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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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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1answer
786 views

Linear Programming: Three variable graphical solution

A small bank offers three type of loans: housing loans at $8.50$% interest, education loans at $13.75$% interest rates, and loans to senior citizens at $12.25$% interest. Further, it needs to ...
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Conditional extreme value of a function

Let $x,y,z$ be the positive real numbers, if $x^2+y^2+z^2=1$, then how can we find the minimal value of this function $f(x,y,z)=\dfrac{xz}{y}+\dfrac{yz}{x}+\dfrac{xy}{z}$.
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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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1answer
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Simple optimization of cylindrical radius for volume

I'm having trouble solving this simple optimization problem, can't work out where I'm going wrong. A brewery wants to make a cylindrical aluminium beer can which will hold 375ml. (This means the ...
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2answers
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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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+50

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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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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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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2answers
48 views

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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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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2answers
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Having trouble forming the initial matrices for a positioning problem

The question asks me to solve the positioning problem where: $$ \dot{x_1} = x_2 $$ $$ \dot{x_2} = u_1 \in U_{bb} $$ $$ x_1(0) = - \text{X} (<0) $$ $$ x_2 (0) = 0 $$ $$ x_1(t_1) = 0$$ $$ x_2(t_1) = ...
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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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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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5answers
65 views

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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35 views

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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1answer
31 views

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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21 views

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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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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25 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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21 views

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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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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33 views

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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1answer
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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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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266 views

Finding minimum sum of absolute differences between heights of $n$ boys and $m$ girls

Given two sets $A$ and $B$, $A$ has heights of $n$ boys and $B$ has heights of $m$ girls, $m \ge n$. We have to find one solution of pairing up $n$ boys with $n$ (out of $m$) girls so that the sum of ...
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88 views

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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+100

Maximizing “log det + log sum exp” function

I'm trying to find a numerical solution to the following optimization problem $$ \text{maximize } f(M) = \frac{1}{2} \log \det(M) + \log \sum_{i=1}^n \exp \left\{ - \frac{1}{2} x_i^T M x_i + a_i ...
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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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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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1answer
295 views

A variation of the Assignment Problem

In the following Wikipedia article about the Assignment Problem in the Example section, it says: Similar tricks can be played in order to allow more tasks than agents, tasks to which multiple ...
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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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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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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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Optimising surface area to volume ratio of a 3d closed surface

How would one prove that the sphere is the 3d closed surface which has the lowest surface area to volume ratio? One could first consider the simpler problem of proving that circle has lowest ...
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The theorem about efficient face… [on hold]

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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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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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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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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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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296 views

Dual cone of a L1 norm cone?

I am listening to convex optimization lectures and I hear that dual cone of a $L1$ norm cone is a $L-\infty$ norm cone. Can anybody please explain how? I understand that every point in the dual cone ...
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Ways to partition first n natural numbers into two sets with equal sums

The problem I have at hand is counting the number of partitions of {1, ..., n} into two sets with equal sums. Eg. for n=7 the possible partitions are: {{7,6,1}, {5,4,3,2}}, {{7,5,2}, {6,4,3,1}}, ...
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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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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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Method of Lagrange multipliers to find all critical points of a function

I am having difficulties in understanding the steps/method required to find the critical points of a function using the method of Lagrange multipliers. I have read through my text book and tried my ...
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How to find the absolute extrema of a function on an elliptical cylinder using Lagrange multipliers?

Optimize the function $ f(x,y) = x^2y $ on the elliptical cylinder $ \ x^2 \ + \ 2y^2 \ \le \ 6 \ $ using Lagrange Multipliers. Well, from what I know that I have to find the gradient then to ...