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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Finding equation of a bent sufficiently flexible cardboard of length $l$ fitting into a gap of width $m<l$

I was thinking about how the walls of a barrel is made then I realized it is someone like fitting a piece of wood of length $l$ in between some "gap" of length $m<l$. This would cause the piece of ...
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Question on optimization algorithm to train peculiar regression

I've been in my operations research course, and we have been working on optimization in particular problems within regression. We hypothesize that for variables $h,s,d,t,$ there is this set ...
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439 views

Minimize Frobenius norm with constraints

As a follow-up on my previous question, I would like to solve the following optimization problem: $\min \Vert MA-B \Vert_F^2-x^HMy\;\;s.t.\;\;M^HM=I$ where $A$ and $B$ are $N\times L$ complex ...
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Finding the minimum of $x_1 + \cdots + x_n$ on ellipsoid

Let $A$ be a positive definite matrix $n \times n$ and $u^T = [1 \cdots 1]$. Use Lagrange multipliers to find the minimum of $f(x) = u^Tx$ on $h(x) = \frac{x^TAx}{2} = 2$ This is what I did. $$L(x,...
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Shortest distance between parallel line and plane

I've been doing questions regarding the shortest distance between lines/planes and points , and I've come across a question asking to find the shortest distance between a line and a plane which are ...
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Can $n$ variables ever have more than $n$ unique satisfiable constraints?

Assuming you have $n$ variables, how many maximum independent satisfiable constraints can you have? What I mean by independent is that the equations all express unique constraints, s.t for example $x +...
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Sort a set of points to minimize the sum of the square distances between two consecutive points

Let $P$ be a finite set of points in $\mathbb{R}^3$. Let the number of points in $P$ be $n\in\mathbb{N}$. I want to sort the points in $P$ to minimize the sum of the distances between two consecutive ...
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702 views

Matlab: need help with optimization

I am trying to minimize the objective function over [x(1),x(2)]: exp(x(1))*(4*x(1)^2+2*x(2)^2+4*x(1)*x(2)+2*x(2)+1)+b subject to constraint ...
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35 views

Minimum of function of $3$ variables

If $xyz = a^3$ then show that the minimum value of $x^2+y^2+z^2$ is $3a^2$. I have tried this problem using the identity $(x + y + z)^2$ but I am not satisfied with my approach. Any other method ...
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Minimum of $|ax-by+c|$

Find the minimum of the function $$ f(x,y)=|ax-by+c|$$ where $a,b,c \in \mathbb N$ and $x,y \in \mathbb Z$. The questions here and here are similar but they are in cases where $x, y$ are ...
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How to find $\sum_{r=0}^k {n\choose 2r}$? [duplicate]

I know that $$\sum_{r=0}^n {n\choose r}=2^n$$ But how do I find the value when r takes only even values till an even number 2k instead of n itself. $$\sum_{r=0}^k {n\choose 2r}$$ An algorithm that ...
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Does convergence of iterates imply convergence of function values?

The question came to my find when I was reading convergence of gradient descent. However, my question is general and does not necessarily stick to GD. Concretely,my question is: \begin{equation} \|x^k-...
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31 views

Estimate original matrix from matrix multiplication result

Let $A$ be $m \times n$ real matrix. Let $B$ be $n \times k$ real matrix. Let $C= A \times B$ ($m \times k$ matrix). Now, the question is: given $C$ and $A$, give an estimate for $B$. Possible ...
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Minimum Value of a linear function [duplicate]

Find the minimum of the function f(x,y)=|ax−by+c| I know that minimum value of ax-bx is gcd(a,b).X and Y are Whole Numbers ...
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Generating equations for this Optimisation problem

Minimize : $ |(Ax + B) - (Cy + D)| $ Such that: $ x \geqslant 0 $ $ y \geqslant 0 $ $a,b,c, d $ are fixed natural numbers and $ x,y $ have integral solutions. I just can't figure out if this can ...
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Finding minimum difference between two linear functions

Given two functions of the form $y = m_1x + c_1$ and $y = m_2x + c_2$ where $m_1,m_2,c_1,c_2$ are positive integers. How to find the absolute minimum difference between the two functions for positive ...
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Petri net analysis.

I have problems with this exercise. First: can the token in place $p_1$ to enable the transitions $t_2$ and $t_3$? The place $p_1$ has a single token, I think it fails to enable $t_2$ and $t_3$. Any ...
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439 views

Gradient-descent and Hidden Markov Models

I would like to use gradient-descent to fit the parameters of a simple 2-state HMM. This paper Levinson, S. E., Rabiner, L. R. and Sondhi, M. M. (1983), An Introduction to the Application of the ...
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1answer
101 views

How to minimize $|Ax+By + C|$ given that $x \geq 0$ and $y\geq 0$ [duplicate]

I am trying to solve problem related to absolute value function, i.e given $Z(x,y) = |Ax + By + C|$ , what is the minimum value of $Z$, if $x \geq 0$ and $y\geq 0$ and x,y belongs to integers
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Minimization of a piecewise affine function of $2$ variables [closed]

How does one minimize the following function? $$f(x,y) = |kx + ly + c|$$ where $x,y \in \mathbb N$.
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I'm walking towards my car - when should I try the remote, in an optimal sense?

