The duality-theorems tag has no wiki summary.
1
vote
0answers
27 views
Is duality theory in optimization as useful as it seems?
I have been reading a lot about nonlinear optimization and duality, and it seems that duality theory is extremely useful. I feel that I am missing some of the negative aspects/difficulties associated ...
0
votes
1answer
80 views
$(U\circ T)^{*} = T^{*}\circ U^{*}$
Let $T : V \longrightarrow W$ and $U : W \longrightarrow Z$ be linear maps. How do I prove that $(U\circ T)^{*} = T^{*}\circ U^{*}$? I'm used to seeing $V^{*}$ not $(U\circ T)^{*}$. Any help is ...
3
votes
0answers
34 views
Duality of $Z(G)$ and $[G,G]$ in representation?
This question and its many wonderful answers illustrate many faces of the duality of $Z(G)$ and $[G,G]$, the centre/ commutator duality of a group.
I was thinking about its manifestation in group ...
0
votes
0answers
23 views
use duality property to find corresponding Fourier or inverse of the function
$g(t)=\frac{1}{a^2+4t^2}$
now i know i should use the transform pair $e^{(-a\mathopen| t \mathclose|)}$ with $\frac{2a}{a^2+\omega^2}$. And with linearity,i can eliminate 2a on numerator. And I stop ...
1
vote
0answers
16 views
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 ...
1
vote
1answer
34 views
Is it true that $C_0^\ast[0,+\infty) = NBV_{loc}[0,+\infty)$
Any function from $BV_{\operatorname{loc}}[0,+\infty)$ defines a continuous linear functional on $C_0[0,+\infty)$. But is it true that any continuous linear functional on $C_0[0,+\infty)$ is given by ...
1
vote
3answers
67 views
What is the dual of this optimization problem?
Consider the points $x_1, \ldots, x_N \in \mathbb{R}^n$, and a (locally bounded, convex) function $f: \mathbb{R}^n \rightarrow \mathbb{R}$.
I am looking for the dual of the following optimization ...
13
votes
1answer
117 views
What is duality?
I have seen some examples of duality. Sometimes applied to theorems, as for example Desargues theorem and Pappus theorem. Sometimes applied to spaces, for example the dual space of a vector space. ...
1
vote
0answers
17 views
about Sobolev imbedding theorem and dual sapce question
Let $D$ be an open and bounded subset of domain $\Omega$, let $f$ be a distribution on $\Omega$. Show that there is an integer $k$ such that the restriction of $f$ to $D$ is in $H^{-k}(D)$.
The hint ...
1
vote
1answer
80 views
Are these convex optimization problems equivalent?
Consider the optimization problem
$$ \mathcal{P}_1: \qquad \min_{x \in \mathbb{R}^n} c^\top x \quad \text{sub. to } \ g(x,y_i) \leq 0 \ \ \forall i = 1,2,...,M$$
where $c \in \mathbb{R}^n$, and ...
1
vote
1answer
42 views
Dual spaces of complex sequences, show the second member is in the dual space
I'm having trouble with some of (ok, most of) the exercises in my 1st-year-master's functional analysis class, so here's one of them, hoping someone can help me out:
If a sequence $(b_n)$ is ...
4
votes
1answer
65 views
If a normed space $X$ is reflexive, show that $X'$ is reflexive.
If a normed space $X$ is reflexive, show that $X'$ is reflexive.
Suppose $X$ is reflexive. Then by definition the Canonical mapping $J : X \to X''$ defined by $x \mapsto g_x$ where $g_x(f) = f(x)$ is ...
0
votes
1answer
55 views
Dual cone of a L1 norm cone?
I am listening to convex optimization lectures and I hear that dual cone of a $L1$ norm cone is a $L-\infty$ norm cone. Can anybody please explain how? I understand that every point in the dual cone ...
1
vote
0answers
47 views
Closed form for Lagrange dual
Can Lagrange dual always be computed in closed form? Can you give me a simple example where the dual is not analytically computable?
4
votes
0answers
86 views
What is the dual space of $C([0,T];X)$ ($X$ Hilbert space)?
What is the dual space of $C([0,T];X)$, where $X$ is a Hilbert space? Is it $\operatorname{BV}([0,T]; X^*)$?
As we know, for $C([0,T])$, the dual space is $\operatorname{BV}([0,T])$, but when it is ...
3
votes
1answer
82 views
Connection between dual space V* and negation P^c
Notice the following similarity between the vector space dual and negation in propositional logic:
$$ V^* \equiv V \rightarrow F $$
$$ P^c \equiv P \rightarrow \bot $$
Is there some general notion ...
