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1answer
24 views

Do lagrangian multipliers converge to dual variables in LPs?

Can anybody clarify the following to me? Consider an LP, say a maximization problem, with solution x* and optimal value Z*. Its dual will have optimal value W*=Z* (by strong duality) and optimal ...
2
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1answer
40 views

Connections of Finite groups and quantum groups

I'm a master's student waiting to start my phd in quantum groups and their represenation theory in march 2015. I love representation theory $\textit{per se}$, and looking for references on this work I ...
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0answers
30 views

How to derive dual of this L1 norm approximation problem?

I am working through a question in Convex Optimization by Boyd and Vandenberghe. I've made an image with the original question, and the part of the solution I don't understand: how the dual is ...
2
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1answer
44 views

Isomorphism between $E$ and $Lin(E)$ : infinite dimensional case.

Does $E$ and $Lin(E)$ where $Lin(E)=\{A : E \rightarrow E ∣ A \quad\text{is linear}\}$ are isomorph if $E$ infinite dimensional case ? I know that if $E$ is finite dimension the result is true ...
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1answer
15 views

Understanding a bilinear form problem from Greub's Multilinear Algebra

I read the following problem from exercise sets of Greub's Multilinear Algebra, Chapter I, Sec. 1 Let $E$, $E^*$ be a pair of dual spaces and assume that $\mathit{\Phi}:E^{*}\times E\to\Gamma$ is ...
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0answers
16 views

Dual problem of SDP

Suppose we have the following optimization problem: \begin{array}{l} \mathop {\min }\limits_{{\bf{X}},{\bf{x}}} \,\mathrm{Tr}\left( {{\bf{XA}}} \right) + 2{{\bf{a}}^H}{\bf{x}} + b\\ ...
2
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1answer
32 views

Why does duality imply, that it is enough to consider p>2.

Let $f\in L^p(\Omega, \nu).$ Let $L$ be a self-adjoint operator on $L^p.$ Suppose we want, for every $p>1,$ to prove an inequality $$||Lf||_p\leq C(p)||f||_p,$$ where $C(p)$ is some function ...
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0answers
12 views

Duality gap of nonconvex problem

I have an optimization problem (presumably) nonconvex but the objective funtion is increasing, continuous, and smooth. I also have a set of linear constraints which are fulfilled with equality, i.e., ...
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0answers
19 views

Duality set for $L~p$ spaces, $1<p<\infty$.

I need to show that, given $f \in L^p$, $1<p<\infty$, the duality set $F(f)$ is equal to the point $$\|f\|_p^{2-p}|f|^{p-2}\overline{f}.$$ I have a hint: this is a consequence of convexity of ...
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0answers
49 views

Representation of a linear functional Lipschitz in total variation

Let $\Omega$ be a Borel space and let $\mathcal P(\Omega)$ be the space of all Borel probability measures on $\Omega$ endowed with the topology of weak convergence. Define the total variation metric ...
2
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2answers
87 views

How to find the polar dual of $\{(x,y):y\geqslant x^2\}$ in $\Bbb R^2$?

How to find the polar dual of $\{(x,y):y\geqslant x^2\}$ in $\Bbb R^2$? If the polar dual of a set $A$ is $A^*=\{x\text{ in }\Bbb R^2:ax\leqslant 1\text{ for all }a\text{ in }A\}$. I study from ...
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2answers
26 views

LP: An algorithm to decide whether a polyhedron is a subst of another polyhedron

I've encountered the following question which I am unable to solve: $$ P = \{\vec x | A\vec x \geq \vec a\} \\ Q = \{\vec x | B\vec x \geq \vec b\}\\ P, Q \subseteq R^n $$ Find an algorithm to ...
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0answers
17 views

Conic programming duality - relative interior

Consider the primal/dual conic programming problems $$ \newcommand{\ip}[1]{\left< #1 \right>} \newcommand{\myvec}[1]{\mathbf{#1}} \newcommand{\bvec}[0]{\mathbf{b}} ...
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0answers
7 views

If a voronoi vertex q is an endpoint of a voronoi edge l, then the delaunay polygon dual to q has a delaunay edge dual to l as one of its edges.

If a Voronoi vertex $q$ is an endpoint of a Voronoi edge $l$, then the Delaunay polygon dual to $q$ has a Delaunay edge dual to $l$ as one of its edges. I understand and know what the ...
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0answers
62 views

The dual simplex algorithm

Following is the dual simplex algorithm, adapted from p. 283 of Daniel Solow's "Linear Programming, An Introduction to Finite Improvement Algorithms", Elsevier Science Publishing Co., Inc., 1984. I ...
2
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1answer
38 views

Does the Duality Theorem of Linear Programming hold only in closed convex cones

I've just read the the Duality Theorem of Linear Programming. Here is the proof from my book (and my questions after it): Duality Theorem of Linear Programming: If the primal or dual linear ...
0
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1answer
15 views

how to show weakly convergence is equivalent to point convergence under a certain condition?

