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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Maximal area in fixed perimeter [duplicate]

An old story I heard starts by two people that was arguing about how much land a man need. So they called to a young man, and said to him: You are stating in that point, and start running all the ...
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1answer
114 views

Definition of stability in the case of Levenberg-Marquardt optimization method

I've come across this guide: Fortunately, it inherits the speed advantage of the Gauss–Newton algorithm and the stability of the steepest descent method. What's a stability in this case? Does it ...
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1answer
36 views

Maximize a sum of sinusoids with comensurable periods

I'm writing a program that requires finding $$\text{argmax}_\theta\sum_{k=1}^na_k\cos(k\theta+b_k),$$ where $a_k$ and $b_k$ can be any real numbers. How can I do this efficiently?
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610 views

Optimization / personalization within clusters

I have the following optimization problem: I have a (random and very noisy) objective function $f(A, P)$, where $A$ is a vector of "observable" parameters of the input and $P$ is the parameters that ...
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1answer
194 views

Optimization of Complex Functions, No Use?

During reading the appendix to an engineering text, I came across the following remark: "Complex cost functions are of no interest, because in the field of complex numbers no ordering (relations ...
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1answer
31 views

Maximization under constrains

I would like to maximize the function: $\frac{1}{2}\sum_{i=1}^{N}\lvert x_i-\frac{1}{N}\rvert$ under the constrains $\sum_{i=1}^{N}x_i=1$ and $0\le x_i \le 1$ $\forall i\in(1,...,N)$ I have done ...
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3answers
1k views

Determining domain interval for optimization problems

This example is from Paul's Online Notes for Calc I. You have $500$ feet of fencing material and you want to enclose a field with a fence. A building is on one side of the field (and so won't ...
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2answers
151 views

Bounded logarithmic function

I am trying to find any function that it grows logarithmically up to a certain point, and after that point it remains constant. Can anyone help me with that
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73 views

Propose suitable algorithm for min-max optimization problem

Consider: \begin{equation}\min_{x, y} \max_{\omega} | f(x, y, \omega) |\end{equation} where $(x , y)\in \mathbb{R}\times \mathbb{R} $ and $\omega \in (0, \infty)$. $f$ is the result of dividing ...
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2answers
83 views

How do I maximize $|t-e^z|$, for $z\in D$, the unit disk?

I guess this question doesn't have a closed form solution for all $t\in \Bbb C$, but I know one for $t=1$ provided by Daniel Fischer in a question I asked. $$\begin{align} \left\lvert ...
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1answer
83 views

Help solving an optimization problem involving inverse square roots

Does anyone know if the following optimization problem has an elegant solution? Let $A=\{a_1, a_2, \ldots, a_n\}$ be positive real numbers. Let $B=\{b_1, b_2, \ldots, b_n\}$ be unknown positive real ...
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1answer
315 views

Max/Min Problem using derivatives

Question: A professional basketball team plays in an arena that holds 20000 spectators. Average attendance at each game has been 14000. The average ticket price is 75 dollars. Market research shows ...
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221 views

Finding extrema of multivariable functions.

A problem asks me to find the absolute extrema of the function given by $f: \mathbb{R}^2 \rightarrow \mathbb{R} ,f(x,y)=(x^2+y^2)e^{-(x^2+y^2)}$. Now, how can I find the critical points?. As far as I ...
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2answers
278 views

Is the hessian negative semi-definite if we have an interior maximum?

Is it true that given a smooth scalar field f on a domain D , if f attains a maximum (minimum) on the interior of D then the hessian of f evaluated at this max (min) is negative (positive) ...
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1answer
102 views

Maximizing the area of rectangle inscribed in triangle

I'd like to ask if someone could help me out with this problem. Let's have a triangle with coordinates $[0,0],[4,0],[1,3]$. Inscribe a rectangle into this triangle, so its area is maximized The base ...
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3answers
377 views

Lagrange multipliers - finding maximum/minimum

I have solved the question, and obtained the critical points, but don't know how to show its a maximum or minimum of a function. I don't understand other answers because symbols confuse me so much and ...
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1answer
128 views

Help with local extrema of $f(x)=x^4-5x^2$

Find the coordinates of any local extreme points and inflection points of the function $f(x)=x^4-5x^2$ My try: Find critical points: $f^{\prime}(x)=4x^3-10x=0$ $f^{\prime}(x)=2x(2x^2-5)=0 ...
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1answer
136 views

Finding minimum of the trace of the matrix equals finding maximum of the trace of the inverse matrix?

