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1answer
63 views

Lagrangian dual for the sum of norms

I would like some help in deriving the Lagrangian dual function of a sum-of-norms minimization problem : $\sum{||A_{i}x-b_{i}||}$ when $A_{i}$ are matrices, and $b_{i},x$ vectors. I understand I can ...
1
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1answer
217 views

Orthogonal Complements property

I have a question about how to prove a certain property of orthogonal complements of vector subspaces. Given $\mathrm E$, a vector space over a commutative field k, define: $$\phi\text{ : } \mathrm E ...
2
votes
2answers
189 views

Recovering the solution of optimization problem from the dual problem

In the context of (most of the times convex) optimization problems - I understand that I can build a Lagrange dual problem and assuming I know there is strong duality (no gap) I can find the optimum ...
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0answers
23 views

how to get the dual of that optimization problem

max $1^\top x$ such that $x^\top M x = 0$ and $x_i^2 = x_i$ for all i For the above problem how can I derive the dual form. My main problem is to choose matrix notation or the element-wise notation ...
0
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1answer
92 views

how to construct the Lagrangian dual problem?

The primal optimization problem is, \begin{align*}\min_x\;&f_0(x)\\ \text{s.t.}\;&f_i(x)\le0\\ &h_j(x)=0\end{align*}, to construct the dual problem, I form the Lagrangian, ...
0
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1answer
65 views

Dual space of a complete vector space

Let $(X, \mathcal{F}, \mathbb{P})$ be a Probability space. Consider the space of all functions with topology induced from convergence in Probability. I am interested in knowing the dual space of it. ...
2
votes
1answer
44 views

Riesz Representation Theorem: isomorph

Riesz' Representation Theorem states that every linear functional can be represented by a vector. This shows that the Dual can be ANTILINEARLY and norm preserving identified with the Hilbert Space ...
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0answers
25 views

how to check slater condition for a constrained optimization problem?

Given any optimization problem that you suppose to solve with Lagrange by thrusting strong duality, you need to be sure the Slater Conditions. And I guess there is no algorithmic way to solve for all ...
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2answers
56 views

How to find dual spaces?

I would greatly appreciate it if you could kindly share how to find dual spaces? For example, let X be the vector space of n-dimensional vectors with the Euclidean norm. Prove that X*=X. I know a ...
0
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1answer
71 views

Linear isoparametrics with dual finite elements

The subject presented here is some content of the Wikipedia page about Platonic solids combined with my own experience on Finite Elements.To start with the latter, there is a standard piece of Finite ...
0
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0answers
26 views

Dual of Linear Program

I was wondering what a $symmetric $ dual is. For example, the following is supposed to be a symmetric primal and dual form of LP. Primal : $$ \max c^Tx$$ subject to $$ Ax \le b $$ $$x \ge 0 $$ Dual: ...
1
vote
1answer
95 views

What is dual representation in plain English?

Can someone please explain what is Dual representation in plain English. I read its definition on wikipedia and at many other places but could not develop an intution for it. Please explain in plain ...
2
votes
1answer
160 views

Serre duality as a right adjoint functor

As stated on the wikipedia page, Grothendieck generalized Serre duality by stating that there exists a right adjoint functor $f^!$ to the functor $Rf_!$ when one works within the correct category. ...
1
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1answer
73 views

Dual of this primal optimization problem?

How would one find the dual of the following problem? $min_x 1/2 ||y-x||_2^2 + \lambda||x||_1 $ Can someone please explain to me how to do this since there are no specific constraints?
5
votes
1answer
298 views

Dual to the dual norm is the original norm (?)

I have the following questions about dual norms : How do you prove that the dual of the dual norm is in fact the original norm? This is what I have so far: If I have $\|y\|_* $ as the norm dual of ...
1
vote
1answer
48 views

inf sup duality in Hilbert spaces

Let $Y$ be a Hilbert space, for all $y \in Y$ and $X$ a closed subspace of $Y$, I want to prove the following duality result: $$\inf_{g \in X} || y -g|| = \sup_{(f,X)=0} \frac{(y,f)}{||f||},$$ where ...
0
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0answers
35 views

Finding the dual of a linear program

I have an exam next week and I would like to make sure I am doing this problem correctly and I would also appreciate if somebody could explain to me the purpose of duality? What is the ultimate goal ...
1
vote
1answer
769 views

Is it logically valid to prove DeMorgan's laws using the duality of boolean algebra?

