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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Deriving optimal time to change

I am working in economics and I am trying to build a model that take into account the fact that indivudal can take a decision once in their life time that changes the value of a parameter R. To be ...
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Maximum of $e^{-x} \sin(x)$, $x \geq 0$

What is the maximum of $e^{-x} \sin(x)$ for $x \geq 0$? Is there a closed-form solution? If not, what is a good approximation $y$ such that $\text{max}_{x\geq 0}e^{-x} \sin(x) \leq y$?
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Comparing a maximization to an integration with economics application

This seemingly simple question has interesting interpretation in economics, but I only state the mathematical problem here. Suppose $B(0)=C(0)=C'(0)=0$, $B'(\cdot)>0,\ B''(\cdot)\leq0,\ ...
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How do I construct the Jacobian for use in a Levenberg-Marquardt algorithm.

I am working on a 3D reconstruction system and I am looking to use a Levenberg-marquardt algorithm to do bundle adjustment. I am not too sure about how LM works and what it requires. The model I am ...
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316 views

Why is Newton's method faster than gradient descent?

Can you provide some intuition as to why Newton's method is faster than gradient descent? Often we are in a scenario where we want to minimize a function f(x) where x is a vector of parameters. To do ...
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What is the range of $y$ if $x+y+z=4$ and $xy+yz+xz=5$ for $x, y, z \in\mathbb{R}_+$

What is the range of $y$ if $x+y+z=4$ and $xy+yz+xz=5$ for $x, y, z \in\mathbb{R}_+$ How to explain the following method? Let $x=z$ then: $$2x+y=4\quad;\quad 2xy+x^{2}=5$$ $$\implies \left( ...
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Newton's optimization: why wouldn't the results be good if we take $x_0$ between $(-\frac{1}{\sqrt 3},\frac{1}{\sqrt 3})$ for $f(x)= x^3- x-1\;?$

I was reading about the numerical method of Newton for finding the roots of $f(x)$ in Thomas' Calculus ; the author presented an example Find the x-coordinate of the point where the curve $f(x)= ...
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64 views

What is the maximum number of boxes that can fit in a rectangular container

I'm looking for an algorithm for the following question: What is the maximum number of boxes with sides a,b,c that can fit in a rectangular container with sides $x$,$y$,$z$. For example, the ...
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29 views

Is closed convex set with finite number of extreme points convex polyhedron

I have this simple question related to convex set and convex polyhedron. As the content in the title, it's basically my question: Is closed convex set with finite number of extreme points convex ...
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Matlab: need help with optimization

I am trying to minimize the objective function over [x(1),x(2)]: exp(x(1))*(4*x(1)^2+2*x(2)^2+4*x(1)*x(2)+2*x(2)+1)+b subject to constraint ...
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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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Multivariable optimization for time to build a ship in a game, and maybe some possible application in “everyday” life

I precise first that english is not my monther tongue and I may will not be as clear as I would like, just ask me question if you need, thank you. I am playing a game (Galaxy Empire) for a while, ...
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35 views

Weakly lower semicontinuous functional on a bounded closed and convex set

Let $J$ be a sequentially weakly lower semicontinuous functional on $C$ with values on the real line. Moreover let $C$ be a bounded, closed and convex subset of a Hilbert space $H$. Is it true that ...
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Explanation of formula

Suppose that we have $M$ production stations $A_1, \dots, A_M$ of a product and $N$ destination stations $B_1, \dots, B_N$ of the product. We suppose that $x_{ij}$ units of the product are ...
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Maximise volume given inequality constraint on its dimensions without using Lagrange, KKT or Linear Programming

The problem (from Calculus for Business, Economics, Life Sciences and Social Sciences 12e): I found this and that, but they use Lagrange/KKT. What I tried: Girth $= 2w + 2h$ Maximise ...
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Find the smallest possible value of $a^4+b^4+c^4-136abc$

Let $a$, $b$, and $c$ be real numbers such that $a+b+c=-68$ and $ab+bc+ca=1156$. The smallest possible value of $a^4+b^4+c^4-136abc$ is $k$. Find the remainder when $k$ is divided by $1000$. I ...
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How to describe minimization of L1 norm error using linear programming?

