Given a Hessian matrix H returns a positive definite matrix B by performing the iterations BH+taul until B is positive definite Here tau is a parameter defined as tau = max (beta, 2 *tau), for initial values of beta = 0.25, tau =。 InputHHessian matrix Output: B- Positive definite matrix. % set initial values for beta, tau, and B beta 0.25: tau0: B-H % p value is positive if the matrix is not positive definite [+, p] = chol (B) ; % Creating the identity matrix I = eye (size (H) ) ; while po tau = max (beta, 2 *tau); B = H+tau*1; (-, p] = chol (B); end end

(c) The result for test case matrix H is given below

>> H [1 3 3 1] >B-hessmod1 (H) B= 2.1250 1.8750 1.8750 2.1250 >> B = hessm0d2 (H) B=