# Tagged Questions

Convex Optimization is a special case of mathematical optimization. It includes Linear Programming and least-squares.

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### How to solve $\min \limits_{\mathbf{x}} \| \mathbf{Ax}-\mathbf{b} \|^2$?

Let $\mathbf{x}=[x_1,\ldots,x_K]$. I have the following optimization problem: \begin{array}{rl} \min \limits_{\mathbf{x}} & \| \mathbf{Ax}-\mathbf{b} \|^2 \\ \mbox{s.t.} & x_k\ge 0, \forall ...
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### Convexity of the product of two functions in higher dimensions

Exercise 3.32 page 119 of Convex Optimization is concerned with the proof that if $f:\mathbb{R}\rightarrow\mathbb{R}:x\mapsto f(x)$ and $g:\mathbb{R}\rightarrow\mathbb{R}:x\mapsto g(x)$ are both ...
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### What is the time complexity of conjugate gradient method

I have been trying to figure our the time complexity of conjugate gradient method I have to solve a system of linear equations given by $$Ax=b$$ where A is sparse and positive definite symmetrix ...
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### prove this is a strongly convex function

The definition of strongly convex from Wikipedia: It is not necessary for a function to be differentiable in order to be strongly convex. A third definition for a strongly convex function, with ...
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### proximal operator of infinity norm

What is the proximal operator of $\|x\|_\infty$? I know we have to take the subgradient and compute it but I am a bit stuck. Can anyone show me steps?
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### If the Minkowski sum of two convex closed sets is a Euclidean ball, then can the two sets be anything other than Euclidean balls?

If for two convex closed sets $S_1$ and $S_2$, the Minkowski sum is a Euclidean ball then can $S_1$ and $S_2$ be anything other than Euclidean balls themselves. I suspect they can be but I haven't ...
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### If this problem is not unbounded, what's wrong in this dual derivation?

In a paper with 100 citation, Robust Support Vector Machine Training via Convex outlier Ablation, a convex relaxation is used. In this paper, a form of robust svm proposed: \begin{align} \min_{0\leq \...
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### 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 ...
On $I = [0, 2π]$ consider the function $f : I → \mathbb{R}$ to be convex. Define: $$a_k\pi := \int_0^{2\pi}f(x) \cos(kx)\,dx$$ Show that the convexity of $f$ implies that $a_k ≥ 0$ when $k ≥ 1$. ...
### Proximal mapping of $f(U) = -\log \det(U)$
This is an assignment problem which I failed to solve in a couple of days. Denote the set of all $n \times n$ symmetric matrices and the set of all $n \times n$ symmetric positive definite matrices ...