For question about the concept of dual, either in the sense of vector spaces, topological group or dual problems.
In mathematics, the term "dual" may have several meaning:
- the dual space of a vector space;
- dual in the sense of convex optimization.
Questions tagged [duality-theorems]
1280 questions
6
votes
3 answers
What is the intuition behind Gordan's theorem?
Gordan's theorem:
Exactly one of the following has a solution:
$y^TA > 0$ for some $y \in \mathbb R^m$
$Ax = 0$ ;$ x \geq 0$ for some non-zero $x \in \mathbb R^n$
I am not looking for the proof. I am looking for a way to wrap my head around the…
VeeKay
- 163
5
votes
2 answers
How are addition and multiplication duals of each other?
I don't understand why, in mathematical discourse, addition and multiplication are so often regarded as duals of each other, considering that, for example, $\forall x, y, z\in \mathbb{Z}$ (say),
$$
x \times (y + z) = (x\times y) + (x \times…
kjo
- 14,334
4
votes
0 answers
Am I have right answer about dual problem?
I have some problems to make from primal problem to dual problem.
If I have the primal problem like down below.
Problem
$$ \text{Min} \qquad 3w_1 + 4w_2 + 5w_3 $$
$$ w_1 - w_2 \le ε_1 $$
$$ w_2 - w_3 \le ε_2 $$
$$ …
김종현
- 51
3
votes
1 answer
Given the optimal solution of the dual problem, how to get the optimal solution of the primal problem?
For linear programming, can we get the optimal solution of the primal problem based on the optimal solution of the dual problem? It seems that for dual problems, people are more concerned about the objective function value rather than the optimal…
J Allan
- 31
2
votes
0 answers
Why do mathematical duals occur?
I'm just curious why duality in mathematics is so common?
To be more concrete, here is an example: the harmonic mean as the dual of the mean. It looks almost like the opposite of the mean, yet it behaves quite similarly.
profPlum
- 337
2
votes
1 answer
Understanding a duality pairing of characters
Reading an old paper of Weil's (translation: On certain groups of unitary operators), I'm confused about what should be a rather basic point.
Let $G$ be a locally compact abelian group. Now in addition he assumes that $G$ is isomorphic to its dual…
user21725
1
vote
0 answers
Prove one program to be another's dual program
This is the Primal problem
$\min f_0(x)$,
subject to $f_i(x) \leq 0$, i = $1,2,\ldots,m$
$f(x)$ is a linear program
and my target is to prove that
$\max g(λ)$
subject to $\lambda \geq 0$
where
$g(λ) = \inf L(x, \lambda)$, which is the Lagrange…
richard
- 111
1
vote
0 answers
dual basis and standaard basis and matrix
let $V$ be a vector space from dimantion $n$ and $V^{\star}$ be a map from $V$ to $R$ ($V^{\star}$: $V$$\mapsto$$R$) and $A$ be a matrix from a bilineare form $T$:$V$$\times$V$^{\star}$ $\mapsto$$R$ with respect to the standaard basis $e_1$...$e_n$…
armin
- 79
1
vote
1 answer
Interpretation of the dual of the minimum spanning tree
I am looking at the dual solution obtained for the minimum spanning tree problem. How can I extract the information on which edges are included in the optimal solution?
The dual formulation that I am using is;
$
\begin{align}
~\max &~ z (|V|-1) +…
whitepanda
- 196
1
vote
0 answers
About the dual variable's space in Fenchel's duality
My friends,
I have a question about Fenchel's duality.
Background: According to Wiki, in Fenchel duality, we have the following theorem:
Let $X$ and $Y$ be Banach spaces, $f: X \rightarrow \mathbb{R} \cup \{+ \infty\}$ and $g: Y \rightarrow…
Vincent
- 11
1
vote
0 answers
The dual of an optimal problem subject to infinity norm of a matrix
The optimal problem comes from paper , section 3.1 is
$$
\hat{y}:= \arg\min_y \{ y^T W^{-1} y: ||y - s ||_{\infty} \leq \lambda \tag{1}
$$
where…
yao
- 11
1
vote
0 answers
Duality, smoothing
I'm currently working on the following problem
$$ \min_y \underset{(i,j) \in E}\max \left | \left \langle \tilde{a_{ij}},y \right \rangle \right | \quad \text{such that } \left \langle f,y \right \rangle = 1 $$
which i minimize using Nesterov's…
Zeb
- 11
1
vote
0 answers
Dual Transformation Problem
Let $T:V\rightarrow W$ be a linear transformation, and $T^*:W^*\rightarrow V^*$ the dual transfomation, which is defined by $T^*(f):=f \ \circ T$ for every linear functional $f\ \in W^*$.
$C{w}:W\rightarrow W^{**}$ and $C{v}:V\rightarrow V^{**}$…
Itay4
- 2,166
1
vote
2 answers
Representing a subspace as intersection of kernels
Let $V$ be a linear space of dimension $n$, and let $W$ be a subspace of $V$ of dimension $k$.
Prove that $W$ can be represented as intersection of the kernels of $(n-k)$ linear functionals.
I tried to use the nullity of dual space, but got…
Itay4
- 2,166
1
vote
0 answers
Using Complementary slackness THM to check optimal solution for dual
I Have the following LP :
$\mathcal Min$ $z= 8x_1+4x_2+3x_3$
s.t.
$\mathcal 2x_1-x_2+x_3+x_4$ $\geq 1$ ($y_1$)
$\mathcal x_1+x_2-x_4$ $\geq 2$ ($y_2$)
$x_1$->$x_4$ $\geq 0$
I wrote the Dual for this problem and it's as follow :
$\mathcal Max$…
dev
- 15