# Questions tagged [duality-theorems]

For question about the concept of dual, either in the sense of vector spaces, topological group or dual problems.

1,123 questions
Filter by
Sorted by
Tagged with
1 vote
8 views

### Is there a result on the maximum value of the dual variable for a parametric LP in terms of the parameters of the LP?

I am working with a linear program of the following kind: \begin{array}[t]{l} \min c^{\top} x\\ s.t.\\ \quad A x = b\\ \quad 0 \le x \le x^{u} \end{array} Can I find out the upper limit of the shadow ...
• 11
1 vote
23 views

### Theorem of Alternatives proof only one of the systems is solvable

Let $A \in R^{nxm}$, $x \in R^n$, $c,y \in R^m$ show that, either I) $Ax=c$ II) $A^Ty=0, c^Ty=1$ is solvable I'm completely new to the theorem of alternatives, so my attempt is: If I is solvable ...
14 views

1 vote
43 views

### Looking for the IBM report: Beckman, Categorical notions and duality in automata theory

I am desperately looking for an old IBM research report, cited at the end of Chapter 3 in Samuel Eilenberg's book Automata, Languages and Machines (1974): Beckman, Categorical notions and duality in ...
• 36.2k
32 views

### Serre duality and symmetric product

Let $X$ be a complex surface, $K$ the canonical divisor, $F$ a rank 2 vector bundle on $X$, $n$ a positive integer, $S^nF$ the $n$-th symmetric product of $F$, $P$ a rational divisor, and $a$ a ...
• 2,034
23 views

### Bidual of a space and Legendre-Fenchel transform

Suppose I have a convex and lower semi-continuous functional $\psi:C_c(\mathcal{X})\to\mathbb{R}$ and that I know of a representation of the form: \Lambda(\mu)= \sup_{f\in C_c(\...
• 805
1 vote
46 views

### Is this duality result true, i.e., $\|f\| := \sqrt{\|f\|_1^2 + \|f\|_2^2} \implies |x|^2 =\inf_{y\in E} \left [ |x-y |_1^2 + |y|_2^2 \right ]$?

Let $(E, |\cdot|_1)$ be a normed vector space (n.v.s) and $(E', \| \cdot \|_1)$ its dual. Let $|\cdot|_2$ be an equivalent norm of $|\cdot|_1$ on $E$, and $\| \cdot \|_2$ its dual norm on $E'$. In a ...
• 1,153
23 views

### Legendre-Fenchel dual on general spaces

I have some questions on duality and Legendre-Fenchel Transform, and I hope some of you can help (perhaps providing some references as well). All the books/references I looked at on functional ...
• 805
17 views

### Proof that the Dual of an LCA Topological Group is LC with the compact-open topology

Let G be a LCA Topological Group, I define it's dual group, $\Gamma$, by the group of continuous characters $$\gamma:G\to\mathbb{T}$$ where $\mathbb{T}$ is the complex unit circle. I want to equip ...
21 views

• 55
25 views

### Convert Dual LP to a Network Flow Problem

Consider the Dual LP: $$\min\sum_{e\in E}c_ey_e$$ s.t.: $$\sum_{e:e\in p}y_e\ge 1\ \ \text{for each}\ \ p\in \mathcal P\\ y_e\ge 0$$ $e$: Represents an edge in $E$ $c_e$: Max flow capacity of an ...
• 477
30 views

### Does a $\mathbb P^1$ with four embedded points have the same dualizing sheaf as $\mathbb P^1$?

Suppose $X$ is a $\mathbb P^1$ with four distinct embedded points, looking locally like $\operatorname{Spec} \mathbb C[x,y] / (x^2, xy)$. I think a subresult of some work I'm doing would be that the ...
• 4,432