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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How to position rectangles such that they are as close as possible to a reference point but do not overlap?

Given a set of rectangles contained within a larger rectangle such that none overlap, what is the most efficient way to determine the position of a new rectangle such that it is as close as possible ...
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94 views

How to optimize a function with several variables

I need to develop code to optimize a set or variables based on the following conditions. I don't have the source of function. The function gets a point (x,y) and generate a mapped point (x',y') ...
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836 views

Minimizing Appreciating Quantities vs. Maximizing Depreciating Quantities

Suppose you have a set $S = \{r_1, ..., r_n :\, r_k \in (1, \infty)\, \forall \,k \in \{1,...,n\}\}$. Find a bijective mapping $f: \{0,...,n-1\}\rightarrow \{1,...,n\}$ that minimizes \begin{align*} ...
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48 views

Edge in a convex polytope

I want to show that a convex polytope $A$ that is an intersection of half-spaces contains an edge if $ A=\{x \in \mathbb{R}^n|Ax=0 \wedge x \ge 0\}$, where x greater equal 0 means, that all components ...
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445 views

How to maximize scalar product using Lagrangian methods

maximize $U(x) = u \cdot x$ with respect to $p \cdot x = w$ given that $u, p, x \in \mathbb{R}^L_+$, $w \in \mathbb{R_+}$. I can solve it classically: \begin{align} u \cdot x &= \sum u_i x_i \\ ...
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57 views

Maximum of $ f(x,y) = 1 - (x^2 + y^2)^{2/3} $

For the function $ f(x,y) = 1 - (x^2 + y^2)^{2/3} $ one has to find extrema and saddle points. Without applying much imagination, it is obvious that the global maximum is at $ (0,0)$. To prove ...
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379 views

The Weierstrass Approximation Theorem Vs The Runge's Phenomenon

I am learning about different interpolation methods in my internship. Today as I was looking this article on Wikipedia to learn about the Runge's Phenomenon exhibited by Polynomial Interpolation. I ...
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240 views

Error on optimization problem, maximize log determinant on CVX

$A$ is an $N \times N$ complex matrix $W$ is an $N \times N$ complex matrix $C$ is an $N \times N$ complex diagonal matrix $u$ is a scalar $V$ is an $N \times N$ complex matrix, whose diagonal elects ...
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57 views

minximum and maximum of $P=x+y+z$

Let $x,y,z \in R;x \ge 1,y \ge 2,z \ge 3$ and $$\sum\limits_{\large{\text{cyc}}} {\frac{{{x^2} - x + 1}}{{x + \sqrt {x - 1} }} = 12} $$ Search minximum and maximum of $P=x+y+z$
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66 views

Loss Function and Probability

Suppose we have some input variables and the output variable is categorical. So the output $G(X)$ can be in one of $k$ classes. Therefore an estimator $\hat{G}(X)$ will also be in one of these $k$ ...
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76 views

Angles inequality in a tetrahedron

Let $ I \in \triangle \text{ABC}$ of tetrahedron $ABCD$. Prove the inequality: $$\angle ADB+\angle BDC+\angle CDA \ge \angle ADI+\angle BDI+\angle CDI$$ Non-Euclidean geometry is very new for me. ...
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442 views

Constrained optimization problem with the integral

I want to maximize the following integral $I = \int_0^\infty {f\left( {P(x),x} \right){p_X}(x)dx} $ subject to the constraint $\int_0^\infty {P(x){p_X}(x)dx} = P_x$, where $p_X(.)$ is a probability ...
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37 views

Why does the result of the Lagrangian depend on the formulation of the constraint?

Consider the following maximization problem: $$ \max f(x) = 3 x^3 - 3 x^2, s.t. g(x) = (3-x)^3 \ge 0 $$ Now it's obvious that the maximum is obtained at $ x =3 $. In this point, however, the ...
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56 views

Converting a linear program into standard form

In especially, I have a question about the demand that if I have $ Ax \leq b$, then I can convert this into $A'x'=b$ for some new $A'$ and $x'$. I have given the system of equations: $20x_1+30x_2 ...
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1answer
31 views

Finding a concave function that minimize the middle value while the boundary values are fixed

This question came to me while I was listening to Dominik's talk this afternoon. First, let me remind you what does f is concave mean. It means f satisfies $pf(x)+(1-p)f(y)\le f(px+(1-p)y)$, $\forall ...
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97 views

Solving a conjecture by bruteforcing

Say we wanted to check the Beal conjecture ["If $A^x+B^y=C^z$, where $[A, B, C, x, y, z \in N] \wedge [x, y, z \gt 2] \to $ A, B and C must have a common prime factor", from the official site]. What ...
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115 views

