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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Finding the largest triangle inscribed in the unit circle

Among all triangles inscribed in the unit circle, how can the one with the largest area be found?
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Optimization problem for a parity-check code

I have $n$ data blocks and $k$ parity blocks distributed across $m$ boxes where each box can contain atmost $b$ blocks. Each parity block is Ex-or of some data blocks (for ease of understanding we can ...
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1answer
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Degeneracy in Linear Programming

Consider the standard form polyhedron, and assume that the rows of the matrix A are linearly independent. $$ \left \{ x | Ax = b, x \geq 0 \right \} $$ (a) Suppose that two different bases lead to ...
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Proving the existence of $b$ such that $\prod_{k=1}^n(1-\cos(a_k-b))=\frac{1}{2^n}$

Let $n>0$ and $a_1,\ldots,a_n\in \mathbb R$. Prove there is some $b$ such that $\prod_{k=1}^n(1-\cos(a_k-b))=\frac{1}{2^n}$ This is motivated by this question Finding a point on the unit ...
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1answer
320 views

Conjugate Gradient Method and Sparse Systems

What is it about conjugate gradient that makes it useful for attacking sparse linear systems. Why would steepest descent be significantly worse? Please keep in mind that I am still trying to fully ...
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2answers
390 views

Derivation of Euler-Lagrange equation

Here is a simple (probably trivial) step in the derivation of the Euler-Lagrange equation. If we denote $Y(x) = y(x) + \epsilon \eta(x) $, I want to know why is $\dfrac{\partial ...
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3answers
721 views

Optimal symmetric rank-1 approximation

I want to find $\mathbf{x}$ that minimizes $\|A-\mathbf{x}\mathbf{x}'\|^2$ where $\|\cdot\|$ is Frobenius norm. Differentiating with respect to $\mathbf{x}$ and setting to $\mathbf{0}$, I get ...
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1answer
198 views

Finding the extrema of $E(\vec{r})=\frac1a x^2+\frac1b y^2+ \frac1c z^2$ with respect to constraints geometrically

I have a function, $$E(\vec{r})=\frac1a x^2+\frac1b y^2+ \frac1c z^2.$$ Where $\vec{r}=(x,y,z)$ and $a>b>c>0$. I wish to find the maximum and minimum of this function with respect to the ...
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480 views

Minimize $x^TAx$, subject to $||x||=1$. Show that ${x^*}^TAx^*$ is the smallest eigenvalue of $A$ in magnitude.

I'm solving constrained optimization exercises for preparing my final exam. I got stuck at this question. $$ \begin{array}{ll} \text{min} & \mathbf{x}^T\mathbf{A}\mathbf{x} \\ \text{s.t.} & ...
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110 views

Find the values of $x$, $y$ and $z$ minimizing $\frac{r^2x}{y+z}+\frac{s^2y}{x+z}+\frac{t^2z}{x+y}$

$$\frac{r^2x}{y+z}+\frac{s^2y}{x+z}+\frac{t^2z}{x+y}$$ $r$, $s$, $t$ are positive coefficients. Find the values of non-negative variables $x$, $y$ and $z$ so that the above expression is a minimum.
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Proof that $E_n = \int_0^1 \left| \ln t - P_n(t) \right|\mathrm{d}t = 1/(n+1)^2$

Assume one wants to minimize the distance between $f(x)=\ln x$ and $P_n(t)$ where $P_n$ denotes a polynomial of degree $n$. Etc $P_1 = ax 0+ b$. One way to judge whether the polynomial is a good ...
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2answers
760 views

Max and min value of $7x+8y$ in a given half-plane limited by straight lines?

So, there are four inequalities: $$\begin{eqnarray*} y &\geq &-3x+15; \\ y &\leq &-11/3x+56/3; \\ x &\geq &0; \\ y &\geq &0. \end{eqnarray*}$$ If we draw all those ...
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1answer
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Which optimization class does the following problem falls into (LP, MIP, CP..) and which solver to use

I have the following optimization problem. I want to solve it using a computer solver. But I am not sure which problem class it falls into or which solver to use. Problem: There is a set of objects ...
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1answer
59 views

Find minimal $\alpha_3$ such that $u\in H^3(\Omega)$ and $u(x,y,z)=x^\alpha(1-x)y(1-y)z(1-z)$

My instructor presented me the quiz below but forgot to define key terms such as minimality and $H^3$. Quiz Let $u(x,y,z)=x^\alpha(1-x)y(1-y)z(1-z)$. Find the minimal $\alpha_3$ such that $u\in ...
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1answer
138 views

Optimizing $x^2+y^2$ from two given equations? [duplicate]

What is the maximum value of $x^2+y^2$, where $(x,y)$ are solutions to: $$2x^2+5xy+3y^2=2$$ and $$6x^2+8xy+4y^2=3$$ Note: Calculus is not allowed. I tried everything I could but whenever I got for ...
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275 views

