All Questions
Tagged with nonlinear-programming or nonlinear-optimization
40 questions
1
vote
1
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124
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Limiting Entropy of deterministic sequences - 1
Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$.
Consider distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Given $\{a_i\}_{i=1}^m$, let $\mathcal{P}_{a,m}$ be distribution at ...
- 13.9k
29
votes
1
answer
1k
views
Can a real quartic polynomial in two variables have more than 4 isolated local minima?
This question: "Can a real quartic polynomial in two variables have at most 4 isolated local minima?" came up in this post on Math SE but with no answer so far.
Finding examples of 4 ...
- 431
27
votes
5
answers
2k
views
Is the matrix $\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}$ nonsingular?
Suppose we have a $(2m-1) \times (2m-1)$ matrix defined as follows:
$$\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}.$$
For example, if $m=3$, the matrix is
$$\begin{pmatrix}6 & 20 & 6& 0 ...
- 1,121
6
votes
1
answer
596
views
Combinatorial equation system with exponentially many equations in quadratic many variables
A certain question on graph theory (about the existance of graphs with a certain coloring inherited by perfect matchings) can be translated into the satisfiability problem of a certain set of ...
- 155
5
votes
1
answer
534
views
Minimiser of a certain functional
Let $f_i \in L^1 ([0, 1])$ be a sequence of functions equibounded in $L^1$ norm - that is, there exists some $M > 0$ such that $\|f_i\|_{L^1} < M$.
Define the functional $F: L^1([0, 1]) \to \...
- 6,155
2
votes
1
answer
203
views
Does this maximisation problem admit a finite upper bound?
Let $\mathcal M_2$ be the space of real $2\times 2$ matrices and $\mathcal S_2\subset \mathcal M_2$ be its subset consisting of positive semidefinite elements, i.e. $A\in \mathcal S_2$ iff $A$ is ...
- 1,389
2
votes
1
answer
128
views
Entropy dominance
Let $0<a<b<c$ be distinct positive reals.
Define four different probability distributions:
$$\mathcal{P}_{ab}:P_{a,ab}=\frac{a}{a+b}=1-P_{b,ab}$$
$$\mathcal{P}_{bc}:P_{b,bc}=\frac{b}{b+c}=1-...
- 13.9k
2
votes
0
answers
111
views
Maximization of an integral functional over a closed convex set
I want to maximize $$F(w):=\sum_{1\le i,\:j\le2}\int\lambda^{\otimes2}({\rm d}(x,y))\left(w_i(x)f_j(x,y)\wedge w_j(y)f_i(y,x)\right)g_{ij}(x,y)$$ over the closed convex set $$S:=\left\{w\in{\mathcal L^...
- 167
1
vote
1
answer
148
views
How can we calculate the generalized gradient of $L^2\ni x\mapsto a\min(x(s),by(t))$?
Let $(T,\mathcal T,\tau)$ be a measure space, $a,b\ge0$, $s,t\in T$ and $$f(x):=a\min(x(s),bx(t))\;\;\;\text{for }x\in L^2(\tau).$$
How can we calculate the generalized gradient $\partial_Cf(x)$ of ...
- 167
1
vote
1
answer
101
views
Entropy difference dominance of sequences
Consider a collection of positive integers $\{a_i\}_{i=1}^m$ and the distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$.
Similarly for the collection $\{a_i\}_{i=1}^{m+1}$ form the distribution $q_i=\...
- 13.9k
14
votes
3
answers
2k
views
How to prove $e^x\left|\int_x^{x+1}\sin(e^t) \,\mathrm d t\right|\le 1.4$?
Related question asked by me on Math SE a few days ago: How to prove $e^x\left|\int_x^{x+1}\sin(e^t) \,\mathrm d t\right|\le 1.4$?
A few days ago, somebody asked How to prove $ \mathrm{e}^x\left|\...
- 1,326
11
votes
2
answers
425
views
Maximization of a cubic form over the $14$-dimensional sphere
For any integers $i$ and $j$ such as $1\le i<j\le6$, let $x_{ij}$ be a nonnegative real number.
Is it true that, given the condition
$$\sum_{1\le i<j\le6}x_{ij}^2=1,$$
the sum
$$\sum_{1\le i<...
- 128k
9
votes
3
answers
2k
views
Smallest root of a degree 3 polynomial
Is it true that the smallest root $t$ of the polynomial
$$
20 t^3 - 30 t^2 + (12 - 4 \cos^2 \alpha - 4 \cos^2 \beta - 4 \cos^2 \gamma) t + \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma - 2 \cos \alpha \...