I'm taking an introductory course in boolean algebra, and have been assigned the task of proving DeMorgan's Laws (so, disclaimer, this is homework). One line of reasoning that I came up with would be ...
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2answers
135 views

Properties of the duals of $\ell^1$ and $\ell^{\infty}$

a) True or false: (i) $(\ell^{1})^* = \ell^{\infty}$ (ii) $\ell^1 \subset (\ell^\infty)^*$ (iii) $(\ell^\infty)^* \subset \ell^1$ (iv) $(\ell^1)^{**} \subset \ell^1$ b) Give the set of dual vectors ...
5
votes
1answer
94 views

Dual of a topological vector space. Is it nontrivial?

In the case of normed spaces we know their duals are nonempty using a quick application of the Hahn Banach Theorem. If we step back to the larger class of locally convex spaces, an enthralling ...
4
votes
1answer
408 views

Help me organize these concepts — KKT conditions and dual problem

This is a long question in which I explain my current understanding of certain ideas. If anyone is interested in reading this and would like to provide any commentary/feedback that may help me ...
0
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1answer
33 views

Convergence weakly to measure?

Let $ w_e (x)=\frac {\partial^2}{\partial x_1^2}\sqrt {x_1^2+e}.$ Show that $ w_e $ converges weakly as $e\to 0$ in the dual of $ C (\bar {B_1}) $ to measure $\mu $ I am that dual $ C (B) $ is borel ...
3
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1answer
95 views

Measure dualization

What ways are known to correspond, or transfer, a Borel probability measure $\mu$ over some Banach space $X$ to a Borel probability measure $P$ over $X^{*}$, the dual space? Of course, if $X^{*} = ...
1
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0answers
49 views

Duality in Chebychev approximation

I got messed up with this problem and can't find any clue to solve this. Hope some one here can help me. Let $A$ be an $m \times n$ matrix an let $b$ be a vector in $R^{m}$. We consider the ...
2
votes
1answer
173 views

Duality and the Minimax Theorem

I review LP duality by reading Lecture 7: The LP Duality Theorem. I get the idea how to find the dual LP from primal LP, but my basic knowledge is not enough for finding dual LP for the LP in chapter ...
3
votes
2answers
151 views

Why do we need duality in linear programming or convex optimization?

I'm learning convex optimization, just get started with linear programming, and there is such a thing as duality in linear programming. Here is my problems, why there is a dual problem for a linear ...
6
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2answers
136 views

How does Liu intend his readers to compute $H^0(X,\Omega^1_{X/k})$ of this scheme?

In the chapter on duality theory in Liu's Algebraic Geometry and Arithmetic Curves, there is the following exercise. I'm having trouble seeing how one can apply the duality theory developed in Liu to ...
5
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0answers
188 views

Dual and completion of metric spaces

Say we have a metric space $(M,d)$, and we want to complete it in the following sense: Definition: A completion of $(M,d)$ is a complete metric space $(\widetilde{M},d')$ together with a Lipschitz ...
2
votes
1answer
666 views

The principle of duality for sets

The Wikipedia article on the algebra of sets briefly mentions the following: These are examples of an extremely important and powerful property of set algebra, namely, the principle of duality for ...
0
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1answer
61 views

Finding the dual of this primal LP.

I am going over sample questions from a sample exam, and I got stuck on the following question. I need to determine the dual of this LP: $min: c^Tx + d^Tu \\ s.t: Ax + Du = b\\ x \ge 0$ $A$ is an ...
0
votes
1answer
51 views

Clarification needed for this linear programming problem

I am stuck on the following problem: I have got only confusion over option (1). The options (2) ,(3) are correct and option (4) is wrong. But how can I check whether the problem has more ...
3
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0answers
109 views

weak duality theorem

Studying duality theory I have not found clear this point considering the primal a minimize problem, if x and p are feasible solution to the primal and to the dual then $p^tb \leq c^tx$ for ...
2
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1answer
33 views

Relationship between $(L|_M)^*:N^*\to M^*$ and $L^*|_{N^0}:N^0\to M^0$?

Suppose $L:V\to W$ is a linear transformation, and $L(M)\subseteq N$ for some subspaces $M\subseteq V$ and $N\subseteq W$. A question I'm reading asks rather open-endedly if there is a relationship ...
3
votes
1answer
61 views

Convergent Series in a dual space

I don't know how solve this problem. Please I need help. Let $X =\mathcal{C}[0,1]$ with the uniform norm and let $\{p_j\}_{j\in\mathbb{N}}$, $\{q_j\}_{j\in\mathbb{N}}\subseteq X$ such that ...
3
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0answers
56 views

Duality between K-theory and K-homology in the non-spin^c case.

