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

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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 ...
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27 views

Proximal-type support function properties - nonnegative & strongly convex (proof)

I am reading the paper of Prof. Yurii Nesterov: Primal-dual subgradient methods for convex problems The following part confuses me: $\\$ $\\$ ${\color{red}{E}}\...
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72 views

LASSO constrained and penalised forms: first - order conditions?

The LASSO problem can be formulated in two ways: 1) Constrained formulation: $$ \|Xw-y\|^2\to\min_{w}\\ \text{s.t. } \sum_{i}|w_i|\leq{t}. $$ 2)Penalised formulation: $$ \|Xw-y\|^2+\lambda\left(\...
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32 views

Density and Dual Spaces

Let $X$ be Banach, and $Y \subset X$ a (strict) linear subspace with the property that for any $f,g \in X^*$, if $f \neq g$ then $f|_Y \neq g|_Y$. What can we say about the density of $Y$ in $X$? ...
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34 views

Do we complement Boolean variables in the Dual?

The Principle of Duality states that starting with a Boolean expression, another Boolean expression can be obtained by : 1. Changing OR to AND 2. Changing AND to OR 3. Changing 0 to 1 4. Changing 1 ...
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31 views

Extending $L^{p}$ Duality to $\sigma$-finite Spaces

Let $1 \leq p < \infty$, $(X,\mathcal{M},\mu)$ be a sigma-finite measure space. Let $L$ be a continuous linear form on $L^{p}(X,\mathcal{M},\mu)$. Then, show that $\exists g \in L^{p'}$ such that: ...
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How does changing the cost vector of a primal linear programming problem affect the solution of the dual?

Say the linear program: max $p'x$ such that $Ax=b$ and $x \geq 0$ is primal and dual feasible, and $\bar{u}$ is known to be the optimal solution to the dual. If the $\lambda \ne 0$ times the $i$th row ...
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34 views

Formulating the Dual of a linear program

I have a linear program: Maximize 18x + 12y subject to: x+y <= 20 x <= 12 y <= 16 x,y >=0 I have found ...
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25 views

Proving that there exists a basis for a given dual basis

I need some guidance with the following proof: Let V be a finite dimensional vector space, and V* its dual. Let C = $(f1, ... , fn)\subset{V*}$ be a basis for V*. Let $w\in{V*}$. Prove that there ...
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Nature of the Hessian of the dual function?

I originally posted this over at MathOverflow but it did not receive much (...any) attention. I'm hoping someone can point me in the right direction over here. Consider a nonlinear optimization ...
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32 views

Is this statement true? (characterize elements of dual group)

Let $\mathbb{K}$ be a local field. Definition of local field: Let $\mathbb{K}$ be a field and a topological space. Then $\mathbb{K}$ is called a locally compact field if both $\mathbb{K}^+$ ...
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57 views

If $W$ is a subspace of the finite-dimensional vector space $V$, show that $W^* \cong V^*/A(W)$.

If $W$ is a subspace of the finite-dimensional vector space $V$, show that $W^* \cong V^*/A(W)$. Conclude that $\dim A(W) = \dim V - \dim W$. Hint: Define a map $\psi: V^* \rightarrow W^*$ by $\psi(\...
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Goldstine theorem

Given the embedding $j:X\to X''$ defined by, $$j=(x\mapsto(\phi\mapsto\phi(x)))\,,$$ according to my interpretation of the wikipedia page, Goldstine theorem says the following: $$\overline{jB_X}^{...
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51 views

Show that $A(W)$ is a subspace of $V^*$. Show also that if $W' \subseteq W$ then $A(W) \subseteq A(W')$.

Given the subset $W$ of the vector space $V$, call $A(W)$ = {$\phi\in V^* | \phi$ annihilates $W$} the annihilator of $W$. Show that $A(W)$ is a subspace of $V^*$. Show also that if $W' \subseteq W$ ...
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62 views

If $\phi \in W^*$, show that we can find a $\widetilde{\phi} \in V^*$ such that $\widetilde{\phi}\Bigr|_{W} = \phi$

Let W be a subspace of the vector space $V$. If $\phi \in W^*$, show that we can find a $\widetilde{\phi} \in V^*$ such that $\widetilde{\phi}\Bigr|_{W} = \phi$ So, this is what I know: For $W$ to ...
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88 views

Why are Duals of Two Equivalent compound propositions Equivalent?