- 171
5
votes
1
answer
332
views
On a certain norm of the identity operator on $\mathbb R^2$
$\newcommand\R{\mathbb R}\newcommand\Q{\mathcal Q}$For mutually orthogonal vectors unit vectors $a=[a_1,\dots,a_n]^T$ and $b=[b_1,\dots,b_n]^T$ in $\R^n=\R^{n\times1}$ (so that $n\ge2$) and for all $x=...
- 128k
5
votes
4
answers
2k
views
Differentiability of eigenvalue and eigenvector on the non-simple case
Let $h:\mathbb{R}^n\to\mathbb{R}^m, n>1$ be a twice continuously differentiable function and $J_h:\mathbb{R}^n\to\mathbb{R}^{m\times n}$ be its jacobian matrix. Let us consider the functions $A(x):=...
- 1,638
5
votes
2
answers
503
views
Minimizing $x_1^2+x_2^2+x_3^2+x_1x_2+x_2x_3+x_3x_1$
Look at the expression
$$
f(x_1,x_2,x_3) = x_1^2+x_2^2+x_3^2+x_1x_2+x_2x_3+x_3x_1.
$$
The numbers $x_1,x_2,x_3$ are non-negative, and I assume that $x_1+x_2+x_3=3$. This is a sum of squares and "...
- 2,197
4
votes
0
answers
287
views
A conjecture about the barycenter of a polytope
Could someone help me with the following conjecture? Thanks a lot!
Suppose I have a polytope $\Delta$ in $\mathbb R^n (n\geq 2)$ with coordinates $(x_1,x_2,\cdots,x_n)$ defined by linear ...
- 1,121
4
votes
1
answer
287
views
Local maxima of the sum of Gaussian functions in *multiple dimensions* are always strict local maxima - prove/disprove/prove conditionally?
This is a follow up of the question in one dimension, that asked to show that the all the maxima of the sum of Gaussian
$$f_n(x):= \sum_{i=1}^{n}e^{-(x-x_i)^2}, x_1 < x_2 < \dots < x_n$$
are ...
- 1,465
4
votes
1
answer
249
views
Does this functional admit an absolute minimizer?
This is a close relative of the following problem.
Let $\Omega$ be an open, bounded subdomain of $\mathbb R^n$ with smooth boundary, and $f_i \in W^{1, \infty} (\Omega)$ a sequence of functions ...
- 6,155
3
votes
1
answer
322
views
Special version of Tonelli’s theorem
I am trying to prove this theorem. I have not found anything similar to it on the internet.
Special version of Tonelli’s theorem
Assume that the functions $f(x,u): [a,b] \times \mathbb{R} \to \mathbb{...
- 159
3
votes
0
answers
214
views
existence of optimal control
I'm looking for an existential result in optimal control for the following class of problems:
Given $T > 0$, $\bar x, \hat x\in\mathbb R^d$, an instantaneous cost function $c:\mathbb R^d\times \...
- 251
3
votes
1
answer
1k
views
Find a line such that sum of perpendicular distances of points to the line is minimized
Given a set of points (column vectors) $S = \{p_1, p_2, \cdots, p_n\} \subset \Re^d$, let $A \in \Re^{n \times d}$ be a matrix of which each row is just $p_i^T$. It is easy to find a unit vector $s_1$ ...
user49129
3
votes
0
answers
255
views
How can we solve this kind of saddle point problem?
I'm trying to solve a saddle point problem of the following form: Let
$(E,\mathcal E,\lambda)$ be a measure space;
$p$ be a probability density on $(E,\mathcal E,\lambda)$ and $\mu:=p\lambda$
$W$ be ...
- 167
2
votes
1
answer
682
views
Maximal minimum for a sum of two (or more) cosines
Please prove (or disprove, and give the correct answer):
$$2 =\mathrm{argmax}_{r\geq 1}\min_{x\in \mathbb{R}}\left[\cos\left(x\right)+\cos\left(rx\right)\right]
$$
In other words, find $r \geq 1$, ...
- 968
2
votes
2
answers
257
views
Reference request on Min-Max theorem
Consider the following min-max problem
$$\inf_{x\in M} \sup_{y\in N} F(x,y),$$
where $F: M\times N\to\mathbb R$ is Lipschitz and $y\mapsto F(x,y)$ is concave for all $x\in M$. Could we derive $\...
user128095
2
votes
1
answer
81
views
Finding a distribution satisfying uncountably many constraints. Any relevant references?
The problem I'm dealing with has the following form. Let $X$ be some uncountable set, and $Y$ be some finite set. Suppose $f: X \times Y \to [0,1]$, and given $\mathcal{H} \subseteq Y^X$, I'm looking ...
- 33
2
votes
0
answers
131
views
Can we conclude $\sup_g\int f_1g\le\sup_g\int f_2g$ from $\int f_1\le\int f_2$ in this situation?
