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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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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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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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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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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25 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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22 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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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
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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+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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33 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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20 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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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 $\log n /\log an$ [on hold]

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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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
20 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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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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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
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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 ...
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When is $f(x,y,z)= \frac{x \cdot y}{z}$ convex?

I would like to know under what conditions $f(x,y,z)= \frac{x \cdot y}{z}$ is convex, pseudo-convex, or quasi-convex. I know that $g(x,y)= \frac{x^2}{y}$ $ \text{when } y >0 $ is convex and ...
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Number of global min cuts in undirected graph

I'm looking at a proof of the following theorem "The number of global minimum cut is $\le \binom{n}{2}$". It says $\forall i$ from $1$ to $n-1$ Find min-cut seperating $\{1,2,\cdots,i\}$ from $i+1$. ...
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Is this expression uniformly bounded?

Given two numbers $x\in (0,1)$ and $y\in (0,1)$, think of the expression $\min_{n\ge 1} \frac{1-x}{(1-xy)x^{n-1}(1-y^n)}$. Does the supremum of this expression, namely, $\sup_{x,y\in ...
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Proof for bounding a function in two variables, one real and one integer

I would like to proof that the function $f(x,k)=2xk^{-4x^2}$, where $x$ is a real variable and $k$ is an integer variable, is always smaller than $1$ for all $k>2$ and all $x \ge 0$. This is my ...
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Using Calculus To Solve Optimisation Problems

I have a question regarding using calculus to solve an optimisation problem which is quite wordy. It is as follows: A researcher has funds to buy enough computing power for 7 years. Computing power ...
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Minimize the sum with regards to $p$

I need to minimize the following quantity with respect to $p$, but I don't know how to go about it. Here it is: $$\frac {x(p+h)}{pb} + \frac{(k-1)(p+h)}{b}$$ According to my textbook the answer ...
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sub gradient of Moreau envelope

Is there an equality formula for the subgradients of the Moreau envelope, $$e_f(x) = \inf_z \frac{1}{2}||x-z||_2^2 + f(z),$$ of a lsc (lower semicontinuous) and proper function $f:\mathbb{R}^p \to ...
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Variation of the opaque forest problem (a.k.a farmyard problem)

I was wondering about the following variation of the opaque forest problem (see here and there for previous questions) : What is the least length set of segments that will intersect every straight ...
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Proving that maximizing a sum of functions of different independent variables is equivalent to maximizing each function

Let $$ \pi = f_1(x_1) + f_2(x_2) + f_3(x_3) + \dots + f_n(x_n) = \sum_{i=1}^n f_n(x_i) $$ where $f_i$ denote different functions and $x_i$ denote different independent variables Would proving that ...
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Creating a configuration of points where each point is away from all other points by a pre-defined distance

Let's assume that the points $\in \mathbb{R}^2$ and there are only C=5 points (in practice, I may have $\mathbb{R}^{800}$ and 1000 points). The first out of the five points is fixed. We also have been ...
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How to “rotate” on $\mathbb{R}^n$ to maximize minimum pairwise “angle” with set of known vectors?

Say I have $n$ vectors $\{{\bf v}_1,\cdots,{\bf v}_n\}$ which are of unit length: $\|{\bf v}_k\|_2 = 1$ and I want to find new vectors $\{{\bf w}_1,\cdots,{\bf w}_n\}$ by "rotation" i.e. some ...
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Maximization and Minimization of $f(x,y)$

Find the extreme values of the function: $z=f(x,y)=x^2+(y-18)^2+90$ subjected to following constraint $x^2+y^2\leq196$ How to solve this? I used Lagrangian function but how to set up constraint ...
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Solving $\max_{x\in\prod_{i=1}^n s_i} \sum_{i=1}^n f(x_i)$ by maximizing for each $i$ individually.

First, I will clarify some of the notation: $$ x_i \in S_i,\; i\in \{1,2,\dots, n\} \quad x\in S, \quad S\equiv \prod_{i=1}^nS_i \text{ (direct product set)} $$ So basically, we have $x \in S$ which ...
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Probability that max cos(φ)x + sin(φ)y according to uniform distribution = (8,5)

max $x_2$ subject to $x_1 - 2x_2 \le 0$ $2x_1 - 3x_2 \le 2$ $x_1 - x_2 \le 3$ $-x_1 + 2x_2 \le 2$ $-2x_1 + x_2 \le 0$ Optimal solution: (8, 5) --> $x_2 = 5$ Now assume that the objective is ...
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138 views

How to minimize $ab + bc + ca$ given $a^2 + b^2 + c^2 = 1$?

The question is to prove that $ab + bc + ca$ lies in between $-1$ and $1$, given that $a^2 + b^2 + c^2 = 1$. I could prove the maxima by the following approach. I changed the coordinates to spherical ...
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Inverting the equality which contains the operation of taking integer part

I was recently presented with the following equality $$ n = \left[\frac{w}{2d+a}\right]\cdot \left[\frac{h}{2d+b}\right] $$ where all participating variables are non-negative integers, and $[\ldots]$ ...