How to explain or understand ''a function's infinity norm''

Dear all,

I am sorry to bother you. Recently, I have been learning convex optimization and The DCP Rulerset. I clearly understand that I must prove the function is convex before using CVX. That’s difficult.

As we all know, p-norm is convex when p\geq 1 and that is easy to prove according to the properties of the norm.

But I have an optimization problem shown as follows:

g_1(w|m_1,c_1,k_1,m_2,c_2,k_2)=\\ \frac{-m_2w^2+k_2+jc_2w}{\{m_1m_2w^4-(m_1k_2+m_2k_1+m_2k_2+c_1c_2)w^2+k_1k_2\} +j\{-[m_1c_2+m_2(c_1+c_2)]w^3+(c_2k_1+c_1k_2)w\}}
\begin{aligned} &\underset{c_2,k_2,h}{min}\ \ h \\ &subject\ \ to:\ \ ||g_1(w|m_1,c_1(i),k_1(i),m_2,c_2,k_2)||_\infty \leqslant h,\ i=1,...,500 \end{aligned}

where w is a constant vector, m_1,c_1(i),k_1(i),m_2 are constants, c_2,k_2 are scalars.

I just cannot understand a function’s infinity norm means in mathematics. I tried to find some helpful information but failed. So I want to know how to learn the a function’s infinity norm and prove it is convex or not.

Thank you very much in advance.

The Markdown code cannot display all the equations, so I paste the picture.

The infinity norm of a vector is the maximum of the absolute value of the elements of the vector.The infinity norm of an affine argument is convex. The infinity norm is so named because it is the limit of the p-norm as p goes to infinity.

If x is a vector, its infinity norm in CVX can be expressed as norm(x,inf)

The infinity norm of a an affine matrix M is also convex, and is accepted by CVX as norm(M,inf) .

Dear Mark,
Thank you very much! I understand what you said. However, I am still confused about my problem. To my shallow understanding, if c2, k2 are not the variables, g1(w) is a vector without question; but right now c2, k2 are variables, g1(w) is a function of c2, k2. I tried to use norm(g1(w), inf) and some error happened.

% number of variable
d = 3;
% confidence parameter
Beta = 1e-10;
% violation parameter
Epsilon = 0.1;
N = ceil(2 / Epsilon * (log(1/Beta)+d));
% define constants
m1 = 10;
m2 = 0.5;
w1 = 2*pi*50;
zeta1 = 0.01;
% sampling of uncertain parameter
c1 = rand(1,N) * (2*m1*w1*zeta1 - 0) + 0;
k1 = rand(1,N) * (1.2*m1*w1^2 - 0.8*m1*w1^2) + 0.8*m1*w1^2;
% discretization of angular frequency
M = (70-30)/0.1;
w = linspace(30*2*pi,70*2*pi,M);
%% CVX part
    variables c2 k2 h
    subject to
        for j = 1:N
            norm(((-m2*w.^2+k2+1i*c2*w)./((m1*m2*w.^4-(m1*k2+m2*k1(j)+m2*k2+c2*c1(j))*w.^2+k2*k1(j))+1i*(-(m1*c2+m2*(c1(j)+c2))*w.^3+(c2*k1(j)+c1(j)*k2)*w))), inf) <= h;
warning: A non-empty cvx problem already exists in this scope.
   It is being overwritten.

error use  .*  (Line 173) 
Disciplined convex programming error:
      Cannot perform the operation: {complex affine} ./ {complex affine} 

error  ./  (Line 19 ) z = times( x, y, './' );  

error Untitled2 (Line 30)                          
inf) <= h;

g1(w) is not an affine expression, hence CVX does not allow its norm to be used. It doesn’t even allow g1(w)` to be formed.

it appears that “your” problem is not convex.