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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Can Three Equilateral Triangles with Sidelength $s$ Cover A Unit Square?

A previous question on the site asked for a short proof of the fact that three equilateral triangles with unit side length cannot be arranged to cover a square with unit side lengths. Given the truth ...
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447 views

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 ...
7
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224 views

Maximal minimum for a sum of two (or more) cosines

Please prove (or disprove, and give the correct answer): $$2 =\mathrm{argmax}_{r\geq 1}\min_{x\in \mathbb{R}}\left[\cos\left(x\right)+\cos\left(rx\right)\right] $$ In other words, find $r \geq 1$, ...
7
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118 views

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) ...
6
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165 views

The minimum of $I_{n,k}=\int_0^{2\pi}\sqrt{3+2\cos(nx)+2\cos(kx)+2\cos(nx+kx)}dx$ is attained for $k=n$

I have the following conjecture: ``For each given $n\in\mathbb{N},\ n\ge 2$ the minimum of the sequence of integrals $I_{n,k}=\int_0^{2\pi}\sqrt{3+2\cos(nx)+2\cos(kx)+2\cos(nx+kx)}dx,\ k=1,2,\dots,n$ ...
6
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102 views

Condition for trigonometric inequality

I want to prove the following statement: Suppose $\frac{1}{4}(\cos(\theta_1)+\cos(\theta_2))^2+\lambda^2(a\sin(\theta_1)+b\sin(\theta_2))^2\leq 1$ holds for all $\theta_1,\theta_2\in[-\pi,\pi]$, then ...
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76 views

How to solve this equation (may be with least squares)?

I have a system of linear equations in the following form. How can I solve it? $$\operatorname*{argmin}_{a,b} \sum_{i,j} \left( \left| X(i,j)-aY(i,j)\right|-b \right)^2$$ Where $X$ and $Y$ are ...
6
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612 views

Optimization / personalization within clusters

I have the following optimization problem: I have a (random and very noisy) objective function $f(A, P)$, where $A$ is a vector of "observable" parameters of the input and $P$ is the parameters that ...
6
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159 views

Selecting a subset from a set such that a given quantity is minimized

Let $A$ be is a set of some $p$-dimensional points $x \in \mathbb{R}^p$. Let $d_x^A$ denote the mean Euclidean distance from the point $x$ to its $k$ nearest points in $A$ (others than $x$). Let $C ...
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93 views

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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229 views

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 ...
6
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240 views

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 ...
6
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203 views

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 ...
5
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44 views

Minimize $f(m)=\sum_{n=1}^\infty n^m / m^n $

For what real value of $m$ such that $\displaystyle\sum_{n=1}^\infty \frac{n^m}{m^n} $ is minimized? I've been told that it's equivalent of solving for $ \text{Li}_{-n}\left(\frac1n\right)$ for the ...
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85 views

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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162 views

Optimal Strategy for “I'm Thinking of a Number” Game

This question is inspired by one of the classic ways of breaking ties: the "I'm thinking of a number" game. In this game, one person thinks of a number in some range, say from $0$ to $100$ ...
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323 views

Cookie Clicker Chocolate Egg strategy

Introduction Cookie Clicker is a silly Javascript based web game. Here is a brief description of what you do: (description taken from this question: Explain a surprisingly simple optimization result) ...
5
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119 views

Measure minimization for a combination of overlapping sets

This problem may have been worked out before but I don't know where to start looking so I hope one of you can help me. The problem is as follows: There are $N$ variable-sized finite sets ...
5
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157 views

optimal road layout problem - how to convert to maths and see the shapes it makes?

I have this puzzle going round my head about optimal road layouts, but I'm a programmer not a mathematician and I don't really know how to specify it as a maths problem. Once it's well-specified I can ...
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608 views

Maximum and minimum of an integral under integral constraints.

