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 are the Bisection and Newton Method for finding roots complementary to each other?

my lecture note states that the bisection and newton method for finding roots are most of the time complementary to each other but I can not figure out why. I have basic understanding of both of the ...
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Am I allowed to do this chain-rule type thing when optimizing wrt to a functional?

So we have a function $f(x) = g(x)h(y(x))$ that is convex in $y$ which we want to optimize by choosing the appropriate $y(x)$. I have seen the following done in engineering books, but it just looks so ...
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25 views

minimize this objective function

I have a function to minimize and I don't understand how I should proceed. The function is coming from a publication. Background: In a 2D image, $P_1$ and $P_2$ represents 2 patches of colors (RGB) ...
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47 views

Find the smallest positive value taken by $a^3+b^3+c^3-3abc$

Find the smallest positive value taken by $a^3+b^3+c^3-3abc$ for positive integers $a,b,c$. Find all integers $a,b,c$ which give the smallest value. Since it is generally hard to find the minimum ...
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15 views

Maximization over minimum function

I want to solve the following optimization problem. Suppose we are given $p_r^i \in [0,1]$ for $r={1,2,...,N}$ and $i={1,2}$ such that $\sum_{r=1}^N p_r^i =1$ for i={1,2}. We want to find $x_r \in ...
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25 views

Optimizing the area of a rectangle with one side against a wall using the am-gm inequality

Given 300 meters of fence, how can I find the dimensions of a rectangle that is built against a wall the encloses the maximum area. I found this question in a calculus book and saw a simple solution ...
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8 views

Using the Fourier Series in Variational Optimization Problems

Say I have a functional $L(f)$ which takes as input the function $f:\mathbb{R}\to\mathbb{R}$, and I want to find the function that optimizes $L$. Unfortunately, there's no way to define a functional ...
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1answer
17 views

Find the positions of $A$ and $B$ which minimizes the length $AB$.

The line $AB$ joins the points $A(a, 0 )$, $B(0, b)$ on the $x$ and $y$ axes respectively and passes through the points $(8, 27)$. Find the positions of $A$ and $B$ which minimizes the length $AB$. ...
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1answer
10 views

Softmax Regression Derivative

This website, http://deeplearning.stanford.edu/wiki/index.php/Softmax_Regression, claims the derivative of a multinomial regression: $$ J(\theta) = -\frac{1}{m}\sum_{i=1}^m \sum_{j=1}^k 1\{y^i =j\} ...
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13 views

Singularities of composite function

Given a smooth, compact manifold $M$ (of dimension much less than $n$) and two maps $f:\mathbb{R}^n \rightarrow M$, $g:M\rightarrow \mathbb{R}$, I want to understand the topology of the critical set ...
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10 views

Derivation of SVM algorithm (Lagrangian)

I have a question about the derivation of the SVM algorithm (for example, page 3 here ). The question is about the math, so that's why I'm asking this here. Suppose I have the following optimization ...
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197 views

Set is Convex regardless of b

Let the function $f$ be convex, $f :\Bbb R^n \rightarrow \Bbb R$ and let $$S = \{x : f(x) \le b\}$$ The proposition states that the set $S$ is convex regardless of $b$. Can someone explain to me how ...
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1answer
51 views

Maximum of polynomial [on hold]

I was studying statics and came across this problem: Find the value $\beta$ such that $P$ has a maximum value in $R^2 - 1000^2 = P^2 + 2000P\cos(75^{\circ}+\beta)$. When $R$ is constant, the ...
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1answer
23 views

How do you find $∠XPC$ + $∠XPB$ such that $PB+PC$ is maximum where $P$ is a point on $f(x) = (x-1)(x-3)(x-5)$?

Problem: $f(x) = (x-1)(x-3)(x-5)$ intersects the x axis at $A(1,0)$, $B(3,0)$ and $C(5,0)$. A point $P(t,f(t))$ is selected on the curve such that $PB+PC$ is maximum and $t \in (3,5).$ Let $PX$ be ...
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7 views

Optimize polling frequency between producer and consumer to achieve minimum waiting time

Background: I am trying to optimize what we call AJAX request polling frequency in the domain of web design, and I wanted to check if I could use some help from math guys to explore a better ...
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1answer
19 views

How can we solve this system of linear inequalities?

Let $c_i$ be a given non-negative integer for all $i\in\{1,\ldots,n\}$. I would like to find the non-negative integers $a_i$ and $b_i$ for all $i\in\{1,\ldots,n\}$ such that: \begin{align} ...
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22 views

Local maximum of $(2^{xy}{z \choose y})^{z+1}$

I have an optimization problem where I need to calculate the maximum of the following function $$ f(x,y,z) = (2^{xy}{z \choose y})^{z+1} $$ where $$ (z+1)(a+y(\lceil{\log_2{(z+1)}}\rceil+x))\leq C $$ ...
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8 views

ADMM Formulation for fractional programs?

