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    filename: A_DecentralizedGradientDescent.m
    path of file: ./repos/Performance-Estimation-Toolbox/Examples/11_Decentralized Optimization Methods
    
    the code of the file until where you have to start completion: function A_DecentralizedGradientDescent
% In this example, we consider K iterations of the decentralized subgradient
% descent (DGD) with N agents, that each holds a local convex function Fi 
% with bounded subgradients, for solving the following decentralized problem:
%   min_x F(x);     where F(x) is the average of local functions Fi. 
% For notational convenience we denote xs=argmin_x F(x), and Fs = F(xs). 
% Agents start all with the same iterate x0 such that ||x0 - xs||^2 <= 1. 
%
% This example shows how to obtain the worst-case performance of DGD with
% PESTO in that case. The performance criterion used is F(xav) - Fs, where
% xav is the average of the iterates over all the agents and all the iterations.
% Communications between the agents can be formulated in two different ways
% in the PEP problem, leading to different types of worst-case solution: 
%   (a) Using a fixed communication network, which leads to exact
%   worst-case results that are specific to the choosen matrix.

    
function A_DecentralizedGradientDescent
% In this example, we consider K iterations of the decentralized subgradient
% descent (DGD) with N agents, that each holds a local convex function Fi 
% with bounded subgradients, for solving the following decentralized problem:
%   min_x F(x);     where F(x) is the average of local functions Fi. 
% For notational convenience we denote xs=argmin_x F(x), and Fs = F(xs). 
% Agents start all with the same iterate x0 such that ||x0 - xs||^2 <= 1. 
%
% This example shows how to obtain the worst-case performance of DGD with
% PESTO in that case. The performance criterion used is F(xav) - Fs, where
% xav is the average of the iterates over all the agents and all the iterations.
% Communications between the agents can be formulated in two different ways
% in the PEP problem, leading to different types of worst-case solution: 
%   (a) Using a fixed communication network, which leads to exact
%   worst-case results that are specific to the choosen matrix.
%   (b) Using an entire spectral class of communication networks: the ones represented by       
%   a symmetric (generealized) doubly stochastic matrix with a given range of eigenvalues
%   This leads to a relaxation of PEP, providing worst-case valid for any 
%   matrix of the spectral class.
% Both formulations can be tested here but (b) is used by default.

% For details, see
%   [1] Colla, Sebastien, and Julien M. Hend