Find the maximum and minimum of the following integral in terms of $f(x),a,C$: \begin{align}I=\int_{0}^{a} \frac{x}{f(x)}p(x)dx \end{align} s.t.: 1) $\int_{0}^{a} p(x)dx=1$ 2) $\int_{0}^{a} ...
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79 views

$\int_{-\infty}^{+\infty}\left|f(g(t))-f(h(g(t)))\right| \,dt=0$ How to find $f$?

Let $h,g$ be given entire functions. Consider $$\int_{-\infty}^{+\infty}|f(g(t))-f(h(g(t)))|\, dt=0,$$ where $|\cdot|$ means modulus. How do I find non-polynomial analytic solutions for $f\,$? ...
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27 views

Shaking a box of rocks (Optimal Packing)

My coworker was telling me that when he plants seeds on his farm, he puts them all in a large container on the tractor and after a period of just driving, the seeds are more densely packed than when ...
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99 views

StackEgg optimal algorithm

What is the minimum number of days that is needed to complete the StackEgg game? (It's on the right if anyone didn't notice.) There are four markers (Questions, Answers, Users, Quality) I believe each ...
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75 views

Calculation of an expression ($\max_{U}\min_i \sum_j |U_{ij}|^2 |e_i^j|^2$)

There is an orthonormal basis $\{e_i\}(i=1,\ldots,n)$ in $\mathbb{C}^n$, each of them is represented in form of column vectors $$\begin{pmatrix} e_i^1\\ \vdots\\e_i^n\end{pmatrix}.$$ My purpose is to ...
4
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44 views

Quadratic optimisation with quadratic equality constraints

I would like to solve the following optimisation problem: $\min_{x} (x'Ax)$ subject to $x'Bx = x'Cx = 1$. Where A is symmetric and B and C are diagonal. Does anyone have a suggestion for an ...
4
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47 views

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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250 views

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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47 views

Is there a name for systems of equations with min and max functions included?

In a big project I'm working on, I'm running into systems of equations that look like the following: $$a = \min(b, c)$$ $$b = d^2 + a$$ $$c = \max(a + b, d)$$ Basically, nonlinear systems of ...
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59 views

Closest Pair between 2 sets of powers

Assume $a$ and $b$ are natural numbers, $A=\{a,a^2,a^3,\cdots\}$ and $B=\{b,b^2,b^3,\cdots\}$, find $\min\left|a_i-b_j\right|$ where $a_i\in A$ and $b_j\in B$. For example, if $a=3$, $b=10$, then ...
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180 views

Measuring how change in input variables contributes to output in non linear equation.

How do we measure how a variable contributes to an output as its value increases, and how it relates to other input variables? Let's say we're playing a video game, where you can buy items to augment ...
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62 views

Selecting k vectors with maximum spread out of a set of n vectors

Given a set $\mathcal{V}$ of $n$ vectors, find a subset $\mathcal{V}_k = \mathcal{V} - \mathcal{V}_{n-k}$ containing $k$ maximally spread vectors. Intuitively, these $k$ vectors should be spread as ...
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86 views

Conditions to satisfy trigonometric inequality

I'm looking for sufficient (and necessary would be good too) conditions on $a,b,c$ such that \begin{align} a\cos\phi + b \cos 3\phi + c \cos 5\phi \geq -1 \hspace{20pt} (\forall \phi) \end{align} ...
4
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205 views

Properties of the projection onto a nonconvex set

Consider a set $\Omega\subseteq\mathbb{R}^n$ being "sufficiently regular", for example being the image of a $C^1$ mapping from $\mathbb{R}^p$ for some $p\ge1$. We may then consider the mapping $$ ...
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How to minimize $\min_k k \frac{b^k/n}{\lfloor b^k/n \rfloor}$

This problem looks familiar, but I don't remember its solution: $$ \min_k \ \ \frac{b^k/n}{\lfloor b^k/n \rfloor}k $$ subject to $$ b^k \ge n \\ b,n,k \in \mathbb{N} $$ Does it have a name? What's ...
4
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81 views

