I'm interested in

$$\sup_{x, y} \left\| A x + B y\right\|_2$$ subject to $$\left\|x\right\|_2 = \left\|y\right\|_2 = 1$$

where $A$, $B$ and $x$, $y$ are real matrices and vectors, respectively, of compatible sizes. (Generalizations to more than two terms are also interesting)

Does this problem have a name in the literature? (If not, does it reduce to a well-known one?)


Note that this is a stronger constraint than $\left\|\begin{bmatrix}x\\y\end{bmatrix}\right\|_2 = \sqrt 2$

  • Do you mean Frobenius norm or spectral norm ? – loup blanc Oct 12 at 8:49
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    @loupblanc The norms are Euclidean (applied to vectors). I tagged "operator-norms", because the problem looks similar (a generalization of induced matrix norm) – Max Oct 12 at 12:02
  • @Max Interesting question which reminds me of eigenvalue of perturbed matrices. See link – BigM Oct 15 at 22:24
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    I feel this can be done by two applications of the S-lemma (see Chapter 5 of Boyd & Vandenberghe's Convex Optimization). – Suvrit Oct 22 at 2:48

The sets $\{Ax : \|x\|=1\}$ and $\{By : \|y\|=1\}$ are ellipsoids. Hence the set $\{Ax+By : \|x\|=\|y\|=1\}$ is the Minkowski sum of two ellipsoids. Googling for these terms returned this paper which may be interesting to you (although browsing through it I did not find an immediate solution to your problem).

Yan, Yan; Chirikjian, Gregory S., Closed-form characterization of the Minkowski sum and difference of two ellipsoids, Geom. Dedicata 177, 103-128 (2015). ZBL1321.65033.

I do not know if there is a name for this problem. However, we can view this in terms of the usual operator norm.

Given $A \in \mathrm{Mat}_{k \times n}(\mathbb{R}), B \in \mathrm{Mat}_{k \times m}(\mathbb{R})$, consider for $x \in \mathbb{R}^n, y \in \mathbb{R}^m$:

\begin{align*} \underset{\substack{ \lvert \lvert x \rvert \rvert = 1 \\ \lvert \lvert y \rvert \rvert = 1}}{\mathrm{sup}} \lvert \lvert Ax + By \rvert \rvert &= \underset{\substack{ \lvert \lvert x \rvert \rvert \leq 1 \\ \lvert \lvert y \rvert \rvert \leq 1}}{\mathrm{sup}} \lvert \lvert Ax + By \rvert \rvert \\ &= \underset{\substack{ \lvert \lvert x \rvert \rvert \leq 1 \\ \lvert \lvert y \rvert \rvert \leq 1}}{\mathrm{sup}} \, \left \lvert \left \lvert \begin{pmatrix}A & B\end{pmatrix} \begin{pmatrix}x \\ y\end{pmatrix} \right \rvert \right\rvert \\ &= \underset{\left \lvert \left \lvert \begin{pmatrix} x \\ y \end{pmatrix} \right \rvert \right \rvert_\infty = 1}{\mathrm{sup}} \left \lvert \left \lvert \begin{pmatrix}A & B\end{pmatrix} \begin{pmatrix}x \\ y\end{pmatrix} \right \rvert \right\rvert, \end{align*} where $\left \lvert \left \lvert \begin{pmatrix} x \\ y \end{pmatrix} \right \rvert \right \rvert_\infty := \max(\lvert \lvert x \rvert \rvert, \lvert \lvert y \rvert \rvert)$ is the supremum norm from viewing $\mathbb{R}^{n+m}$ as the product $\mathbb{R}^n \times \mathbb{R}^m$. Observe then that the last expression is $\lvert \lvert (\begin{smallmatrix}A & B\end{smallmatrix}) \rvert \rvert_{\mathrm{op}}$, where $\lvert \lvert \cdot \rvert \rvert_{\mathrm{op}}$ is the operator norm induced from $\lvert \lvert \cdot \rvert \rvert_\infty$ on $\mathbb{R}^{n+m}$ and $\lvert \lvert \cdot \rvert \rvert$ on $\mathbb{R}^k$.