Disclaimer: Please bear with me, the question isn't as complicated as it looks like, but I wasn't able to find any simplification for which no counterexample comes to my find.
Let $(E,\mathcal E,\...
- 167
2
votes
1
answer
1k
views
Can one maximize the spectral norm of a matrix via semidefinite programming?
Consider the following optimization problem:
Maximize $\|X\|_2$, subject to $X$ being Hermitian (or symmetric) and a bunch of semidefinite constraints on $X$. Here, $\|X\|_2$ is the spectral norm of ...
- 1,332
2
votes
1
answer
154
views
Is the optimum of this problem convex in the constraint parameter?
Let $f:\mathbb R^+ \to \mathbb R$ be a smooth function, satisfying $f(1)=0$, and suppose that
$|f|$ grows with the distance from $1$: $|f(x)|$ is strictly increasing when $x \ge 1$, and strictly ...
- 6,741
1
vote
1
answer
271
views
Can we invoke "almost supermartingale" Theorem for deterministic sequences?
Perhaps stupid question.
Question: Can "almost supermartingale" theorem be equally applicable to prove the convergence of some algorithms solving non-random optimization problems?
Attempt ...
- 45
1
vote
1
answer
190
views
Proof of extended version of non-random "almost supermartingale"
In this question, a non-random version of "almost supermartingale" theorem is proved.
Here, I would like to extend/apply the non-random version to the slightly different situation. I wonder ...
- 45
1
vote
0
answers
56
views
Extension of this maximisation problem : finite or not?
$\mathcal M$ is the space of real $d\times d$ matrices and $\mathcal S\subset \mathcal M$ is its subset consisting of positive semidefinite elements. We consider the distance the product space $\...
- 1,389
1
vote
1
answer
174
views
Solve a 2-dimensional optimal control problem via Riccati nonlinear equation
Consider the 2-dimensional optimal control problem of the LQR kind
$$
\min_u \int_0^\infty (x^T Q x + u^TRu) \, dt \quad\text{such that}\quad \begin{cases}\dot x(t) = Ax(t)+Bu(t) \\ x(0) = \...
- 207
1
vote
1
answer
309
views
Numerical estimation of partial derivatives of convolved functions when closed forms do not exist
Summary: Some peak functions are convolutions which may not have a closed form solution. A classical example can that of a Voigt which is a convolution of a Lorentzian and a Gaussian, followed by ...
- 879
1
vote
1
answer
232
views
Maximize a Lebesgue integral subject to an equality constraint
I want to maximize $$\Phi_g(w):=\sum_{i\in I}\sum_{j\in I}\int\lambda({\rm d}x)\int\lambda({\rm d}y)\left(w_i(x)p(x)q_j(y)\wedge w_j(y)p(y)q_i(x)\right)\sigma_{ij}(x,y)|g(x)-g(y)|^2$$ over all choices ...
- 167
1
vote
1
answer
70
views
Entropy dominance of certain restricted sequenes
Say you have positive $\{a_i\}_{i=1}^n$ and you have $p_i=\frac{a_i}{\sum_{i=1}^na_i}$, then assume you have a $C$ such that $C<2a_n\ll\sum_{i=1}^na_i$ (that is $C$ is not very large), then define $...
- 13.9k
1
vote
0
answers
49
views
Limiting Entropy of deterministic sequences - 2
Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$ and the distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Given $\{a_i\}_{i=1}^m$, let $\mathcal{P}_a$ be limiting distribution $m\...
- 13.9k
1
vote
0
answers
99
views
Minimize $\langle(1-\kappa)^{-1}f,f\rangle$ for a parameter-dependent integral operator $\kappa$
I've got a contractive self-adjoint linear integral operator $\kappa$ of the form $$(\kappa g)(x):=g(x)+\int\lambda({\rm d}y)k(x,y)(g(y)-g(x))\;\;\;\text{for }g\in L^2(\mu),$$ where $k$ depends on the ...
- 167
1
vote
1
answer
132
views
Local maxima of the sum of Gaussian functions in *one dimension* are always strict local maxima - proof?
Motivated by this question asked earlier, I was wondering whether one can prove easily that the local maxima of the sum of Gaussians:
$$f_n(x):= \sum_{i=1}^{n}e^{-(x-x_i)^2}, \quad x_1 < x_2 < \...
- 1,465
0
votes
0
answers
63
views
A maximisation problem : finite or not?
Let $\mathcal M_2$ be the space of real $2\times 2$ matrices and $\mathcal S_2\subset \mathcal M_2$ be its subset consisting of positive semidefinite elements, i.e. $A\in \mathcal S_2$ iff $A$ is ...
- 1,389