All of the examples can be found in Jupyter notebook form here.

# Robust approximate fitting

Section 6.4.2 Boyd & Vandenberghe "Convex Optimization" Original by Lieven Vandenberghe Adapted for Convex by Joelle Skaf - 10/03/05

Adapted for Convex.jl by Karanveer Mohan and David Zeng - 26/05/14 Original cvx code and plots here: http://web.cvxr.com/cvx/examples/cvxbook/Ch06_approx_fitting/html/fig6_15.html

Consider the least-squares problem: minimize $\|(A + tB)x - b\|_2$ where $t$ is an uncertain parameter in [-1,1] Three approximate solutions are found:

1. nominal optimal (i.e. letting t=0)
2. stochastic robust approximation: minimize $\mathbb{E}\|(A+tB)x - b\|_2$ assuming $u$ is uniformly distributed on [-1,1]. (reduces to minimizing $\mathbb{E} \|(A+tB)x-b\|^2 = \|A*x-b\|^2 + x^TPx$ where $P = \mathbb{E}(t^2) B^TB = (1/3) B^TB$ )
3. worst-case robust approximation: minimize $\mathrm{sup}_{-1\leq u\leq 1} \|(A+tB)x - b\|_2$ (reduces to minimizing $\max\{\|(A-B)x - b\|_2, \|(A+B)x - b\|_2\}$ ).
using Convex, LinearAlgebra, SCS

Input Data

m = 20;
n = 10;
A = randn(m, n);
(U, S, V) = svd(A);
S = diagm(exp10.(range(-1, stop = 1, length = n)));
A = U[:, 1:n] * S * V';

B = randn(m, n);
B = B / norm(B);

b = randn(m, 1);
x = Variable(n)
Variable
size: (10, 1)
sign: real
vexity: affine
id: 602…404

Case 1: Nominal optimal solution

p = minimize(norm(A * x - b, 2))
solve!(p, SCS.Optimizer; silent_solver = true)
x_nom = evaluate(x)
10-element Vector{Float64}:
4.224395017569452
0.929857403253579
-0.5618460254017501
-2.688972143152298
0.7253067923133656
-0.3854338264710944
-0.24029323916934917
-0.6079615531215272
-2.641692125057168
0.1966035656949765

Case 2: Stochastic robust approximation

P = 1 / 3 * B' * B;
p = minimize(square(pos(norm(A * x - b))) + quadform(x, Symmetric(P)))
solve!(p, SCS.Optimizer; silent_solver = true)
x_stoch = evaluate(x)
10-element Vector{Float64}:
2.829962700615146
0.3281533897690989
0.2472011896004751
-0.9401436049349646
2.0235346205582245
0.9674425564687203
0.08282676729342943
-0.5616217848520628
-1.8393284681838489
0.26738477296005614

Case 3: Worst-case robust approximation

p = minimize(max(norm((A - B) * x - b), norm((A + B) * x - b)))
solve!(p, SCS.Optimizer; silent_solver = true)
x_wc = evaluate(x)
10-element Vector{Float64}:
1.2729462504148357
-0.3929731447243677
0.9549760407247866
0.32400809810805187
2.278097210371635
1.6072812179640021
0.2590784897656687
0.033356243523547754
-2.0106560631010857
0.16891872664011406

Plot residuals:

parvals = range(-2, stop = 2, length = 100);

errvals(x) = [norm((A + parvals[k] * B) * x - b) for k in eachindex(parvals)]
errvals_ls = errvals(x_nom)
errvals_stoch = errvals(x_stoch)
errvals_wc = errvals(x_wc)

using Plots
plot(parvals, errvals_ls, label = "Nominal problem")
plot!(parvals, errvals_stoch, label = "Stochastic Robust Approximation")
plot!(parvals, errvals_wc, label = "Worst-Case Robust Approximation")
plot!(
title = "Residual r(u) vs a parameter u for three approximate solutions",
xlabel = "u",
ylabel = "r(u) = ||A(u)x-b||_2",
)