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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Minimize a matrix function with constraints

Let $A, X\in\mathbb{R}^{n\times n}$. The scalar objective function is $$J=\mathrm{tr}(AX)$$ If no constraints, let the derivative of $J$ with respect to $X$ be zeros, then we have $$A=0$$ Suppose $A$ ...
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Maximizing the sum of two numbers, the sum of whose squares is constant

How could we prove that if the sum of the squares of two numbers is a constant, then the sum of the numbers would have its maximum value when the numbers are equal? This result is also true for ...
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Dynamic programming - A type of balanced 0-1 matrix

I was reading the Wikipedia article of Dynamic programming, however, I'm having a hard time understanding the explanation given in the example for a type of balanced 0-1 matrix. The problem is stated ...
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586 views

Optimally combining samples to estimate averages

Suppose I have two tables, each of unknown size, and I'd like to estimate the average of their true sizes. I hire 2 contractors: one guarantees good precision (i.e., her measurement ...
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2answers
340 views

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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Determine the smallest number P

I have here a hard problem, which I couldn't solve. Denote $M$ the set of all functions $f:[0,1]\to\mathbb{R}$ with the following properties: $f(x)\ge0, \forall x$ in $[0,1]$, $f(1)=1$, $f(x+y)\ge ...
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4answers
68 views

Optimization with a constrained function

Okay so I understand how to find points of extrema when for example, We have $3x^2 + 2y^2 + 6z^2$ subject to the constaint $x+y+z=1$. I followed the method of the Lagrange multiplier and resulted in ...
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1answer
47 views

Nonconvex set converging to a convex set despite holes

I'm looking at the example in Figure 4-7 of "Variational Analysis" (Rockafellar and Wets). Basically, there's a sequence of sets $C_{\nu}$ riddled with holes, and it states that the sequence ...
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significance of zeros of a transfer function?

In control theory, the poles of a transfer function give information about the stability and behavior of a system. I'm not sure and can't find anywhere what the significance of the zeros of a ...
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Find the maximum or minimum value of the quadratic function.

Find the maximum or minimum value of the quadratic function by completing the squares. Also, state the value of $x$ at which the function is maximum or minimum. $y=2x^2-4x+7$ $x^2$ has a coefficient ...
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Compute the minimum distance between the centre to the curve $xy=4$.

I wish to solve the following problem: Compute the minimum distance between the center to the curve $xy=4$. But I don't know where to start from?
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84 views

Effecient way to find optimal solution in a 2 player game

I have a function: \begin{equation*} f(a_1,\ldots,a_7,b_1,\ldots,b_4)=-14-7 a_1+30 a_1 a_2-7 a_4-2 a_4 a_5+21 a_6+21 a_7+16 a_1 b_1-24 a_1 a_2 b_1+6 a_4 b_1-6 a_4 a_5 b_1+6 a_1 b_2-6 a_1 a_2 b_2+8 a_4 ...
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137 views

Optimization for constrained problem

I'm reading about Lagrange multipliers from a Pattern recognition book appendix and on one point the following is stated: $\begin{align} ...
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464 views

How to find the minimum value of the expression?

Let $x$, $y$, $z$ be three nonegative real numbers and $x^2 + y^2 + z^2 = 5.$ Find the minimum of the expression $$E=\dfrac{1}{2}(x^2 y^2 + y^2 z^2 + z^2 x^2) + \dfrac{96}{x + y + z + 1}.$$ What ...
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Optimizing non linear programs of two variables

The scenario is; We've got $n$ stationary 360$^{\circ}$ sensors in an confined area (each sensor is located at some arbitrary $\left(x,y\right) = \left(x_{n},y_{n}\right)$), once a unit $t$ enters ...
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2answers
170 views

How many points to find a polynomial?

