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

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Duality between Thom space and a manifold embedded into a sphere

In a document https://www.math.purdue.edu/~gottlieb/Bibliography/53.pdf (s. 19) it is mentioned that there is a map $S^n \to M^+ \wedge Th(\nu (M, S^n))$, which gives a Spanier-Whitehead duality ...
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34 views

Dual of a vector space of continuous functions

Let $X=C_{\partial}([a,b])$ be the Banach space of continuous real-valued functions on $[a,b]\subset \mathbb{R}$ such that $f(a)=f(b)=0$ equipped with the supremum norm. I want to now what is its ...
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16 views

Principle of Duality on digraphs: dual properties?

Given an arc $uv$ of a digraph $D$, the dual $D'$ of the digraph $D$ has the arc $vu$. I am trying to find dual properties for digraphs. I could find a page 301 of document on Principle of Duality for ...
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16 views

Demonstration of Cycle-cut duality on elementary graphs?

I want to see examples on the Duality theorem between cycles and cuts on the page 26 of Graph Theory Electronic Edition 2005 by Reinhard Diestel. How to demonstrate the duality theorem between ...
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29 views

min cost flow: getting primal solution from dual

Let $(N,A)$ be directed acyclic graph with arc weights $w: A\rightarrow \mathbb{N}$. I want to solve the following LP: $$ \text{min} \sum _{(i,j)\in A} x(j) - x(i) $$ subject to $x(j) - x(i) \geq w(a)...
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1answer
14 views

Relation between the dual module and the dual of a vector space.

I need to know if there is a relation between the dual module of a subspace U of a finite dimensional vector space V, looked as a G-module, and the dual vector space of U. In this case, G is a finite ...
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1answer
25 views

Connection between complementarity problem and optimization problem?

I do not understand the connection between complementarity problems and optimization problems. I have tried to look at other definitions for complementarity problem to see if that would help me with ...
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1answer
24 views

Linear independence of functionals imply null coeficients on sum?

Let $V$ be a vector space over a field $K$, $V^*$ be it's dual (it's linear functionals), $\{\alpha_1,...,\alpha_n\}$ be a basis for $V$ and $\{f_1,...,f_n\}$ be the dual basis. Any subset $S^*$ of ...
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2answers
51 views

Relation between the dual space, transpose matrices and rank-nullity theorem

Summing up, how can one use linear functionals, transpose matrices, row and column rank equality and annihilators to prove the rank-nullity theorem? While studying linear algebra, I'm trying to get ...
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1answer
66 views

Dual statements involving functors

I know how to construct the dual of a statement concerning objects and morphisms of a category, and understand the duality principle associated, but I am having trouble when various categories and ...
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19 views

How can I prove algebraic and topological dual spaces do not coincide in infinite dimensional normed vector spaces?

I've heard it's enough to give an example of a non continuous linear functional, but I'm kinda confused, because some definitions ask for "bounded" at infinite spaces, does bounded mean continuous in ...
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19 views

To prove an algebraic dual space is diferent from a topologial dual space?

I've got a normed space of infinite dimension, I've proved that if the space is finite dimensional, then the topological and algebraic dual spaces coincide, but the other way is harder, I can't do ...
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1answer
37 views

Diference between dual spaces

What is the diference between Algebraic Dual Space and Topologic Dual Space in Normed Vector spaces with $dim=\infty$
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33 views

I don't understand how the adjoint operator is used in a book that I'm reading

I'm reading Stochastic Differential Equations in Infinite Dimensions and don't understand what the authors do in Chapter 2.3.1. Let me introduce the necessary objects: Let $K$ and $H$ be real ...
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21 views

How can I show the corresponding dual solution is unique when the given primal solution is nondegenerate, basic feasible?

the given problem is to show that if $x_1,...,x_n$ is a nondegenerate basic feasible solution of the primal LP max $\sum_{j=1}^{n}c_jx_j$ s.t. $\sum_{j=1}^na_{ij}x_j\leq b_i, \forall i\in\{1,...,...
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1answer
16 views

Duality theorem between Cycle Space and Cut Space in terms of Matrices?

