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

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35 views

Trying to visualize and understand double dual space

Currently I am reading "Finite-dimensional vector spaces" by Paul Halmos. I would have a question regarding the theorem on page 25. It says: If $V$ is a finite-dimensional vector space, then ...
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63 views

Is this an Example of a Dual Space? [on hold]

Is the set of possible bases that I describe $∀(e_1,e_2,e_3)$justSlash$∀(e_1,e_2,F(e_1, e_2))$ F defined V, \times. v=e_1 \times e_2*for any linear vector space of dimension 3* and their linear ...
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19 views

Geometric interpretation of linear programming dual

Is there a geometric interpretation of the linear programming dual in terms of the primal? I feel like without some sort of intuition of it, I don't truly understand it.
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21 views

Finding dual of certain problem

Could anyone help me finding lagrangian function and lagrangian dual of the following problem: \begin{equation} \begin{split} \max_{X}\quad & \operatorname{trace}(H X H^T)\\ \text{s.t} \quad &...
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38 views

Finding shortest vertical segment connecting two sets of intersecting half-planes

Consider two sets of $n$ half-planes each. Denote the sets by $A$ and $B$. How can we find a vertical segment $s$ of a minimum length such that the upper end of $s$ is in the intersection of $A$ and ...
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25 views

If $U$ is a vector subspace of a Hilbert space $H$, then each $x∈H$ acts on $U$ as a bounded linear function $〈x〉$. Is $x↦〈x〉$ injective?

If $H$ is a $\mathbb R$-Hilbert space, then the duality pairing $$\langle\;\cdot\;,\;\cdot\;\rangle_{H,\:H'}:H\times H'\;,\;\;\;(x,\Phi)\mapsto\Phi(x)$$ can be considered as being a mapping $H\times H\...
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39 views

Question about linear functional where x=a*y holds

I have a question regarding linear functionals. In "Finite-dimensional vector spaces" from Paul Halmos there is an exercise which I would like to visualize better. Given two functionals $y$ and $z$ on ...
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6 views

Getting dual values for a constraint in java using cplex

I am trying to get the dual values for a constraint in java using cplex. And I am getting the correct value for a positive co-efficient. But whenever I am multiplying my objective function with a ...
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1answer
49 views

Definition of the Laplacian as an operator from $H_0^1(\Omega)$ to $H_0^1(\Omega)'$

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ $f\in L^2(\Omega)$ and $$\langle f\rangle:=\left.\langle\;\cdot\;,f\rangle_{L^2(\Omega)}\right|_{\...
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20 views

Dual polyhedron & dual cone

From Wiki: Def. of dual of polyhedral (polytope): polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. EX: 2. Def. of dual ...
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Finding Dual of non-standart programming problem

I am working in optimization field. My programming problem is not of the standart form, however it is convex. Objective is nonlinear but concave (log of product). I do maximization. Constaints: ...
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27 views

Can a binary integer linear program be converted to a linear program?

I need to solve the following binary integer linear program: $$ \max \textbf{c}^T \textbf{x} $$ Subject to: $$ \textbf{Ax} \le \textbf{b} $$ Where $ \textbf{b} \in \mathbb{Z}^n $, $\textbf{A} \in \...
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1answer
14 views

Riesz map between Sobolev space and its dual

I'm asked to find the Riesz's map: $$R:H^1_0(\Omega) \rightarrow H^{-1}(\Omega) $$ $$R: u \mapsto F_u \quad \text{s.t.} \quad <F_u,v>_*= (u_F,v)_{H_0^1} \ \ \forall v \in H_0^1 $$ I chose $(u,...
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1answer
42 views

Solve the closed form solution for argmax of $ x^Ty - x^T\ln(x)$

Let $y \in \mathbb{R}^n, \ln(x) = \begin{bmatrix} \ln(x_1) \\ \vdots \\ \ln(x_n) \end{bmatrix} \in \mathbb{R}^n$ Show that $$x^* = \text{argmax}_{x \in \mathcal{D}} \quad x^Ty - x^T\ln(x)$$ ...
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1answer
29 views

Reference: Gaussianity of linear functional of Gaussian process

My question is similar to this one, but I'm looking for a reference rather than derivation. I've been told, inserting my own commentary in square brackets, If you take $X$ in $C([a,b])$ [i.e., $X$...
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12 views

Tannaka Krein duality for finite groups, explicit

Tannaka-Krein duality theory says that the natural mapping $G\rightarrow Aut^{\otimes}(F)$ (see http://mathoverflow.net/questions/155743/can-one-explain-tannaka-krein-duality-for-a-finite-group-to-a-...
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8 views

Why is the duality gap zero for non convex quadratic programming with a single constraint?

From what I read here Convex Optimization, Appendix B and Perfect Duality §2.3, strong duality holds for quadratic programs of the form: $$ \min_x x^\top Ax+x^\top a+\alpha \ni x^\top Bx+x^\top b+\...
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210 views

Which concepts in Differential Geometry can NOT be represented using Geometric Algebra?

1. It is not clear to me that linear duals, and not just Hodge duals, can be represented in geometric algebra at all. See, for example, here. Can linear duals (i.e. linear functionals) be ...
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32 views

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 $S^n \to M^+ \wedge \mathrm{Th}\left(\nu \left(M, S^n\right)\right)$, which gives a ...
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18 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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30 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
15 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
26 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
64 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
67 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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20 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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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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34 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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23 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
17 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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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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17 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
19 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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25 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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13 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
64 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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37 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 ...