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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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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2answers
119 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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15 views

Maximum of polynomial

I was studying statics and came across this problem: Find the value $\beta$ such that $P$ has a maximum value in $R² - 1000^2 = P^2 + 2000Pcos(75+\beta)$. $R$ is constant, the resultant of $P$ and ...
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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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Closed form expression for the constrained least squares problem

$\underset{A}{\min} \hspace{15mm} \sum_{k=1}^{N-1} ||\textbf{x}_{k+1}-A\textbf{x}_{k}||^2_2$ $s.t \hspace{15mm} W.A = 0 $ where (.) is the element by element product and $W$, $\textbf{x}_{i}$ are ...
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6 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
15 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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19 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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1answer
42 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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35 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
23 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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21 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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1answer
26 views

Understand a compact Interval

For example , if I define a function of several variables $f : X \rightarrow Y$ , for example $f (a , b, c , x, y , z )$, then by putting the constraint $x + y + z = 2$ and $a + b + c = 3$ , for $ ...
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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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17 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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155 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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1answer
23 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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63 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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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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27 views

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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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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1answer
19 views

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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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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53 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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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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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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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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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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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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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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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 ...
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1answer
29 views

set up Linear programming problem

How do I set up this problem ? A product can be made in three sizes, large, medium, and small, which yield a net unit profit of $12, 10$ and $9$ respectively. The company has three centers where ...
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Is my proof correct? Convex optimization

There's a theorem that says that if $C \subset \mathbb R^n$ is a convex set, then $x^* \in C$ is the closest point in $C$ to $y \notin C$ if and only if $(y-x^*)\cdot(x-x^*)\leq 0$ for all $x \in C$. ...
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1answer
21 views

Maximising the logarithmic expectation in coin bets

We are throwing a coin $N$ times and for some reason the probability that we get heads in the $n$-th toss is $p_n\geq\frac 12$. Now starting with capital $X_0$, before each toss we decide to bet a ...
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1answer
99 views

Is this limit finite, or infinite?

Is $$\lim_{x\uparrow 1}\left(\frac{1-x}{x}\max\{nx^n|n\in\mathbb{N}\} \right)$$ infinite, or finite? ($\mathbb{N}$ is the set of the natural numbers). According to Mathematica, it looks like ...
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True or false? “sum of an m-strongly convex and a convex function is m-strongly convex”

I would like to know if the following conjecture is true or false? If $f(x) = g(x) + h(x)$ where $g$ is m-strongly convex and $h$ is convex, then $f$ is m-strongly convex. NOTE: For a ...
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Is this supremum infinite or finite?

Given two numbers $x\in (0,1)$ and $y\in (0,1)$, think of the expression $\min_{n\ge 1} \frac{1-x}{(1-xy)y^{n-1}(1-x^n)}$. Does the supremum of this expression, namely, $\sup_{x,y\in ...
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1answer
43 views

Project allocation optimization with tricky constraint

I have an allocation problem that should be straightforward, except that it has very specific constraints. I want to assign approximately 300 students to 170 projects in pairs - so that each project ...
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1answer
35 views

What is the difference between linear and integer programming?

Recently I tried to solve a maximization integer programming problem using linear programming by flooring the max point - but got the wrong answer. I'm wondering if someone can explain mathematically ...