# 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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### Is the layout of Burning Man's city “grid” optimal?

My intuition is that the layout of Burning Man's city "grid" optimizes for the smallest sum of all distances between any two points on the map. Am I correct? Is the proof obvious? Or is there another ...
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### Are any tools or techniques available to solve the “placement of safety points” problem?

Definition 0. Given a metric space $X$ and subsets $H$ and $S$ thereof, define: $$d(H,S) = \sup_{h \in H} \inf_{s \in S}d(h,s)$$ (This as an asymmetric version of the Hausdorff distance.) Here's ...
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### Optimal decomposition of discrete function into sum of factorised terms

I am trying to solve the following optimisation problem. Let $x_i \in \{1, \ldots, N_i\}$ be discrete variables, and $f(x_1, \ldots , x_n)$ any real-valued function. I want to decompose $f$ into a ...
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### Consider the problem minimize $f(x)= x^4 −1.$

I am studying for a test and I found this problem in the textbook that I'm using, there may be a conceptual problem that I'm coming across. My test is on unconstrained optimization (multi-variable) ...
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### Optimization of relative entropy

Wondering if my following question is an application of information theory: Lets say we have a factory and ship boxes of stuff outside. If a competitor stands outside my factory, observes the stream ...
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### A graph problem

Consider the following graph problem. We are given a set of vertices $A_i$, $B_i$, and $C_i$ where $i \in \{1,2,3 \}$. For each vertex, there is a corresponding weight where the weight of vertex $A_i$ ...
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### Construct a matrix of polynomials to optimize condition-like score

I'm a physicist currently working on my PhD. Within my studies, my colleagues & I encountered a (strictly mathematical) problem that baffles us (and anyone else we've talked to so far) and is also ...
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### Heuristics for topological sort

I have a number of modules connected in a Directed Acyclic Graph. My problem is to find an optimal execution order (minimize the total execution time). Any topological sort suffices for a valid ...
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### Find distance to a set (subspace) without computing closest point

General setup: we have a finite-dimensional normed linear space $(V, \| \cdot \|)$, a subspace $U \subset V$, and a fixed vector $v_0 \in V$. We want to find the distance between $v_0$ and $U$. (No ...
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### Response Surface Methodology using Moving Least Squares Method

I would like to obtain the response surface of a mathematical function for reliability-based design optimization (RBDO). To obtain a reliably response surface, I learned that moving least squares ...
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### Do balls optimize the boundary area for a fixed volume?

I recently saw this question, asking if the circle is the planar figure which has the least perimeter given the area. As far as I know, this is a classical problem, and the answer is affirmative. I am ...
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### Subgradient of convex minimization duality

$$\min(f_0(x))$$ $$\text{s.t. }f_i(x) \le y_i \forall i, i = 1 ,\ldots, m$$ $$f_i : \text{convex};\quad x : \text{variable}$$ It is also considered that $g(y)$ is the optimal value of the problem ...
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### Is this minimization problem NP-Complete?

We are given an $n\times(n+k)$ matrix $A$, with entries in $GF(2)$, of the form $A=\begin{pmatrix}I_n & B\end{pmatrix}$, where $I_n$ is the $n\times n$ identity matrix, and $B$ has no "zero" rows ...
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I would like to solve the following optimisation problem: $$\text{minimize} \quad x'Ax \qquad \qquad \text{subject to} \quad x'Bx = x'Cx = 1$$ Where $A$ is symmetric and $B$ and $C$ are diagonal. ...
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### Find a function that maximizes $\int_{0}^{1}f(x)\,\rm dx$ with given constraints

Find a function $f(x)$ that maximizes the following integral $$\max\int_{0}^{1}f(x)\,\rm dx\quad \text{s.t.}\quad \frac{d}{dx}ln(f(x))<0$$ $f(x)$ also continues, $f:[0,1]\rightarrow R$ and we ...
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### Making sense of the big world of gradient methods

There are many extensions of gradient descent: stochastic-, Nesterov accelerated-, proximal-, conjugate-, dual-, mirrored-, splitted-, coordinate- gradient descend and more. It also appears that many ...
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### mathematics of chemical stoichiometry

I would like to better understand the mathematical description of chemical stoichiometry and thermodynamic chemical equilibrium. This problem has many features and I know my description might be too ...
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