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• Norm approximation problem
• • cvx code
  • yalmip code
  • cvx code
  • yalmip code
• Reformulation
• • cvx code
  • yalmip code
• cvx: dual
• yalmip: dual
• Reference

Norm approximation problem

We consider the unconstrained norm approximation problem
min ⁡ ∥ A x − b ∥ \min \lVert Ax-b\rVert min∥Ax−b∥
where ∥ ⋅ ∥ \lVert\cdot\lVert ∥⋅∥ is any norm.
Reformulation 1:
min ⁡ ∥ y ∥ s . t . A x − b = y \min \lVert y\rVert\\ s.t.\quad Ax-b=y min∥y∥s.t.Ax−b=y

cvx code

%% input data
rng(729);
n = 4;
m = 2*n;
A = randn(m, n);
b = randn(m, 1);
p = 2;
q = p/(p-1);

% reformulation 1: original problem
cvx_begin quiet
    variable x(n)
    minimize (norm(A*x-b, p))
cvx_end

opt1 = cvx_optval;

yalmip code

%% yalmip
x = sdpvar(n, 1);
obj = norm(A*x-b, p);
F = [];
optimize(F, obj);

Reformulation 2:
max ⁡ b ′ v s . t . { ∥ v ∥ ∗ ≤ 1 A ′ v = 0 \max b'v\\ s.t. \begin{cases} \lVert v\rVert^*\leq1\\ A'v=0 \end{cases} maxb′vs.t.{∥v∥∗≤1A′v=0​

cvx code

cvx_begin quiet
    variable nu(m)
    maximize (b'*nu);
    norm(nu, q)<=1;
    A'*nu == 0;
cvx_end
opt3 = cvx_optval;

yalmip code

%% yalmip: reformulation 3
nu = sdpvar(m, 1);
F = [norm(nu, q)<=1, A'*nu==0];
obj = b'*nu;
optimize(F, -obj);

Reformulation

The above problem can be reformulated as:
min ⁡ ( 1 / 2 ) ∥ y ∥ 2 s . t . A x − b = y \min (1/2)\lVert y\rVert^2\\ s.t.\quad Ax-b=y min(1/2)∥y∥2s.t.Ax−b=y
Here we have introduced new variables, and replaced the objective by half its square. The dual of the reformulated problem is
max ⁡ − ( 1 / 2 ) ∥ ν ∥ ∗ 2 + b T ν s . t . A T ν = 0 \max -(1/2)\lVert \nu\rVert_*^2+b^T\nu\\ s.t.\quad A^T\nu=0 max−(1/2)∥ν∥∗2​+bTνs.t.ATν=0
where we use the fact that the conjugate of ( 1 / 2 ) ∥ ⋅ ∥ 2 (1/2)\lVert\cdot\rVert^2 (1/2)∥⋅∥2 is ( 1 / 2 ) ∥ ⋅ ∥ ∗ 2 (1/2)\lVert\cdot\rVert_*^2 (1/2)∥⋅∥∗2​

cvx code

cvx_begin
    variables x(n) y(m)
    minimize (0.5*square_pos(norm(y, p)))
    A*x-b==y;
cvx_end
opt4 = (2*cvx_optval).^(0.5);

yalmip code

%% yalmip: reformulation 4
x = sdpvar(n, 1);
y = sdpvar(m, 1);
F = [A*x-b==y];
obj = 0.5*norm(y, p)^2;
optimize(F, obj);
opt4_yal = (2*value(obj))^(0.5);

cvx: dual

%% cvx: reformulation 5
cvx_begin quiet
    variable nu(m)
    maximize (-0.5*square_pos(norm(nu, q))+b'*nu);
    A'*nu == 0;
cvx_end
opt5 = (2*cvx_optval)^(0.5);

yalmip: dual

%% yalmip: reformulation 5
nu = sdpvar(m, 1);
F = [A'*nu==0];
obj = -0.5*(norm(nu, q))^2+b'*nu;
optimize(F, -obj);
opt5_yal = (2*value(obj))^0.5;

Reference

Convex Optimization S.Boyd Page 268

标签：lVert,cvx,min,0.5,approximation,Lp,nu,norm
来源： https://blog.csdn.net/qq_18822147/article/details/112644556