Optimal Mix / constrained optimization

I'm looking to solve a constrained optimization problem. I'm running into trouble with the number of inputs: Say $Z$ is the output I want to maximize, subject to a budget constraint of $\Sigma x ...
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41 views

Finding the sequence that maximizes a constrained sum

Let $0 < a < 1$ and let $S_k$ be a unknown sequence of such that $S_k > 0$ and $$ S_n + S_{n-1} + \ldots + S_1 = C = constant. $$ What should be $S_k$ so that the sum $$ S_n + aS_{n-1} + ...
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275 views

Finding the smallest subset of a set of vectors which contains another vector in the span

Consider a set $S=\{ \underline{v_1},\dots , \underline{v_n} \} $ of vectors of dimension $d<n$. Suppose for some vector $\underline{b}$ that the solution space for the matrix equation $\left[ ...
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maximum area of soap film bound by wire of unit length

A soap film has zero mean curvature at any point, and the area of any soap film bordered by wire is the surface of least area that spans the wire. What is the maximum total surface area of soap film ...
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48 views

Simple question of maximum value a part can have?

We have to partition n chocolates among m children. Children will be happy if max and min a child has got is less than 2. What is the max a child can get?? For n=6 m=3 ,the partition will be 2 2 2 ...
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619 views

Lasso - constraint form equivalent to penalty form

We know that there are two definitions to describe lasso. Regression with constraint definition: $$\min\limits_{\beta} \|y-X\beta\|^2, \sum\limits_{p}|\beta_p|\leq t, \exists t $$ Regression with ...
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Levenberg-Marquardt - Is forcing Hessian to be positive definite OK?

I am often doing parameter estimation using Levenberg-Marquard method which involves solving the following linear system at each step: $$(H+\lambda I)\delta=r_{i}$$ where $H$ is a square Hessian ...
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24 views

Extremal of a function -Euler equation

I have to calculate $J(t+h)-J(t)$ where $J(x)=\int_0^1 x'^3 dt$, $x=x(t)$, $h\in C^1[0,1]$, $h(0)=h(1)=0$ I have solution, I will write it below, and I will write my question. $J(t+h)-J(t) ...
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Minimizing area of a triangle with two fixed point and a point on parabola

A triangle is made up of three points, $A, B$, and $P$. $A(-1, 0)$ $B(0, 1)$ $P$ is a point on $y^2 = x$ Minimize the area of Triangle $ABP$. My approach is far too complicated, which ...
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129 views

Gradient descent/ nonlinear optimization intuition needed

all. I'm taking an introductory AI class, and we're using the gradient descent algorithm to find the optimized/ lowest cost of a set of thetas (variable coefficients) to best fit a regression line. In ...
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positive semidefiniteness: a psd matrix substracted by another rank 1 psd matrix

Given that $A$ is a positive semidefinite matrix, $x$ is a vector, $\lambda_0 \in [0, +\infty) $ is a real non-negative number. I want to know the answer to the following optimization problem. $$ ...
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114 views

A hard multivariate optimization problem in $n-1$ variables

For $n>1$, I want to find the smallest value, and corresponding $x_i$ values, of $f(x_2,\dots,x_n) = \prod_{k=2}^n (x_k+1)^k$ subject to the constraints $x_j > 0$ for all $j$ and $\prod_{k=2}^n ...
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Maximize area of a triangle with fixed perimeter

If perimeter of a triangle is $2d$, what is the length of sides so the triangle has maximal area? I found some solution using circle and angles, but I think I have to use derivatives. I need ...
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105 views

Maximum and Minimum Value of $f(x)$

$$f(x)=\sin(x)+\int_{-\pi/2}^{\pi/2}\left(\sin(x)+t\cos(x)\right)f(t)\,\mathrm dt$$ Find maximum and minimum values of $f(x)$. I tried to simplify this expression by checking even or odd ...
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Optimum fitting for flanges in a rectangular plate

I have a $2500~\text{mm}\times6300~\text{mm}\times25~\text{mm}$ (width $\times$ length $\times$ thickness) steel plate I want to cut flanges of diameter $235~\text{mm}$ can anyone please suggest $1)$ ...
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114 views

Finding Maximum Under Constraint

Suppose $a$,$b$,$c$ satisfy $a+b+c=1$ and $a$,$b$,$c\in [0,1]$ Find the maximum value of $(a-b)(b-c)(c-a)$
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389 views

How to find the shortest path between two points under the restriction?

Let two different points $M_1(x_1,y_1,z_1)$, $M_2(x_2,y_2,z_2)$ and two nonintersecting lines $l_1$, $l_2$ be given. How to find the shortest path between $M_1$ and $M_2$ which intersects both $l_1$ ...
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211 views

Is the gradient of this summation correct?