Fastest curve from $p_0$ to $p_1$

I'm trying to solve a problem in path planning: Given points $p_0$ and $p_1$ and vectors $v_0$ and $v_1$, find a function $p(t)$ st. $p(0) = p_0$, $p(T) = p_1$, $p'(0) = v_0$ and $p'(T) = ...
14
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1answer
1k views

Maximum total distance between points on a sphere

What is the configuration (set of locations) of $n$ points on the surface of a sphere such that the sum of distances is maximum for $n=1,2,3,...$? The sum of distances is measured by summing the ...
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1answer
382 views

Largest $n$-vertex polyhedron that fits into a unit sphere

In two dimensions, it is not hard to see that the $n$-vertex polygon of maximum area that fits into a unit circle is the regular $n$-gon whose vertices lie on the circle: For any other vertex ...
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Generalization of the Sultan's dowry problem

We know the solution of the Sultan's dowry problem: To reject the first $n/e$ candidates and then to select the first who exceeds the best of the sample. How to find the best strategy if we want ...
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4answers
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Finding minimum $\frac{x+y}{z}+\frac{x+z}{y}+\frac{y+z}{x}$

I would appreciate if somebody could help me with the following problem Q. Finding maximum minimum $$\frac{x+y}{z}+\frac{x+z}{y}+\frac{y+z}{x}(\text{where} ~~x,y,z>0)$$
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Decomposing a discrete signal into a sum of rectangle functions

Hello math@stackexchange community ! I have a simple question that seems to have a non trivial answer. Given a discrete one dimensional signal $w(x)$ defined in a finite range, and the boxcar ...
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Determine the minimum of $a^2 + b^2$ if $a,b\in\mathbb{R}$ are such that $x^4 + ax^3 + bx^2 + ax + 1 = 0$ has at least one real solution

I just wanted the solution, a hint or a start to the following question. Determine the minimum of $a^2 + b^2$ if $a$ and $b$ are real numbers for which the equation $$x^4 + ax^3 + bx^2 + ax + 1 = 0$$ ...
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1answer
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Maximizing the volume of a rectangular prism

A rectangular prism has a surface area of $300$ square inches. What whole number dimensions give the prism the greatest volume? This is a math olympiad problem. It involves the volume and surface ...
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795 views

Multilinear optimization

Are there any efficient algorithms to solve, multi-linear objective and multi-linear constraint optimization problems? The multilinear functions are sums of bilinear, trilinear (and so on) terms ...
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1answer
393 views

Newton's method vs. gradient descent with exact line search

tl;dr: When is gradient descent with exact line search preferred over Newton's method? I simply don't understand why exact line search is ever useful, and here's my reasoning. Let's say I have a ...
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1answer
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Converting sum of infinity norm and L1 norm to linear programming

So I'm trying to convert this minimization problem, min $\parallel Ax-y \parallel_{\infty}$ + $\parallel x \parallel_{1}$ where $A$ is $m$ by $n$, $y$ is $m$ by $1$ and $x$ is $n$ by $1$. into a ...
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1answer
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Angular alignment of points on two concentric circles

I have two concentric circles $C_1$ and $C_2$ with radii $r_1,r_2$ such that $r_1< r_2$and a set of finite points $P=\left \{ p_1,p_2...p_n \right \}$ and $Q=\left \{ q_1,q_2...q_n \right \}$ are ...
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1answer
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Maximizing distance between points

I asked a similar question on SciComp, but it is a little out of the domain, so I thought I'd give it a try here as well. Give n points, I would like to place them in a periodic box (periodic such ...
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Convergence of Gauss-Newton method for piecewise linear functions

Notation for Gauss-Newton method Non-linear least squares problems are often solved by the Levenberg-Marquardt algorithm, which can be viewed as a Gauss–Newton method using a trust region approach. ...
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2answers
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Sparsest matrix with specified row and column sums

Given a sequence of row sums $r_1, \ldots, r_m$ and column sums $c_1, \ldots, c_n$, all positive, I'd like to find a matrix $A_{m\times n}$ consistent with the given row and column sums that has the ...
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2answers
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Least sum of distances

Problem: Let $A, B, C, D$ be points in a $3$-dimensional space. Find the point $X$ that minimizes the sum of the distances $AX+ BX + CX + DX$. Context: During a course, I was assigned a ...
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2answers
881 views

Gradient-descent algorithm always converges to the closest local optima?

