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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Minimize Energy in Image processing - Geodesic active contours

I've read some papers in Geodesic active contours (Image processing), which use the minimization of an Energy, consist of Internal Energy and External energy, for example, in the paper of Kass (Snake: ...
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Every polyhedron $P \ne \mathbb{K}^n$ equals an intersection of finitely many half spaces.

Currently, I am reading some lecture notes on linear optimisation. I cannot see why the following (seemingly trivial) proposition holds. (How could I understand/proove it?) Every polyhedron $P \ne ...
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35 views

Difference between Half Quadratic vs Quadratic

Half quadratic minimization/penalty/optimization, I am unable to find any related material/resources. If anyone can point to some useful resources, it will be great
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Largest number of pairs that can be added while keeping the population at least 60% male

I'm doing problems from the AoPS Algebra Beginner's book. There's this problem that states the following, At her ranch, Georgia starts an animal shelter to save dogs. After the first three days, she ...
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Optimization with Lagrange multipliers

I am new to Lagrange multipliers. Could some one show me how to minimize the following function: \begin{align} f(x,y)=ax+by-\sqrt{cxy} \end{align} subject to: \begin{align} 0 &\le x\\ 0 &\le y ...
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Optimal Strategy for “I'm Thinking of a Number” Game

This question is inspired by one of the classic ways of breaking ties: the "I'm thinking of a number" game. In this game, one person thinks of a number in some range, say from $0$ to $100$ ...
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finding the closest matrix of a given form

let's say I have a vector $(a_1\dots a_n)$, where each component is between $-1$ and $1$. Now from this vector I define a $n\times n$ matrix $M$ such that $$M_{ij} = \begin{cases} 1&\,& i = ...
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Minimizing the expression $(1+1/x)(1+m/y)$ over positive reals such that $mx+y=1$

Let $x$ and $y$ be positive real numbers such that $mx+y=1$. Find the positive $m$ such that the minimum of: $$\left( 1 + \frac{1}{x} \right)\left( 1 + \frac{m}{y} \right).$$ is $81$. I have ...
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28 views

Cookie Clicker Chocolate Egg strategy

Introduction Cookie Clicker is a silly Javascript based web game. Here is a brief description of what you do: (description taken from this question: Explain a surprisingly simple optimization result) ...
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Solving optimization problem over time

I'm not an expert in optimization and would therefore like to get some good starting points about a particular problem. Suppose we want to solve an optimization problem over time, eg $$\max_x{c^T_K ...
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Convex optimization approximation

Consider the optimization problem $\mathcal{P}_0$ $$ \min_{x \in \mathbb{R}^2} \left\| x-p \right\|^2 $$ $$ \text{sub. to: } \ A x \leq b, \ \ x_1^2 + x_2^2 = 1 $$ where $p \in \mathbb{R}^2$ is a ...
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Constrained Optimization of a function of two variables.

I was given the following tutorial problem, and I'm having a bit of trouble seeing how it works. I've been asked to find the four critical points of this system, with two of these being degenerate ...
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5answers
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Minimize $\cos(t)\cos(t-\alpha)$

How can I minimize $f(t)=\cos(t)\cos(t-\alpha)$? I guessed that the minimum is precisely halfway between the adjacent roots $\pi/2$ and $\pi/2+\alpha$. However, I'm not sure how to prove this. Is ...
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2answers
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Local minimum of $\sqrt[x]{n\over v-x+1}$

I'm trying to find the local minimum of $\sqrt[x]{n\over v-x+1}$ with respect to $x$. The restrictions on $x$ are that it must be $\le v$ and $\ge 1$. Also, $v$ and $n$ are fixed, and $v<n$. My ...
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26 views

Linear problem: maximizing net income

Problem: A company produces and sells two different products. The demand for each product is unlimited, but the company is constrained by cash avaliable and machine capacity. Each unit of the first ...
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When does a polynomial have finitely many critical points on a level set of another polynomial?

