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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Do dynamic programming and greedy algorithms solve the same type of problems?

I wonder if dynamic programming and greedy algorithms solve the same type of problems, either accurately or approximately? Specifically, As far as I know, the type of problems that dynamic ...
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Optimum solution to a Linear programming problem

If we have a feasible space for a given LPP (linear programming problem), how is it that its optimum solution lies on one of the corner points of the graphical solution? (I am here concerned only with ...
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323 views

Graph theory problem (edge-disjoint matchings)

Find the smallest number $x$ so that if an $n$-vertex simple graph has at least $x$ edges then it contains $k$ pairwise edge-disjoint perfect matchings* ($k$ is a positive integer, $n$ is an even ...
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1answer
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$a,b,c>0,a+b+c=21$ prove that $a+\sqrt{ab} +\sqrt[3]{abc} \leq 28$

$a,b,c>0,a+b+c=21$ prove that $a+\sqrt{ab} +\sqrt[3]{abc} \leq 28$ I have tried to use AM-GM inequality, but get no result as follows: $$a+\sqrt{ab}+\sqrt[3]{abc}\leq ...
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4answers
819 views

Calculating the max and min of $\sin(x)+\sin(y)+\sin(z)$

I took the partial derivatives of $\sin(x)+\sin(y)+\sin(z)$ and it didn't work out, so I am trying to use Lagrange's method (with the constraint: $x+y+z=\pi$)... I am not sure how to set this up. ...
10
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5answers
707 views

Why is convexity more important than quasi-convexity in optimization?

In the mathematical optimization literature it is common to distinguish problems according to whether or not they are convex. The reason seems to be that convex problems are guaranteed to have ...
10
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Best fit ellipsoid

Given a collection of points $P \subset \mathbb R^3$, a crude characterization of the "shape" of $P$ is sometimes given by the principal components. We construct a covariance matrix, e.g., if $P$ is ...
7
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1answer
301 views

Maximum subset sum of $d$-dimensional vectors

This is a $d$-dimensional generalisation of the post Inequality with Complex Numbers. (See my comment under Robert Israel's answer.) Generalising Potato's proof for $d$-dimensions, we can show the ...
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1answer
939 views

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 ...
2
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2answers
102 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.
10
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73 views

Using a compass and straightedge, what is the shortest way to divide a line segment into $n$ equal parts?

Sometimes I help my next door neighbor's daughter with her homework. Today she had to trisect a line segment using a compass and straightedge. Admittedly, I had to look this up on the internet, and ...
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292 views

Explain a surprisingly simple optimization result

The following optimization problem came to my attention as an idealization of the silly browser game Cookie Clicker, but is representative of a range of strategy games: You have an initial ...
7
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388 views

Arithmetic mean is less than geometric mean (Spivak Calculus 3rd Chapter 2 Problem 22)

If $a_1, \ldots, a_n \ge 0$, the arithmetic mean $$A_n={a_1 + \cdots + a_n \over n}$$ and the geometric mean $$G_n = \sqrt[n]{a_1 \cdots a_n}$$ satisfy $G_n \le A_n$. As a first step to prove this ...
7
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406 views

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 ...
7
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585 views

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 ...
4
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1answer
126 views

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
72 views

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 ...
3
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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) ...
3
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1answer
147 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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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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1answer
526 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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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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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
56 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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295 views

How to prove equality from poincare inequality?

Let $$D = \{y \in C^1(0,1) : y(0) = y(1) = 0\}$$ Suppose there exists a $C_0$ such that $$\int_{0}^{1} y^2 \ dx \leq C_0 \int_{0}^{1} (y')^2 \ dx$$ for all $y \in D$, and for all $C < C_0$ ...
0
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1answer
121 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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Gerrymandering/Optimization of electoral districts for one particular party

I'm asking this on behalf of Zach Weiner (actually it's my own initiative in order to promote this site). Original text is here, and is as follows: Hey-- This is Zach from SMBC, and I have a math ...
14
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1answer
245 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
626 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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How to prove the sum of squares is minimum?

Given $n$ positive values. Their sum is $k$. $$ x_1 + x_2 + \cdots + x_n = k $$ The sum of their squares is defined as: $$ x_1^2 + x_2^2 + \cdots + x_n^2 $$ I think that the sum of squares is ...
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3answers
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What do curly less than sign mean?

