Questions tagged [duality-theorems]
For question about the concept of dual, either in the sense of vector spaces, topological group or dual problems.
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How does Liu intend his readers to compute $H^0(X,\Omega^1_{X/k})$ of this scheme?
In the chapter on duality theory in Liu's Algebraic Geometry and Arithmetic Curves, there is the following exercise. I'm having trouble seeing how one can apply the duality theory developed in Liu to ...
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Dual and completion of metric spaces
Say we have a metric space $(M,d)$, and we want to complete it in the following sense:
Definition: A completion of $(M,d)$ is a complete metric space $(\widetilde{M},d')$ together with a Lipschitz ...
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The principle of duality for sets
The Wikipedia article on the algebra of sets briefly mentions the following:
These are examples of an extremely important and powerful property of set algebra, namely, the principle of duality for ...
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Finding the dual of this primal LP.
I am going over sample questions from a sample exam, and I got stuck on the following question. I need to determine the dual of this LP:
$min: c^Tx + d^Tu \\
s.t: Ax + Du = b\\
x \ge 0$
$A$ is an $m$...
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Clarification needed for this linear programming problem
I am stuck on the following problem:
Consider the following optimization problem:
$$\text{Maximize }3x+4y+2z\text{ subject to}$$
$$x+y+z\le12$$
$$x+2y-z\le5$$
$$x-y+z\le2$$
$$\text{where } x,y,z \ge ...
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votes
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answer
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weak duality theorem
Studying duality theory I have not found clear this point
considering the primal a minimize problem, if $x$ and $p$ are feasible solution to the primal and to the dual then $p^tb \leq c^tx$
for ...
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1
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Relationship between $(L|_M)^*:N^*\to M^*$ and $L^*|_{N^0}:N^0\to M^0$?
Suppose $L:V\to W$ is a linear transformation, and $L(M)\subseteq N$ for some subspaces $M\subseteq V$ and $N\subseteq W$.
A question I'm reading asks rather open-endedly if there is a relationship ...
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Convergent Series in a dual space
I don't know how solve this problem.
Please I need help.
Let $X =\mathcal{C}[0,1]$ with the uniform norm and let
$\{p_j\}_{j\in\mathbb{N}}$, $\{q_j\}_{j\in\mathbb{N}}\subseteq X$ such
that the ...
user70195
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Duality between K-theory and K-homology in the non-spin^c case.
Let M be a closed manifold. Then there is a cap product $K^\ast(M) \times K_\ast(M) \to K_\ast(M)$ between the K-theory of M and its K-homology. For a definition of it one could see my prior question ...
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How to visualize duality in Linear Programming
In my course of linear programming, we are given the following definitions of a primal/dual model. However, I cannot really get my head around what it actually is? Are we simplifying the problem? Are ...
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Is there a bijection there?
Let $X$ be a normed vector space and $T$ a subset of $X^{\prime} = \mathcal{L}(X,\mathbb{R})$. Then define the set:
$$^{\circ}T\ :=\ \{\;x\in X\ :\ F(x)=0,\ \forall\ F\in T\;\}.$$
(When) Is possible ...
user70195
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vote
1
answer
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How to construct an LP problem that makes a (partial) theorem fail?
I am following a course on linear programming, and one of the exercises calls for an example, that may show that a theorem fails, if a assumption is omitted from the theorem.
The theorem is Theorem 5....
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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 ...
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$(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 ...
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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 ...
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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 \...
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vote
1
answer
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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 ...
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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 ...
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answers
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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. ...
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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 ...
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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 $g:\...
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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 ...
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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 ...
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1
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Dual cone of a $L^1$ norm cone?
I am listening to convex optimization lectures and I hear that dual cone of a $L^1$ norm cone is a $L^{\infty}$ norm cone. Can anybody please explain how? I understand that every point in the dual ...
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Motivation for the development of the double dual of a vector space [duplicate]
I was reading on the double dual of a vector space $V$ recently. I was wondering what applications (within mathematics) there are for this concept and/or what was the motivation for the development of ...
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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?
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What is the dual space of $C([0,T];X)$ ($X$ Hilbert space)? [duplicate]
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 ...
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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 ...
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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 ...
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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, ...
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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 \\
s.t.~\...
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Global min-max optimization
When is
\begin{equation}
\min_X \max_Y f(X,Y)
\end{equation}
globally solvable? I.e., when can we find global solution for the optimization problem?
I am not looking for reformulations. Is it only ...
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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 ...
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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)$, ...
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Induced Exact Sequence of Dual Spaces
So given a short exact sequence of vector spaces $$0\longrightarrow U\longrightarrow V \longrightarrow W\longrightarrow 0$$ With linear transformations $S$ and $T$ from left to right in the non-...
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Understanding a duality pairing of characters
Reading an old paper of Weil's (translation: On certain groups of unitary operators), I'm confused about what should be a rather basic point.
Let $G$ be a locally compact abelian group. Now in ...
user21725
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intuitive explanation of Primal-Dual algorithms
I've recently heard of Primal-Dual algorithms and I was wondering if someone could give me an intuitive explanation of it. I searched online, but did not find an intuitive explanation. I'd be glad if ...
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Consequences of Pontryagin Duality?
What are some interesting corollaries and consequences of the Pontryagin Duality theorem? My question can be taken as broadly as you'd like, even up to including any philosophy introduced specifically ...
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Double Dual of $ \ell^\infty$
For my quetion in MO is $\forall X$, $X^{**}$=X$\oplus Y$ for a $Y$ another set I am not really sure in Thomas answer why the first assumption saying that such a $Y$ exist iff the sequence
$0 \to X \...
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Maximizing and Minimizing a function
Let $f(x,y)$ be a function such that $f:\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$.
Now we have to maximize $f$ over $x$ and minimize it over $y$ $i.e.\ $
$$\underset{x}{\text{max}}\: \...
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Motivation to understand double dual space
I am helping my brother with Linear Algebra. I am not able to motivate him to understand what double dual space is. Is there a nice way of explaining the concept? Thanks for your advices, examples ...
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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 ...
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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:
$E^...
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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),
$$
...
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definition: dual of a vector field
Let $X:\mathbb{R}^3\rightarrow T\mathbb{R}^3$ be a vector field, what is the definition of its dual ? I know that the set of vector fields on $\mathbb{R}^3$ forms an $\mathbb{R}$-vector space.
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Underlying assumption in a Primal/Dual table
I just read in one of the questions answered by @MikeSpivey that the following table is provided in Sierksma's Linear and Integer Programming: Theory and Practice, Volume 1, page 144.
...
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Multiple solutions for both primal and dual
If matrix $A$ in an LP (or $A^T$ in its dual) has full row (column- in dual) rank, is it possible that both primal and dual have multiple solutions?
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Duality and the Fourier transform
Regarding Fourier transform, I read that the translation property and frequency-shift property are a duality. What does that mean and why is it true? Is there a physical implications? Thanks.
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What is the topological dual of $C_b(\mathbb{R})$
Consider the Banach space $C_b(\mathbb{R})$ of continuous bounded functions on $\mathbb{R}$ equipped with the sup-norm.
1) Do we know a precise description of its topological dual $C_b(\mathbb{R})^*$...
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Is it possible to solve the argument maximization problem $\arg\max_x \langle x,l\rangle - f_1(x) - f_2(x)$ via convex duality?
I am attemping to solve the argument maximization problem
$$\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 ...
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