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 prove the sum of squares is minimum?

Given $n$ positive values. Their sum is $k$. $$ x_1 + x_2 + \cdots + x_n = k $$ The sum of their squares is defined as: $$ x_1^2 + x_2^2 + \cdots + x_n^2 $$ I think that the sum of squares is ...
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3answers
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Newton's method in higher dimensions explained

I'm studying about Newton's method and I get the single dimension case perfectly, but the multidimensional version makes me ask question... In Wikipedia Newton's method in higher dimensions is ...
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1answer
426 views

Convert Semidefinite program forms

How do I convert the following SDP problem (written in the standard inequality form): $$\min c^T x$$ $$\text{s.t. }F(x)\succeq0$$ When $F(x)\equiv F_{0}+\sum_{i=1}^{m}x_{i}F_{i}$ when $F_{i}\in S^{n}...
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1answer
79 views

Find minimum of $P=\frac{\sqrt{3(2x^2+2x+1)}}{3}+\frac{1}{\sqrt{2x^2+(3-\sqrt{3})x +3}}+\frac{1}{\sqrt{2x^2+(3+\sqrt{3})x +3}}$

For $x\in\mathbb{R}$ find minimum of $P$. $P=\dfrac{\sqrt{3(2x^2+2x+1)}}{3}+\dfrac{1}{\sqrt{2x^2+(3-\sqrt{3})x +3}}+\dfrac{1}{\sqrt{2x^2+(3+\sqrt{3})x +3}}$ Source : Viet Nam national test for high ...
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0answers
184 views

Reformulation of Goldbach's Conjecture as optimization problem correct?

Question I think I managed to reformulate a stronger version of Goldbach's conjecture as an optimization problem: $$ \frac{\partial F_n}{\partial a_n} = \frac{\partial F_n}{\partial \overline a_n} =...
15
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1answer
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Maximum total distance between points on a sphere

What is the configuration (set of locations) of $n$ points on the surface of a sphere such that the sum of distances is maximum for $n=1,2,3,...$? The sum of distances is measured by summing the ...
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3answers
762 views

Explain a surprisingly simple optimization result

The following optimization problem came to my attention as an idealization of the silly browser game Cookie Clicker, but is representative of a range of strategy games: You have an initial ...
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3answers
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Primal- degenerate optimal, Dual - unique optimal

Simple question- Is it possible for a linear programming optimization problem possible to have a degenerate optimal solution whereas the dual has a unique optimal solution? I can't find a scenario ...
6
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5answers
908 views

Arithmetic mean is less than geometric mean (Spivak Calculus 3rd Chapter 2 Problem 22)

If $a_1, \ldots, a_n \ge 0$, the arithmetic mean $$A_n={a_1 + \cdots + a_n \over n}$$ and the geometric mean $$G_n = \sqrt[n]{a_1 \cdots a_n}$$ satisfy $G_n \le A_n$. As a first step to prove this ...
5
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1answer
927 views

Why does SVD provide the least squares solution to $Ax=b$?

I am studying the Singular Value Decomposition and its properties. It is widely used in order to solve equations of the form $Ax=b$. I have seen the following: When we have the equation system $Ax=b$, ...
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1answer
70 views

Show that if $ A(t) \in \mathbb{R ^{n×n}}$ is a differentiable function of $t$ then $\frac{d(det A(t))}{dt}$ [duplicate]

Let $\mathbb{R^{ n×n}}$ be the space of n × n real matrices. Show that if $ A(t) \in \mathbb{R ^{n×n}}$ is a differentiable function of $t$ then $\frac{d(det A(t))}{dt}$ is the sum of the determinants ...
2
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1answer
184 views

Control on Conformal map

Let $\Omega$ be smooth simply connected open set of $\mathbb{R}^2$ such that $\overline{\Omega}$ is compact. We know that there exists a conformal diffeomorphism $\psi$ from $\mathbb{D}$ to $\Omega$. ...
2
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1answer
143 views

Maximum of the sum of cube

(1) $-2\leq a_{i} \leq 2$ $~(i=1,2,3,4,5)$ (2) $\displaystyle\sum_{cyclic}a_{i}=0$ then, find the maximum value of $\displaystyle\sum_{cyclic}a_{i}^{3}$ also, can it be generalized as for $\...
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2answers
1k views

satisfy the Euler-Lagrange equation

Two circles of unit radius, each normal to the line through their centers are a distance d apart. A soap film is formed between themas shown below; energetic considerations require the filem to ...
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1answer
42 views

Maximize the sum of numbers subject to an upper bound on their sum of squares

I want to find a maximum for $\sum_i^n x_i$ subject to $\sum_i^n x_i^2 \leq c$, where c is some constant. I've done some searching and that's hinted at using the Cauchy-Schwarz inequality, but I ...
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1answer
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The composition of two convex functions is convex

Let $f$ be a convex function on a convex domain $\Omega$ and $g$ a convex non-decreasing function on $\mathbb{R}$. prove that the composition of $g(f)$ is convex on $\Omega$. Under what conditions is $...
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5answers
467 views

How to find the minimum of $a+b+\sqrt{a^2+b^2}$

let $a,b>0$, and such $$\dfrac{2}{a}+\dfrac{1}{b}=1$$ Find this minimum $$a+b+\sqrt{a^2+b^2}$$ My try: since $$2b+a=ab$$ so $$a+b+\sqrt{a^2+b^2}=\sqrt{a^2+b^2+2ab}+\sqrt{a^2+b^2}=\sqrt{a^2+b^2+4b+...
5
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2answers
489 views

How to find the shift that minimizes the difference between two vectors?