Given a set of $n$ pair points $(x_1, y_1), ..., (x_n, y_n)$ in the plane, I need to find a line $ax + by = c$ that fits the points of the L1 norm error points as closely as possible. I need a linear ...
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50 views

Maximum and minimum of the function $xy+z^2$

Find the maximum and minimum values of the function $f(x,y,z)=xy+z^2$ in the circumference obtained by intersections between the sphere $x^2+y^2+z^2=4$ and the plane $y-x=0$. I did Lagrange and found ...
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Can lagrange multiplier(Kuhn tucker multipliers?) change in corner solution?

If we want to maximize $f(x)$ subject to two constraints, one which says that $x< c$ $c>0$, and another that says that $x\geq 0 $. Assume there are no problems with either $x=0, x>0$ or $\mu ...
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Using Lagrange multipliers to identify the Extremes of function $f(x, y)=x-y$, under condition $g(x,y)=x^2 + y^2 - 4=0$

I'm studying in preparation for a Mathematical Analysis II examination and I'm solving past exam exercises. If it's any indicator of difficulty, the exercise is Exercise 3 of 4, part $b$ and graded ...
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Shortest distance between parallel line and plane

I've been doing questions regarding the shortest distance between lines/planes and points , and I've come across a question asking to find the shortest distance between a line and a plane which are ...
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Existence of non-negative solution to a diagonally dominant tridiagonal system

Let $D \in \mathbb{R}^{n \times n}$. having only non-negative entries, strictly diagonally dominant (both row-wise and column-wise), tridiagonal. Show that $$\exists\; x \in \mathbb{R}^n \quad ...
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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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Maximal distance of a segment

Let a path enclosed by lines as illustrated in this figure Fig. knowing that the widths of the two paths are $\ell$ and $\ell^{\prime}$ respectively. What the maximum distance $x^{\star}$ to be able ...
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KKT conditions for nonlinear problem

I need to state the KKT conditions for the following problem: Minimise $x_1^2 + 2x_2^2$ subject to $(x_1-1)^2 + x_2^2 \le 1$ and $x_2 = 1$. I have that these conditions are: $f(x^*) \le 0$ ...
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Convex Optimization of quadratic function with inequality constraints

How would I solve the following problem? $$\min_{x\in\mathbb{R}^n} x^T A x$$ subject to the constraints $$x_i\geq 1,\,i=1,\dots,n,$$ where A is positive semidefinite and symmetric. Is it possible to ...
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Find min $ax+by+cz$ subject to $0 \le y \le 1, 0\le z \le 1$ and $\max(0,y+z-1) \le x \le \min(y,z)$

I am seeking an elegant way to solve the following problem. Let $a,b,c$ be constant real numbers. Find min $ax+by+cz$ subject to $0 \le y \le 1, 0\le z \le 1$ and $\max(0,y+z-1) \le x \le \min(y,z)$. ...
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Feasible set and level sets

Consider the following problem: Minimise $x_1^2 + 2x_2^2$ subject to $(x_1)^2 + x_2^2 \le 1$ and $x_2 = 1$. Sketch the feasible set and the level sets of the objective function, and determine an ...
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Optimization on a grid

I worked a lot on defining the problem so I will be grateful to get input if i'm not clear enouth and I will fix the question. We have a grid made out of uniform points on $[x,y],$ $x,y\in[0,1],$ ...
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what is the maximum value of $x(x+y)^3$ given that $x^2+y^2/d=1$?

Without losing generality, we can assume $x,y\geq 0$ and then use $x$ to replace $y$. This is complicated. Instead I use $x=\sin\theta$, $y=\sqrt{d}\cos\theta$, and then I only need to get the ...
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Maximizing the sum of the products of endpoints of edges in a graph

Let $G$ be a graph with vertex set $V=\{v_1,v_2\dots v_n\}$ and edge set $E$. Let $f:V\rightarrow \mathbb [0,\infty)$ be a real valued function such that $\sum\limits_{i=1}^n f(v_i)=A$. What is the ...
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Solving SVM classifier with two weight vectors

I am trying to implement a paper that basically proposes the following way to train two classifiers on some data with two types of labels. I do not know how to tweak existing solvers for SVM to do the ...
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Design a circuit for a function

I am so confused on this problem. We are given a function $f$ and told to design a circuit that has four inputs labeled $b_3,...,b_0$, and an output $f$, where $f = 1$ if the 4-bit input pattern is a ...
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Envelope theorem for Conditional value at risk

