All Questions

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 ...

I have a minimization problem as follows: $\min\left( \int_0^1\int_0^1\beta(t)\beta(s)G_1(t, s)dtds\right)^{1/2}+\left( \int_0^1\int_0^1\beta(t)\beta(s)G_2(t, s)dtds\right)^{1/2} $ $\texttt{s.t.}\;\;\;...

Given real numbers $y_i's$, consider the following convex optimization problem: $$ \min_{x_i's} \sum_{i=1}^N(y_i-x_i)^2 + \lambda\sum_{i=1}^{N-1}|x_{i+1}-x_{i}|. $$ The paper A Direct Algorithm for 1D ...

Let $(E,\mathcal E,\mu)$ be a probability space, $I$ be a finite nonempty set, $\gamma:(E\times I)^2\to[0,\infty)$ be measurable, $$F_1(g,w):=\sum_{i\in I}\int\mu({\rm d}x)w_i(x)g(x)\sum_{j\in I}\int\...

In Theorem 10.27 of the book Functional Analysis, Calculus of Variations and Optimal Control, there is the following gradient formula: ($\operatorname{co}$ deotes the convex hull). Is there an ...

Say I have a function $L : (W_1,..,W_{H+1}) \rightarrow \mathbb{R}$ i.e it takes a tuple of $n$ matrices of different dimensions and computes a number from them. Then I see being defined a ...