Let $K$ be a positive definite, symmetric matrix. Let $C$ be a nondegenerate matrix of the same order. Elements of $K$ and $C$ depend on some parameter $a.$ Is it true to say that $$ ...
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1answer
62 views

Simplifying This Boolean Expression? (A Little Rusty)

I have the Boolean expression: F = A'B'C'D + A'BC'D' + ABC + AB'C'D' + ABCD'. Note that the ' indicates the negation of the variable by my convention. I am trying to show that F = BC + A'C' + B'D' is ...
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57 views

Strictly convex self-concordant function

Some definitions: A function $f:R^n\rightarrow R$ is convex[strictly convex] if for every $\lambda\in[0,1]$ [$\lambda\in(0,1)$] and for every $x,y$ [$x\neq y$] in $R^n$ we have $f(\lambda ...
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1answer
198 views

L1-norm minimization

This is undoubtedly a trivial question but might as well ask: Why is the L1 norm minimization a heuristic for finding the sparsest vector? What I mean is that if the L1 norm sums the elements of a ...
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1answer
74 views

Direction of steepest descent and minimization?

I have the following linear function: $min$ 1/2 $<x, x>$ + $r^Tx$ for every x belonging to $R^n$, $r^Tx$ belongs to $R^n$ Now, = $x^TAy$ and A is symmetric positive definite. = $x^TAy$ is ...
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1answer
335 views

Sum of weighted squared distances is minimized by the weighted average?

Let $x_1, \ldots, x_n \in \mathbb{R}^d$ denote $n$ points in $d$-dimensional Euclidean space, and $w_1, \ldots, w_n \in \mathbb{R}_{\geq 0}$ any non-negative weights. In some paper I came across the ...
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45 views

Effect of approximating a non-differentiable function on optimisation of minimisation

I am looking at a problem of constrained minimization, where the function to be minimized contains the Heaviside function, and as such is not twice continuously differentiable. My question is what ...
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1answer
139 views

Find $x, y$ such that $\left | \frac ab -\frac xy \right |$ is minimal

Given positive integers $a, b, D$. How to find $x, y \in \mathbb{Z^+}$ such that $$M =\left | \frac ab -\frac xy \right |$$ is minimal and $x + y \le D$? For a solution, I can get it by ...
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1answer
55 views

Range Of Quartic Polynomial Of Two Variables

$a$,$b$ are real numbers such that $~3\leq a^{2}+ab+b^{2}\leq 6$. I would like to find the range of $~a^{4}+b^{4}$. Is it possible to find it with (well-known) AM-GM, CS, etc...?
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43 views

Convex hull of a 0/1 set in $ \mathbb{R}^d$

Reading in my textbook, I found the following example: $$ $$Let S $ \subseteq $ {0,1}$^d$ be an arbitrary 0/1 set in $ \mathbb{R}^d$ and the Polyhedron Q = conv(S). It can be shown easily that the set ...
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1answer
47 views

Minimise Total $x$, Maximum $x$ or $|x|$ in integer/linear programming

Suppose we have a linear program (it may be integer, it probably doesn't matter). Suppose the variables $t_j$ give the tardiness for job $j$, and we want to minimise something to do with this ...
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1answer
136 views

Maximize $ax + by + c$

Working on a problem of comparative advantage of the economist David Ricardo, I've gone into solving a more general case of that study in which I stumbled over this question : how do we maximize the ...
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36 views

Representing a 2D function as a sum of rectangles of arbitrary shape and orientation

Suppose I am given a non-negative function $f(x,y)$ defined for $x \in [0,1]$ and $y \in [0,1]$. I'd like to represent this function as a weighted sum $w_i$ of a small number of rectangular apertures. ...
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Measure minimization for a combination of overlapping sets

This problem may have been worked out before but I don't know where to start looking so I hope one of you can help me. The problem is as follows: There are $N$ variable-sized finite sets ...
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1answer
28 views

Zisserman Lecture and $x_{MLE}$

In the Zisserman Lecture below http://www.robots.ox.ac.uk/~az/lectures/est/lect34.pdf page 36, he derives $x_{MLE}$ for Gaussian sensor fusion. There are two noisy measurements $z_1$ and $z_2$ ...
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1answer
72 views

Need (solid) proof for finding a maximum value

Can anyone please verify my logic to find maximum value of a function? here is my work: Goal is to find $x,y$ which maximize $f(x,y)$ ($f(x,y)$ is a function developed by myself) only $y$ has a ...
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1answer
24 views

First order necessary conditions for $\max_{x_1}f(x_1,g(x_1)).$

$$\max_{x_1}f(x_1,g(x_1)).$$ And, let $f$ attends max at $x_1^*$, so first order necessary conditions imply that $$\dfrac{\partial f(x_1^*,g(x_1^*))}{\partial x_1}+\dfrac{\partial ...
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2answers
650 views

Finding global minimizer and maximizer

Let $f(x,y)=x^4-8x^2+y^4-18y^2$ Find the set of global minimizers of f? Does f have a global maximizer?Justify? I first calculated the gradient of f and then let partial derivative of x and y to ...
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72 views

minimum strength of the edges occurring in any path P

Let $G=(V,E)$ be a graph and let $s: E \to \mathbb{R}^+$ be a function. Let us call $s(e)$ the strength of the edge $e$. For any path $P$ in $G$, the reliability of $P$ is, by definition, the minimum ...
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What (if anything) can I say about the inverse of the matrix product B'AB if B is not square?