4
votes
1answer
138 views
Cech cohomology on Riemann Surfaces (serre duality)
I'm trying to give a more or less easy proof of Serre duality on Riemann surfaces (if you have any hint, a part from Otto Forsters book, go ahead).
I have some notes where it says that Cech cohomology ...
1
vote
1answer
81 views
Directly from primal to dual when primal not in standard form
This is a simple problem, but after spending some hours with linear programs in the primal and its dual form, I still can't do it quite intuitively for LPs which are not in the standard form. I know, ...
2
votes
3answers
79 views
Dual of a Linear Program
\begin{align}
\min_{x} c^Tx \\
s.t.~Ax=b
\end{align}
Note that here $x$ is unrestricted. I need to prove that the dual of this program is given by
\begin{align}
\max_{\lambda} \lambda^Tb \\
...
0
votes
1answer
140 views
Global Min-Max Optimization
When is
\begin{equation}
\min_X \max_Y f(X,Y)
\end{equation}
globally solvable? (i.e. we can find global solution for the optimization problem?)
I am not looking for reformulations.
Is it only when ...
2
votes
1answer
135 views
Weak convergence and weak$^*$ convergence question
Let $X$ be a Banach space and $X^*$ be its dual space. Let $\phi_n\in X^\ast$ and for all $x\in X$ we have $\phi_n(x)\to c\in\mathbb{C}$ as $n\to\infty$. I want to show that the sequence $\phi_n$ has ...
0
votes
0answers
150 views
Dual of max-flow
I have a hard time understanding dual of max flow problems. Can experienced thinkers solve the problems below and possibly give reasoning? Thank you.
$$S\rightarrow A\quad 1$$
$$S\rightarrow B ...
6
votes
1answer
129 views
How does Pontryagin duality fit into the general cohomology theory framework?
Pontryagin duality implies the isomorphic relation of the function space $C(G)$ on a locally compact group $G$ to the function space on it's dual group $\hat G \overset{\sim}{=}\text{Hom}(G,T)$, ...
0
votes
3answers
257 views
Induced Exact Sequence of Dual Spaces
So given a short exact sequence of vector spaces $$0\longrightarrow U\longrightarrow V \longrightarrow W\longrightarrow 0$$ With linear transformations $S$ and $T$ from left to right in the ...
2
votes
1answer
43 views
Understanding a duality pairing of characters
Reading an old paper of Weil's (translation: On certain groups of unitary operators), I'm confused about what should be a rather basic point.
Let $G$ be a locally compact abelian group. Now in ...
2
votes
0answers
113 views
intuitive explanation of Primal-Dual algorithms
I've recently heard of Primal-Dual algorithms and I was wondering if someone could give me an intuitive explanation of it. I searched online, but did not find an intuitive explanation. I'd be glad if ...
4
votes
1answer
96 views
Consequences of Pontryagin Duality?
What are some interesting corollaries and consequences of the Pontryagin Duality theorem? My question can be taken as broadly as you'd like, even up to including any philosophy introduced specifically ...
0
votes
0answers
99 views
Double Dual of $ \ell^\infty$
For my quetion in MO is $\forall X$, $X^{**}$=X$\oplus Y$ for a $Y$ another set I am not really sure in Thomas answer why the first assumption saying that such a $Y$ exist iff the sequence
$0 \to X ...
3
votes
5answers
690 views
Max Min of function less than Min max of function
I can't understand why max min of a function is less than equal to min max of that function i.e
Why
$$\underset{x}{\text{max}}\:\underset{y}{\text{min}} f(x,y) \leq ...
0
votes
1answer
322 views
Maximizing and Minimizing a function
Let $f(x,y)$ be a function such that $f:\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$.
Now we have to maximize $f$ over $x$ and minimize it over $y$ $i.e.\ $
$$\underset{x}{\text{max}}\: ...
5
votes
2answers
387 views
Intersection Pairing and Poincaré Duality
Let $M$ be an $n$-dimensional compact and oriented manifold. Then one can define the intersection pairing $H_k(M,\mathbb Z) \times H_{n-k}(M,\mathbb Z) \to \mathbb Z$. One possible formulation of the ...
2
votes
1answer
342 views
Dual of $C[0,1]$, Hilbert space and Riesz representation.
Let $E=C[0,1]$, space of all real-valued continuous functions on $[0,1]$, $\mathcal{E}$ be its Borel $\sigma$-algebra and $\mu$ a Gaussian measure on $E$. I need help proving the following claim:
...
2
votes
2answers
394 views
How are addition and multiplication duals of each other?