Consider the following summation: $$f(P_e,P_R) = \sum\limits_{i \neq R}\left(\delta_{i}-h(P_{e},P_{R},P_{i})\right)^2$$ where $\delta_{i}$ is a real number and $h$ is the function: ...
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A question about the second derivative test.

Suppose we're given a function $f : X \rightarrow \mathbb{R}$ with $X \subseteq \mathbb{R}.$ Then by the second derivative test, we have that for all points $x \in X$ such that $f$ is twice ...
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88 views

Projections onto closed and convex sets

I have to prove that if $A$ is convex and closed set, then $z=P_A(x)$ for all $z\in A$ if and only if $\langle x-z, z-y\rangle \geq 0$ for all $y\in A$ I have following proof which is not much ...
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52 views

Optimization of entropy for fixed distance to uniform

Suppose that I know that a probability distribution with $n$ outcomes is very close to being uniform (that is: $\forall i,p_i=\frac{1}{n}$), and in particular for $n\epsilon\ll 1$ the distribution ...
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46 views

formulas to maximize the output

Good day, I have a math problem in my game development like this: I have two numbers (football player skills): Attack skill (AS) and Defend skill (DS). They are in range from 1.0 to 8.0... now I want ...
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1answer
125 views

The relationship between fisher information and EM algorithm?

I wonder what is the relationship between fisher information and EM algorithm? When I read papers about EM algorithm, people sometimes discussed about fisher information, and there are algorithms ...
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96 views

maximization of a Strictly convex function

The things we know, usually minimization of a convex function, unique solution will exist. My question is, maximization of a strictly convex function, will that give an unique maximum? If so how we ...
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Extreme values of a function

$x_1,x_2,...,x_m\in R^n$ are given. How to find $u\in R^n$ such that $\sum^{m}_{i=1} (d(u,x_i))^2$ is minimal. I tried this way: If $J(u)=\sum ^{m}_{i=1} (d(u,x_i))^2$ ...
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137 views

Prove $x^2+y^2+z^2 \ge 14$ with constraints

Let $0<x\le y \le z,\ z\ge 3,\ y+z \ge 5,\ x+y+z = 6.$ Prove the inequalities: $I)\ x^2 + y^2 + z^2 \ge 14$ $II)\ \sqrt x + \sqrt y + \sqrt z \le 1 + \sqrt 2 + \sqrt 3$ My teacher said the ...
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Fundamental matrices

Find the fundamental matrix for the two-dimensional system defined by $x_1' = x_1 + tx_2$, and $x_2'=x_2$. And determine the solution for which $x_1(0)=c_1$, and $x_2(0)=c_2$. I am stuck because of ...
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356 views

Are derivatives defined at boundaries?

Given a differentiable function $f : [-5,5] \rightarrow \mathbb{R},$ I was under the impression that the derivative $f'$ has domain $(-5,5).$ However, according to Wikipedia ...a differentiable ...
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Using math for interior decorating with lamps

When I was in college, I owned three lamps and had a dark apartment. I kept trying to position them in different areas of the room, but it was still dark. Then I decided to model the problem with ...
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175 views

formulate this scheduling problem as linear programming problem

Sorry if this very silly, but i am something new to optimization theory: We have $m$ identical Machines and $n$ jobs. A job $j$ can be done in any of the identical machines in $p_{j}$ time units. ...
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73 views

How to define an objective function that conveys the concept of selecting the best elements in a set

Consider a set of tasks $\mathcal{T} = \{t_1, \ldots, t_I\}$. Consider also a set of workers $\mathcal{W} = \{w^1, \ldots, w^J\}$, where each worker $w^j \in \mathcal{W}$ is associated with a value ...
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4answers
79 views

Maximum value of a product

How to write the number $60$ as $\displaystyle\sum^{6}_{i=1} x_i$ such that $\displaystyle\prod^{6}_{i=1} x_i$ has maximum value? Thanks to everyone :) Is there a way to solve this using Lagrange ...
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72 views

Extreme values of a function with conditions

What is a way to find extreme values of a function $u(x,y,z)=xy+yz+xz$ with conditions $x+y=2, y+z=1$?
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81 views

Prove inequality: $74 - 37\sqrt 2 \le a+b+6(c+d) \le 74 +37\sqrt 2$ without calculus

Let $a,b,c,d \in \mathbb R$ such that $a^2 + b^2 + 1 = 2(a+b), c^2 + d^2 + 6^2 = 12(c+d)$, prove inequality without calculus (or langrange multiplier): $$74 - 37\sqrt 2 \le a+b+6(c+d) \le 74 ...