3 votes
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Is the set of continuous, monotonic functions on $[0, 1]$ convex?

Let $f$ and $g$ be monotonely increasing functions mapping $[0,1]\rightarrow\mathbb{R}$ and $\lambda,x,y\in[0,1]$ be arbitrary, with $x<y$. Then, $$ \lambda f(x)+(1-\lambda)g(x)\leq \lambda f(y)+(1-...
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2 votes
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Dual of a simple constrained least-squares problem

The lagrange function that you wrote is slightly incorrect. You have to exchange $b^T$ and $b$. To obtain the dual that you are looking for: \begin{align} \max_{u\in \mathbb R}\min_{x\in \mathbb R^n} \...
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1 vote

Is the set of continuous, monotonic functions on $[0, 1]$ convex?

Given $x,y\in [0,1]$, call $\phi_{x,y}\in C^0[0,1]'$ the linear functional $\phi_{x,y}(f)=f(y)-f(x)$. Then your subset is $\bigcap\limits_{1\le x<y\le 1} \phi_{x,y}^{-1}[0,\infty)$, and therefore ...
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