Questions tagged [duality-theorems]
For question about the concept of dual, either in the sense of vector spaces, topological group or dual problems.
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The vector space duality functor from Erdman's Elements of Linear and Multilinear Algebra
In Example 3.2.9 of Elements of Linear and Multilinear Algebra by John Erdman , given a linear map $T \in {\cal L}(V,W)$ where $V, W$ are vector spaces over a field $\mathbb{F}$, the pair of maps $V \...
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Understanding Feasibility, Constraints in the Lagrange Dual Function: A Query on Boyd's Least-Squares Example
I am confused by a basic problem with the Lagrange dual function. As Boyd states,
$g(\lambda, \mu) = \underset{\scriptsize \text{$x \in D$}}{inf}L(x, \lambda, \mu) = \underset{\scriptsize \text{$x \in ...
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Concave maximization over $d$-dimensional simplex.
Can either an analytic solution or the dual be characterized for the following concave maximization:
$v:= \underset{w \in \Delta_d}{\max} \sum^{d}_{i=1}\frac{1}{\sqrt{1+b_i/w_i}}$
where $\Delta_d$ ...
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Isomorphism as $\Bbb{F}_p$ vector space and Proof of local Tate--Duality $H^1(G_K,E)[p]\cong (E(K)/pE(K))^*$
This is a question regarding Theorem $1.4$ of
https://kskedlaya.org/kolyvagin-seminar/duality.pdf.
Let $E/K$ be an elliptic curve over number field $K$.
Let $p$ be a prime number.
The goal of Theorem $...
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Slack Variables and Duality in Convex Optimization
In context of convex optimization the slack variable $\vec{s} \ge 0$ can be used to convert inequality $A \vec{x} \le \vec{b}$ to the equality $A \vec{x} + \vec{s}= \vec{b}$.
Now in wikipedia is ...
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What is the value of dual/lagrangian variable of an infeasible problem?
I found some materials saying that: if the primal is unbounded, then the dual is infeasible; If the dual is unbounded, then the primal is infeasible; but it's possible for both the dual and the primal ...
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Lattice with supermodular height function is lower semimodular
Question
Let $(L,\leq)$ be a lattice of finite length and let its height function $h$ be supermodular, meaning that
$$h(x \wedge y) + h(x \vee y) \geq h(x) + h(y) \quad \forall x,y\in L.$$
Does it ...
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finding dual problem of the function below
I am trying to solve a question from Amir Beck's book "Introduction to nonlinear optimization"
the problem is to find the dual problem with one decision variable of the next minimization ...
- 49
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Is the solution of the dual problem feasible?
Thank you for reading my question.
Assume we have a problem,
$$
\begin{align}
&\min_x f_0(x)\\
s.t.\quad &h_i(x)=0, i= 1,\dots, p\\
&f_i(x)\leq 0, i =1,\dots,m
\end{align}
$$
which is ...
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Duality Results for Convex SDP Programming
Suppose we have an SDP program with a convex (but nonlinear) objective $f(X)$, where $X$ is a positive semidefinite matrix. All other constraints are linear. Does there exist a dual program for such a ...
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Determine the condition of $\lambda$ so that the given linear programing problem has infinitely many optimal solutions [closed]
I'm having the following linear programming problem:
$f = 2x_1 + 4x_2 + \lambda x_3 \text{( to max )} \text{ with constraints } x_i \ge 0\, \forall i = 1,2,3,4$ and
$\left\{\begin{aligned}{}
...
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Cohomology of number fields, Theorem (8.7.9), $Ш^1(G_S,A)\to H^1(G_S,A)\to \prod_{p\in S} H^1(k_p,A)\to \hat{H^1(G_S,A')}$
I'm reading a book 'Cohomology of number fields' Second edition by J.Neukirhi A.Schmidt, K.Winberg.
In the page 511, theorem (8.7.9), there appears suddenly.
$Ш^1(G_S,A)\to H^1(G_S,A)\to \prod_{p\in S}...
- 6,351
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Dual of Quadratic Programming with inequality constraints
I am new to duality concepts and I was reading a document that dualizes the following problem:
\begin{equation}
\min_{x,y} \ ||x-y||^2
\\s.t. \ A_x x \leq b_x, \ A_y y \leq b_y
\end{equation}
into:
\...
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Getting arbitrarily close to $L^1$ norm using $L^\infty$ functions and duality
Let $f\in L^1(\mathbb{R}^d)$ be complex-valued with norm
$$\| f\|_{L^1} >\varepsilon$$
for some $\varepsilon >0$. We know that we can write this using duality as
$$\sup_{\| g\|_{L^\infty=1}}\...
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Proof of Modified Farkas lemma: $y\ge0,A^Ty=0,y^Tb<0$ or $Ax\le b$ has a solution
The proof of Farka's lemma is known.
An important corollary of Farkas lemma is stated as
Modified Farkas Lemma. Let $A$ be an $m\times n$ matrix with values in $R$ and $b\in R^m$. Then exactly one of ...
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