I am reading "Convex Optimization" by Stephen Boyd. He is using a curved greater than and curved less than equal to signs. $f(x^*) \succcurlyeq \alpha$ or $f(x*) \preccurlyeq \alpha$ Can someone ...
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Optimization problem (in linear algebra course!)

Let $a_1, a_2, \ldots, a_n$ be real numbers such that $a_1 + \cdots + a_n = 0$ and $a_1^2 + \cdots +a_n^2 = 1$. What is the maximum value of $a_1a_2 + a_2a_3 + \cdots + a_{n - 1}a_n + a_na_1$? I'd ...
8
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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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1answer
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On problems of coins totaling to a given amount

I don't know the proper terms to type into Google, so please pardon me for asking here first. While jingling around a few coins, I realized that one nice puzzle might be to figure out which $n$ or so ...
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1answer
553 views

Why are additional constraint and penalty term equivalent in ridge regression?

Tikhonov regularization (or ridge regression) adds a constraint that $\|\beta\|^2$, the $L^2$-norm of the parameter vector, is not greater than a given value (say $c$). Equivalently, it may solve ...
5
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1answer
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dual problem of a Semidefinite programming in a non-standard forme

I have a problem with calculating the dual problem of : $$ \mbox{Minimize } tr(Y) + \frac{1}{\eta} tr(Z) $$ $$ \begin{pmatrix} Y & X \\ X & Z+\varepsilon I \end{pmatrix} \succeq 0 ...
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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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1answer
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Fundamental Optimization question consisting of two parts.

A) Find all extrema of $$f(x)=\sum_{k=1}^{n} x_{k}^{2} $$ subject to the constraint $\sum_{k=1}^{n}\vert x_k\vert^p=1$ B) prove that $$\frac{1}{n^{(2-p)/(2p)}}(\sum \vert x_k\vert^p)^{(1/p)}\le (\sum ...
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1answer
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How to compute the Pareto Frontier, intuitively speaking?

I'm working on a multi-objective optimization problem and we have 'alternatives' that are quantified on two dimensions - value and cost. Now the question is 'how does one compute a pareto frontier'? ...
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2answers
441 views

Generalizing Lagrange multipliers to use the subdifferential?

Background: This is a followup to this question: Lagrange multipliers with non-smooth constraints Lagrange multipliers can be used for constrained optimization problems of the form $\min_{\vec x} ...
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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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How is $L_{2}$ Minkowski norm different from $L^{2}$ norm?

I am reading the book Multidimensional Particle Swarm Optimization for Machine Learning and Pattern Recognition. They use $L_{2}$ Minkowski norm (Euclidean) as the distance metric in the feature space ...
3
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332 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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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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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 ...
3
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2answers
249 views

Lagrange multipliers with non-smooth constraints

I read in a textbook a passing comment that Lagrange multipliers are not applicable if there are points of non-differentiability in the constraints (even if the constraints are continuous). For ...
2
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1answer
43 views

Set convergence and lim inf and lim sup

I'm a bit confused with the general concept of convergence of a sequence of sets. I'm well aware that the limit of a sequence $\{C^{\nu}\}$ exists iff $$\liminf_{\nu \rightarrow \infty} C^{\nu} = ...
2
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3answers
62 views

Maximal area of a triangle

What would be the most elementary proof of the following: A triangle has been drawn inside the circle with radius $r$ and its area is as large as possible. Prove that the triangle is equilateral. I ...
2
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4answers
142 views

Three Variables-Inequality with $a+b+c=abc$

$a$,$b$,$c$ are positive numbers such that $~a+b+c=abc$ Find the maximum value of $~\dfrac{1}{\sqrt{1+a^{2}}}+\dfrac{1}{\sqrt{1+b^{2}}}+\dfrac{1}{\sqrt{1+c^{2}}}$
2
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2answers
94 views

Mminimize the integral and obtaining the constants $a$ and $b$

Determine the constants $a$ and $b$ for the integral $$ \int\limits _{0}^{1}(ax+b-f(x))^{2} dx$$ take the smallest possible value if $f(x)=(x^{2}+1)^{-1}$ thanks