Let $\{f_n\}$ in $C[0,1]$ satisfy $\sup_n \sup_{x\in[0,1]}|f(x)|<\infty$. Show that $$f_n \ \ \text{converges weakly to 0} \Longleftrightarrow \lim_{n}f_n(x)=0 \ \ \text{for all }x\in[0,1].$$ ...
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0answers
30 views

dual form of an optimization

Consider the following optimization in primal form $\displaystyle\max_{x_1, \ldots, x_n}\sum_{i=1}^n d_ix_i -\sum_{i=1}^n x_i\log(x_i)$ subject to $a_i\leq x_i\leq b_i$ and $\sum_{i=1}^n ...
1
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1answer
35 views

Why $(1,\textbf{0}) \not\in \{(r,\textbf{w}): r=tz_0-\textbf{c}^T\textbf{x}, \textbf{w}=t\textbf{b}-\textbf{Ax}, \;\textbf{x}\geq\textbf{0}, t\geq0\}$

I'm learning about linear and nonlinear programming and on the chapter about duality I have the following statement and proof I can't understand: minimize $\textbf{c}^T\textbf{x}$ subject to ...
3
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1answer
35 views

Trouble seeing why this is the dual of an LP

$A$ is an $m \times n$ matrix. Using the notation $x=(x_1, \ldots, x_n)$, $z=(z_1, \ldots, z_m)$, and $y=(y_1, \ldots, y_m)$, I'm reading that if the primal LP is $$ \min 0x_1 + 0x_2 + \cdots + 0x_n ...
1
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1answer
31 views

Functional defined on the space of functions with compact support

Let $X=C_c(\mathbb{R})$, the space of functions with compact support, normed with the max norm. Define $\Gamma: X \rightarrow \mathbb{R}$ as: $$ \Gamma f= \int_{- \infty}^{\infty} f(t)dt ...
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0answers
31 views

Dual of a polyhedra vs. dual of an optimalization problem

There are lot of fields where the term duality appear. Is there any relationship between dual of an optimalization problem and dual of a polyhedra?
2
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1answer
40 views

Hecke equivariance in Poincaré duality.

Consider the first singular homology and cohomology groups of a modular curve, $H^1(X,\mathbb{Z})$ and $H_1(X,\mathbb{Z})$. The Hecke algebra acts on both of them and they are dual to each other under ...
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0answers
10 views

Dual representations of affine span of a set of affine transformers.

I am not well versed in the literature of affine transformers or Farkas' Lemma. I just know the basics of the two concepts. Are there any dual representation of affine span of a set of affine ...
5
votes
1answer
81 views

Categorical proof of Pontrjagin Duality?

I would like to ask if there is any reference in which Pontrjagin Duality is proved in a categorical context: I started reading the Pontrjagin Dual entry in nLab, ...
0
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0answers
22 views

dual feasibility question in augmented Lagrangians and the method of multipliers

I am going through Boyd's tutorial on ADMM. My question is basically from Sec 2.3. Consider the optimization problem $$\min.~f(x)~~~~\text{s.t.}~~~Ax = b.$$ Then the Lagrangian is ...
1
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0answers
27 views

Dual Vector Spaces - Evaluation at a Point of a Polynomial Gives a Basis

My question is the following: Let $\{a_0,a_1,...,a_n\}$ be (pairwise) distinct, real numbers. Let $V$ be the vector spaces of all polynomials of degree at most $n$, ie $V = \Bbb P_n$. Let $\phi_j : ...
3
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0answers
39 views

Understanding the duals of $L^{\infty}(K)$ and $C(K)$ and weak-* compactness

Let $(K\subset \mathbb{R}^n,\mathcal{B}(K),\lambda)$ be a measure space, where $\lambda$ is the Lebesgue measure and $K$ is compact. According to Wikipedia (with adapted notation), The dual ...
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1answer
310 views

Dual Simplex Method Example Problem

I have tried to solve this Linear Program: max z = −2*x1 − x2 s.t. −2*x1 + x2 + x3 ≤ −4 x1 + 2x2 − x3 ≤ −6 x1,x2,x3≥0 Choosing -6, ...
1
vote
1answer
60 views

The first Weyl algebra is Calabi-Yau

Why is the Weyl algebra $A_1(k)$ over a field $k$ Calabi-Yau? (My definition of Calabi-Yau is Ginzburg's)
2
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2answers
71 views

Dual Vector Space embedding

Is there an embedding of any vector space $V$ into $V^*$? As far as I know it is not true. The statement that I know of is that there is natural embedding of $V$ into $V^{**}$ Is there any ...
2
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2answers
44 views

