E_GradientMethod.m

function E_GradientMethod
% In this example, we use a fixed-step gradient method for
% solving the L-smooth convex minimization problem
%   min_x F(x); for notational convenience we denote xs=argmin_x F(x).
%
% We show how to compute the worst-case value of F(xN)-F(xs) when xN is
% obtained by doing N steps of the gradient method starting with an initial
% iterate satisfying ||x0-xs||<=1.
%
% Result to be compared with that of
% [1] Yoel Drori. "Contributions to the Complexity Analysis of
%     Optimization Algorithms." PhD thesis, Tel-Aviv University, 2014.

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

% (1) Set up the objective function
param.mu = 0;     % Strong convexity parameter
param.L  = 1;     % Smoothness parameter

F=P.DeclareFunction('SmoothStronglyConvex',param); % 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)
P.InitialCondition( (x0-xs)^2 <= 1); % Initial condition ||x0-xs||^2<= 1

% (3) Algorithm
h = 1/param.L;		% step size
N = 10;             % number of iterations

x = x0;
for i = 1:N
    x = x-h*F.gradient(x);
end
xN = x;

% (4) Set up the performance measure
fN = F.value(xN);
P.PerformanceMetric(fN-fs);      % Worst-case evaluated as F(x)-F(xs)

% (5) Solve the PEP
P.solve()

% (6) Evaluate the output
double(fN-fs)   % worst-case objective function accuracy

% The result should be (see [1])
% param.L/2/(2*N+1)
end