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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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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32 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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1answer
105 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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118 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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44 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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352 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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60 views

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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1answer
50 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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1answer
891 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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1answer
441 views

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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1answer
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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2answers
103 views

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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1answer
153 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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110 views

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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1answer
129 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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2k views

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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1answer
113 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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2answers
122 views

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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3answers
120 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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2answers
485 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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1answer
258 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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149 views

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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95 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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1answer
53 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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50 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
155 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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1answer
116 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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1answer
41 views

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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139 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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1answer
2k views

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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464 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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300 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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75 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
83 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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82 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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86 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 ...
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0answers
36 views

Optimal paths between two closed lines

Imagine I have an outer shape and an inner shape (that may overlap like the following picture) Is there any algorithm or mathematical property I can use to find a third set of points which will be ...
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1answer
346 views

Integer Linear Programming (ILP): NP-hard vs. NP-complete?

I was thinking about examples where a problem is NP-hard but was not NP-complete and ILP came to mind. It is obviously NP-hard but is it NP-complete? I.e., is it in NP? Given a certificate (the ...
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136 views

Maxima and minima of multivariable function $f(x,y)=6x^3y^2-x^4y^2-x^3y^3$

$$f(x,y)=6x^3y^2-x^4y^2-x^3y^3$$ $$\frac{\delta f}{\delta x}=18x^2y^2-4x^3y^2-3x^2y^3$$ $$\frac{\delta f}{\delta y}=12x^3y-2x^4y-3x^3y^2$$ Points, in which partial derivatives ar equal to 0 are: ...
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807 views

Closest point in $y = \sqrt{x}$ to the origin is at $x=-1/2$?

When I solve for the point in $y = \sqrt{x}$ closest to the origin using calculus, I get $x = -1/2$. And this is the case for ALL functions $y = \sqrt{x + c}$ using the distance formula $d^2 = x^2 + ...
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Confusion related to optimization of log(det(X)) function

I have this confusion related to optimization of the log(det(X)) function. I didn't get how it implicitly maintains the constraint of X being positive definite. For eg if I have a matrix ...
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226 views

Prove inequality $(x+y+z-2xyz)^2 \le 2$

Problem: Prove inequality $(x+y+z-2xyz)^2 \le 2\ (1)$ with $x^2+y^2+z^2 = 1 \land x,y,z \in \mathbb R$ I tried expand $LHS$ and have: $$(1)\iff 1 - 2 (xy+yz+xz) + 4 xyz(x+y+z)-(2xyz)^2 \ge 0$$ ...
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177 views

Distance between two $m \times n$ matrices

Consider two $m \times n$ matrices $A$ and $B$, where $m > n$. The Singular Value decomposition of $A$ and $B$ can be given as: $A = U_A\Sigma_AV_A^T$ and $B = U_B\Sigma_BV_B^T$ The left and ...
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175 views

Prove that a local min is also a global min

Let Q $\in \mathbb{R}^{d \times d} $ and A $\in$ $\mathbb{R}^{d' \times d} $ be two matrices. Let b $\in \mathbb{R}^d$ and c $\in \mathbb{R}^{d'}$ be two vectors. Suppose that d' < d. I want to ...
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1k views

Constrained infinity norm minimization

I have a problem like this: $$\min_x |Ax|_\infty \text{ s.t. } \sum_i x_i = c$$ That is, I want to find the vector $x$ whose elements sum to a constant $c$ that minimized the infinity norm of $Ax$. ...
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40 views

Train and horseman velocity

This is a more or less easy exercise but there is one point I do not understand. We have a train starting at $(0|0) $with velocity $v_t=20 m/s$ straight in the $y$-direction and a horseman at ...
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71 views

Positioning Problem

Please help me with how to start with this question: A particle of unit mass moves along the x-axis subject to a force $u(t)$. We want to transfer the particle from rest at the origin, to rest as x=1 ...
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199 views

Not so easy optimization of variables?

What is the maximum value of $x^2+y^2$, where $(x,y)$ are solutions to $2x^2+5xy+3y^2=2$ and $6x^2+8xy+4y^2=3$. (calculus is not allowed). I tried everything I could but whenever I got for example ...