Also note that this characterization $\underset{\lvert \lvert x_i \rvert \rvert = 1}{\mathrm{sup}} \, \lvert \lvert \sum_i A_i x_i \rvert \rvert = \lvert \lvert (\begin{smallmatrix} A_1 & \ldots & A_\ell \end{smallmatrix}) \rvert \rvert_{\mathrm{op}}$ works for any finite sum.

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    The usual (Euclidean-induced) norm reduces to singular value decomposition. Are there similarly useful insights one can draw from the fact that this is an induced-norm problem (for a rather esoteric norm)? (BTW I don't think the 0s are needed in the block matrices) – Max Oct 15 at 4:32
  • The supremum norm isn't so esoteric! If both vector norms being considered are supremum norms, then the operator norm is just the row-sum maximum $\max_{i} \sum_{j = 1}^k a_{i,j}$. However, since we have both the Euclidean norm and the supremum norm, I'm not sure what the resulting operator norm should be. (You're right about the zeroes; I've just edited the expression in the answer to remove them. I must have had square matrices in my mind when writing that!) – robinz16 Oct 15 at 21:59
  • Sorry, the last comment should have $\max_i \sum_{j=1}^k \lvert a_{i, j} \rvert$ for the row-sum norm. – robinz16 Oct 16 at 0:53

That follows is a detailed answer to a comment of Federico Poloni.

Since the OP does not seem interested in a successive approximation resolution, I consider the problem from an algebraic point of view. In the sequel, we assume that $n\geq 2$.

Let $A=[a_{i,j}],B=[b_{i,j}]$ be generic $n\times n$ real matrices (the $(a_{i,j}),(b_{i,j})$ are parameters that are mutually transcendental over $\mathbb{Q}$) . We consider the quotient field $K=\mathbb{Q}((a_{i,j}),(b_{i,j}))$.

We consider the problem $(\mathcal{P})$: search the maximum of the function

$f:(x,y)\in \mathbb{R}^n\times \mathbb{R}^n\rightarrow (Ax+By)^T(Ax+By)$ under the conditions $(1)$ $x^Tx=y^Ty=1$.

$\textbf{Lemma 1}$. The Lagrange condition for a local extremum of $f$ under $(1)$ is

$(2)$ $A^T(Ax+By)-ux=0,B^T(Ax+By)-vy=0$ where $u,v\in\mathbb{R}$.

$\textbf{Proof}$. There are real $u,v$ s.t. for every vectors $h,k$, one has

$h^TA^T(Ax+By)+k^TB^T(Ax+By)-uh^Tx-vk^Ty=0$,

That implies $(2)$. $\square$

$\textbf{Lemma 2}$. The absolute minimum of $f$ is obtained on $U=\{(x,y);Ax+By=0,x^Tx=y^Ty=1\}$, an algebraic set of dimension $n-2$.

$\textbf{Proof}$. If $(x,y)\in U$, thet $A(x)=B(-y)\in V=A(S^{n-1})\cap B(S^{n-1})\subset \mathbb{R}^n$. When $A,B$ are in general position, $V$ is an algebraic set of dimension $2(n-1)-n=n-2$ which is diffeomorphic to $U$. $\square$

Therefore the system $\mathcal{S}=\{(1),(2),Ax+By\not= 0\}$ provides all the candidates for $\max(f)$. We go to see that the associated ideal has a Hilbert dimension equal to $0$.

$\textbf{Proposition}$. When $n\leq 6$, $\mathcal{S}$ has generically at most $2n(n+1)$ real solutions; moreover, the complexity of solving $\mathcal{S}$ is the same as the complexity of solving a polynomial of degree $n(n+1)$ with Galois group $S_{n(n+1)}$ over $K$. In particular, the generic problem $\mathcal{P}$ is non-solvable by radicals.

