Code covered by the BSD License

# Augmented Lagrangian

### Mayowa Aregbesola (view profile)

Augmented Lagrangian methods are a certain class of algorithms for solving constrained optimization.

prob1.m
```close all
clear all
clc
diary('results1.txt')
n=0;            %initialize iteration counter
e = 1e-6;       %epsilon

%% problem 1a
f = @(x) 2*x(1) - 3*x(2);				%f(x)
G = @(x) [2; -3];						%Gradient of f(x)
H = @(x) [0, 0; 0, 0];					%Hessian of f(x)
g = @(x) x(1)^2 + x(2)^2 - 25;			%g(x)
Dg = @(x) [2*x(1); 2*x(2)];				%gradient of g(x)
Hg = {@(x) [2, 0; 0, 2]};				% Hessian of g(x)
x0 = [-2.8; 4.2];        				% set starting value
xt = [-10/sqrt(13); 15/sqrt(13)]; 		%true value
disp('Problem 1a');
[x, itr] = newton_constrained(x0, f, g, G, H, Dg, Hg, e, xt, 1);

%% Problem 1b
f = @(x) x(1)^2 + 2*x(1)*x(2) + x(2)^2;			%f(x)
G = @(x) [2*x(1) + 2*x(2); 2*x(2) + 2*x(1)];    %Gradient of f(x)
H = @(x) [2, 2; 2, 2];                          %Hessian of f(x)
g = @(x) 3*x(1)^2 + x(2)^2 - 9;                 %g(x)
Dg = @(x) [6*x(1); 2*x(2)];                     %gradient of g(x)
Hg = {@(x) [6, 0; 0, 2]};               		% Hessian of g(x)
x0 = [-1.6; 1.6];                           % set starting value
xt = [-1.5; 1.5];                               %true value
disp('Problem 1b');
[x, itr] = newton_constrained(x0, f, g, G, H, Dg, Hg, e, xt, 1);

%% Problem 1c
f = @(x) 3*x(1)^3 + 2*x(2)^3 + x(3)^3 + x(4)^3;                                           %f(x)
G = @(x) [9*x(1)^2; 6*x(2)^2; 3*x(3)^2; 3*x(4)^2];                                        %Gradient of f(x)
H = @(x) [18*x(1), 0, 0, 0; 0, 12*x(2)^2, 0, 0; 0, 0, 6*x(3), 0; 0, 0, 0, 6*x(4)];        %Hessian of f(x)
g = @(x) [x(1)^2 + x(2)^2 + x(3)^2 + x(4)^2 - 4; x(1) + x(2) + 2*x(3) + 3*x(4) - 1];      %g(x)
Dg = @(x) [2*x(1), 1; 2*x(2), 1; 2*x(3), 2; 2*x(4), 3];                                   %gradient of g(x)
Hg = {@(x) eye(4)*2; @(x) eye(4)*0};                                                      % Hessian of g(x)
x0 =[-1.84; 0.21; 0.42; 0.59];        %set starting value                                 % set starting value
xt = [-1.84931; 0.20806; 0.41612; 0.603003];                                              %true value
disp('Problem 2a');
[x, itr] = newton_constrained(x0, f, g, G, H, Dg, Hg, e, xt, 1);
diary off```