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For two known matrices $H_p$ and $H_s$, both of dimensions $N \times N$, Can we find two $N \times 1 $ vectors $X_0$ and $X_1$, such that $\| H_p (X_1-X_0)^2 \|$ is maximised and simultaneously $\lVert H_s (X_1-X_0)^2 \rVert$ is minimised? Known constraints are N= total sum of elements of vectors $X_0$ and $X_1$ ($1^TX_0 + 1^TX_1=P$, $1^T$ is row vector of all 1's), (and $P_0$= sum of elements of $X_0$ and $P_1$ = sum of elements of $X_1$), and all elements of $X_1$ and $X_0$ are assumed to be positive. I am also interested in knowing how the solution would change if some thresholds (ex. $\epsilon$) is put to the problem (ex. $\| H_p (X_1-X_0)^2 \| \geq \epsilon ?$)

Aravind Muraleedharan
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  • The solution $X_0$ and $X_1$ should satisfy the constraints. if $H_p^TH_p=A$, then the threshold constraint whould look something like $(X_1-X_0)^TA(X_1-X_0)=\epsilon$. Now this equation has 2N variables. But total equations (constraints) available are only 2. Thats why I think infinitrly many solutions exist now. I was thinking of some available constraint matrix with 2N equations so as to help solve this optimization. If not, then only set of known solutions could be the eigen vectors of Matrix A and epsilon the corresponding eigen values. – Aravind Muraleedharan Jan 12 '24 at 07:57
  • Some related questions: – Aravind Muraleedharan Jan 18 '24 at 17:40
  • Some related questions: [1] https://math.stackexchange.com/questions/623428/solve-for-x-in-y-xtax [2] https://mathoverflow.net/questions/341784/does-the-perron-vector-maximize-xtax-in-the-simplex [3] https://math.stackexchange.com/questions/2935423/constraints-xtbx-1-whats-the-maximum-xtax – Aravind Muraleedharan Jan 18 '24 at 18:10
  • [4] https://math.stackexchange.com/questions/3943210/solve-xtax-k-with-k-cte – Aravind Muraleedharan Jan 18 '24 at 18:21
  • I was thinking of selecting the eigen vector corresponding to the largest eigen value as the solution for X1-X0 and then impose another constriant of X1+X0 to solve X1 and X0. But I was thinking of a strong analytical argument to why choose the largest eigen value's eigen vector. I also see its related to Perron vector – Aravind Muraleedharan Jan 18 '24 at 18:24

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