$\textbf{Proof}$. We use Grobner basis theory, the unknowns being $x=[x_i],y=[y_i]$; that's why I guess $n\leq 6$ (the time of calculation for $n=6$ is $2$ minutes).

Generically, the system reduces to a system in the form

$\{P_{2n(n+1)}(x_1)=0,x_i=Q_i(x_1),y_j=R_j(x_1)\}$ where $P_{2n(n+1)}$ is an even polynomial of degree $2n(n+1)$ and $Q_i,R_j$ are polynomials of degree $<2n(n+1)$, all being given explicitly by the software (note that, for $n=6$, the size of the coefficients is huge!). The parity of $P(x_1)=\tilde{P}(x_1^2)$ comes from $f(x,y)=f(-x,-y)$.

Clearly, the complexity lies entirely in the search for roots of $\tilde{P}$. To show that the Galois group of $\tilde{P}$ is $S_{n(n+1)}$, it suffices to use the "specialization theorem" that says that if we choose explicit values for the $(a_{i,j}),(b_{i,j})$, then the Galois group of the obtained $\tilde{P}_0$ is a subgroup of the Galois group associated to the generic $\tilde{P}$.

That can be done (with random choices in $\mathbb{Z}$) for $n\leq 6$. $\square$

$\textbf{Conjecture}$. The result of the above Proposition is true for every $n$.

$\textbf{Remarks}.$ 1. When we know the $O(n^2)$ candidates $(x^i,y^i)$, to obtain the required maximum, it suffices to test the associated values of $f(x^i,y^i)$, that has a total complexity in $O(n^4)$.

  1. However, for every $n\geq 2$, the generic polynomial $\tilde{P}$ is non-solvable (by radicals). That implies that, if we randomly choose $A,B$ (the $(a_{i,j}),(b_{i,j})$ are independent and follow a normal law), then the problem $\mathcal{P}$ is non-solvable (by radicals) with probability $1$. We can calculate an approximation of the roots of $\tilde{P}$ with complexity $O(n^3)$.

  2. Of course, there are couples $(A,B)$ s.t. $\mathcal{P}$ is solvable. For example, when $A,B\in O(n)$, $\max(f)=2^2=4$.

Longer than a comment:(A complete answer is essentially given in the link provided by F. poloni)

$2\times2$ matrices: Let say A and B are given by
\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} a' & b' \\ c' & d' \end{pmatrix} respectively and $(x,y)=(\cos\theta,\sin\theta)$ and $(x',y')=(\cos\phi,\sin\phi)$. You would like to maximize the following

$$\|A(x,y)+B(x',y')\|^2=(a\cos\theta+b\sin\theta+a'\cos\phi+b'\sin\phi)^2+(c\cos\theta+d\sin\theta+c'\cos\phi+d'\sin\phi)^2.$$

Using $-\sqrt{a^2+b^2}\leq a\cos\theta+b\sin\theta\leq\sqrt{a^2+b^2}$ and so on we obtain $$\|A(x,y)+B(x',y')\|^2\leq\big( \sqrt{a^2+b^2}+\sqrt{a'^2+b'^2}\Big)^2+ \Big(\sqrt{c^2+d^2}+\sqrt{c'^2+d'^2}\Big)^2=(a^2+b^2+c^2+d^2)+(a'^2+b'^2+c'^2+d'^2)+2(\sqrt{a^2+b^2},\sqrt{c^2+d^2})\cdot(\sqrt{a'^2+b'^2},\sqrt{c'^2+d'^2})=\|A\|_2^2+\|B\|_2^2+2\big(\|(a,b)\|_2,\|(c,d)\|_2\big)\cdot\big(\|(a',b')\|_2,\|(c',d')\|_2\big) $$ Remark. This upper bound is not necessarily attained. See below the comments.

