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

learn more… | top users | synonyms

0
votes
0answers
28 views

Lefschetz Duality for Simplicial Homology

What are the duality theorems for compact, orientable simplicial complexes (possibly the triangulation of a manifold with boundary)? Is there a good way to calculate this boundary just from a list of ...
0
votes
1answer
42 views

Infinite dimensional transpose?

I know that if $L$ is a linear transformation from $V$ to $W$ where $V,W$ are finite dimensional, then we can conclude that the dimension of image (rank) of $L$ is same as that of its transpose, i.e., ...
2
votes
1answer
79 views

Why the dual of some results are true while others are false?

In mathematics, many results have their "dual" versions. In many cases, if a result is true, then its dual is true as well. However, there are some examples while the dual of a true statement is ...
3
votes
1answer
57 views

Integrals of compactly supported functions of positive type

Consider a continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$, supported on $[-1,1]$, of positive type. Assume $f(0) = 1$; what is the "largest" area $\int f\,dx$ that can be achieved? To be ...
1
vote
0answers
23 views

Show that it is a element of $(H^1(\Omega))'$

Let $\Omega$ a open regular subset of $\mathbb{R}^n$, $n\ge 1$. I want to prove that if $u\in L^2(\Omega)$, the distribution $\partial_{x_i} u $ is an element of $(H^1(\Omega))'$ (for $1\le i \le ...
1
vote
0answers
40 views

Legendre transform and Minimax Theorems.

Denote the class of lower-semi-continuous convex functions $f:\mathbb{R}^n\to \mathbb{R}\cup\{\pm\infty\}$ by $Lscx(\mathbb{R}^n)$ ( so that only function attaining the value $-\infty$ is the constant ...
3
votes
1answer
93 views

Question on how to get back “classical” Serre-duality from its derived functor formulation

I'm really new to derived categories, so i hope this isn't a stupid question. I'm trying to understand how the duality described as for example in Residues and Duality of R. Hartshorne, using the ...
0
votes
0answers
29 views

Second derivative of Bregman divergence

Suppose I define an exponential family distribution: $$ f(x; \theta) = \exp \left( \langle x, \theta \rangle - h(x) - \psi(\theta)\right) $$ where the log-partition function is: $$ \psi(\theta) = ...
1
vote
0answers
87 views

p-direct sum and dual spaces

I have a problem with my assignment of Linear Analysis. It should be rather easy and straight-forward, but I have problems =(. Let E and F be normed spaces. For $p \in [1,\infty]$, define the ...
1
vote
1answer
16 views

Solving the euclidian distance squared to kernelize a Lagrangian dual

Homework question, looking for a hint on the following problem: I'm trying to solve this dual lagranging form (which could potentially be wrong already, but let's assume it is right) $\boldsymbol{x}$ ...
0
votes
1answer
61 views

Dual simplex doubt (unrestricted)

I have this two problems and i only want to find the dual form: $\begin{gather} max\hspace{.1cm}z =5x_1+6x_2\\ s.t\hspace{.1cm}x_1+2x_2=5\\ -x_1+5x_2 \ge 3\\ x_2 \ge 0\\ x_1\hspace{.1cm} ...
0
votes
0answers
34 views

Equivalence of Two Statements (Duality Theory, Optimization)

Let $a$ and $a_{1}, ... , a_{m}$ be given vectors in $\mathbb{R^{n}}$. Prove that the following two statements are equivalent. $a)$ For all $x \geq 0$ we have $a^tx \leq\max\limits_{i}(a_{i}^tx)$. ...
0
votes
0answers
87 views

Fenchel dual vs Lagrange dual

Consider the Fenchel dual and the Lagrangian dual. Are these duals equivalent? In other words, is using one of the these duals (say for solving an optimization), would give the same answer as using ...
0
votes
0answers
22 views

Show piecewise linearity of value function

Define, $$v(z) := \text{min}_{y\in\mathbb{R}^p_+} \{qy:Wy=z \}$$ where $W$ is a complete recourse matrix and $q$ is sufficiently expensive, i.e. $q \geq \lambda W$. Then show that $v$ is piecewise ...
1
vote
1answer
49 views

Dual of concave function is convex

If $U(x)$ is strictly increasing and strictly concave and $lim_{x \rightarrow \infty}$ U'(x) = 0, prove that its dual: $$U^{*}(y) = max_x \{U(x) - xy\}$$ is convex. Does anyone know how to prove ...
1
vote
1answer
55 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
votes
1answer
61 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 ...
0
votes
0answers
100 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
votes
1answer
54 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 ...
0
votes
1answer
37 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 ...
0
votes
0answers
28 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
votes
1answer
41 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 ...
0
votes
0answers
18 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., ...
0
votes
0answers
42 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 ...
1
vote
0answers
57 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
votes
2answers
132 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 ...
0
votes
2answers
35 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 ...
0
votes
0answers
30 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}} ...
0
votes
0answers
10 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 ...
1
vote
0answers
101 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
votes
1answer
66 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
votes
1answer
17 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 $$f_n \ \ \text{converges weakly to 0} \Longleftrightarrow \lim_{n}f_n(x)=0 \ \ \text{for all }x\in[0,1].$$ ...
1
vote
1answer
37 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
votes
1answer
40 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
vote
1answer
41 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 ...
0
votes
0answers
33 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
votes
1answer
61 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 ...
0
votes
0answers
21 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
84 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
votes
0answers
38 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
vote
0answers
90 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
votes
0answers
50 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 ...
1
vote
1answer
1k 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
71 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
votes
2answers
93 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
votes
2answers
96 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
votes
0answers
33 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 ...
1
vote
0answers
26 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 ...
1
vote
0answers
37 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$ ...
2
votes
1answer
126 views

Show that $L^1\subsetneq (L^\infty)^*$ [duplicate]

How does one show that $L^1\subsetneq (L^\infty)^*$? I am having trouble in this. Any help would be appreciated.