I have a fractional program $$ \min \frac{f(x)}{g(x)} $$ where $f(x)$ and $g(x)$ are both quadratic functions (i.e. they are convex). It has been shown that this can be alternatively represented by ...
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47 views

minimum possible value of a linear function of n variables

Suppose $x_1,x_2,\ldots,x_n$ are unknowons satisfying the constraint $a_1x_1 + \cdots + a_nx_n ≥ b$, where $a_1, \ldots , a_n, b ≥ 0$. Then the minimum possible value of the expression $c_1x_1 + ...
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41 views

How to prove that the following function has a unique mode?

I am trying to prove that the function $$f(\alpha)=n\ln \alpha-n\ln\Big(\sum_{i=1}^{n}t_i^\alpha+\int_{a}^{b}x^{\alpha+\beta-1}e^{-\lambda x^\beta}\,dx\Big)+(\alpha-1)\sum_{i=1}^{n}\ln t_i,$$ where ...
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2answers
25 views

SVD with Sparisty Penalties

I am trying to solve a problem like $$\min_{u,v} \Vert X-uv^T \Vert_F^2 + \beta \Vert v \Vert_1 \mbox{ s.t. } \Vert u \Vert_2 \le 1$$ where $X \in \mathbb{R}^{N\times N}$, $u, v \in ...
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26 views

Show that any vertex $v$ of $P$ is half-integral.

Let $G$ be an undirected graph and define $$P=\{x \in R^{V}: x(u)+x(v) \leq 1 \:\:\text{for all edges}\:\: e=uv,\:\: x \geq 0\}$$ Show that any vertex $v$ of $P$ is half integral.
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Optimizing functions with a complex domain and a real codomain

In general I want to understand if it makes sense to optimize a function of the following form $f: \mathbb{C} → \mathbb{R}$ for my specific problem $f(z) = | z | ^{2} $ (wich I is not analytic since ...
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Removing variables from convex linear program

I am solving linear program (possibly non-convex). Then we know that dual is always convex. Then I noticed that depending on objective functional I can sometimes remove particular variables from this ...
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18 views

Negative eigenvalue of a hessian matrix entails a local decrease in function value?

I was reading up on non-convex optimization, and I can across this sentence: "Since Hessian(f(w)) has a negative eigenvalue, there is always a point that is near w which has smaller function value" ...
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156 views

How do you find the maximum value of $|z^2 - 2iz+1|$ given that $|z|=3$, using triangle inequality?

Problem: How do you find the maximum value of $|z^2 - 2iz+1|$ given that $|z|=3$, using triangle inequality? My attempt: $$|z^2 - 2iz+1|\le|z|^2+2|i||z|+1$$ $$\implies |z^2 - 2iz+1|\le16$$ ...
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24 views

Win/Lose ratios and selection strategies

Imagine the following scenario: You're on a TCG tournament which allowed you to bring N decks with you. After each game, you might select another deck for your next game. You are allowed to keep ...
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65 views

Making a Matrix singular

During my research I came across the following problem. Intuitively this should be an easy one. However, the simplest version of it looks like this: Let $C \geq \frac{1}{2}$ be some fixed ...
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1answer
28 views

Inequality with three unknowns

Consider: \begin{equation} \Big(e^x-1\Big)\mathbb{1}_{(x\geq0)} \leq \lambda_1+\lambda_2e^x + \lambda_3x^2 \end{equation} where $\lambda_1$, $\lambda_2$, $\lambda_3$ are three unknown constants. By ...
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Gradient Descent with multiplication term

Say I have the objective: $\arg \min_{R, T} \|y - RTx\|^2_2$ where, R and T are matrices (not necessarily square) and y and x are known vectors. I wish to try and optimize R and T using Gradient ...
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Understanding ADMM: how is it applied to this particular problem?

While reading Boyd's paper on ADMM I encountered an issue. Consider the following problem: Problem. Minimize $f(u) + g(v)$ subject to $Au + Bv = c$, where $f$ and $g$ are closed, proper, convex and ...
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1answer
24 views

Continuity of Parameterized Optimal Solution

Suppose for every $y$, $f(x,y)$ is strictly convex in $x$. Further, $f(x,y)$ is continuous in $y$. Let $\mathcal X$ be compact (in my problem, $\mathcal X$ is an interval). Can anyone suggest any ...
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Setting the right restriction in a simple linear optimization task

here is the task: One factory produces 3 types of cars: small, midsize and big. There are 6000 tons of steel and 60000 total time available. For each type of car produced, there must be 1000 cars of ...
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What is the maximum of the following function?