The book Graphs and Matrices by Bapat formulates linear algebra on graph theory, yet I cannot find important theorems such as Duality theorem between the cycle space and the cut space (Diestel p.26, ...
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1answer
22 views

Extending a bounded linear operator of finite rank

Let $X$ and $Y$ be normed spaces and let $W$ be a subspace of $X$. Assume that $T$ is a bounded linear operator from $W$ to $Y$, that is of finite rank. Show that $T$ can be extended to a bounded ...
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18 views

Represent of multilinear map [duplicate]

Let $V_1,V_2$ be vector space and $\{e_i\},\{\overline e_i\}$ are basis respectively. $\forall ~l\in L(V_1,V_2; F)$ ,why $l$ can be represented as $$ l=\sum\limits_{ij} a_{ij} \omega^i\otimes \...
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2answers
45 views

linear programming infeasibility, dual & primal relation

By the strong duality theorem we know that LP can have 4 possible outcomes: dual and primal are both feasible, dual is unbounded and primal is infeasible, dual is infeasible and primal is unbounded, ...
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16 views

Dual of a point which is in the convex cone of a set, contains the dual cone of that set

Let $\Lambda\subseteq R^n$ contains $m$ elements, where $\lambda_i$ is the $ith$ element, and $co(\Lambda)$ is the smallest convex cone contains $\Lambda$. Also, consider any point $u\in R^n$. Now, I ...
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13 views

Convex solver returns disordered dual variables, how to re-order?

I have the following convex optimization problem: $$ \begin{align*} \min_{x} &f(x) \\ \text{subject to} \\ x_{k+1} &= x_k+\cdots\qquad k=0,1,2,\ldots,N-1 \end{align*} $$ I managed to solve ...
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1answer
18 views

Why the dual and the canonical dual may have the same optimal solution?

For instance let be \begin{cases} \max & 3x_1 &+2x_2 &+4x_3 & -x_4\\ &x_1 &+x_2 &-2x_3&&\le4\\ &2x_1&+3x_3&&-4x_4&\ge 5\\ &2x_1&+x_2&...
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24 views

References for Duality Theory

I was wondering if anyone had any recommendations for Duality Theory. I've touched on Duality before in various courses but it's coming up quite a lot in my studies at the moment. I guess what I'm ...
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34 views

Some intuition about the second dual of $l^1$

In Functional Analysis we treated the Hahn-Banach theorem, and if I understood correctly, the dual space of $l^1$ (space of all absolutely summable sequences) is isomorphic to the space of all bounded ...
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12 views

Dual of cartesian product of duals of polytopes

I am working on the following problem. Let $P\subseteq \mathbb{R}^d, Q\subseteq \mathbb{R}^e$ be full-dimensional polytopes, both with the origin in the interior. Describe $(P^{\circ }\times Q^...
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1answer
33 views

Characterization of elements of $X^*$ via the Radon-Nikodym theorem

I am reading Lindenstrauss' Classical Banach Spaces II and I am having trouble with the following characterization of integrals. First a couple of preliminary definitions: Let $(\Omega, \Sigma, \...
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1answer
35 views

Converting generic linear problems into their dual

I'm revising how to do dual problems in linear algebra. I'm very weak in Linear programing but I struggle to cope with the topic during lectures and assignements. I have to convert the following ...
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1answer
36 views

Linear Optimization proof. Duality proof.

I need help with this problem. The exact problem is in this link http://d2vlcm61l7u1fs.cloudfront.net/media%2F959%2F959d289e-6f26-4e21-875e-bb71f3f5a49f%2Fphprimn1q.png Sorry for the poor formatting. ...
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1answer
40 views

If $H$ is a Hilbert space, are we able to identify the derivative ${\rm D}f(x)$ at some $x\in H$ of a differentiable $f\in H'$ with an element of $H$?

I'm confused about some equation I've seen in a book and want to write down some thoughts. I would appreciate, if somebody could tell me whether I'm terribly mistaken or not: Let $(H,\langle\;\...
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1answer
63 views

Solving a linear program thanks to complementary slackness theorem

Using the complementary slackness theorem, say if the following basis optimal: $$x_1*=0=x_5*,x_2*=4/3,x_3*=2/3,x_4*=5/3$$ \begin{cases} \max & 7x_1 &+6x_2&+5x_3&-2x_4&+...
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7 views

Solving the primal problem via dual, the solution of the Lagrangian function L(x,λ∗) has to be unique.

Solving the primal problem via dual, the solution of the Lagrangian function L(x,λ∗) has to be unique. This is a subtlety requirement made by convex optimization book written by Prof. Boyd. Who can ...
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31 views

If the primal is unbounded, then the dual is infeasible.

In the context of duality in linear programming, prove that If the primal is unbounded, then the dual is infeasible. My book says that this is a corollary to complementary slackness. What's ...
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1answer
17 views

Perfect Pairing, non-degeneracy and dimension.