I know that if we have two equivalent propositions p and q then p* and q* will also be equivalent where p* and q* are duals of p and q respectively. I am looking for some explanation to why duals of ...
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37 views

Weak closure and dual space

Let $X$ be a normed space and let $W\subset X^*$ be a subspace which separates the points in $X$. Let $\psi \in X^*$ such that $\ker \psi $ is $W$-weakly closed. Show that $\psi \in W$. Any ideas?
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23 views

Lagrange Duality clarification

For a given Linear programming problem \begin{align} max \ c^Tx \\s.t\ Ax \leq b \end{align} and for lagrange multiplier $p\geq0\\$ \begin{align} g(p):= max \{ c^Tx + p^T(b-Ax): x\in \mathbb{R}^n\} ...
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35 views

Dense surjection into double dual

Since not all Banach spaces $E$ are reflexive, $$\{(E^\ast\ni f \mapsto f(x)): x\in E\}$$ is not necessarily the whole of $E^{\ast\ast}$. However, is it always dense in $E^{\ast\ast}$?
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36 views

Continuous inculsion of the dual of continuous included Banach spaces

If $B$ and $C$ are Banach spaces and $B \subset C$ with the inclusion being continuous. If it true that the set of continuous linear functionals on $C$, $C'$, is continuous included in the set of ...
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84 views

Symmetric bilinear forms and (continuous) dual spaces

Let $V$ be an infinite dimensional locally compact vector space over a field $k$ (the field $k$ has the discrete topology and on $V$ we fix the linear topology ). Moreover suppose that on $V$ is ...
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30 views

Is there a non-trivial character on any locally compact Abelian group?

Let $G$ be a locally compact Abelian group. Is there a non-trivial character on $G$? indeed, i want the proof of the existence of a non-trivial character on a locally compact Abelian group.
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Nonzero linear transposition

Let $U,V$ be vector spaces and $U^\intercal, V^\intercal$ their duals. Let $T \in \operatorname{Hom}(V,U)$(A linear map) Denote by $T^\intercal \in \operatorname{Hom}(U^\intercal,V^\intercal)$ the ...
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Duality between $Ax<b$ and another system, but Gordan's just misses

I am trying to show that $$Ax < b$$ Is feasible iff $$ A^T y =0 , b^Ty + s = 0, (y,s) \ge 0, (y,s) \ne 0$$ Is infeasible. Work So Far Now when I try to hit this with Gordan's Lemma I seem ...
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54 views

Prove that the Pontryagin dual of $\mathbb{R}$ is $\mathbb{R}$.

From Wikipedia: the group of real numbers $\mathbb{R}$, is isomorphic to its own dual; the characters on $\mathbb{R}$ are of the form $r \to e^{i\theta r}$. How can I prove this assertion?
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Cartier dual of finite flat group scheme

Let $F$ be a finite flat group scheme over an algebraically closed field $k$ of characteristic 0. Note that I'm not assuming $F$ to be commutative. Does the Cartier dual $F^D$ of $F$ exist? That ...
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Dual maps and Differentiation

Let $D: F[t]_{n} \rightarrow F[t]_{n}$ be the differential map. Let $f \in (F[t]_{n})^{*}$ be the functional assigning to each $y(t) \in F[t]_{n}$ the integral $\int_{0}^{1}y(t) dt$. Compute the ...
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dual basis and existence of non-zero functional

Consider the following statements, given $V$ is a vector space. (A): Given $v \in V, v \neq 0$, there exists $f \in V'$ such that $f(v) \neq0$. (B): Given a set of linearly independent vectors $\{...
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88 views

Show that $T^*$ is one to one $\implies$ $T$ is onto

Let $V$ and $W$ be $F$-vector space, and $V^*$ and $W^*$ be the dual space of $V$ and $W$, respectly. Let $T:V \to W$ be a linear transformation. Define $T^*:W^* \to V^*$ by $T^*(f)=f\circ T$ for all $...
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22 views

knapsack problem using duality

I have this knapsack problem \begin{align} \max {z}=10&x_1+8x_2-13x_3 +10x_4+10x_5 + 5x_6\\[0.4cm] \text{s.t. }\quad\,\,\,7&x_1+6x_2+10x_3+\,\,\,8x_4 +\,\,\,9x_5 + 5x_6\le39 \\[0.2cm]& ...
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Proving that$ B^*$ is a dual basis

Suppose we have: $$ E= \mathbb{R^2}[t]=\{p(t)=a_0+a_1t+a_2t^2\} $$ $$m_0, m_1, m_2$$ Where $m_0, m_1, m_2$ are different real numbers. We define $$F_i:E\rightarrow \mathbb{R}, i = 0,1,2$$ By $$F_i(p)=...
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24 views

How is $v^T(Ax-b) = -b^Tv+(A^Tv)^Tx$?

