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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Least wasteful use of stamps to achieve a given postage

You have sheets of $42$-cent stamps and $29$-cent stamps, but you need at least $\$3.20$ to mail a package. What is the least amount you can make with the $42$- and $29$-cent stamps that ...
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Optimizing a functional with a differential equation as a constraint

I am working on solving the following optimization problem. I think it is well-poised but, if not, please give me some pointers that could make the question make more sense. We have a parametric ...
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(easier version)What is a working example for finding the maximum of a algebraic function $f(a,b,c,d,e)$ with 4 equality …

edit 2: If you are founding an answer would be interesting to you, please upvote this question, because i may use those point in order to start a new bounty. Edit 3: Can somebody answer? Edit 4:I ...
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Find extremes of function $f(x,y,z) = x^2y + y^2z + x - z$

I am preparing for an exam tuesday morning and I would like to ask you, if someone could please review my solution for the following excercise. I don't have the correct answer so I am unable to check ...
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2answers
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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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1answer
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Find minimum in a constrained three-variable equation

After my last question I have worked through the math quite a bit and now I'm stuck again. This time my question is less wordy. I have two equations for $t$, one with respect to each $a_{x}$ and ...
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1answer
77 views

Spacing of fence posts with minimal distance to other fence posts

Definition 1: A "fence" is a set of "fence post positions", where each pair of adjacent positions has the same difference (the spacing), e.g. $\{1,2, 3, 4\}$. A fence is described by three values ...
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What exactly are the curves that are a best fit to the Harmonic Cantilever?

Let's start with a few references to get an idea: Daniel Goldwater: Harmonic Cantilever Book Stacking Problem Block-stacking problem Harmonic Series and Bricks Interesting related issues: Maximum ...
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287 views

Minimize Frobenius norm with unitary constraint

I am trying to find a unitary tramsformation, $M$, that minimizes $\Vert MA-B \Vert_F^2$ where $A$ and $B$ are $N\times L,\;L\ge N$. I know how to solve it without the unitary constraint. I thought ...
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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
114 views

How find this minimum $\sum_{i=1}^{n}a^2_{i}-2\sum_{i=1}^{n}a_{i}a_{i+1},a_{n+1}=a_{1}$

let $a_{1},a_{2},\cdots,a_{n}\ge 0$,and such $a_{1}+a_{2}+\cdots+a_{n}=1$. Find this follow minimum $$I=a^2_{1}+a^2_{2}+\cdots+a^2_{n}-2a_{1}a_{2}-2a_{2}a_{3}-\cdots-2a_{n-1}a_{n}-2a_{n}a_{1}$$ My ...
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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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4answers
642 views

Approximate a function over the interval $[0, 1]$ by a polynomial of degree $n$ (or less).

To approximate a function $G$ over the interval $[0,1]$ by a polynomial $P$ of degree $n$ (or less), we minimize the function $f:R^{n+1} \to R$ given by $F(a) = \int_0^1 (G(x) - P_a(x))^2\,dx$, where ...
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3answers
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Prove $ax - x\log(x)$ is convex?

How do you prove a function like $ax - x\log(x)$ is convex? The definition doesn't seem to work easily due to the non-linearity of the log function. Any ideas?
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Computing the volume of a set efficiently

Given a set of vectors $\mathbf v_i$ for $i=1,\dots,k$, $\mathbf v_i \in \{0,1\}^n$, is that possible to efficiently find the volume of the set, $$\left\{\mathbf x \in [0,1]^n:\mathbf x \le ...
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1answer
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Is $f(t)=t^\alpha$ for $\alpha\in(0,1)$ a sub-additive function? [duplicate]

Possible Duplicate: Does $|x|^p$ with $0<p<1$ satisfy the triangular inequality on $\mathbb{R}$? Is the function $$f(t)= t^{\alpha},\quad \alpha\in (0,1)$$ a subadditive ...
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Minimizing weighted sum of distances

given a set of coordinates and the following function: cost = $\sum \sqrt{(x_i−X)^2+(y_i−Y)^2}w_i$ I would like to find the point (X, Y) for which this function is minimal. A simple example shows ...
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1answer
673 views

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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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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1answer
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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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3answers
709 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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1answer
152 views

binary quadratic optimization problem

I am trying to solve the following binary quadratic program. $$ \min_{\Delta} \Delta^T H \Delta + c^T\Delta \\ \text{Such that:} ~~~\Delta\in \{0,1\}^n ~~\text{and}~~ \sum_{i=1}^n \Delta_i \leq \Gamma ...
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Online stochastic convex optimization.

I need to find/approximate the argument that minimizes a stochastic convex function $F(\theta, Z)$: $$ {\arg\min_{\theta}} E_{Z}[ F(\theta, Z) ]$$ Where $Z$ is some random variable (we could assume ...
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How to maximize area of two circles inside a rectangle without overlapping?

Two circles have to be drawn inside a rectangle of dimensions $W\times H$ such that the area of both circles is to be as large as possible without overlapping. Let the radii of the circles be $r_1$ ...
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385 views

A primitive function of $ e^{x^{2}} $

I made some efforts to set a closed form of primitive function of $ e^{x^{2}} $ i find this function : $ f(x)=\frac{x}{2x^{2}-1}e^{x^{2}} $ where : $f'(x)=(\frac{x}{2x^{2}-1}e^{x^{2}})'$= ...
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Minimizing Frobenius norm for two variables

I need to minimize squared Frobenius norm: $\|\mathbf{A} - \mathbf{x}\mathbf{y}^T\|_F^2$. Namely I need to prove that for this norm to reach minimum $\mathbf{x}$ should be eigenvector of ...
3
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4answers
160 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
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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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1answer
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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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How to maximize (baking) surface area?

I like eating crust, so I am trying different baking molds to try to get the most crust per dough. More generally, I'm interested in the reverse of this more specific question — how to maximize the ...
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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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1answer
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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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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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The Weierstrass Approximation Theorem Vs The Runge's Phenomenon

I am learning about different interpolation methods in my internship. Today as I was looking this article on Wikipedia to learn about the Runge's Phenomenon exhibited by Polynomial Interpolation. I ...
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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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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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1answer
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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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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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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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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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1answer
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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 ...
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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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2answers
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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 ...
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2answers
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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
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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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1answer
344 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 ...