Suppose I have two polynomial functions $f$ and $g$ and I am interested in the critical points that $f$ has on a level set of $g$, i.e. $\{x\in \mathbb R^n : g(x)=a_1\}$ for some $a_1\in \mathbb R$ . ...
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Objective function with two variables

A factory produces jointly two articles, and it has the problem to decide their prices in order to maximize the monthly income, knowing that the demand d1 (in hundreds of units) of the first article ...
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Finding the critical points in a constrained optimization problem using the Lagrangian

I've been given the following constrained optimization problem, but I'm having trouble even getting the critical points out - the numbers just seem way too complicated... Find the local maxima and ...
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Uniqueness of solution to quantile minimization problem

I read here: http://librarum.org/book/11685/31 (p. 51, Ex. 3) that quantiles are solutions to certain minimization problem. Here is the proof: ...
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Determining extrema of $f(x,y,z)=(xyz)^{\frac{2}{3}}$ on $x^2+y^2+z^2=1$

Determine where on the sphere $x^2+y^2+z^2=1$ the function $f(x,y,z)=(xyz)^{\frac{2}{3}}$ attains its maximum and minimum. Using Lagrange multipliers one gets the solutions ...
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Model $\min \frac{1}{2} \parallel Ax-B \parallel_2 + \lambda_1 \parallel Cx \parallel_1 + \lambda_2 \parallel Dx \parallel_\infty $ into standard form

I need to solve the following convex optimization problem: $\min \frac{1}{2} \parallel Ax-B \parallel_2 + \lambda_1 \parallel Cx \parallel_1 + \lambda_2 \parallel Dx \parallel_\infty$ s.t $x ...
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29 views

Minimize $\ell_1$ norm subject to $\ell_2$ constraint

I am trying to solve the following optimization problem: $$\min_{\|Px\|_2=1} \|x\|_1$$ I know it is non-convex. But some non-convex problems are still solvable. Update $P$ is 2x3. $x$ is a ...
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Optimize rate of collection in counters

Suppose you have $K$ counters. The value of these $K$ counters are all $0$. Every second, each counter has a $J$ chance of incrementing itself, up to a max value of $I$. Every second, you may choose ...
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Proof of Second Partials Test

How does one rigorously prove the second partials test without firstly assuming that $D(a,b)=AC-B^2$ that states the following: $ A=\frac {\partial^{2}f(a,b)}{\partial x^{2}},B=$$\frac ...
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Ternary balance with unknown weight

Main references: Ternary (Wolfram MathWorld) Balanced ternary (Wikipedia) Weighing scale: Balance (Wikipedia) <quote> Balanced ternary has other applications besides computing. For example, a ...
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Is the following graph having two local minima

https://www.desmos.com/calculator/abuvb1zdkb I think yes, the main question i think is of the definition of neighbourhood For a function with domain $(-\infty, -3)\cup (3, \infty)$ $ $ Is -3 in ...
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constrained optimization and differential equation

Consider the following differential equation system (cylindrical coordinate system): $\frac{dP_x}{dz} = P_x C \int\limits_0^{2\pi}\int\limits_0^a \frac{f(r, \theta)}{g(r, \theta, z)} r dr d\theta$ ...
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The theorem about efficient face… [closed]

i need to proof of the theorem below "if x is a efficient solution that is in relative interior of the face then face is efficient..." that programming is MOLP. please help me...
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Nonlinear equations with boolean variables

Let $i = 1, \dots, v$, $j = 1, \dots, v$ and $n = 1, \dots, N$. $i$ and $j$ indicate origin and destination nodes in a graph, respectively. An individual is denoted by $n$. Also let $0 < \alpha, ...
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Notation for the set of all arguments corresponding to local minima.

The notation $$\mathop{\mathrm{arg\, min}}_{x \in X} f(x)$$ is sometimes used for the set of all $x \in X$ corresponding to global minima of the function $x \in X \mapsto f(x).$ Is there notation for ...
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The minimum number of circles in order to obtain a COVER of a specific square

Suppose a unit square $X$, with side length $l=1$ as below, which is COVERed by a set $Y$ of circles with the same constant radius of $r=\dfrac{\sqrt{2}}{10}$, where a ...
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A basic question on stochastic gradient descent

Consider a stochastic gradient iteration: $$\theta_{k+1} = \theta_{k} - \gamma_k F(\theta_k)$$ where $F$ is a noisy estimate of the gradient $\nabla f$ Now, a book says that it converges in the ...
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alternating direction method of multipliers for nonlinear inverse problems?

I have a standard inverse problem with L1 regularization: $\|F(\mathbf{x})-\mathbf{y}\|^2_2+\alpha\|\mathbf{x}\|_1$, where $F(\mathbf{x})$ is nonlinear. I am wondering if this is a good problem to use ...
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Finding optimal velocity profile using Dynamic Programming

This question has been asked on scicomp but I thought maybe it's more a mathematical problem of how Bellman's idea is to be applied here. The main problem for me is: How to introduce the time ...
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3answers
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Is this optimization problem solvable?