I am looking for a efficient way to find the value of k that minimizes $\sum(s_t - b_{t+k})^2$ where $s$ and $b$ are N-dimensional vectors and the values are wrapped around like this: $b_{t+k} := ...
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3answers
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How to interpret Hessian of a function

I know that gradient of a function gives the direction in which the directional derivative of the function is maximum. Is there any similar interpretation of Hessian ?
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0answers
369 views

Real approximation to the maximum using Laplace's method integral

The Laplace's Method states that under some conditions, it holds that: $ \sqrt{\frac{2\pi}{M(-g''(x_0))}} h(x_0) e^{M g(x_0)} \approx \int_a^b\! h(x) e^{M g(x)}\, dx \text { as } M\to\infty$ Where $...
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1answer
354 views

Positive semidefinite cone is generated by all rank one matrices.

The positive semidefinite cone is generated by all rank one matrices $xx^T$ . They form the extreme raysof the cone. The positive definite matrices lie in the interior of the cone. The positive ...
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1answer
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Prove the supremum of the set of affine functions is convex

Let $\langle f_i \rangle _{i \in I}$ be a family of affine functions on a convex and compact set $\Omega \subset \mathbb{R^d}$ such that $f_i = a_i.x +b_i$ for $x \in \Omega$. Prove that f, defined by ...
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1answer
544 views

Is Minimax equals to Maximin?

Consider a loss funcation $\ell(x,y)$ with a penalty $g(x,y)$ If I want to consider the worst case robust scenario, that is \begin{equation} \min_x \max_y \ell(x,y) + g(x,y) \end{equation} Is it ...
2
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1answer
73 views

Conjugacy relation in the primal and dual problem

The following is my derivation in the Conjugacy relation in the primal and dual problem. I am shaky in it; so hope for some advices. Consider the following problem, $f(x),g(x)$ are convex ...
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1answer
187 views

Martingale formulation of Bellman's Optimality Principle

Related question: Deducing an optimal gambling strategy (using martingales). What I tried: For no 2, if $\ln Z_n - n \alpha$ is a supermartingale, then for $m < n$, $$E[\ln Z_n - n \alpha | \...
2
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4answers
180 views

Three Variables-Inequality with $a+b+c=abc$

$a$,$b$,$c$ are positive numbers such that $~a+b+c=abc$ Find the maximum value of $~\dfrac{1}{\sqrt{1+a^{2}}}+\dfrac{1}{\sqrt{1+b^{2}}}+\dfrac{1}{\sqrt{1+c^{2}}}$
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2answers
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Multiple solutions for both primal and dual

If matrix $A$ in an LP (or $A^T$ in its dual) has full row (column- in dual) rank, is it possible that both primal and dual have multiple solutions?
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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: (3,2),...
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2answers
114 views

Minimizing $f(x,y,z)=\dfrac{|x|+|y|+|z|}{xyz}$ on a sphere

I need to find the minimum of the function: $$f(x,y,z)=\dfrac{|x|+|y|+|z|}{xyz}$$ with the condition: $$x^2+y^2+z^2=r^2$$ Using numerical methods it's quite easy to solve the problem. How can I ...
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Maximize volume of box in ellipsoid

I need to find the dimensions of the box with maximum volume (with faces parallel to the coordinate planes) that can be inscribed in ellipsoid $$\frac{x^2}{4} + \frac{y^2}{9} + \frac{z^2}{16} = 1$$ ...
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3answers
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How to plot a phase portrait for this system of differential equations?

I beg your help.. I'd like the phase portrait for this system. I don't know how to use Mathematica/Matlab ... :( If anyone can make this portrait and post a print screen here, I would thank you ...
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How to optimize a singular covariance-weighted residual?