Let $X$ be a Gaussian random variable and suppose $f(p,X)$ is a strictly increasing and continuous function in $p \in \mathbb R$. Conditional value at risk is defined in the following way ...
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MInimum value of the sum of three numbers

if product of three numbers is 1, how do you find the minimum value of the sum of those three numbers? i tried to find the possible values of the numbers that would give a product of one but I'm not ...
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primal to dual solution conversion ??

i have an optimization problem $$\text{ maximize } z=3x+4y$$ $$\text{ such that: } x+y ≤ 450 \text{ and } 2x+y ≤ 600$$ the optimal solution to this problems comes to be $x=0$; $y=450$; $p=150$ ...
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Implementation of EM algorithm for Gaussian Mixture Models using Matlab

Using the EM algorithm, I want to train a Gaussian Mixture model using four components on a given dataset. The set is three dimensional and contains 300 samples. The problem is that after about 6 ...
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Minimising logistic loss function to find optimal matrix

Please take a look at this paper on classifying triples (re link prediction): http://arxiv.org/pdf/1510.04935v2.pdf The question is about how to solve equation 2 using stochastic gradient descent. It ...
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Optimising volume of a truncated cone

Given a slant height h and radius r1 how can I find a truncated cone with largest volume?. Is there any calculus involved in ...
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Intuition behind eigen values in Optimization Problems

The question may seem very simple, I am not able to understand the intuition behind the solution of following problem min $A\vec{v}$ where $A$ is some matrix The solution is the eigen vector ...
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Best way to get combination of elements to fill given area

I am application developer and I came across interesting mathematical problem. Let's assume we are given: dimensions of space we would like to fill: a = 3m; b = 4m; set of elements: Items = { {a0, ...
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On a maximization exercise in $\mathbb R^2$.

I am given a set $A = \{ (x,y) \in \mathbb R^2 \mid x^2 + y^2 \le 1 , x+y \ge 0 \}$ and a function $f(x,y) = (x-y)^2 (x+y)$ I am tasked with finding $f(A)$. So because $f$ is a continuous function ...
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What's the largest possible volume of a taco, and how do I make one that big?

Let $f$ be a continuous, even function over some interval $I=[-a,a]$ such that the total arc length of $f$ over $I$ is at least $2$, $f(0)=0$, and $f$ is increasing on $(0,a)$. [You might imagine ...
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Can I use Lagrange multipliers with redundant constrains?

Can I use Lagrangian multipliers with redundant constrains? For example, suppose I have the following problem: Find the maximum of $F(x,y,z)$, subject to $f(x,y,z)=0$ and $g(x,y,z)=0$. But you also ...
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Model Linear-Programming Problem

A factory needs to complete $n$ jobs by using $m$ machines. To complete each job $j, j=1,\dots,n$, an amount of $r_j\geq 0$ processing units is required. Each machine $i$ has a processing speed ...
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How to find a sequence that maximizes a ratio

Given positive parameters $n$, $P$ and $Q$, what is a sequence $a_1,\dots,a_n$ such that for every $k$: $$ \frac{1}{k}\leq a_k \leq 1 $$ which maximizes the ratio: $$ R = \frac{P + \sum_{k=1}^n ...
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Effect of marginalization on Gauss-Newton equations

Consider the problem of minimizing the cost function $f(x)=\eta(x)^TW\eta(x)$, where $\eta(x)=z-h(x)$ is an error function between the observations (measurements $z$) and their prediction $h(x)$, and ...
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Minimizing a quadratic function of 2 variables in quadratic region

Let $f$ be a real valued quadratic function of 2 real variables: $$f(x,y) = ax^2 + by^2 + cxy + dx + ey + f$$ How to minimize it? Subject to constraints: $$ 0\leq x \leq 1, \quad 0\leq y \leq 1 $$ ...
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Scaled proximal operator for proximal Newton method

The scaled proximal operator was introduced as an extension of the (regular) proximal operator: $prox^H_h(x) = \arg\min_y h(y) + \frac{1}{2}\|y-x\|^2_H$. (See ...
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391 views

Maximizing a function while minimizing one part of the same function

I have a function with two variables say $f(x,y)=f_1(x)-f_2(x,y)$ where $f_1(x)$ is the well known quadratic-form function in x while $f_2(x,y)$ is also a quadratic function in both x and y but not ...