I have the following optimization problem: $$ \min_\mathbf{x}\|\mathbf{a+Bx}\|^2 ~~\text{s.t}~~ \|\mathbf{y+Ax}\|_\infty \leq \beta\|\mathbf{y}\|_\infty ~~,~~ \|\mathbf{x}\|^2 \leq \alpha^2$$ where ...
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What exactly are the curves that are a best fit to the Harmonic Cantilever?

Let's start with a few references to get an idea: Daniel Goldwater: Harmonic Cantilever Book Stacking Problem Block-stacking problem Harmonic Series and Bricks Interesting related issues: Maximum ...
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Finding Optimal Threshold Values For Ensemble Predictor

I have a range of eight models (each providing a p-value as measure of significance of a certain property of an instance), which, for a final prediction (binary, with 1 being a positive hit) I combine ...
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Minimum of function of x and y

any ideas how to find minimum of the following function: $f(x,y)=a-b\frac{x}{y}-c\frac{y}{x}+\frac{d}{x}+\frac{e}{y}-\frac{g}{xy}$. Assume that $a,b,c,d,e,g>0$. We can also assume that $x,y \ge 1$. ...
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System of equations in Lagrange multiplier problem

Continuing from Confounding Lagrange multiplier problem: I'm having trouble solving the system of equations below arisen from a Lagrange multiplier problem where we are to optimize $f(x,y,z) = 4x^2 + ...
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Function optimization with parameters [closed]

I have a function $ f(x) = [SSE, \alpha, \beta, \gamma] $ where SSE - is sum of squared errors, $ \alpha, \beta, \gamma $ - are parameters, which I need to find, them must satisfy condition $\alpha ...
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How to solve this optimization problem? (may be gradient descent?)

I have the following optimization problem. $$\operatorname*{argmax}_{w} \|(1-w)\boldsymbol{X} -w\boldsymbol{Y}\|^2 \\ s.t. \quad 0<w<1 $$ How can I find the solution of this problem? May be ...
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How to solve this equation (may be with least squares)?

I have a system of linear equations in the following form. How can I solve it? $$\operatorname*{argmin}_{a,b} \sum_{i,j} \left( \left| X(i,j)-aY(i,j)\right|-b \right)^2$$ Where $X$ and $Y$ are ...
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Holt's linear trend method and optimal smoothing parameters

I am learning about time series and forecasting and I stumbled across exponential smoothing and other derived methods. In an exponential smoothing model, each prediction is given by a level equation ...
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Confounding Lagrange multiplier problem

Optimize $f(x,y,z) = 4x^2 + 3y^2 + 5z^2$ over $g(x,y,z) = xy + 2yz + 3xz = 6$ According to the theorem the gradients must be parallell, $\nabla f = \lambda \nabla g$, so their cross product must ...
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Numerical nonconvex optimization problem

I have numerical data for the mapping $w:\mathcal{S}^{2+}\to\mathbb{R}$, where $\mathcal{S}^{2+}$ is $\{\mathbf{x}\in\mathcal{S}^2:x_3\ge0\}$, the 2-hemisphere on or above the $x_1-x_2$ plane. I ...
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Alternating Direction Method of Multipliers (ADMM) application

$\newcommand{\argmin}{\operatorname{argmin}}$ Recall, that ADMM algorithm solves the problem of the form: $\min \text{ } f(X) + g(Z)$ $\text{s.t. } AX + BZ = C$ where $X$, $Z$ and $C$ are real ...
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Gradient descent vs. Newton's method — which one requires more computation?

Broadly speaking, when numerically minimizing a d-dimensional objective function: Gradient descent generally requires more iterations, but each iteration is fast (we only need to compute 1st ...
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Evolutionary algorithm

Can someone provide me a good reference for the CMA-ES algorithm? I'm new in the world of optimization and just reading the author reference doesn't help me a lot. I know the basic idea of a genetic ...
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“Rank-K Correction” of a matrix and significance?

Today my studies led me to read about the matrix inversion lemma, which Wikipedia introduces as follows: In mathematics (specifically linear algebra), the Woodbury matrix identity, named after Max ...