# 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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### Can we use the duality notation such that the second variable is an element of the measure space?

I read article New Sequential Compactness Results for Spaces of Scalarly Integrable Functions by Erik J. Balder, on page 8 : Author defined the function $a: T\times E\to \mathbb{R}$ by the usual ...
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### Converting primal to dual

\begin{align} &\text{minimize } &z = 7x_1 – 4.2x_2 – 2x_3 + x_4 \\ &\text{subject to } &x_1 + 2x_3 + 2x_4 &\leq 20 \\ &&x_1 + 2x_2 &= 18 \\ &&x_1 + 2x_3 – 3x_4 &...
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### Projective applications and projective duality

Let $P$ be a projective plane, and $p_{1},p_{2}$ different points of $P$. Consider now a projective line $L\subset P$ not passing for the aforementioned points. Let $F(p_{i})\,,\, i \in\{1,2\}$ be the ...
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### Deriving the dual linear program.

The last optimization subject I took was so long ago, so I just grabbed a guide online to derive and verify the following claim from a lecture note This is the guide: http://pierrepinson.com/31761/...
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### Theorem dual to trivial kernel $\iff$ injective in $\boldsymbol{Grp}$

Cokernels are "dual" to kernels. I've been told that if X dual to Y and there's a theorem about X then there's a "dual" theorem about Y. Theorem: In $\boldsymbol{Grp}$ a homomorphism has trivial ...
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### How to prove a solution not optimal using duality?

I have been given an LPP problem and asked to prove the basic solution (x1, x2) is not optimal. But every time I am solving it using simplex and the duality method the objective function keeps coming ...
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### Dual problem to SDP problem

I'm having problem with formulating dual problem to Semidefinite programing problem: $$\max\;\;tr(X)$$ $$s.t.\;\; \left[ \begin{array}{cc} A & X \\ X & B \end{array} \right]\succeq0$$ where ...
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### Difference between Lagrangian and Wolfe Dual

Does anyone know when the Wolfe dual is preferred over the Lagrangian Dual? I'm reading up on the optimization for SVMs and the presentation used the Wolfe Dual. Why can't we use the Lagrangian Dual?
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### Doubt regarding the optimality condition of a duality problem

In a LP problem does max z = min w indicate that the solution is not optimal? Or the opposite? Here min w is the dual of the primal max z
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### Prove that if a canonical minimization problem is unbounded $\Leftrightarrow$ the dual canonical maximization problem is infeasible

I'm not sure exactly how to prove this given $g ≥ f$. I'm able to understand the proof for "if a canonical maximization linear programming problem is unbounded then the dual canonical minimization ...
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### How to solve the dual problem?

Derive the dual problem of the problem by using the Lagrange dual function. Min. f(X)=x1^2 + x2^2 s.t 4-x1-x2^2<=0 3x2-x1<=0 x1, x2>=0
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### Generalizing linear duality via the $Hom$ functor

I am a beginner in category theory, so the following question could be either obvious or of no interest. Anyhow, I cannot find a statisfactory framing, and possibly an answer for the following ...
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### Dual definition of self-duality

We can define the notion of self-duality on lattices as follow : A lattice $L$ is self-dual if there is a permutation $\pi$ such that : for all $a \in L$, $\ \downarrow \pi(a)$ is antiisomorphic ...
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### Strong duality as optimality condition

My understanding of strong duality (for linear programming) is that if the dual form is optimal, then the primal is also optimal, and vice versa. Therefore, we only need to optimize one problem (...
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### Verifying dual transform properties

I am reading a computational geometry book and it gives some claims about a dual transform, which I am trying to verify via linear algebra (linear operator) theory. Let $p:=(p_x, p_y)$ be a point ...
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### On the monotonicity of angles between dual basis vectors

I want to show that if I make the angles between basis vectors $e_1,...,e_n\in\Bbb R^n$ smaller, then the angles between the dual basis vectors $e_1^*,...,e_n^*\in\Bbb R^n$ become larger. More ...
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### How to find the dual problem of this LP?

I want to find the dual problem of the following LP: $\min c'x$ s.t. $Ax=b,x\ge a$ where a>0. I'm considering substitute $y=x-a$ so that it becomes: $\min c'y+c'a$ s.t. $Ax=b-Aa,y\ge 0$. I know I ...
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### Dual Theorem Problem

Suppose that a column for the primal tableau involves a slack variable, s, and we have a column of the identity, say ⃗ei corresponding to s in matrix A. Show that in the optimal row 0, the value under ...
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### Derivation of Lagrangian dual problem

I am new to Lagrangians, and I am not sure if what I am doing is correct. The original problem was to find $\min\limits_{\theta}-log(\theta(1-\theta)^2), .5 \leq \theta \leq 1$, write the Lagrangian, ...
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### Primal and dual feasibility and boundedness

Suppose we have a primal maximization LP: $$\text{maximize } c^Tx \\ \text{subject to } Ax \le b, x \ge 0$$ If $x$ is a feasible solution to the primal LP, and $y$ is a feasible solution to the ...
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Given this problem $\min c^Tx + \frac{1}{2}x^tHx$ subject to $b \leq Ax \leq b+r$ $l \leq x \leq u$ by adding slacks the subject becomes: $Ax - w = b$ $x - g = l$ $x + t = u$ $w + p = r$ $g, w, ... 1answer 112 views ### Can the Fenchel conjugate be characterized through the Fenchel-Young inequality? It is known (see for example this question), that the Fenchel conjugate of a smooth function$f: \mathbb{R}^d \to \mathbb{R}$verifies the identity $$f(x) + f^\star(\nabla f(x)) = \langle x, \nabla f(... 0answers 21 views ### Finding the dual problem with exponential equation I need to find the dual problem of this: but I have no idea how to work with the norm or the exponential function 0answers 31 views ### Finding Optimal Dual Solution without solving de Dual Problem (knowing the Optimal Primal Solution) I've been asked to solve this problem using the Dual Simplex Method.$$ \begin{array}{ll} \min: & x_1+6x_2+3x_3 \\ \text{s.t}: & -4x_1-4x_2-2x_3+x_4=-18 \\ & 2x_1+2x_2-4x_3+x_5 = -16 \\ ... 2answers 75 views ### Equality in trace duality For$A,B\in\mathbb{R}^{n\times m}$we have the trace duality property $$|\langle A, B \rangle|\leq \|A\|_1 \|B\|_{\infty}$$ where$\|A\|_p$is the Schatten$p$-norm (i.e.$\|\cdot \|_1$is the ... 0answers 16 views ### Linear programming - Dual I am trying to derive the dual of the following problem: \begin{equation} \underset{\mathbf{g}, \mathbf{r}, \boldsymbol{\alpha}, \underline{\mathbf{t}}_j, \underline{\mathbf{u}}} {\text{minimize}}~~ \... 0answers 30 views ### If there exists a$\phi \in W'$so$\text{null}(T') = \text{span}(\phi)$, prove that$\text{range}(T) = \text{null}(\phi)\$

I've already proven this question, however, my proof is much more complicated than an answer I saw posted on another website. However, I cannot seem to make sense of why the answer on the website is ...