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    filename: B_ProjectedGradientMethod.m
    path of file: ./repos/Performance-Estimation-Toolbox/Examples/02_Methods for composite convex minimization
    
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% In this example, we use a projected gradient method for
% solving the constrained smooth strongly convex minimization problem
%   min_x { F(x) = f_1(x) + f_2(x) } 
%   for notational convenience we denote xs=argmin_x F(x);
% where f_1(x) is L-smooth and mu-strongly convex and where f_2(x) is
% a convex indicator function.
%
% We show how to compute the worst-case value of F(xN)-F(xs) when xN is
% obtained by doing N steps of the method starting with an initial
% iterate satisfying F(x0)-F(xs)<=1.

% (0) Initialize an empty PEP
P = pep();

% (1) Set up the objective function
paramf1.mu = .1;    % Strong convexity parameter
paramf1.L  = 1;     % Smoothness parameter

f1 = P.DeclareFunction('SmoothStronglyConvex', paramf1);
f2 = P.DeclareFunction('ConvexIndicator');
F  = f1 + f2; % F is the objective function

% (2) Set up the starting point and initial condition
x0      = P.StartingPoint();        % x0 is some starting point
[xs,fs] = F.OptimalPoint();         % xs is an optimal point, and fs=F(xs)
[g0,f0] = F.oracle(x0);

    
function B_ProjectedGradientMethod
% In this example, we use a projected gradient method for
% solving the constrained smooth strongly convex minimization problem
%   min_x { F(x) = f_1(x) + f_2(x) } 
%   for notational convenience we denote xs=argmin_x F(x);
% where f_1(x) is L-smooth and mu-strongly convex and where f_2(x) is
% a convex indicator function.
%
% We show how to compute the worst-case value of F(xN)-F(xs) when xN is
% obtained by doing N steps of the method starting with an initial
% iterate satisfying F(x0)-F(xs)<=1.

% (0) Initialize an empty PEP
P = pep();

% (1) Set up the objective function
paramf1.mu = .1;    % Strong convexity parameter
paramf1.L  = 1;     % Smoothness parameter

f1 = P.DeclareFunction('SmoothStronglyConvex', paramf1);
f2 = P.DeclareFunction('ConvexIndicator');
F  = f1 + f2; % F is the objective function

% (2) Set up the starting point and initial condition
x0      = P.StartingPoint();        % x0 is some starting point
[xs,fs] = F.OptimalPoint();         % xs is an optimal point, and fs=F(xs)
[g0,f0] = F.oracle(x0);
P.InitialCondition( f0-fs <= 1);    % Initial condition f0-fs<=1

% (3) Algorithm
gamma = 1/paramf1.L;    % stepsize
N = 1;                  % number of iterations

x = x0;
for i = 1:N
    xint = gradient_step(x, f1, gamma);
    x    = projection_step(xint, f2);
end