On this wikipedia entry https://en.wikipedia.org/wiki/Bilinear_form#Different_spaces it tells us that if $B: V \times W \to K$ is a bilinear map, then In finite dimensions, [a perfect pairing] is ...
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28 views

Quadratic dual help

I'm trying to figure out the whether the quadratic dual of the following algebra is anti-commutative. My algebra is $A=T(V)/J$ where $J=\langle I\rangle$ and $I\subseteq\bigwedge^2(V)$. Firstly I ...
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Variant of conjugate function: $V(s) = \underset{x}\max \{\langle s,x-x_0\rangle-\beta f(x)\}$

Consider one variant of conjugate function: $$V(s) = \underset{x}\max \{\langle s,x-x_0\rangle-\beta f(x)\}$$ You can think $s$ as a linear functional. If I do the following steps: \begin{...
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15 views

Unbounded variables and dual of a linear program

I have to find the dual of \begin{cases} \max & -x_1 &-2x_2+x_3\\ & -3x_1 &+x_2&\le-1\\ & x_1 &-x_2&\ge 1\\ & -2x_1 &+7x_2&\le6\\ & -5x_1 & +...
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28 views

How to form a dual problem in convex optimization (in a broad view)

After reading some papers, this problem confuses me. There are different forms of dual problem to the primal problem: $$\underset{x}\min \ \ f(x)$$ where $f(x)$ is a convex function. By ...
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1answer
30 views

The dual space of a dual space

Eh, I don't quite understand the first question. Can someone explain it? And for the second question, can I say that they have the same dimension. And since the kernel is ${0}$, it is injective. ...
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3answers
25 views

vProve that the $\phi's$ form a basis for $V^*$

Hey guys. I've learned linear algebra before, but I kinda forget the part about dual space. For this problem, I think that because $V^*$ has the same dimension as V, which is n in this problem. And ...
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2answers
38 views

Find the dual of the lp problem

The problem given is and I need to find the dual: Min $Z=x_1$ st. $x_1+x_2 \leq 4$ $x_2 \geq 0$ So this is what I did, I said: Let $x_1=x_1'-x_1''$ where $x_1',x_1'' \geq 0$ So now ...
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46 views

Dual curve of an algebraic curve in affine coordinates

$F$ is an irreductible algebraic curve and we consider the application that sends a non-singular point $(a : b : c)$ in homogenius coordinates, to $\phi(a, b, c) = \bigl(f_x(a, b, c), f_y(a, b, c), ...
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10 views

Convert the Matching Polytope LP to Dual Program

For my course in discrete optimization I am studying about Polytopes and their dual programms. They state that the convex hull of Perfect Matchings in grahph $G=(V,E)$ is given by: $$ x\geq 0\\ x(\...
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1answer
46 views

Is this dual transform incidence and order preserving?

I am trying to understand duality explained in the book Computational Geometry Algorithms and Applications, 3rd Ed - de Berg et al. Unfortunately, I have some problem of solving a question in this ...
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23 views

Where does the duality comes from in linear programing and can we get the optimal basis from it?

$$\begin{cases} \max & c^Tx\\ & Ax\le b\\ & x\ge 0 \end{cases}\Leftrightarrow \begin{cases} \min & y^Tb\\ & y^TA\ge c^T\\ & y^T\ge 0 \end{cases}$$ Then we come to the ...
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9 views

Finding the lower bound of a linear program with the duality method

The issue I have some difficulties understanding the lower bound of a program when applying the duality method. It seems that it comes from $$c^T\underbrace{\le}_{x\ge 0\\y^TA\ge c^T} y^TAx\...
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9 views

Lower bound of averaging gradient method (Prof. Yurii Nesterov's paper)

I am reading the paper of Prof. Yurii Nesterov: Primal-dual subgradient methods for convex problems The last inequality confuses me: (p.231) Note: 1. The ...
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1answer
18 views

What varialbes enter the $\min/\max$ in dual problem?

Having the following linear program: \begin{cases} \max & -x_1 & -2 x_2&+x_3\\ & -3 x_1 &+x_2 & &\le -1\\ & x_1 &-x_2 & &\ge 1\\ &-2x_1 & +7 x_2 &...
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1answer
16 views

Relationship between Primal and Dual problems

Considering the following program: \begin{cases} \max & 8x_1 & + 3x_2\\ & x_1 &-6x_2&\ge2\\ & 5x_1 +&7x_2&=-4\\ &x_1&&\le 0\\ && x_2&\ge 0 \end{...
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15 views

Primal + Dual relation with Complementary Slackness.

If let's say there exist an optimal solution to the primal with $x_1 = 0$, what can we deduce about the dual? Here is my attempt to answer this particular question: Since there exist an optimal ...