  • Your upper bound is not attained because, in general, $[\cos(\theta),\sin(\theta)]^T$ cannot be parallel to $[a,b]^T$ and $[c,d]^T$. On the other hand, the coordinates of a solution that realizes the $\max$ are algebraic numbers that are roots of non-solvable polynomials (as a consequence of my answer that I deleted). I see that my censors who downvoted my answer courageously (without giving their name) selectively act on the answers. – loup blanc Oct 22 at 4:21
  • What do you mean by "in general....."?? You have a unit vector. Its equal and so parallel to something with sine and cosine. That's called polar coordinates. max is attained on the right hand side. And in the first line things are equal. And regarding that non math part, I cannot recall if I downvoted your answer or not. Usually people in here dont shout their names if they do so. You have every right to upvote or downvote. Not a big deal. – BigM Oct 22 at 5:16
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    If $a\cos(\theta)+b\sin(\theta)=\sqrt{a^2+b^2}$ then, $c\cos(\theta)+d\sin(\theta)<\sqrt{c^2+d^2}$ except when $\det(A)=0$. – loup blanc Oct 22 at 5:45
  • @loupblanc I agree with you that the upper bound is not attained, but not when you write that the solution are roots of non-solvable polynomials. Why should it be that way? Just because a polynomial has degree 16, it does not mean that it is not solvable with radicals. Especially if that polynomial has a special form (obtained by repeated squarings). – Federico Poloni Oct 29 at 9:19
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    @BigM , yes I agree. Say that it's only an upper bound. – loup blanc Oct 29 at 15:23

I write a new post (again) to show how the great idea of Suvrit (cf. above) allows to calculate the required maximum without using any algorithm; I don't calculate the $(x,y)$ that realizes the maximum.

Let $A,B\in M_n(\mathbb{R})$; we search the maximum of the function

$f:(x,y)∈\mathbb{R}^n\times\mathbb{R}^n→(Ax+By)^T(Ax+By)$ under the conditions (1) $x^Tx=y^Ty=1$.

Let $\alpha=\max(f)$; then one has the implication linking the following $3$ quadratic functions of $[x,y]^T$

$f_1=1-x^Tx\geq 0,f_2=1-y^Ty\geq 0$ $\implies$ $g=\alpha-(Ax+By)^T(Ax+By)\geq 0$.

According to the S-lemma, there exist $u,v\geq 0$ s.t., for every $x,y$, one has $g\geq uf_1+vf_2$.

Let $M_{u,v}=\begin{pmatrix}uI_n-A^TA&-A^TB\\-B^TA&vI_n-B^TB\end{pmatrix}$. The above condition is equivalent to

for every $x,y$, one has $[x^T,y^T]M_{u,v}[x,y]^T\geq u+v-\alpha$.

Necessarily, the symmetric matrix $M_{u,v}$ is $\geq 0$ and, in particular, $u\geq \rho(A^TA),v\geq \rho(B^TB)$. Moreover $x=y=0$ implies that $\alpha\geq u+v$. We search the smallest of the $\alpha$'s realizing these conditons; then $\alpha=u_0+v_0$ where $u_0+v_0$ is the minimal $u+v$ s.t. $M_{u,v}\geq 0$.

Of course, when $u,v>0$ are great enough, $M_{u,v}>0$. Then we seek $u+v$ minimal s.t. $M_{u,v}\geq 0$ and $\det(M_{u,v})=0$, that is, we search the maximal $a$ s.t. the hyperplane $u+v=a$ is tangent to the hypersurface $\det(M_{u,v})=0$. Many experiments "show" that the method works. Clearly, a rigorous proof will be welcome (in particular, in relation to the convexity of the function $\det(M_{u,v})=0$).

$\textbf{Conclusion}$. We obtain $a=\max(f)$ as follows.

i) Calculate the polynomial $g(u,a)=\det(M_{u,-u+a})$.

ii) Calculate the polynomial $h(a)=discrim(g(u,a),u)$, the discriminant of $g$ w.r.t. $u$.

iii) The required $\max$ is the greatest root of $h$.

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