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See the Michaelis-Menten equation in biochemistry.

For one source of nonuniqueness, note that $a^2 + b^2 = c^2 + d^2$ is equivalent to $(a+c)(a-c) = (d+b)(d-b)$. For any polynomials $s,t,u,v$ we can take $a+c = st$, $a-c = uv$, $d+b=su$ and $d-b = tv …

If $z_k = x_k y_k$, the quantity you're looking at is $$ y^T D Q D y = \left(\sum_k z_k\right)^2 - \sum_k z_k^2$$ where $Q$ is the $n \times n$ symmetric matrix with diagonal terms $0$ and off-diagon …

In any particular instance, we could in principle compute the probability, though in practice for a nontrivial problem it would be a very difficult computation. Assuming an objective function that is …

Newton's is not really a "minimizing method". If you're using Newton's method to find a root of $f'$, the root you find might be a local minimum, local maximum or neither. To remove the root $x=0$ fr …

By the Poincaré–Bendixson theorem, if $\overline{A}$ is (EDIT: compact and) contained in an open set with no fixed points or periodic orbits, then every trajectory starting in $A$ must eventually exit …

No, because this could easily have several local maxima.

If I understand correctly, you have a linear programming problem $P$ and a basic solution $x^*$ with corresponding basic solution $y^*$ of the dual problem $D$ such that $y^*$ is feasible for $D$ but …

Consider the example $$ A = \pmatrix{0 & 0\cr 1 & 0\cr},\ B = \pmatrix{0 & 1\cr 0 & 0\cr} $$ $A$ and $B$ have eigenvalues $0$, but $1$ is an eigenvalue of $A+B$. I'm not sure in what sense you mean …

The usual linear programming problem requires at least some variables to be nonnegative or has inequality constraints. Here you only have equality constraints and $p \in \mathbb R^n$, so your feasibl …

The gradient is $0$ iff $[x,y] = [1/D-1, 1/C-1]$, at which point the objective value is $1/(C+D-CD)$. If this is in the feasible region $[0, A] \times [0,B]$, it may be optimal. Compare to the optima …

For continuous problems, minimizing a convex function on a convex domain is considered an easy problem, because there is only ever one local minimum, and a local minimum is the global minimum. Findin …

A local minimum of a quasi-convex function is a global minimum. At a critical point where the function is $C^2$, the Hessian matrix is positive semidefinite; if such a critical point is not a local …

The quadratic form is positive semidefinite, so minimizing it requires solving a system of sparse linear equations corresponding to the gradient. The paper mentions that TAUCS was used for solving …

Let $A$ be the event that all points are actually in a square of side $c/\beta$. Then $\text{Pr}(L_n \le c \sqrt{n}|A) = \text{Pr}(L_n \le \beta \sqrt{n})$. Since $P(A) \sim n^2 (c/\beta)^{2n-2} (1-c …