Let M be a closed manifold. Then there is a cap product $K^\ast(M) \times K_\ast(M) \to K_\ast(M)$ between the K-theory of M and its K-homology. For a definition of it one could see my prior question ...
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0answers
965 views

From a primal problem optimal solution to a dual problem optimal solution

Having this linear programming problem: $minimize$ $ 2x_1 + 9 x_2 + 3x_3$ subject to $-2x_1 + 2 x_2 + x_3 ≥ 1$ $x_1 + 4 x_2 - x_3 ≥ 1$ $x_1, x_2, x_3 ≥ 0$ and its dual ...
2
votes
0answers
66 views

Is there a bijection there?

Let $X$ be a normed vector space and $T$ a subset of $X^{\prime} = \mathcal{L}(X,\mathbb{R})$. Then define the set: $$^{\circ}T\ :=\ \{\;x\in X\ :\ F(x)=0,\ \forall\ F\in T\;\}.$$ (When) Is possible ...
1
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1answer
50 views

How to construct an LP problem that makes a (partial) theorem fail?

I am following a course on linear programming, and one of the exercises calls for an example, that may show that a theorem fails, if a assumption is omitted from the theorem. The theorem is Theorem ...
6
votes
1answer
157 views

Is duality theory in optimization as useful as it seems?

I have been reading a lot about nonlinear optimization and duality, and it seems that duality theory is extremely useful. I feel that I am missing some of the negative aspects/difficulties associated ...
0
votes
1answer
96 views

$(U\circ T)^{*} = T^{*}\circ U^{*}$

Let $T : V \longrightarrow W$ and $U : W \longrightarrow Z$ be linear maps. How do I prove that $(U\circ T)^{*} = T^{*}\circ U^{*}$? I'm used to seeing $V^{*}$ not $(U\circ T)^{*}$. Any help is ...
6
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0answers
68 views

Duality of $Z(G)$ and $[G,G]$ in representation?

This question and its many wonderful answers illustrate many faces of the duality of $Z(G)$ and $[G,G]$, the centre/ commutator duality of a group. I was thinking about its manifestation in group ...
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0answers
42 views

About the dual variable's space in Fenchel's duality

My friends, I have a question about Fenchel's duality. Background: According to Wiki, in Fenchel duality, we have the following theorem: Let $X$ and $Y$ be Banach spaces, $f: X \rightarrow ...
1
vote
1answer
54 views

Is it true that $C_0^\ast[0,+\infty) = NBV_{loc}[0,+\infty)$

Any function from $BV_{\operatorname{loc}}[0,+\infty)$ defines a continuous linear functional on $C_0[0,+\infty)$. But is it true that any continuous linear functional on $C_0[0,+\infty)$ is given by ...
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3answers
166 views

What is the dual of this optimization problem?

Consider the points $x_1, \ldots, x_N \in \mathbb{R}^n$, and a (locally bounded, convex) function $f: \mathbb{R}^n \rightarrow \mathbb{R}$. I am looking for the dual of the following optimization ...
14
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1answer
368 views

What is duality?

I have seen some examples of duality. Sometimes applied to theorems, as for example Desargues theorem and Pappus theorem. Sometimes applied to spaces, for example the dual space of a vector space. ...
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0answers
35 views

about Sobolev imbedding theorem and dual sapce question

Let $D$ be an open and bounded subset of domain $\Omega$, let $f$ be a distribution on $\Omega$. Show that there is an integer $k$ such that the restriction of $f$ to $D$ is in $H^{-k}(D)$. The hint ...
1
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1answer
166 views

Are these convex optimization problems equivalent?

Consider the optimization problem $$ \mathcal{P}_1: \qquad \min_{x \in \mathbb{R}^n} c^\top x \quad \text{sub. to } \ g(x,y_i) \leq 0 \ \ \forall i = 1,2,...,M$$ where $c \in \mathbb{R}^n$, and ...
1
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1answer
86 views

Dual spaces of complex sequences, show the second member is in the dual space

I'm having trouble with some of (ok, most of) the exercises in my 1st-year-master's functional analysis class, so here's one of them, hoping someone can help me out: If a sequence $(b_n)$ is ...
4
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1answer
246 views

If a normed space $X$ is reflexive, show that $X'$ is reflexive.

If a normed space $X$ is reflexive, show that $X'$ is reflexive. Suppose $X$ is reflexive. Then by definition the Canonical mapping $J : X \to X''$ defined by $x \mapsto g_x$ where $g_x(f) = f(x)$ is ...
0
votes
1answer
299 views

Dual cone of a L1 norm cone?

I am listening to convex optimization lectures and I hear that dual cone of a $L1$ norm cone is a $L-\infty$ norm cone. Can anybody please explain how? I understand that every point in the dual cone ...