Definitions: $$v(x)\equiv\{g_1(x),g_2(x),\ldots,g_n(x)\}^T$$ $$C\equiv \operatorname{cov}(v)=\langle vv^T \rangle -\langle v\rangle \langle v^T \rangle =\int f(x)v(x)v(x)^T \, dx-\int f(x)v(x) \, dx ...
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1answer
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Find the Lagrange multipliers with one constraint: $f(x,y,z) = xyz$ and $g(x,y,z) = x^2+2y^2+3z^2 = 6$

Where $f(x,y,z) = xyz$ and the constraint is $g(x,y,z) = x^2+2y^2+3z^2 = 6$ I have tried this problem like three or four times and not gotten the solution, I even asked this question once and got the ...
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5answers
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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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2answers
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Augmented Reality Transformation Matrix Optimization

i am a software developer, i'm working on an Augmented Reality system. I'd like to receive some advice in order to optimize my math model. My program has to be slim and fast. Here's the situation: ...
6
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2answers
147 views

Find the minimum value of $A=\frac{2-a^3}{a}+\frac{2-b^3}{b}+\frac{2-c^3}{c}$

Let $a, b$ and $c$ three positive real numbers such that $a+b+c=3$. Find the minimum value of $$A=\frac{2-a^3}{a}+\frac{2-b^3}{b}+\frac{2-c^3}{c}.$$ Here is my attempt. By symmetry we can assume that ...
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Math notation for location of the maximum

My question is about notation. I have maximum of the function $f(x)$. This can be expressed as $\max(f)$ How can I express in compact form that $x_0$ is the location of that maximum.
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Maximizing the sum of two numbers, the sum of whose squares is constant

How could we prove that if the sum of the squares of two numbers is a constant, then the sum of the numbers would have its maximum value when the numbers are equal? This result is also true for ...
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1answer
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Simple optimization trick

Let $f,g:X\to\Bbb R$ be two functions where $X$ is any set. Then $$ \left|\sup_x f(x) - \sup_x g(x)\right|\leq \sup_x|f(x) - g(x)|. $$ This fact is fairly easy to prove, but it seems to be a ...
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3answers
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Proving the regular n-gon maximizes area for fixed perimeter.

It is often assumed that, given $n$, the regular $n$-gon will make the most efficient use of perimeter for area. I have never seen this proven. Anyone have something slick? (That is, how can we ...
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2answers
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Finding the widest angle to shoot a soccer ball from the sideline using optimization

I'm trying to do an independent project for my Math class, but I was stuck and couldn't figure out how to use optimization to find position along the sideline that gives the widest angle to shoot. As ...
2
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1answer
252 views

Optimisation Problem on Cone

The problem I've got here is to prove that semi vertical angle of a cone with maximum volume with total surface area constant is equal to $arcsin(\frac{1}{3})$ I am trying to do that by making some ...
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0answers
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$4$ or more type $2$ implies $3$ or less type $1$

I'm having difficulties with the logic with the last part of the reformulation part of the problem below. Let $x_i$ be the the number of ships of type $i$ to purchase. For $4a:$ (the ...
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2answers
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If a nonnegative function of $x_1,\dots,x_n$ can be written as $\sum g_k(x_k)$, then the summands can be taken nonnegative

Suppose that $f:\mathbb{R}^n\to [0,\infty)$ is a function that can be written in the form $$f(x_1,\dots,x_n)=g_1(x_1)+g_2(x_2)+\dots+g_n(x_n) $$ Can we also choose all $g_k$ to be nonnegative? ...
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Global maxima/minima of $f(x,y,z) = x+y+z$ in $A$

Find the global maxima/minima of $f(x,y,z) = x+y+z$ for points inside of $A = \{ (x,y,z) \in \mathbb{R}^3: x^2-y^2 = 1 \wedge 2x+z = 1 \}$ I renamed the conditions of $A$ to a function $g(x,y,z) = x^...
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290 views

Applying the Lagrangian function to find critical points

So I have the following function $$ f(x,y) = x^2+y^2 $$ subject to $$ g(x,y) = x+y-1 = 0. $$ And I have to use the Lagrangian to find the critical points, and determine wether they are ...
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0answers
93 views

Trace minimization when some matrix is unknown

The problem is as follows: $\displaystyle\min_{V}$ trace($V^TH^T\Phi HV$)$\\$ s.t. $V^TV=I_d$ in the case when $H$ is not known. When $H$ is known, the solution is given by the eigenvectors ...
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1answer
80 views

Armijo rule intuition and implementation

I am minimizing a convex function $f(x,y)$ using the steepest descent method: $$\mathbf{x}_{n+1}=\mathbf{x}_n-\gamma \nabla F(\mathbf{x}_n),\ n \ge 0$$ My function is defined over a specific domain $...
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1answer
43 views

Convexity of set

If $C\subset\mathbb{R}^m$ is a convex set, $A$ is an $m\times n$-matrix and $b\in\mathbb{R}^m$, how do I prove that the set $S=\{x\in\mathbb{R}^m|Ax+b\in C\}$ is convex? I know that the definition of ...
0
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2answers
112 views

Constraint minimization of sum of Non-symmetric matrices

I am trying to find closed form solution to following problem \begin{equation} \begin{array}{c} \text{min} \hspace{4mm} \big(\lambda_1\left( \mathbf{y}^T V^{(1)}\mathbf{x} \right)^2 + \lambda_2\...