Assume $f(\vec x)$, which is Lipschitz continuous, has two local optima $\vec x_1^*$ and $\vec x_2^*$( $\vec x_1^*$ is the global minimum). We start the gradient-descent algorithm from $\vec x_0$ and ...
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1answer
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How to calculate lim inf and lim sup for given sequence of sets

Let the indicator function be defined as $$I(x) \triangleq \begin{cases} 1, & \quad x \geq 0 \\ 0, & \quad x < 0 \end{cases}$$ and $I_{\nu}(x) \in [0,1]$ be a continuous approximation of ...
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Convergence of sequences of sets

Assume we have the indicator function $$I(x) \triangleq \begin{cases} 1, & \quad x \geq 0 \\ 0, & \quad x < 0 \end{cases}$$ and the sequence of its approximation functions $\{ I_{\nu}(x) ...
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2answers
536 views

Minima of symmetric functions given a constraint

If $f(x,y,z,\ldots)$ is symmetric in all variables, (i.e $f$ remains the same after interchanging any two variables), and we want to find the extrema of $f$ given a symmetric constraint ...
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Why gradient descent works?

On Wikipedia, this is the following description of gradient descent: Gradient descent is based on the observation that if the multivariable function $F(\mathbf{x})$ is defined and differentiable ...
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Is this “theorem” true in Optimization Theory?

If I have a function $f(x,y)$ subjected to a region $D$ on the xy-plane, then the extreme values of $f(x,y)$ occurs at the extreme "corners" points of $D$? I remember waaaaaaay back in calculus, if ...
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What is the best approximate of points on a sphere?

I have a unit radius sphere with a set $S$ of $n$ points on it. How can I find a map $f:S\to \mathbb{R}^4$ which minimizes $$\sum_{x,y\in S} \bigg( d_{\text{geodesic}} (x,y)^{2} - ...
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2answers
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Gradient and Swiftest Ascent

I want to understand intuitively why it is that the gradient gives the direction of steepest ascent. (I will consider the case of $f:\mathbb{R}^2\to\mathbb{R}$) The standard proof is to note that the ...
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1answer
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Constrained infinity norm minimization

I have a problem like this: $$\min_x |Ax|_\infty \text{ s.t. } \sum_i x_i = c$$ That is, I want to find the vector $x$ whose elements sum to a constant $c$ that minimized the infinity norm of $Ax$. ...
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1answer
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minimum lines, maximum points

There are $P$ points in the 2-dimensional plane. Through each point, we draw two orthogonal lines: one horizontal (parallel to x axis), one vertical (parallel to y axis). Obviously, some of these ...
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1answer
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When $\min \max = \max \min$?

Let $X \subset \mathbb{R}^n$ and $Y \subset \mathbb{R}^m$ be compact sets. Consider a continuous function $f : X \times Y \rightarrow \mathbb{R}$. Say under which condition we have $$ \min_{x \in ...
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$\beta_k$ for Conjugate Gradient Method

I followed the derivation for the Conjugate Gradient method from the documents shared below http://en.wikipedia.org/wiki/Conjugate_gradient_method ...
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1answer
912 views

Solution Technique to Optimize Sets of Constraint Functions with Objective Function that is Heaviside Step Function

I have the following constraint inequalities and equalities: $$Ax \leq b$$ $$A_{eq}x = b_{eq}$$ The problem is that the objective function, which I am asked to minimized, is defined as ...
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Constraint optimization with Calculus of Variations. How to handle positive function constraint?

the I am attempting to maximize the functional $F[f]$ with a constrain that $f$ has to be non-negative and some other integral constraints. More, specifically, \begin{align*} &\max F[f]\\ ...
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1answer
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Maximisation problem

I am trying the following question: If$$a+b+c+d=0,\;a^2+b^2+c^2+d^2=1$$ Then what is the maximum value of $ab+bc+cd+da?$ By the rearrangement inequality I can get $ab+bc+cd+da\leq 1$ but I am ...
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1answer
142 views

A problem on positive semi-definite quadratic forms/matrices

Suppose $a+b+c=0$ and (without loss of generality) $a\leq b\leq 0\leq c$, $a^2+b^2+c^2=1$, is the following quadratic form positive semi-definite? Thank you very much. \begin{equation*} \begin{split} ...
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Clarification on optimization problem

While reading a combinatorics paper about packing densities in compositions, I encountered the following optimization problem. Maximize $$f(\alpha_1, \dots, \alpha_5) = \sum_{1 \le i < j ...
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Sequence of solutions of approximated problem converging to stationary solution of the original problem

Let $I(x)$ be the indicator function defined as $$I(x) \triangleq \begin{cases} 1, & x \geq 0 \\ 0, & x < 0 \end{cases}$$ and let $L(x,\alpha) \in [0,1]$ be a smoothing function that ...
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0answers
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Why these two problems lead to same answers?

Suppose these two problems: Problem 1: $$\min_{X,P} \quad\max_{1\leq l\leq L-1} \quad {|\sum_{1\leq i\leq N_p}^{N_p}x_ie^{\frac{2\pi l}{N}p_i}| \over {\sum_{i=1}^{N_p} x_i^2}} \quad \equiv \quad ...