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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Find point on a line that is nearest to the origin

Can you help me with this exercise? Find the nearest point to the origin $(0,0,0)$ in the line given by the intersection of planes $x+y+z=2$ and $12x+3y+3z=12$. The intersection of the planes is ...
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1answer
71 views

Upper bound of $x_1x_2x_3+x_2x_3x_4+\dots+x_{n}x_1x_2$

Let $n\geq 3$ be a positive integer and let $x_i$'s be non-negative real numbers with $x_1+x_2+\dots+x_n=1$. What is the maximum of $x_1x_2x_3+x_2x_3x_4+\dots+x_{n}x_1x_2$? If the sum were symmetric ...
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Relationship between Newton's method in root finding and optimization

In both root finding and optimization, there are Newton's method. Wikipedia has 2 links here and here. Root finding is using first order derivative and optimization is using Hessian. What's the ...
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21 views

How to cover a sphere with caps removed, with equidistant points?

I have a sphere with the caps removed, so: $$x^2 + y^2 + z^2 = R^2$$ for $|x|, |y| \leq R$ and for $|z| \leq R_z <R$. This creates a sphere with the top and bottom cap cut off. $R_z$ will be ~ $2R/...
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+200

A trigonometric problem when calculating distance to the boundary of a convex hull

Suppose we have a sphere and a point outside of the sphere. We denote the point outside as $v$ and the origin of the sphere as $x$. The convex hull of the sphere and $v$ should be like an ice cream ...
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Finding solution to Calculus of Variation of linear functional whose domain consists of vector valued function

Problem Statement: Find $x^*$ such that it solves the optimization problem $$\max_{x \in \Omega} \quad f(x) = e_i^TAx$$ $$ \Omega = \{x: t \to \Delta^{n}|x \in C^1, x(0) = x_o\}$$ Where $\Delta^...
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1answer
70 views

Find maximum value of a function [closed]

$a$, $b$, and $c$ are real numbers, and $a+b+c=0$ and $a^2+b^2+c^2=2$. I need help finding the maximum value of: $$\big|a^2b^2(a-b)+b^2c^2(b-c)+c^2a^2(c-a)\big|$$ To be honest, I don't know where ...
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61 views

For which point of $x+y+3z+k=10$, the expression $x^2+y^2+9z^2+4k^2$ is minimal? [closed]

For which $x,y,z,k \in \mathbb R$ of $x+y+3z+k=10$ is the expression $x^2+y^2+9z^2+4k^2$ minimal?
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how to minimize $tr (X^T W - Y^T W_p)(X^T W -Y^TW_p)^T$ in closed form

Assume we are dealing with matrices. Then how to minimize $$ E(W,W_p) = tr (X^T W - Y^T W_p)(X^T W -Y^TW_p)^T $$ w.r.t both $W, W_p$ simultaneously? I can calculate the derivatives of $W$ and $W_p$ ...
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25 views

Scheduling grid optimization

I am trying to optimize the programming of multiple TV channels for a given week. For each show (a day, a time and a TV show) it is possible to forecast in advance the number of people that will watch ...
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2answers
41 views

Maximizing a convex quadratic function in CVX and Matlab

I understand that a convex function can not be maximized as there is no such value. However, consider the following function: $$\begin{array}{ll} \text{maximize} & 3x^2 + 5y^2\\ \text{subject to} ...
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26 views

related to biconcave optimization

I have a bivariate function $f(x,y)$ both $x,y$ can assume values within closed interval i.e. $x_1\leq x\leq x_2$ and similarly $y_1 \leq y \leq y_2$. I know that for a fix value of $x$ the function ...
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2answers
36 views

L1 minimization problem with nested sums as LP problem

I've been trying to solve this problem but I have an issue with the fact that there is a sum under each absolute value. I'm trying to convert this minimization problem (with respect to $x, y_1, \dots,...
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1answer
37 views

Why isn't Linear Programming less convoluted? [Soft Question]

Just a quick question. So I'm taking a course in linear optimization, and one of the things that we're going over obviously is the simplex method. I just started the class so I may not be seeing the ...
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29 views

Linear program with ceiling or floor functions

Is it possible to solve a linear program where constraints have ceiling or floor functions applied to variables (with maybe some constants)? For instance: $$\lceil (x_1 + a)/b \rceil + \lceil (x_2 + c)...
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42 views

Convex optimization with $\ell_0$ “norm”

I have an optimization problem of the form $$\begin{align*}\text{minimize }\;&f(x)\\ \text{subject to }\;&||x||_0 \le t,\end{align*}$$ where $t$ is a given constant and $f:\mathbb{R}^d \to \...
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1answer
23 views

Prove this property of the Hessian

I have been reading about the hessian for a scholar work about optimization and I find this property: Let be $H_{P_0}$ the determinant of the hessian matrix for the Lagrangian function $\mathscr{L}(x,...
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1answer
56 views

Working with Lagrange multipliers, reducing gradients is okay, right?

