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Q:

Maximizing F(A) subject to constraints

This looks like a typical problem of LP, but the constraints are not $A$ but $\bf A$:
Let $A$ be a symmetric matrix with real entries and denote by $\mathbf{A}$ the corresponding bilinear form on $V=\mathbb{R}^n$. Find the matrix $X\in\mathbb{R}^{n\times n}$ such that
$$\langle{} \mathbf{A}{{X}}\rangle\leq{}F(A), \quad \lVert{}{X}\rVert{}\leq{}\mu,$$
where $\langle\cdot,\cdot\rangle$ denotes the inner product, $F(\cdot)$ the cofactor matrix, and $\mu$ a positive real.
Any ideas?

A:

Same as here: Solving constrained LPS: one variable, proof available in Wlodek’s book.
One can indeed conclude that $A=XA’$ and $X$ is a multiple of the identity.
Consider the linear (homogeneous) function $\varphi$ whose gradient is $A’$:
$\displaystyle \varphi(x)=\sum_{i=1}^n x_{i} \lambda_{i}-F(A)$.
Set ${\displaystyle A’}x=C,\; x\geq 0$; now $x$ is a feasible point for the unconstrained problem and its first-order optimality condition is $\varphi_x'(x)=0$. If we show that $\varphi_x'(x)>0$, we will be done.
We have $\varphi’_x(x)=\sum_{i=1}^n \lambda_{i}-2 F'(A) \leq 0$ hence by continuity of $F$ and $A\geq{}0$, $F'(A) \leq

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Personnel – Set off on the adventure of a lifetime as a paramilitary Ranger or a specialized operative to investigate and fight on the frontlines of the apocalypse.

Co-Op – Work together with up to three other players to save humanity in this intense cooperative experience.

First-Person Shooter – Evade the enemy and destroy them with a variety of highly-effective firearms, including submachine guns, shotguns, sniper rifles and more.

Sequel: Apocalyptica

Will we ever see a sequel to Out There on