# Tagged Questions

Optimization is the process of choosing the "best" value among possible values. They are often formulated as questions on the minimization/maximization of functions, with or without constraints.

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### Constant such that $\max\left(\frac{5}{5-3c},\frac{5b}{5-3d}\right)\geq k\cdot\frac{2+3b}{5-c-2d}$

What is the greatest constant $k>0$ such that $$\max\left(\frac{5}{5-3c},\frac{5b}{5-3d}\right)\geq k\cdot\frac{2+3b}{5-c-2d}$$ for all $0\leq b\leq 1$ and $0\leq c\leq d\leq 1$? The right-hand ...
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### Optimization: constructive solution

Consider the following program \begin{align} \max_{x,y \geq 0} f(x,y)\tag{1} \end{align} I wanna construct a solution with $y^* = 0$ and $x^* > 0$. Suppose FOCs satisfy \begin{align} f_x(x^*,y^*) ...
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### Constrained optimization problem using Largange multipliers: ellipsoid collision detection and response

This one is purely for the mathematics so the result is far less important than the method itself. My task is to implement a fast and efficient ellipsoid collision detection and response algorithm. ...
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### Value of $F(r)$ to maximize $\frac{\int_r^1xf(x)dx}{2-F(r)}$

Consider a continuous distribution on $(0,1)$ with probability distribution function $f$ and cumulative distribution function $F$. Define $$g(r)=\frac{\int_r^1xf(x)dx}{2-F(r)}$$ and let $r_M\in(0,1)$ ...
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### how to find lower bound function?

In Expectation Maximization algorithms, first step is to find lower bound function of the objective function that is equals to objective function at current estimate. What are the ways to find lower ...
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