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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Why Am i standing in a global minimum?

I`been asked the following in optimization If I am located in a point where all the possible factible directions turn out to be worse for the function, Am I located in a global minimum? The answer is ...
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How to orthogonalize a set of 2x2 matrices?

I have set of 2D affine transformations of images and I need to modify the transformations such way that they become as close to rotations as possible to minimize distortions of images. Let the ...
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14 views

Optimize the distribution if it is left unsmoothed

I have a question about distribution. Let see my problem The paper said that the distributions p and q are left unsmoothed, so we can ignore Kernel density. But I don't understand what is left ...
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4answers
98 views

Minimizing $\tan^2 x+\frac{\tan^2 y}{4}+\frac{\tan^2 z}{9}$

Given that $\tan x+2\tan y+3\tan z=40 , \ \ \ x,y,z \in \left(\dfrac{\pi}{2},\dfrac{3\pi}{2}\right),$ We need to find the minimum value of $ \tan^2 x+\dfrac{\tan^2 y}{4}+\dfrac{\tan^2 z}{9}$ ...
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The projected global optimum of a funtion onto the simplex to obtain the optimum in the simplex? [on hold]

One determines the global optimum of a funtion in the space by skipping the constraints. Now by projection the founded optimum into the simplex, which is define in such a way that the initial ...
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minimising total cost

A publishing company sells 75000 books during a year It costs a publishing company 0.6 dollars to store a book for a year. Each time they print additional copies, setting up the printers cost $2500. ...
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2answers
50 views

minimal value of $x^2+2y^2+5z^2$ with constraint.

$x,y,z>0$, and $xy+yz+zx=1$. I need to find the minimum value of $x^2+2y^2+5z^2$ In general what can we say about the minimal value of $\frac{ax^2+by^2+cz^2}{xy+xz+yz}$, over all positive numbers ...
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46 views

Solving LP with two $L_1$ inequality constraints

Is there a "fast" way to solve the following LP formulation with the following constraints: $$ \max_{\mathbf{f}} \mathbf{f}'.\mathbf{g} \\ \mathbf{1}'\mathbf{f}=1\\ \|\mathbf{f}-\mathbf{h}\|_1\le ...
2
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1answer
41 views

min : sum of L2 norm and squared-L2 norm.

Is there a closed-form solution of the following convex problem: $$\min_x \| x - u \| + C \| x - v \|^2$$ where $\| \cdot \|$ is the L2 norm.
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2answers
78 views

How do you maximize a polynomial over an integer domain?

I am working on maximizing the polynomial $f:\mathbb{R}^N \rightarrow \mathbb{R}$ $$f(v):=\prod_{i=1}^{N}( v_i+\alpha_i^2)$$ over integer $n$-partitions of $P$, $n\leq N$: $$\left\{v \left|\, v_i \in ...
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1answer
19 views

Solving first order constraints; lagrangian function and utility maximisation

I am supposed to find the demand curve if the following is given; $U(x,y) = xy$ price of $x * x$ + price of $y * y = m$ (so a general case, and I will be adding certain prices and income levels ...
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19 views

Lagrange multipliers for minimax optimization

Does the equally constrained minimax optimization can be solved by using Lagrange multiplier method? Thanks in advance.
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34 views

Vertices of Polyhedral

Suppose there are matrix $A\in\mathbb{R}^{n \times m}$ and vector $b\in\mathbb{R}^n$. Consider a non-empty polyhedron $P = \{Ax \leq b\} $. Then, there exists a vector $\bar{x}\in P $ such that ...
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3answers
38 views

linear solution of curve fitting on multiple linear functions differing by a multiplier

I recently posted this question here but I thought this could be of interest also in mathematics, given I found a partially related question here I am facing the following problem. I know nonlinear ...
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2answers
21 views

How to mathematically prove the optimality conditions for a univariate function?

Consider a univariate function $f(x)$. I know the graphical intuition behind why $f'(x)=0$ at the extrema of $f$. But how do you prove it mathematically? I start with the assumption of $x^*$ being a ...
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1answer
33 views

Why don't we go beyond the Hessian in multivariate optimization?

In univariate optimization, we perform the first derivative test to identify stationary points and the second derivative test to classify the stationary points as minima, maxima and inconclusive. When ...
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How does the Lagrangian multipliers equation for multiple equality conditions follow?