I don't understand why, in mathematical discourse, addition and multiplication are so often regarded as duals of each other, considering that, for example, $\forall x, y, z\in \mathbb{Z}$ (say),
$$
...
0
votes
1answer
115 views
definition: dual of a vector field
Let $X:\mathbb{R}^3\rightarrow T\mathbb{R}^3$ be a vector field, what is the definition of its dual ? I know that the set of vector fields on $\mathbb{R}^3$ forms an $\mathbb{R}$-vector space.
0
votes
1answer
173 views
Underlying assumption in a Primal/Dual table
I just read in one of the questions answered by @MikeSpivey that the following table is provided in Sierksma's Linear and Integer Programming: Theory and Practice, Volume 1, page 144.
...
1
vote
1answer
93 views
Multiple solutions for both primal and dual
If matrix $A$ in an LP (or $A^T$ in its dual) has full row (column- in dual) rank, is it possible that both primal and dual have multiple solutions?
1
vote
2answers
297 views
Duality and the Fourier transform
Regarding Fourier transform, I read that the translation property and frequency-shift property are a duality. What does that mean and why is it true? Is there a physical implications? Thanks.
1
vote
2answers
164 views
What is the topological dual of $C_b(\mathbb{R})$
Consider the Banach space $C_b(\mathbb{R})$ of continuous bounded functions on $\mathbb{R}$ equipped with the sup-norm.
1) Do we know a precise description of its topological dual ...
1
vote
0answers
27 views
Is it possible to solve the argument maximization problem $\arg\max_x \langle x,l\rangle - f_1(x) - f_2(x)$ via convex duality?
I am attemping to solve the argument maximization problem
$$\arg\sup_x \{\langle x,l\rangle - f_1(x) - f_2(x)\}\qquad\qquad\qquad\qquad (1)$$
where the functions $f_1$ and $f_2$ are concave but ...
3
votes
1answer
86 views
What are some nice examples to illustrate that a basis for $V^\ast$ induces a basis for $V$
Let $V$ be a $n$-dimensional vector space over $\mathbf{C}$ and let $(v_1^\ast,\ldots,v_n^\ast)$ be a basis for $V^\ast$. Then, there is a unique basis $(v_1,\ldots,v_n)$ for $V$ such that ...
0
votes
0answers
85 views
Is the dual cone of the dual cone equal to the original cone? [duplicate]
Possible Duplicate:
Dual of a dual cone
I try to prove the following statement:
Let $V$ be a finite-dimensional ordered topological vector space ($V^{**} \cong V$) with a closed positive ...
6
votes
1answer
137 views
Difficulties in Writing the Dual of a Primal Program
I am a student and I am studying the following problem during my spare time. Your comments and suggestions would be helpful.
Given the following primal program:
(Decision variables are $\xi_{v}$, ...
4
votes
1answer
293 views
Conditions for no duality gap in quadratic programming?
Assume $Q \in \mathbb{R}^{n\times n}$, and $b,c,d \in \mathbb{R}^n$. A quadratic programming problem is:
$$ \min_{x \in \mathbb{R}^{n}} \tfrac{1}{2} x^T Q x + c^T x,$$
subject to $A x \leq b, E x ...
3
votes
0answers
61 views
Questions about no duality gap and optimal solutions
My questions are regarding a constrained problem,
$$\min_{x \in X \subseteq \mathbb{R}^n} f(x),$$
subject to $g(x) \leq 0 \in \mathbb{R}^m, h(x) =0 \in \mathbb{R}^k$. Its dual problem is $$\sup_{u ...
1
vote
1answer
183 views
Questions about weak duality theorem
Following are some corollaries regarding the weak duality theorem.
Consider a constrained problem,
$\min_{x \in X} f(x),$ subject to
$g(x) \leq 0$ and $h(x) =0$.
Its dual problem is $\sup_{u \geq ...
1
vote
1answer
343 views
Weak duality theorem and false corollary
Let $A\in\mathbb{R}^{m\times n}, \ c\in \mathbb{R}^n, \ b\in\mathbb{R}^m$ and consider the linear program $$\max \{ c^Tx : Ax\le b\} \ (1)$$
Its dual is $$\min \{ b^Ty : A^Ty=c, \ y\ge 0\} \ (2)$$ The ...
1
vote
2answers
351 views
boolean algebra - theorems
I have a homework question "Show the following is true using theorems. State which theorem you use at each step." This is just one of many problems I have! So, if you can help me with this one problem ...
10
votes
2answers
980 views
Precise connection between Poincare Duality and Serre Duality
The statements of Poincare duality for manifolds and Serre Duality for coherent sheaves on algebraic varieties or analytic spaces look tantalizingly similar. I have heard tangential statements from ...