Suppose I have: a matrix $A$ with dimension $n \times n$ a matrix $B$ with dimension $n \times m$. $C = B^{T}AB$. I'm interested in finding an expression for $C^{-1}$ when $m < n$. The ...
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2answers
100 views

Help finding local extrema of $f(x)=\frac{x}{\sqrt{2}}-3\sin\frac{x}{2}$

Find the local extrema of $f(x)=\dfrac{x}{\sqrt{2}}-3\sin\dfrac{x}{2}$ on the interval $0 \leq x \leq 2\pi$ $f^{\prime}=\dfrac{1}{\sqrt{2}}-3\cos \left(\dfrac{x}{2}\right) \left ( ...
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0answers
179 views

Measuring how change in input variables contributes to output in non linear equation.

How do we measure how a variable contributes to an output as its value increases, and how it relates to other input variables? Let's say we're playing a video game, where you can buy items to augment ...
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0answers
264 views

Area optimization: Packing rectangles inside rectangle

Background: I am scrambling to figure out the optimization algorithm to build sprite image, which is essentially a big container rectangular image, with multiple rectangular images. I have found an ...
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2answers
92 views

Lagrange multiplier. What happen when gradient of boundaries is null

Suppose that you have to maximize the function $f(x)$ ($f : \mathbb{R} \rightarrow \mathbb{R}$), continuous and differentiable for each $x \in A = \left\{ x |g(x) =c \right\}$, where $g : \mathbb{R} ...
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1answer
83 views

Strictly positive solution of linear equations

Suppose $A\in\mathbb{R}^{m\times n}$, $b\in\mathbb{R}^m$, and $b\in \mathcal{R}(A)$. Show that there exists an $x$ satisfying $x \succcurlyeq 0$, $Ax = b$ if and only if there exists no ...
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Optimization problem of quadratic function - Compressive sensing

I got the optimization problem in Compressive sensing in form $f =arg min \ \frac{\mu}{2} ||\Phi.f-y||^2 + \frac{\lambda}{2} ||f-v-w||^2$ where $\Phi$ is orthogonal Gaussian sensing matrix ...
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1answer
147 views

Maximising with multiple constraints

I have $$Z=f(x_1 ,x_2 ,x_3 ,... ,x_n)$$ function and $$\left[\begin{array}{r}c_1=g_1(x_1 ,x_2 ,x_3 ,... ,x_n) \\c_2=g_2(x_1 ,x_2 ,x_3 ,... ,x_n)\\c_3=g_3(x_1 ,x_2 ,x_3 ,... ,x_n) \\...\\c_m=g_m(x_1 ...
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1answer
55 views

Weighted Set covering problem with a fixed number of colors

I have a set of elements U = {1, 2, .... , n} and a set S of k sets whose union form the whole universe. Each of these sets is associated with a cost. I have a fixed number of colors, C = {1 , 2, ...
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1answer
150 views

Free software or algorithm for Second-Order Cone Program

I need to solve the following optimization problem: $$ \mathbf{x}^\ast = \operatorname{argmin}_{\mathbf{x}} \Vert \mathbf{Rx} \Vert_2^2 \;\;\; \mathrm{s.t.} \;\;\; \mathbf{s}^\mathrm{H} \mathbf{x} = ...
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1answer
76 views

Basic Question about Newton's Method for Optimization

This is a very basic question about Newton's method for optimization, but I cannot seem to find the answer in any of my searches. If we are using Newton's method (or gradient descent), how do we find ...
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48 views

find conditions on input data such that a linear system has (no) feasible points

As a result of the apllication of Farkas' lemma I obtained the following problem: Let $ m,n,q \in \mathbb{N} $, $ b \in \mathbb{N}^m, l \in \mathbb{N}^m $ with $ l_i \mid q$ for all $ i=1,\ldots,m $. ...
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1answer
124 views

pairwise disjoint subsets of divisors of $ n $ (maximum number)

Let $ n \in \mathbb{N} $, $ n>1 $ and $ a_1,\ldots,a_k \in \mathbb{N} $ (not necessarily distinct!) with $ a_i \mid n $ for all $ i=1,\ldots,k $ be given. Assume that $ \sum_{i=1}^k a_i = K\cdot n ...
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1answer
509 views

Maximum area of a rectangle inside a triangle

I recently came across a problem where it gave a triangle with integer side lengths, and it asked you to find the maximum area of a rectangle of a triangle. I solved the problem correctly, but it ...