I understand the intuitive narrative that wikipedia gives. I understand until the part that says: $\triangledown f \in S$, which means $\triangledown f$ is also an "illegal" direction, along with the ...
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3answers
72 views

Maximum value of $a+b$ given that $\frac{1}{a} + \frac{1}{b} = \frac{1}{20}$

What is the maximum value of $a+b$ given that $\frac{1}{a} + \frac{1}{b} = \frac{1}{20}$ here $a,b \in \mathbb{Z^+}$? What I have gotten so far: From the above, $\frac{a+b}{ab} = ...
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1answer
39 views
+100

Framing a travelling salesman problem

I have an optimization(optimisation) problem, I think it is travelling salesman, where I want to find an answer to the question: "What is the best coffee shop for person x within a 50km radius?" The ...
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24 views

Cost optimization subject to constraints

I am having trouble coming up with a solution (I don't even know where to start) for a cost optimization scenario. I need to minimize the cost of purchasing x number of different products (prices of ...
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35 views

Constrained Optimization : Minimize sum of dot products

I am working on a problem to minimize sum of dot product. The problem can be stated as following. Given a matrix where each element is either 0 or 1. $$ \ A_{ij} = \{0,1\}; $$ with the constraint ...
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21 views

Reconstruct a vector with a known vector and residual

I observe $\vec y \in \mathcal R^n$ and know $\vec x$. I assume that $\vec y$ mostly consists of $\vec x$, with some added residual $\vec r$. This gives me the problem $\vec y = a\vec x + \vec r$, ...
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Traveling Salesman Problem with soft Time Windows and IBM ILOG CPLEX [closed]

I'm trying to set up a traveling salesman problem with soft time windows in OPL and IBM ILOG CPLEX to solve. I have the Probelm after all, as a smaller version installed, but can not find my mistake, ...
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Cost per item. Diminishing marginal discount, if you will. (Bigger discount for first few items) Optimal number of units to buy?

The graph above shows price per unit. Say they are cupcakes. When you buy a higher quantity, you get a lower price per unit. Say it levels off like this graph. Obviously, buying 2 nets a nice ...
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Nonlinear programming-separable programming

i have this function: $$5x_1x_2+4x_2x_3$$ and i need yo know if is separable or not I guees is not separable, because i can´t write the function in form: $$f_1(x_1)=x_1$$ and $$f_2(x_2)=x_2$$ ...
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How can I find the unit vector that minimizes the number of nonzero projections that a set of points has on it?

$\underset{\mathbf{w}}{\min} ~ \|\mathbf{X}^T\mathbf{w}\|_1~~~\text{subject to:}~ \|\mathbf{w}\|_2^2=1$ where $\mathbf{X}\in\mathbb{R}^{d\times m}$ is a set of $d$-dimensional points and $m>d$. ...
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Can a polygon with minimal perimeter self-intersect?

Recipe. Do the following. Throw $N$ random points $(x_0,y_0),(x_1,y_1),x_2,y_2),\cdots,(x_{N-1},y_{N-1})$ in the plane.Define $(x_N,y_N)=(x_0,y_0)$ : enumeration is $\mod N$ . These points are joined ...
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60 views

The minimum of $x^2+y^2$ under the constraints $x+y=a$ and $xy=a+3$

I solved the following problem: If $x,y,a \in \mathbb{R}$ such that $x+y=a$ and $xy=a+3$, find the minimum of $x^2+y^2$ Here is my solution. $x^2+y^2=(x+y)^2 -2xy= a^2-2a-6$. The minimum value is ...
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Compactness in minimax theorem

According to Von Neuman's minimax theorem we have $$\max_{x\in X} \min_{y\in Y}f(x,y)=\min_{y\in Y} \max_{x\in X} f(x,y)$$ for some compact sets $X$ and $Y$ and a convex (in $y$), concave (in $x$) ...
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Can you provide an alternative formula for the minimum of two numbers?

Today I've done a wonderful discovery. I've found out that the following operation between two real numbers actually is the maximum of those two numbers: $$ \max(a,b) = \log(\log(\exp(\exp(a))) + ...
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How can we find $\frac{2^m}{e^n}$ with an accuracy of $10$ decimal digits?

