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8 votes
2 answers
2k views

What is Young measure?

I read about Young measures from the book, Weak convergence methods for nonlinear partial differential equations by L.C. Evans. He introduces the concept by the following theorem: Theorem. Assume ...
Devashish Sonowal's user avatar
5 votes
0 answers
199 views

measure of an image under an argmax function

I am trying to find any techniques to analyze the measure of an image of a set under an argmax function. For example, let $\Omega\subset\mathbb{R}^n$ be compact and $\phi:\Omega\to\mathbb{R}$ be ...
Christopher Miller's user avatar
3 votes
1 answer
420 views

measurable selection and values of optimization problem

In general, my problem can be formulated as follows: Let $X$ be a random variable with value in $\mathbb R^2$, and let $G:\mathbb R^2 \times \mathbb R\rightarrow \mathbb R$ be a function which is ...
Ryan's user avatar
  • 325
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 ...
0xbadf00d's user avatar
  • 167
3 votes
0 answers
106 views

Dependency of the Wasserstein distance on the parameter: a differential perspective

Let $\mu(dx)=\sum_{i=1}^np_i\delta_{x_i}(dx)$ and $\nu(dy)=\rho(y)dy$ be two probability measures on $\mathbb R^d$. Consider the $2-$Wasserstein distance below: $$W_2(\mu,\nu)^2 \quad := \quad \inf_{\...
user111097's user avatar
2 votes
1 answer
133 views

Optimal-score partitions

The question about throwing darts asked on the MathOverflow page Sacred Geometry of Chance was not well received, apparently because of "[t]oo much noise around the actual math", as stated in a well-...
Iosif Pinelis's user avatar
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,\...
0xbadf00d's user avatar
  • 167
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^...
0xbadf00d's user avatar
  • 167
2 votes
0 answers
45 views

Maximizing the sum of a decreasing function over a separated set

Fix $d>0$. Let $f:[0,\infty)\to(0,\infty)$ be a decreasing function of $x$ for $x\geq d$. Let $S_d\subset\mathbb{R}^n$ represent a set of points containing the origin such that the (Euclidean) ...
brett1479's user avatar
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 ...
0xbadf00d's user avatar
  • 167
1 vote
0 answers
240 views

Maximize a smooth integral functional by pointwise maximization of the integrand

Let $(E,\mathcal E,\lambda),(E',\mathcal E',\lambda')$ be measure spaces, $I$ be a finite nonempty set, $p,q_i$ be probability densities on $(E,\mathcal E,\lambda)$, $\varphi_i:E'\to E$ be bijective ...
0xbadf00d's user avatar
  • 167
1 vote
0 answers
106 views

Show a Poincaré inequality for a Markov kernel and minimize the Poincaré constant

Let $\tilde\kappa$ denote the transition kernel of the Markov chain generated by the Metropolis-Hastings algorithm with proposal kernel $\tilde Q$ and target distribution $\tilde\mu$ (see definitions ...
0xbadf00d's user avatar
  • 167
0 votes
1 answer
65 views

Is there a general guideline for minimizing $\sup_{y\in H}F(\;\cdot\;,y)$?

Let $H$ be a $\mathbb R$-Hilbert space and $F:H^2\to\mathbb R$. Is there a general guideline for minimizing $\sup_{y\in H}F(\;\cdot\;,y)$? Since the question is rather abstract, feel free to impose ...
0xbadf00d's user avatar
  • 167