I would like to fit a formula $ax^b + cx^d+ e$ to a set of points. I have two questions. If my data were perfect, how many points do I need in the worst case to get $a,b,c,d,e$ exactly? If my data ...
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706 views

Shortest distance between two shapes

This is the scenario of my problem. I have an image of two objects ( of arbitrary shape, not convex, not touching or crossing each other, kept a few space apart). And I am supposed to find the ...
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Minimize $\| ACE \|$ by geometrical means

I have the following figure Where $AB=10$m, $BD=12$m and $DE=12$m. The point C can slide along the segment BD. Now the problem is to minimize the distance from A to D going along the dashed line. ...
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82 views

Maximum of an expression with four variables

Assume $0 < p_1 \le p_2\le p_3 \le p_4$. What is the maximum of the following expression? $$ \frac{\left(p_1+p_4\right)\left(p_2+p_3\right)}{\left(p_1+p_3\right)\left(p_2+p_4\right)} $$ Is that ...
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244 views

What is the complexity of computing the minimum distance between two convex polyhedra that both have $n$ faces?

EDIT: (in response to what deinst said) sometimes using a sledgehammer for some menial task is rather convenient - especially if it also has the complexity $O(n)$ (which is what my question is about) ...
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Least Square Method with Positive Parameters

this is my first post here in the Stack Exchange. A friend told me about this forum and I'm giving it a try. I searched a bit past threads, but couldn't find what I wanted, so I'm posting the problem ...
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80 views

a plausible maximum or minimum

Is the following statement true? Let $a_1\ge a_2\ge \cdots \ge a_n>0$, $b_1\ge b_2\ge \cdots \ge b_n>0$, then $$\max\limits_{\sigma\in S_n}\;\;\prod\limits_{i=1}^n(a_i+b_{\sigma ...
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Question about the simplex method complexity

So I know that in general the simplex method for linear and convex quadratic programming can require exponential time. But assuming a positive semidefinite quadratic program that is solvable by the ...
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Optimizing $a+b+c$ subject to $a^2 + b^2 + c^2 = 27$

If $a,b,c \gt 0$ and $a^2+b^2+c^2=27$, find the maximum and minimum values of $a+b+c$. How to solve this one? (Here's the source of inspiration for the problem.)
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linear least squares minimizing distance from points to rays - is it possible?

I'm writing a tool whose purpose is to process data from a sensor that provides the true bearing to a target, and combine measurements taken at various times into an estimate of the target's position ...
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1answer
358 views

Construct dual network for conversion of min-cut problem to shortest path problem

I was wondering if there is some typo in the following description from Section 8.4 p263 of Network Flows: Theory, Algorithms, and Applications by Ravindra K. Ahuja, Thomas L. Magnanti, and James B. ...
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290 views

Classifying singular points as local min, max or saddle points

I want to determine if a singular point is a local min, max or saddle point. We are dealing with singular points so we cannot use the hessian matrix. What I have written, and I think I must of missed ...
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252 views

Extremal curve passing through a set of points

I'm having trouble recasting the following question in a form amenable to the calculus of variations. Question: Given a set of $n$ points $P=\{(x_1,y_1),..(x_n,y_n)\}$ what is the curve passing ...
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Clarification on optimization problem, continued

Background This is a follow-up to this question. The problem statement is the same: Maximize $$f(\alpha_1, \dots, \alpha_5) = \sum_{1 \le i < j < k \le 5} \alpha_i \alpha_j \alpha_k$$ ...
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How to maximize an entropy function?

I'm very novice in optimization and have a convex optimization function of form $\sum_{i,k} p_{k,i}*\log{p_{k,i}} $ to minimize with the following constraints: $\forall i, a_i = \sum_{k=1}^{m} b_k. ...
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$f$ is convex function iff Hessian matrix is nonnegative-definite.

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, $f \in C^2$. Show that $f$ is convex function iff Hessian matrix is nonnegative-definite. $f(x,y)$ is convex if $f( \lambda x + (1-\lambda )y) \le ...
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How to maximize $\left({a+b \choose a} 2^{-a-b}\right)$?

How can you maximize $\left({a+b \choose a} 2^{-a-b}\right)$ assuming, $a,b \geq 0$ and $0< (a+b) \leq n$, where all the variables are non-negative integers? Is the maximum when $a=b=n/2$, ...
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1answer
135 views

Previous table of simplex algorithm

I have this problem of simplex method which shows cycle . Problem : Maximize $Z=10x_1-57x_2-9x_3-24x_4$ Constraint to : $ 0.5x_1-5.5x_2-2.5x_3+9x_4<=0 $, $ ...
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Proving the regular n-gon maximizes area for fixed perimeter.

