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My first approach with duality was in projective geometry, where it it is a very powerful tool: we can proof a theorem for points and we have a dual theorem for straight lines ! But duality is a powerful tool in many fields of math applications. Also in chemistry I suppose are used involutions that are an example of duality. There are really many ...


3

It is very often the case that on ehas a vector space $V$ over some field $k$ and one is interested in linear maps $V\to k$. What is more natural that to collect them into a set? Every time we have some objects we are interested in the first thing we do is it collect them in a set. In this case, we write $V^*$ the set of all linear maps $V\to k$. Now linear ...


1

It's actually easy. Let's do it from first principles. First observe that \begin{eqnarray} \underset{\mu \ge 0}{\text{sup }}\mu^T(g(x)-u) = \begin{cases}0, &\mbox{ if }g(x) \le u,\\+\infty, &\mbox{ otherwise.}\end{cases} \end{eqnarray} Now, \begin{eqnarray} \begin{split} \text{LHS of 1.47} &= \underset{u \in \mathbb{R}^r}{\text{inf }}p(u) + P(u) ...


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Given any coalgebra $C$ over a field ${\mathbb k}$, the category of $C$-comodules fully embeds into the category of $C^{\ast}$-modules by sending a $C$-comodule $(M, \nabla: M\to C\otimes M$) to the $C^{\ast}$-module having $M$ as the underlying ${\mathbb k}$-vector space, and with $C^{\ast}$-action given by $C^{\ast}\otimes M\to C^{\ast}\otimes C\otimes ...


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It's good you're learning Fenchel-Rockefellar duality by doing-it-yourself. Question: Does my whole derivation make sense ? Answer: Yes! Qestion: Is the way to form the conjugate function in the second step correct? Answer: Almost there... Indeed, let me recover your results from first principles (without assuming any knowledge of the concept of "dual", ...


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A1: Yes, by definition your dual function for the original constrained problem is unique. Moreover, it is by definition concave and its supremum (or maximum, when it's reachable) gives you a lower bound on the optimal value of the original problem. A2: Yes, you are right, but you have to be careful here. I'm referring to Luenberger, Ch.14, p.442, eq.13. The ...


0

There are many (but of course closely related!) notions of "duality". By far the most powerful is the Fenchel-Rockafellar. This this 'technology', you can convert rather complicated problems into dual form in just one line of calculation. It some cases, the problem so-obtained is much easier to attack than the original. To get the ball rolling, start with ...


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Boyd and Vandenberghe is a good place to start. It makes the discussion of duality about as simple as possible, and it certainly has many good examples. The book is free online.


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Show that if $y\not=0$ then the null space of $y$ has co-dimension $1$. That is, fix $z$ where $y(z)\not=0$. Observe that any $x$ is $az+w$ for some scalar $a$ and some $w$ satisfying $y(w)=0$. Let $a=y(x)/y(z)$ and $w=x-az$. Hence if $B$ is a vector-space basis for the null space of y, then $Bu\{z\}$ is a basis for $V$, so B has $n-1$ members.


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The square bracket notation is rather unusual. Most mathematicians tend to use the triangular brackets for this pairing: $\langle x,y\rangle$, if $x\in V$ and $y\in V^*$. (The star notation is also the usual way to denote the dual space.) Let me answer in the most general form; that is, when the base field is an arbitrary field $F$. To answer your ...



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