Find optimal measure

Let $\Omega$ be a convex compact set in $\mathbb{R}^n$, $f\colon \Omega \to \mathbb{R}$ be a convex function. Consider an optimization problem $$ \int\limits_{\Omega}f(x)\,\mu(dx) \to ...
4
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119 views

Solving a system of $3 \arg \max$ equations

Considering these 3 equations: $$ \alpha_{1} = \arg \max\limits_{\alpha \geq 0} \quad \Big \{ (\alpha - 1) \big (\frac{1/\alpha}{1/\alpha + 1/2 \alpha_{2} + 1/2 \alpha_{3}} \big)^2 \Big \} \\ ...
4
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300 views

(easier version)What is a working example for finding the maximum of a algebraic function $f(a,b,c,d,e)$ with 4 equality …

edit 2: If you are founding an answer would be interesting to you, please upvote this question, because i may use those point in order to start a new bounty. Edit 3: Can somebody answer? Edit 4:I ...
4
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272 views

Universal Correlation measure — ranking correlations

I have time series data of experimental observations for two related processes. I want to measure correlation for use in further analysis. Correlation of the series changes over time and across ...
4
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392 views

Constraint of a Lagrange multiplier

My question concerns Lagrange multipliers and the possibility to impose constraints on the multipliers themselves. I have a Stokes flow which is solved using the Finite Element Method on a domain ...
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921 views

Convergence of Gauss-Newton method for piecewise linear functions

Notation for Gauss-Newton method Non-linear least squares problems are often solved by the Levenberg-Marquardt algorithm, which can be viewed as a Gauss–Newton method using a trust region approach. ...
4
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116 views

When can the optimal value of a SDP be achieved?

Looking at semidefinite programs, are there any sufficient conditions for the solvability (i.e. the optimal value can be achieved, that is infimum=minimum)? Obviously if the problem is unbounded, the ...
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86 views

Questions about no duality gap and optimal solutions

My questions are regarding a constrained problem, $$\min_{x \in X \subseteq \mathbb{R}^n} f(x),$$ subject to $g(x) \leq 0 \in \mathbb{R}^m, h(x) =0 \in \mathbb{R}^k$. Its dual problem is $$\sup_{u ...
4
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308 views

$3$D oriented bounding box optimization

Given: a set $S$ of points in $\mathbb{R}^3$. Find: the smallest oriented bounding box that contains all the points. Note, the bounding box is "oriented" and thus need not be axis-aligned. Can this ...
4
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285 views

Extremal curve passing through a set of points

I'm having trouble recasting the following question in a form amenable to the calculus of variations. Question: Given a set of $n$ points $P=\{(x_1,y_1),..(x_n,y_n)\}$ what is the curve passing ...
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503 views

How to Minimize A Function Where The Number of Variables is Unknown

I have a standard linear programming problems I want to solve: $$ \min_x f^T x \text{ such that } \left\{ \begin{aligned} A\cdot x &\le b, \\ A_{eq}\cdot x &= b_{eq}, \\ lb \le x &\le ub. ...
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66 views

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 ...
3
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50 views

Auction Design : Multiple lots, one win max per bidder, not regret

This is a real life game theory problem. I have to organize an auction. There is a finite number of lots, which are not equivalent. There is a finite number of bidders; the number of bidders is ...
3
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77 views

How to solve the Few Scientists Problem (big word problem) in its general form?

I'm trying to figure out how to solve this word problem. I'm pretty sure it involves calculus or something even harder, but I don't know how to solve the general form. Let me start with the concrete ...
3
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32 views

What does it mean for a problem to be time-homogenous?

(This is an associated question to Scaling in utility maximisation. $c_t$, $w_t$, $n_t$, $A$ are defined there.) I am reading that because of time homogeneity $$\sup_{(n,c)\in ...
3
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36 views

Shortest system of roads between 4 cities

You have $4$ cities placed on the vertices of a square of side length $1$ km. You have to come up with a system of roads such that you can reach any city from another (directly or through another ...