This might seem a silly question but I am thoroughly confused. Given $c^Tx+v^T(Ax-b) = -b^Tv+(c+A^Tv)^Tx$ I get that it can be written as the second term in the RHS, but what about the first one ...
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73 views

linear transformation $T$ is onto if and only if $T^*$ is one to one

Let $V$ and $W$ be $F$-vector space, and $V^*$ and $W^*$ be the dual space of $V$ and $W$, respectly. Let $T:V \to W$ be a linear transformation. Define $T^*:W^* \to V^*$ by $T^*(f)=f\circ T$ for all $...
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SVM: How to normalize |WX| > 0 into |WX| = 1

Question What are the reason/basis/rationale and the actual steps and design/mechanism behind to do the normalization to convert |WX| > 0 into |WX| = 1 in the process of getting the optimal W for the ...
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38 views

How is the dual of a LPP defined?

The dual of a cone is the set of vectors which have a non-positive dot product with any vector in the original cone.This definition does not seem to be valid in the LPP formulation wherein the ...
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29 views

Why is one of the following always true in a given matrix A?

Consider A, a given matrix. I want to show that exactly one of the following holds: 1) $\exists x\neq 0$ such that $Ax=0$ 2) $\exists p$ such that $p^TA>0$ I tried proving that this is ...
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Dual of the following non-linear program

I am new to optimization and understanding some concept. I understood how duality work and tried applying it some linear programs. I followed the same for non-linear programs but I end up wth a ...
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Dual map is zero if and only if map is zero

A problem from Linear Algebra Done Right (Third Ed): Suppose $W$ is finite dimensional and $T \in \mathcal{L}(V, W)$. Prove that $T=0$ if and only if its dual $T' \in \mathcal{L}(W', V')=0$. I am ...
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Can the non-uniqueness of a linear program's dual feasible set be exploited?

I was originally under the impression that a primal LP had a single corresponding dual feasible set. However, it is possible to alter the primal to an algebraically equivalent form which has a ...
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107 views

Is $C[0,1]$ reflexive?

I.e. is the embedding $C[0,1]\hookrightarrow \left( C[0,1] \right)^{**}$ surjective? I am having a hard time answering that question. It would be enough to find a closed subspace of $C[0,1]$ which ...
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Primal/Dual Simplex methods clarification

I have several questions regarding these methods. Primal Simplex Method Does the pivot element always have to be a positive entry in the table? Does the RHS always have to be positive in the pivot ...
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Does Slater's condition for both primal and dual imply compactness of dual solution set?

Consider a convex optimization problem (P) and its dual problem (D). If the solution set for (P) is compact and Slater's condition holds for both (P) and (D). Is the solution set for (D) compact? My ...
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31 views

problem on annihilators on finite dimensional spaces

Suppose $V$ and $W$ are subspaces of a finite-dimensional vector space $U$. Show that if $V^0 \subset W^0$ then $W \subset V$ This is an exercise problem in Linear Algebra Done Right, 3rd ...
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Is there any comprehensive definition of the opposite category $\mathcal C^{op}$ of a category $\mathcal C$?

In the definition of the opposite category $\mathcal C^{op}$ of a category $\mathcal C$, it is said that : objects remain the same and arrows' directions are changed (that is, and arrow $f:A \to B$ in ...
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Explain the dual problem to D-optimal design problem

Given the following D-optimal design problem $$ \text{minimize } \log \det (\sum_{i=1}^p x_i v_i v_i^T)^{-1}\\ \text{subject to } x \geq 0, {\bf{1}}^T x = 1 $$ Find the dual problem. I don't ...
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Gradient of the dual function for a nonlinear program

I'm attempting to find a proof for a property from Floudas' Nonlinear and Mixed-Integer Optimization book. Consider a nonlinear optimization problem of the form \begin{align} \min_{{\bf x}}&\quad ...
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Duality for modules

I'm doing exercises given in Maclane "Homology", and I have problems with the following exercise. The task is to state the dual of this proposition: "For submodules $S_t\subset B, \ t\in T$ the ...
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38 views

Properties about reflexive space

I'm studying fuctional analysis and specifically reflexive spaces. My textbook has a introductory level, so don't cover so many things. My questions are: 1) If $X$ and $Y$ are isomorphics and $X$ is ...
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73 views

Minimum infinity norm control problem

I am having trouble understanding Example 2 of section 5.9 of Luenberger's Optimization by Vector Space Methods. The problem is to select a current $u(t)$ on $[0,1]$ to drive a motor governed by $$ \...