# 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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### Optimizing potential

Let $X$ be a vector field on a Riemmanian manfiold $M$. I recently read that solving: $$\operatorname{argmin}_{\phi}\int_M (\nabla \phi - X)^2 d\mu,$$ where $\mu$ is ...
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### Finding the global minimum

Let $f~:~\Bbb R^2\to \Bbb R$ be defined as: $$f(x)=\left\|\begin{bmatrix}2&1\\3&1\\4&2\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix} - \begin{bmatrix}2\\1\\7\end{bmatrix}\right\|_2^2$$ ...
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### Least-squares problem with quadratic equality constraint

I want to find the solution of a Lagrange equation whose inputs are matrices. First I have the equation Ax=0. By decomposing $A$ into $A_3$ (columns 9 to 11 of A), $A_9$ (the rest of the columns), ...
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### Is Frank Wolfe a descent algorithm?

A colleague was explaining to me that the Frank-Wolfe algorithm is a descent algorithm (i.e. its objective value decreases monotonically at each iteration). However, when I tried simulating it, my ...
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### Maximum value of $(1-F(t))t$ for probability distribution

Consider a continuous distribution on $(0,1)$ with cumulative distribution function $F$. For the value of $t\in(0,1)$ that maximizes $$P(t)=(1-F(t))t,$$ what is the lower bound of $P(t)$? For example, ...
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### Smooth approximation of maximum using softmax?

Look at the Wiki page for Softmax function (section "Smooth approximation of maximum"): https://en.wikipedia.org/wiki/Softmax_function It is saying that the following is a smooth approximation to the ...
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### Minima of symmetric polynomials subject to two symmetric constraints

The homogeneous symmetric polynomial of degree $k$ in $n$ variables is $$f_k(x_1,x_2,\dots,x_n) = \sum_{i_1<i_2<\cdots<i_k}x_{i_1}x_{i_2}\cdots x_{i_k}.$$ Consider the following ...
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### Deciding on supermarket daily orders given a fixed amount of money

I have completely no idea even where to start from, and I am sure this question has been discussed already, simply I need some key words and methods that solve this problem so that to start my survey. ...