Dual Mapping Preserves Linear Independence if and only if Original Mapping is Surjective

Here is my question: Let $V$ and $W$ be finite-dimensional vectors spaces over a field $F$ and $f:V \rightarrow W$ a linear map. Show that $f$ is surjective if and only if the image under ...
2
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0answers
31 views

Reference Request: Algebraic Serre's Duality Theorem for Curves

Serre's Duality Theorem is well known and well studied and, as far as I know, there is a "big" algebraic proof for the general case, which is now kind of standard, and can be found in Hartshorne and ...
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0answers
25 views

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 ...
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0answers
23 views

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$ ...
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0answers
29 views

flabbiness of hyperfunctions

Let $A(\mathbb R^n)$ be the real analytic functions and $\mathscr B(\mathbb R^n)$ the hyperfunctions, dual to $A(\mathbb R^n)$. Further let $W\subset \mathbb C$ be a cone in the complex plane with ...
0
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1answer
66 views

Natural map from vector space to double dual not surjective example

I recognize the fact that the natural map from an infinite dimension vector space $V$ to it's double dual space $V^{**}$ need not necessarily to be surjective because we don't have that the $\dim V$ = ...
0
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1answer
36 views

Maximization with the Dual using the Simplex Method.

I have an exam in a few hours. I need to understand the solution to the following question Find the Maximal to the the following $2 x_1 + 3 x_2$ is the objective function. The constraints are ...
1
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1answer
38 views

How to obtain primal problem from Lagrangian?

If you're trying to optimize $\min_x f_0(x)$ subject to $f_i(x) \leq 0$ then the Lagrangian would be $$L(x, \lambda) = f_0(x) + \sum_i \lambda_i f_i(x)$$ The dual problem is $\max_\lambda g(y)$ ...
1
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1answer
73 views

Existence of a function in Radon Measure

I am having trouble to construct a function $f$ in the following problem. Let $X$ be Locally Compact Hausdorff topological space. Let $\mu$ is a positive Radon measure on $X$ with $\mu(X)=\infty$ ...
3
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1answer
74 views

Proof of Poincare duality in Bott and Tu's book

In Bott and Tu's book, "Differential forms in Algebraic Topology", page 45, Section 5 of Chapter one, he tried to prove the Poincare duality. But I find one step mysterious, namely when he describes ...
3
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1answer
25 views

cofree comodules and embedding

For an $R$-coalgebra C, is it possible for every C-comodule M to be embeded into a C-comodule of the form $\underset{i \in I}{\bigoplus} C$?
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1answer
58 views

Proof of properties of dual cone

Show that if C $\subseteq$ D then $D^*$ $\subseteq$ $C^*$ where * is dual cone operation. Can somebody explain it.
1
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1answer
23 views

Calculating the dual module

Is the dual $\mathbb{Z}$-module of $\mathbb{Z}/n\mathbb{Z}$, that is ${\rm Hom}_{\mathbb{Z}}(\mathbb{Z}/n\mathbb{Z},\mathbb{Z})$ isomorphic to $\mathbb{Z}/n\mathbb{Z}$. Looking at it briefly I think ...
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1answer
84 views

The dual space of normed vector space $X$ is isomorphic to the dual of its completion

Let $\overline{X}$ be Banach space and $X$ be dense subset of it. Show that dual space of X and dual space of $\overline{X}$ are isomorphic. Why these are isomorphic? I don't know how to prove ...
4
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1answer
95 views

Duality in algebraic de Rham cohomology

I am trying to prove that the following is a short exact sequence $$ 0 \rightarrow H^0(X,\Omega_X) \rightarrow H^1_{\text {dR}}(X/k) \rightarrow H^1(X,\mathcal O_X) \rightarrow 0, $$ where $k$ is an ...
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2answers
70 views

Constructing Dual notions via Duality

First of all, I do not have much mathematical background and I have minimal category theory knowledge. I am just trying to understand one or two things about category theory because the concept sounds ...
2
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1answer
56 views

Auto-Langlands dual gruops.

Consider a semisimple Lie group $G$. We define the Langlands dual $\hat{G}$ of $G$ as the group which has as a root system, the root system generated by the coroots of $G$. Recall that given a root ...
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1answer
23 views

Duality in a finite dimension

Let $E$ and $F$ be two finite dimension vector spaces over the same field $K$, $V$ is sub-space of $E$, $L_V(E,F)$ is the set of linear maps from $E$ to $F$ which vanish on $V$. And let $W$ such that ...
1
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1answer
93 views

The convex conjugate of a quadratic form with positive semi definite matrix

I want to find the convex conjugate (Legendre transform) of a quadratic for $1/2x^{t}Qx$ when $Q$ is positive semi-definite. If Q is non-singular, the solution is easy - the gradient is $y-Qx=0$ so ...