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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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 ...
10
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1answer
103 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 ...
8
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3answers
428 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 ...
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5answers
625 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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3answers
687 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 ...
5
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2answers
525 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 ...
5
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1answer
6k views

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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2answers
184 views

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 ...
4
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1answer
169 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 ...
3
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1answer
228 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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2answers
203 views

find maximum area

Consider a problem here : There is a wall in your backyard. It is so long that you can’t see its endpoints. You want to build a fence of length L such that the area enclosed between the wall and the ...
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0answers
84 views

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
171 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 ...
3
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1answer
115 views

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 ...
2
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1answer
606 views

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 ...
2
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1answer
658 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 ...
1
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1answer
48 views

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 ...
1
vote
1answer
77 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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0answers
50 views

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 ...
0
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1answer
57 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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2answers
467 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 $C < C_0$ ...
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1answer
131 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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4answers
725 views

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
264 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) = ...
8
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6answers
915 views

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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2answers
555 views

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 ...
8
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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 ...
6
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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 $$\frac{x+y}{z}+\frac{x+z}{y}+\frac{y+z}{x}(\text{where} ~~x,y,z>0)$$
5
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1answer
130 views

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 ...
5
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3answers
123 views

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 $$x^4 + ax^3 + bx^2 + ax + 1 = 0$$ ...
5
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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 ...
5
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1answer
7k views

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'? ...
5
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2answers
509 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} ...
4
votes
3answers
634 views

Monotonicity of $\ell_p$ norm

Consider a $n$ dimensional space, it is known (Wikipedia) that for $p>r>0$, we have $$ \|x\|_p\leq\|x\|_r\leq n^{(1/r-1/p)}\|x\|_p. $$ I have two questions about the above inequality. $(\bf ...
4
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0answers
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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. ...
3
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2answers
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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 ...
2
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3answers
124 views

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 ...
2
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2answers
85 views

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 ...
2
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3answers
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Find out minimize volume (V) of tetrahedral

I have this problem: On space $ (Oxyz)$ given point $M(1,2,3)$. Plane ($\alpha$) contain point $M$ and ($\alpha$) cross $Ox$ at $A(a,0,0)$; $Oy$ at $B(0,b,0)$; $C(0,0,c)$. Where a,b,c>0 Write the ...
2
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1answer
111 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
votes
3answers
92 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
votes
2answers
103 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
2
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1answer
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Does the uniqueness of solutions to convex optimization with linear constraints hold in n>3 dimensions?

This is a repost of an earlier question, where I think I was not clear enough in what I was asking: I am examining the following optimization problem, for which I would like to know if, when a ...
2
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2answers
165 views

Shortest ternary string containing all ternary strings of length 3?

How can we find/construct the shortest ternary string that contains all ternary strings of length 3? For instance, $120011$ contains $120$, $200$, $001$, and $011$. (The shortest such a string could ...
2
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3answers
868 views

How to interpret Hessian of a function

I know that gradient of a function gives the direction in which the directional derivative of the function is maximum. Is there any similar interpretation of Hessian ?
2
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1answer
183 views

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 ...
2
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1answer
395 views

Preconditioning for an LBFGS

I am working on a high dimensional (N ~ 1000-60000) optimization problem which is currently solved with an LBFGS algorithm. I have experimented with different diagonal preconditioners as I know that ...
2
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1answer
130 views

Control on Conformal map

Let $\Omega$ be smooth simply connected open set of $\mathbb{R}^2$ such that $\overline{\Omega}$ is compact. We know that there exists a conformal diffeomorphism $\psi$ from $\mathbb{D}$ to $\Omega$. ...
2
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2answers
214 views

$\beta_k$ 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 ...
1
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1answer
26 views

how to find the solution of this cost function?

I have the following cost function. $J = \sum_{i=1}^N a\, Trace(W^TX_iW) - b\, Trace(W^TY_iW)$ Where $X_i$ and $Y_i$ are symmetric matrices, $a$ and $b$ are scalars. How can I find W?