If $n$ and $m$ extremely large (1000 digits) and $1 <\frac{2^m}{e^n} < e$, how can we create an effective algorithm to find $\frac{2^m}{e^n}$ with an accuracy of $10$ decimal digits (10 digits ...
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Efficient calculation of minimal expected number of inversions

Problem: I have an array of size n with Z inversions initially and I am allowed to perform K operations where each operation can be decrease the number of inversions by 1. make a random shuffle of ...
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1answer
29 views

Exchanging max and expectation

If $X$ is a random variable and $\rho$ is a parameter, and $L$ is a concave function of $(\rho,X)$, under what conditions is the following statement true? $$\mathbb{E}\max_{\rho} L(\rho,X) ...
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Adding a constant to a list of numbers so that the sum of distances to integer values is minimal

I have a list of numbers {$x_i$} and I want to shift them (add a constant $\delta$) so that they are as close as possible to integer numbers in the sense that the summed distance to integer numbers ...
2
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1answer
23 views

Shared groceries expenses between roommates to be divided as per specific consumption ratio and attendance

My apologies if this question is in the wrong section. Couple of my roommates & I (total 5 people) share the groceries expenses. We record the purchases in an Excel sheet, and also have the ratio ...
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1answer
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Optimization with probability densities - Lagrange multipliers

This question is concerned with the paper "A Lower Bound for a Probability Moment of any Absolutely Continuous Distribution with Finite Variance" by Sigeiti Moriguti appeared in Ann. Math. Statist. ...
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2answers
164 views

Prove that if all edge-costs are different, then there is only one cheapest tree.

Prove that if all edge-costs are different, then there is only one cheapest tree (minimum spanning tree). (Use contradiction and make sure to keep track of the costs of the different trees involved.) ...
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Question on simple quadratic word problem regarding weekly revenue when price of merchandise is lowered

A store owner sells headphones at 24 dollars a piece with roughly 1000 sold per week. The store owner finds that for every 1 dollar decrease on the price per headphone he sells 100 more headphones per ...
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2answers
66 views

How to write such a constraint?

I have the following constraint that I need to write in an optimization problem but I failed to do it. Let $x_{ij}$ be a binary variable. So that: $$ x_{ij} = \begin{cases} 1, & \text{if $i$ ...
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27 views

closed form vs gradient descent baseed methods

I am a beginner to optimization. Could anybody give me a simple example to illustrate when I should use closed form and when I should use iterative methods like gradient descent? Thanks in advance.
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Simple Optimization Problem with linear Algebra

I'm asked to find that the solution of $\displaystyle S(\mathbf{c})=\max_{\mathbf{c}}\frac{\mathbf{X' Z c}}{||\mathbf{X}||\cdot||\mathbf{Z c}||}$, where $\mathbf{X}$ is a $n\times1$ vector, and ...
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Minimizing a non-convex rational function of two variables

I need to minimize the following function $$f(x,y)= \frac{a}{x}+\frac{bx}{y}+\frac{cy}{x}+dy+\frac{e}{y}$$ where $a,b,c,d$, and $e$ are positive constants, and $x$ and $y$ are both strictly positive. ...
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How to find a separating hyperplane?

I know about support vector machine, and it's quadratic programming approach which delivers the best separating hyperplane. My question is: is there a relatively simple algorithm to find a ...
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1answer
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On the optima of probabilistic bounding functions

I have a function $f(x)$ for which finding the optimum (maximum) appears to be analytically intractable and numerically difficult. I have simple expressions for upper and lower bounds on this ...
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20 views

Are iterations involving quantization going to converge?

For $i = 1,2,3$, let $~f_i(y_i)~$ be a convex and differentiable function and $y_i$ a scalar variable. Consider the following iteration $$\left[ \begin{array}{c} \nabla f_1(y_1^{k+1}) \\ \nabla ...
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1answer
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Application of a derivative [closed]

A spherical projectile 40 cm in diameter and weighing 32kg is shot directly upward from ground level at 196m/sec. Ignoring air resistance during its flight, what is the max height the ball will ...
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57 views

What is the 'optimal' equal-area partition of a circle?

What is the (an?) n-partition of a circle that meets the following criteria: The boundaries of each partition can be represented as a union of finitely many finite-piecewise-smooth simple closed ...
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1answer
37 views

Minimum of $f(x,y)=|ax-by|$.

What is the minimum of function \begin{align} f(x,y)=|ax-by| \end{align} subject to constrains \begin{align} 0 \le x \le C_1\\ 0 \le y \le C_2 \end{align} I found a similar question here Minimum of ...
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Finding extremal of a fixed end point problem. Optimisation

I want to find the extremal of the fix-end point problem $\int_1^2 \frac{\dot{x}^2}{t^3}$ with $x(1)=2,x(2)=17$ First I check the euler-lagrange equation is equal to $0$. We have: ...
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Minimizing matrix norm with quotient constraint.

Is there a systematic method of solving the following? $$ \begin{equation*} \begin{aligned} & {\text{minimize}} & & \left\|A \right\|_2 \\ & \text{subject to} & ...