Remote Quadratic Programming

This example considers a strongly convex optimization problem

\[\beta^* \in \argmin_{\norm{\beta}_1 \leq \tau} \sum_{i=1}^{5} f_i(\beta),\]

where each function \(f_i\) is a quadratic in \(S_{m_i, L_i}\).

The overall composite optimization problem is described using the constants

\[\begin{split}\begin{align*} m &= \mat{cccccc}{0, 1, -2, 1, 1, 0}\\ L &= \mat{cccccc}{5, 2, 1, 6, 1, \infty}.\\ \end{align*} \end{split}\]

The Synthesized algorithm to solve this problem must have a block-lower-triangular \(\Dcl\) matrix. Synthesis of an algorithm with convergence rate \(\rho < 0.7679\) is accomplished by using the code

Composite Quadratic Programming

%form the operators
m = [0, 1, -2, 1, 1, 0];
L = [5, 2, 1, 6, 1, inf];
s = length(m);

ops= cell(s, 1);
for i = 1:s
    ops{i} = op_sml(m(i), L(i));
end

%form the systems
sys = opt_system(ops);

%solve the problem
man = opt_synthesis(sys);
sol = man.bisect();  %0.7679

Next, the same optimization problem must be solved in a remote setting. The network dynamics separating the oracles \(\partial f\) include time-delays and channel memory. The following code generates an algorithm with \(\rho < 0.9341\) in the networked setting.

Composite Quadratic Programming with Network Dynamics

%form the operators
m = [0, 1, -2, 1, 1, 0];
L = [5, 2, 1, 6, 1, inf];
s = length(m);

ops= cell(s, 1);
for i = 1:s
    ops{i} = op_sml(m(i), L(i));
end

%describe the network
channel = bridge_pass_through(s);
z = tf('z', 1);
channel.P(5, s+2) = z^(-1);
channel.P(s+2, 5) = z^(-1);

channel.P(s+4, 4) = z^(-2);
channel.P(s+3, 5) = z^(-1);

channel.P(1, s+1) =  0.5/(z-0.5);
channel.P(s+1, 1) = 0.5/(z-0.5);

channel.P(3, s+3) =  -1/(z+0.4);
channel.P(s+3, 3) = -1/(z+0.4);

%form the systems
sys_channel = opt_system(ops, channel);


%information structure for the channel case
%enforces block-lower-triangularity of the closed-loop system
config = opt_config();
config.syn.D_mask = [1, 0, 1, 1, 0, 0;
    1, 1, 1, 1, 0, 0;
    1, 1, 1, 1, 1, 1;
    1, 1, 1, 1, 0, 0;
    1, 1, 1,1 ,1 ,0;
    1, 1, 1, 1, 1, 1];

%solve the problem
man_channel = opt_synthesis(sys_channel, config);
sol_channel = man_channel.bisect(); 
% 0.9357 with block-triangular
% 0.9341 with relaxed info structure