I am employing the method of Lagrange multipliers to determine a maximum. As part of this, I arrive at the following equation involving two gradients and the parameter $\lambda$, as is common for ...
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21 views

Finding dual of certain problem

Could anyone help me finding lagrangian function and lagrangian dual of the following problem: \begin{equation} \begin{split} \max_{X}\quad & \operatorname{trace}(H X H^T)\\ \text{s.t} \quad &...
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Economics maximization problem linear activity

Consider the vectors: $a_1 = \begin{pmatrix} 0 \\ -1 \\ 1 \\0 \end{pmatrix}, a_2 = \begin{pmatrix} 0 \\ 0 \\ -1 \\1 \end{pmatrix}, a_3 = \begin{pmatrix} 2 \\ 0 \\ 0 \\ 1\end{pmatrix}$ Find a single ...
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20 views

Convex set equals convex functions within optimization?

Can optimizing a convex function subject to convex constraints be written as optimizing the function subject to a convex set? Does the intersection of convex nonlinear ineualities necessarily describe ...
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34 views

What is the easiest way to optimize the weighted sum of L2 norms?

I have the following cost function (solving for $M$ - the $x_i$s are known): minimize $\sum_i\sum_j(w_{ij} \cdot (x_i-x_j)^T\cdot M\cdot(x_i-x_j))$ ($w_{ij} \in [-1,1] $) subject to: $M \succeq 0$ (...
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When is this argmax-based function continuous?

Let $w: \mathbb{R}^+\to \mathbb{R}$ be a continuous, strictly-increasing and strictly-concave function. define the following function: $F: \mathbb{R}^+\times\mathbb{R}^+ \to \mathbb{R}$: $$F(s,t) = \...
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1answer
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Proof: $\underset{\|q\|=1}{\max} q^TAq = \lambda_{\max}$ with $q$ the corresponding eigenvector ($A$ symmetric)

This problem is quite old and there should be similar problems. I know the following technique: \begin{equation} \begin{aligned} q^TAq=q^TU\Lambda U^Tq=(U^Tq)^T\Lambda (U^Tq) \end{aligned} \...
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1answer
15 views

Advantage of multi-objective optimization over single objective

What are the advantages of multi-objective optimization over single objective? I am specifically thinking about MO and SO in Genetic Algorithm. I have surfed the net and found many articles talking ...
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1answer
41 views

Linear programming with a product term in the objective function

The title might sound a little weird. I actually want to ask if this problem can be solved as a LP. And if so, how to convert the product term? set $P=\{1,2,3,\ldots,n\}$ for index $i$. Variables $...
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119 views

Two Approaches Two Different Solutions: Optimal Controls vs. Different Method

If I try to solve a problem two different ways, I get two different answers which generally means I am committing some horrible sin! Given the problem, \begin{align} \min_u\ S &= \int dt\ L(x, u) ...
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Interpolate Initial step length for line search methods

I was learning interpolation techniques in initial step length guess. Below is an approach from Nocedal and Wright's book, Numerical Optimization. Interpolate a quadratic to the data $f(x_{k-1})$, $f(...
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Model cost for a state change in an integer program

I have a problem involving tool selection I am trying to model right now. (I am fairly new to this). I have a series of manufacturing operations I need to perform for $i \in \{1,\dots,n\}$. Each ...
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Minimize a huge two-variable logarithmic-trigonometric-radical expression (MSU entrance early July 2016)

Minimize \begin{align}R(a,x)&=\sqrt{13+\log_a\left(\cos\left(\frac xa\right)\right)^2+\log_a\left(\cos\left(\frac xa\right)^4\right)}+\sqrt{97+\log_a\left(\sin\left(\frac xa\right)\right)^2-\...
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What is the minimum value for $(\frac{1}{a}-1)(\frac{1}{b}-1)(\frac{1}{c}-1)$ if $a+b+c=1$ and $a,b,c\in\mathbb{R}^+$?

The primary question was: What is the minimum value for $(1-\frac{1}{a})(1-\frac{1}{b})(1-\frac{1}{c})$ if $a+b+c=1$ and $a,b,c\in\mathbb{R}^+$? $\color{red}{\text{But sorry guys! I messed it up! my ...
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How to find (GEV) distribution parameters with optimization?

I'm currently trying to replicate this study with python. http://pages.stern.nyu.edu/~sfiglews/Docs/RND_draft7.pdf The section I'm currently working on is between p.17-20 in the study. The study ...
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Find the maximum of $U (x,y) = x^\alpha y^\beta$ subject to $I = px + qy$

Let be $U (x,y) = x^\alpha y^\beta$. Find the maximum of the function $U(x,y)$ subject to the equality constraint $I = px + qy$. I have tried to use the Lagrangian function to find the solution for ...
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1answer
32 views

Finding $\min f(x)$ where $f(x)=\int_0^1 |t-x|t\,dt \quad \forall x \in \mathbb{R}$

Can I write the integral as $f(x)=\int_0^{x} |t-x|t\,dt + \int_{x}^1 |t-x|t\,dt$ so that $f(x)=\frac{2x^3-x}{2}+\frac{1-2x^3}{3}$ But here I'm restricting $x$ to the interval $(0,1)$ and I need $x$ ...
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Under which conditions discretization of convex\concave function is submodular?

