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Results tagged with nonlinear-optimization
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user 6101
Nonlinear objectives, nonlinear constraints, non-convex objective, non-convex feasible region.
2
votes
nonlinear equation problem
Here is an existence argument on the lines of the proof of the Perron-Frobenius theorem via the Brouwer fixed point theorem.
Note that from the equation, since by assumption $a_i>0$ and $K_{ji}\ge0$, …
- 55.5k
4
votes
Accepted
How to find the minimum of the integral?
Consider the quadratic functional $J_T$ on the Hilbert space $H^1(0,T)$
$$J_T(u):=\int_0^T(\dot u+u)^2dt\ ,$$
and let $0<m< M$ be given. The complete picture for the minimization problem of $J_T$ on …
- 55.5k
0
votes
Does this non-negative function, with no stationary points, have only descent directions clo...
Set for instance, in polar coordinates:
$$P(r,\theta):=e^{-\frac1{4r}}\Big(2+\cos\big(\theta+\frac1r\big)\Big).$$
It is quickly checked that this defines a $C^\infty$ function on $\mathbb{R^2}$ which …
- 55.5k
2
votes
Optimization problem restricted to a smaller field?
You are facing the classical optimal transport problem, on which there is a huge literature. Here is a recent comprehensive treatise by Cédric Villani (Warning: 1K pages).
- 55.5k
1
vote
Maximizing a pseudoconcave function in a box
I assume $a$ and $b$ are not linearly dependent: if they are, the objective takes a form $f(s)=\phi(a^Ts)$ and the problem reduces to a linear optimization.
Writing the gradient of $f(s):={\sqrt …
- 55.5k
6
votes
Accepted
Convex Sets and Nearest Neighbors
This is the celebrated Chebyshev problem. The answer is positive in $\mathbb{R}^n$, and still open in the Hilbert space.
- 55.5k
1
vote
Accepted
Partial results on composition of operators such that overall composition is monotone
For $0\le \theta\le\pi/2$, say that $T:H\to H$ is $\theta$-monotone (therefore monotone) iff for all $x$, $y$ in $H$, $(x-y,Tx-Ty)\ge \|x-y\|\,\|Tx-Ty\|\cos\theta$, that is, $x-y$ and $Tx-Ty$ make an …
- 55.5k
0
votes
Solving a quadratic matrix equation with fat matrix
Let $A:=\sqrt X$, which is positive definite by the assumption. So, if $T$ solves $T^TT=X$ then $U:=TA^{-1}$ verifies $U^TU=A^{-1}T^TTA^{-1}=A^{-1}(T^TT)A^{-1}=I$, that is, $T$ writes $T=UA$ with $U$ …
- 55.5k