Let $f(x,y) = \frac{xy^\alpha}{x+y},\alpha\in(0,\infty)$. How to compute $$\sup_{(x,y)\in[a,b]\times [0,c]}\frac{xy^\alpha}{x+y},$$ with $b>a>0$?
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One-dimensional deblurring

I just begun studying image deblurring on my own, and I have a question. Most books I found say that I can see the images as arrays, and that I can "vectorize" the arrays of the images by stacking the ...
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1answer
16 views

Does optimal solution from primal problem follow from optimal solution to dual?

In a linear programming context, does the primal optimal solution yield an explicit way to find the primal dual solution? I vaguely remember something like this from an optimization class but can't ...
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Biorthogonal (discrete wavelet) noise bases?

I am slightly interested in discrete wavelet transforms (DWT), but so far I have mostly used already-derived and existing well-known wavelets, such as Daubechies, Cohen-Daubechies-Faveau, Symlets and ...
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3answers
55 views

How to do a regression which includes reciprocals?

I'm trying to find an interpolating formula for a set of coefficients (I have $80$ at the moment). I tried first to find an interpolating polynomial, but that was not useful: using the first ...
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51 views
+100

Interpolation and mapping between scattered vectors in two unequally dimensioned spaces

Imagine two spaces: An ‘input’ space with dimension $m$. An ‘output’ space with dimension $n$. $m \geq n$ There are points in each of these spaces defined such that some characteristic is ...
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35 views

Basic optimization question

A teacher put this problem up the other day and I'm confused about how he got to the answer. Can you explain it to me? Job $X$ provides $20$ vacation days and $143,000$ euro annual salary. Job $Y$ ...
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What the minimum of infinite tetration divided by $\sqrt{x}$?

For some values of $x$ the limit of infinite tetration converges. For example when $x=\sqrt{2}$ this is fixed point $$\lim_{n\rightarrow \infty} \sqrt{2} \uparrow \uparrow n = ...
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22 views

Solve logistic problem with graph - fitting boxes

Suppose you have $n$ boxes, each of which falls into one of the $k$ sizes, and you want to nest smaller ones into larger ones, such that no two boxes $A$ and $B$ are nested inside the same box, if ...
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Distance between point and ellipse - explanation of a paper

EDIT: I notice that the link is hidden, but this post is made with reference to THIS PAPER I'm trying to solve quite an old problem (once again) - to find the distance between a point (in 3d space) ...
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1answer
35 views

Translate this problem to graph theory

Say I have a size $k$ set called $S_k$ with elements that are natural numbers (repetitions are allowed). For instance $\{2, 8, 6, 6, 1, 3\}$ is a valid set for $k = 6.$ I am trying to find the least ...
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Where does the name “tracking type problem” come from?

In PDE-constrained optimization problems, the distributed constrol problem $$ \begin{array}{ll} \displaystyle \min_{y,u} & J(y,u) = \frac{1}{2}\|y-y_d\|_{L^2(\Omega)}^2 + ...
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2answers
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Solve $\min_{A \subset \mathbb{R}} \int_{A} (f(t)-g(t))dt$

Consider the functions $f:\mathbb{R}\rightarrow \mathbb{R}$ and $g:\mathbb{R}\rightarrow \mathbb{R}$ both integrable on any measurable set $A \subset \mathbb{R}$. Consider $$\min_{A \subset ...
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On a max-min problem from an exam.

I have asked a different question on the same exercise (from an exam) a couple weeks ago, I hope it is acceptable to have a different question on the same exercise, I searched the Meta and it seems ...
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32 views

Derivative of a variable across multiple functions

I'm just starting out learning about optimization. Not sure if this makes sense but... Given a variable $x$ that is passed to two separate functions, $f(x)$ and $g(x)$, why do we add the derivatives ...
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2answers
13 views

MLE of a discrete random variable

For some reason I am having difficulty understand how to calculate the mle of a discrete rv. The pmf is: $$p(k;\theta) = \left\{\begin{array}{lcc}\frac{1-\theta}{3}&\mbox{if}&k=0\\ ...
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15 views

Find min and max of $\log n /\log an$ [closed]

I would like to find min and max of $$\log n /\log an$$ where $a$ is a parameter such that $a>\log n/n$ and $n>e$. I think the min is $\sqrt{n}$, but i am not success to prove it. Sorry to ...