Say, I have $f(x)$ with $x \in [0,1]$, then by discretization I mean $f(x_h)$ with $x_h \in \{0, h, 2h, \dots, 1\}$. I know about Lovasz extension, but it works in other way: given submodular ...
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2answers
70 views

How to covert min min problem to linear programming problem?

I have the following problem: set $P=\{1,2,3...,n\}$ for index $i$, set $K=\{1,2,3,...,m\}$ for index $k$. Value $B_i^k$ is indexed by both $i$ and $k$, while value $l_i$ is indexed by only $i$. Here ...
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random pursuit without function evaluations

Assume we want to minimize a convex function $f(x)$ with $x\in \mathbb{R}^n$. Function $f(x)$ represents cost of a system which we cannot compute directly but can observe if system is at state $x$. My ...
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Find the highest point of intersection

Find the highest point of intersection of the sphere $x^2+y^2+z^2=30$ and the cone $x^2+2y^2-z^2=0$. Am I supposed to use the Lagrange multiplier for this? EDIT: So this is what I've tried... $z^2=...
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1answer
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Find a minimum of a quantity

Let $n$ a positive integer. I have the following quantity: $$Q = 3205 \cdot 3^{i-1} + 64i + 64i(3^{i-1}-1)-64(3^{i-1}-1)-32\times3^{i-1}(2i-5)+3$$ I would like to find the integer $i$ which ...
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Optimization with L_infinity norm regularization

I'm trying to solve an optimization problem of the form $$\text{minimize } \; f(x) + \|x\|_\infty$$ where $x$ ranges over all of $\mathbb{R}^n$ and $f:\mathbb{R}^n \to \mathbb{R}$ is a nice, smooth, ...
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How do I model the optimization of haircuts (price vs. frequency vs. satisfaction)?

My goal is to use a simple, real world situation to apply optimization algorithms to and find optimal choices based on computation. What information do I need to gather and what questions do I need to ...
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1answer
29 views

Properties on proximal term

If the equation $x_i$-subproblem showed below is not strictly convex $\arg \min_{x_i}=f_i(x_i)+\frac{\rho}{2}\|A_ix_i+\sum_{j\neq i}A_jx_j^k-c-\frac{\lambda^k}{\rho}\|_2^2$ Why adding the proximal ...
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Find the smallest $\alpha$ such that, for all $x,y,z$, $\alpha\,\left(x^2-x+1\right)\left(y^2-y+1\right)\left(z^2-z+1\right)\ge(xyz)^2+|xyz|+1$.

Find the smallest $\alpha\in\mathbb{R}$ such that, for all $x,y,z\in\mathbb{R}$, the following inequality holds $$\alpha\,\left(x^2-x+1\right)\left(y^2-y+1\right)\left(z^2-z+1\right)\ge(xyz)^2+|xyz|+1\...
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54 views

Optimization with box constraints - via nonlinear function

I have the following convex optimization problem over $\mathbb{R}^n$ with box constraints: $$\begin{align}\text{minimize }&\;f(x)\\ \text{subject to }&\;x \in [-1,1]^n\end{align}$$ I can see ...
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1answer
32 views

how to find closely related values from a set?

I have a set of values, for eg. {20, 1, 1, 21, 8, 22, 11, 40, 5, 21} and will need to find n closely related values. If n is 4 in the given example, the result should be {20, 21, 21, 22} because these ...
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1answer
40 views

Is there a way to measure how (non)convex a function is, maybe analogous to condition number?

Consider the functions $f(x) = \sin x$ and $g(x) = (x+1)^2 (x-1)^2$. We know that $f$ has an infinite number of local minimizers and is nonconvex on a non-compact subset of $R$. We know that $g$ has ...
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1answer
110 views

How can we find minimum of $f(x,y,z)?$

Let $k\in\mathbb{N}$ and $x,y$ and $z$ are positive real number such that $x+y+z=1$. How can we find minimum of $f(x,y,z)$ where $$f(x,y,z)=\frac{x^{k+2}}{x^{k+1}+y^{k}+z^{k}}+\frac{y^{k+2}}{y^{k+1}+z^...
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1answer
20 views

Optimization: Via manifolds point of view of Lagrange multipliers method

My basis on differential manifolds calculus and differential geometry being very superficial, I'm trying to understand this section on WP's article. I'm not being able to realize why most of the ...
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2answers
162 views

Constant such that $\max\left(\frac{3}{3-2c},\frac{3a}{3-2d},\frac{3b}{3-2e}\right)\geq k\cdot\frac{2+3a+4b}{9-c-2d-3e}$

What is the greatest constant $k>0$ such that $$\max\left(\frac{3}{3-2c},\frac{3a}{3-2d},\frac{3b}{3-2e}\right)\geq k\cdot\frac{2+3a+4b}{9-c-2d-3e}$$ for any $0\leq b\leq a\leq 1$ and $0\leq c\...
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3answers
44 views

Calculus 1 - Optimization of a Box

Can you guys help me out with it? i try to solve it but my answer is so weird that i think im wrong... Question- Someone want to build cardboard box with rectangular base. Knowing thatthe rectangle ...