# Tagged Questions

Convex analysis is the study of properties of convex sets and convex functions. For questions about optimization of convex functions over convex sets, please use the (convex-optimization) tag.

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### Biggest ellipse included in a convex polygon

Considering a N edges convex 2D polygon called P. Let's name its vertices $\{p_1, p_2, ..., p_N\}$ described in a counter-clockwise order, with $p_i = (x_i, y_i)$ What would be, and how would one ...
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### Is this a convext set?

Is this one a convex set? how to prove it? I failed to prove it through the definition of convex set. Thank you. $$\left\{(x_1,x_2)\mid\sqrt{x_1^2+x_2^2}+|x_1|+|x_2|\le 1\right\}$$
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### proof of convex set problem

I had a homework problem which asks to prove or disprove that A is a convex set where $A=\{x: g(x) \le c\}$. and $g(x)$ is a convex function. I went ahead in this way: Assume $x_1$ and $x_2$ $\in$ A. ...
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### The most efficient algorithm to solve the following problem

Is there an efficient optimization algorithm to solve the following problem? $(\alpha,\beta,\gamma,\cdots) =$ argmax $\sum_{i}\log(\alpha a_i+\beta b_i+\gamma c_i+\cdots)$, s.t. ...
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### KKT Conditions for Minmax Problem

Let $\mathbf{x}\in\mathbb{R}^n$ and $\mathbf{y}\in\mathbb{R}^m$. Now $$f\left(\mathbf{x}, \mathbf{y}\right):\mathbb{R}^n\times\mathbb{R}^m\rightarrow\mathbb{R}$$ is convex in $\mathbf{x}$ and concave ...
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### Prove a function is non convex

I have a simple function dependent on two variables $x_1$ and $x_2$: $$f(x) = \ln\left(\frac{x_1}{x_2}\right)$$ where $x_1, x_2 > 0$ (strictly positive). I know this function is non convex as, ...
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### How to prove that the product of a decreasing monotonic function and a strictly increasing monotonic function is a concave function?

Given that I have a strictly increasing monotonic function $f$ and a decreasing monotonic function $g$, are there any nice properties to show that the product function $h(x) = f(x)g(x)$ is a concave ...
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### Testing for Convexity for a function

Please any one can help figure out if this funcion is concave or convex, any help is greatly appriciated. Any links on how to test fo convexity for such a function is also greatly appriciated. I tried ...
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### The convex function proof.

Can anyone help me prove convexity conditions? I know convexity conditions. The second derivative function is greater 0 first order convexity conditions. convex function conditions Because my ...
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### Is $f(x,y) = x^\beta/y$ quasi-convex for positive $x,y$ for any real $\beta \geq 1$?

A multivariate function $f:{\mathbb R}^d \to {\mathbb R}$ is quasi-convex on a convex set $S \subset {\mathbb R}^d$ if $f(\lambda z + (1-\lambda)z') \leq \max\{f(z),f(z')\}$ for all $z,z' \in S$ and ...
Consider a $n$-simplex, $n \geq 2$ with vertices $x_i,i=1,...,n+1$. For each edge $(i,j)$, consider $n$-ball $B_{ij}$ such that vertices $x_i$ and $x_j$ are antipodal on this ball. Fix a point $x_0$ ...
Let $x_{1},x_{2},y_{1},y_{2},z_{1},z_{2},w_{1},w_{2}$ are all positive numbers, and such $$x_{1}y_{1}z_{1}-w^3_{1}>0,\; \text{ and }\;x_{2}y_{2}z_{2}-w^3_{2}>0.$$ show that ...