It is often assumed that, given $n$, the regular $n$-gon will make the most efficient use of perimeter for area. I have never seen this proven. Anyone have something slick? (That is, how can we ...
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2answers
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Minimum of $\prod_{1\le i\le n} (1+a_i)$ when $a_1a_2\cdots a_k = M.$

I asked a question here, and also got its generalization:(see tc1729's answer) $$\prod_{1\le i\le n} (1+a_i)\ge 2^{n}\sqrt{a_1a_2\cdots a_n}=2^n\cdot\sqrt M,$$for$$a_1a_2\cdots a_k = M.$$ But I ...
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Minimisation of a distance sum

I have a list $L$ of $N$ numbers, and I want to choose $k$ numbers $\{x_1,x_2, \ldots,x_k\} \subseteq L$ in such a way value $S$ of the those K numbers is minimum. $$ S = \sum_{0< i < j <= k} ...
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1answer
177 views

Minimize $\|Ax-b\|$ where $x$ is a binary vector

For a software project I'm involved on, I have a situation where I have a large vector that is the sum of some smaller vectors. I know all the possible small vectors, and I know that no two of them ...
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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 ?
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1answer
111 views

What is the minimal isoperimetric ratio of a polyhedron with $5$ vertices?

I'm asking and answering this question to provide a partial answer to this question and a comment on this answer at MO. The isoperimetric ratio $\mu$ of a solid is the ratio $A^3/V^2$, where $A$ is ...
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1answer
74 views

Use of low rank approximation of a matrix

I am trying to figure out why do we need a low rank approximation of a matrix. Why is it used and where? Any insights?
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Show that Minimum Spanning Tree is unique

Show that MST is unique in case the edge weights are pairwise different $(w(e)\neq w(f) \text{ for } e\neq f)$. I thought that the proof can be done for example by contradiction, saying that we ...
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1answer
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Algorithm to find optimal cuts of pipe

I have varying lengths of pipe in inventory. When a customer requests various lengths I want to find the optimal way of cutting what I have in inventory. I need to make a program that does this. This ...
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How to minimize $\| y- Ax\|$ subject to $\|x\|=1$ and $x \geq 0$?

Given $y \in \mathbb R^n$ and $A \in \mathbb R^{n \times n}$, whis is some way for $$\min_x \| y- Ax\|$$ subject to $\|x\|=1$, and $x \geq 0$ (which means every components of $x$ is nonnegative)? ...
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1answer
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Prove the supremum of the set of affine functions is convex

Let $\langle f_i \rangle _{i \in I}$ be a family of affine functions on a convex and compact set $\Omega \subset \mathbb{R^d}$ such that $f_i = a_i.x +b_i$ for $x \in \Omega$. Prove that f, defined by ...
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3answers
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Optimization of the Area of a rectangle with regards to an Ellipse

The actual problem reads: Find the area of the largest rectangle that can be inscribed in the ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ I got as far as coming up with the equation for the ...
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2answers
105 views

Discrete optimization

I'm having troubles with searching for analytical solution of following problem. Let we work in 3-D space and have the set of points (uniform net at cube's facets): ($-1,\hspace{2mm} ...
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1answer
212 views

Proof of Non-Convexity

Am looking for a proof of non-convexity of the quotient of two matrix trace functions as given by $\frac{\operatorname{Tr}X^TAX}{\operatorname{Tr}X^TBX}$, when $TrX^TBX>0$ for two different ...
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2answers
196 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 ...
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1answer
359 views

Is this functional weakly lower semicontinuous?

Define $F\colon L^2([0,1]) \to {\mathbb R}$ by $$ F(R) = \int_0^1 \int_0^1 R(t) R(t') \exp\left(-|t-t'| - \left|\int_t^{t'} R(s)\,ds\right|\right) \,dt\,dt'.$$ Is $F$ weakly lower semicontinous, that ...
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1answer
435 views

Derivative of function including matrix logarithm

Is the following equation a first order approximation or incorrect for general matrix Lie groups? And what are the higher order terms? $$\frac{\partial}{\partial\mathbf x} (\log(\mathtt ...