I'm interested to learn about how discrete/'event' based elements are incorporated into optimisation problems. Hopefully this is an interesting problem in its own regard, it's inspired by a daily ...
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Lagrangian Multipliers exercise

Let $M = \{(x, y, z) \in {\rm I\!R}^3 : F(x,y,z) = 0\}$ and let $F(x,y,z) = (3x^2z + y^2 + z^3-1, \, x + z-1)$ . Does the function $f(x, y, z) = x$ have any extrema in $M$? We are asked in advance ...
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Maximum distance from closest vertex of rhombus

Consider the unit rhombus formed by joining following coordinates $A(0,0), B(1,0), C(\frac{3}{2}, \frac{\sqrt{3}}{2}), D(\frac{1}{2}, \frac{\sqrt{3}}{2})$ What is the largest possible distance from ...
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Find the area of largest rectangle that can be inscribed in 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 ...
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Optimization inside integral

I want maximize the integral $$\int_a^b \left( 2 cx y(x) - e y(x)^2 \right) \, \mathrm{d}x$$ with respect to to $y(x)$. If I discretize the problem, I get $$ \frac{b-a}{n}\sum_{i=1}^n 2c(i/n(b-a)+...
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'Finding' a normally distributed random variable

Let a random variable $Z$ have a standard normal distribution. Suppose that we start at $0$. We 'walk' right, along the number line, till we reach $a$. We then turn around, walk back, past $0$, till ...
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275 views

When does a variable leave a basis (in linear programming)?

In the simplex algorithm in linear programming, what are conditions for a variable to leave a basis (not necessarily basis for the/an optimal solution)? I'm supposed to list as many sufficient and ...
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Create a strictly increasing sequence following criterias

Problem Let y be a sequence of real numbers (of length $n$) bounded in the range [0,1]. I am trying to calculate the sequence x ...
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Smallest vector that solves a specific linear system

I am looking for the smallest vector $z$ (w.r.t. the Euclidean norm) that solves the linear system, \begin{equation}\begin{pmatrix} 1&1&1&1\\2&3&5&7\\-2&-1&1&3 \end{...
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648 views

Dual cone of a L1 norm cone?

I am listening to convex optimization lectures and I hear that dual cone of a $L1$ norm cone is a $L-\infty$ norm cone. Can anybody please explain how? I understand that every point in the dual cone ...
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How to describe minimization of L1 norm error using linear programming?

Given a set of $n$ pair points $(x_1, y_1), ..., (x_n, y_n)$ in the plane, I need to find a line $ax + by = c$ that fits the points of the L1 norm error points as closely as possible. I need a linear ...
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Given a number N, how to construct a set of different numbers that has a maximal product, and the sum of these numbers equal N?

Note that: N is positive integer. The set also consists of positive integers. The set consists of different integers. (The thread suggested by @hardmath doesn't have this constraint.) For example: ...
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How to optimize $\min_w\frac{1}{2}\|w-w_t\|^2 : w^\top(y-\hat y)\ge 1$?

I have some trouble minimizing the following problem: $$ \min_w\frac{1}{2}\|w-w_t\|^2 : w^\top(y-\hat y)\ge 1 $$ Considering $w$, $w_t$ are vectors, so are $y$ and $\hat y$. The idea is to find a $w$ ...
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What operation is this in maths

I need to develop an algorithm for a problem then translate it into code, but I am sure someone would had done it before, my question is that what is the name of operation in below sudo code ? ...
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How to solve this problem through bisection search or any other method?

I have an optimization problem in the form $$\text{Minimize}\hspace{1mm}D$$ $$\text{subject to}$$ $$\sigma_1+\sigma_2=\sigma$$ $$\rho_1+\rho_2=\rho$$ $$\epsilon\le\rho_i\le c_i\hspace{1mm},i=1,2$$...
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optimize pasting text

Someone asked me how can he paste a string 1000 times in Windows notepad. While this can be done easily using editors like Vi, I'm trying to answer his question using notepad only. So the problem goes ...
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1answer
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Maximize sum of logs subject to constraints

I have the optimization problem $$\begin{array}{ll} \text{maximize} & \displaystyle\sum_{i=1}^n \log(c_i + x_i)\\ \text{subject to} & \displaystyle\sum_{i=1}^n x_i = 1\\ & x_i\ge0\end{...
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Absolute value of eigenvalues

It is well known that if $A$ is a real symmetric square matrix of size $n$ with eigenvalues $\lambda_1 \leq \cdots \leq \lambda_n$, then $$ \lambda_k = \min \{ \max \{ \langle Ax, x \rangle : \|x\| = ...
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normalization of constraints $ 0 \leq x \leq 1 $ in Lagrangian KKT

With Lagrangian we have an objective function and a set of equality constraints of form $ g_{i}(x_{j}) = 0 $ . With KKT we can have another set of inequality constraints of the form $ h_{i}(x_{j}) \...
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Directional derivative and lagrange multipliers

Find the points $(x,y)\in \mathbb R^2$ and unit vectors $\vec u$ such that the directional derivative of $f(x,y)=3x^2+y$ has the maximum value if $(x,y)$ is in the circle $x^2+y^2=1$ My attempt: ...
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Problem with finding Karush-Kuhn-Tucker points and checking for global or local minima.

I need to solve the following optimization problem $$\begin{align*} & \mathrm{Min}:\quad f(x_1,x_2)=x_1-10x_2\\ & \mathrm{subject \ to}: \quad x_1^2 -x_2 \geq 0\\ & \qquad \qquad \...
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discrete convexity arising in a simple discrete optimization problem

Let $S$ be a fixed integer satisfying $S \ge 1$, let $a$ range over the integers between $1$ and $S$ inclusive, and for $i = 1, \dotsc, a$, let each $x_i$ range over the nonnegative integers, such ...
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Index of a stationary point of constrained optimization

For an unconstrained optimization problem with objective function $F(x,y,z)$ the index of a stationary point is well-defined: If $(x^*, y^*, z^*)$ is a point where the gradient of $F(x,y,z)$ vanishes, ...
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How to solve this equation (may be with least squares)?

I have the optimization problem $$\arg\min_{a,b} \sum_{i,j} \left( \left| X(i,j)-aY(i,j)\right|-b \right)^2$$ Where $X$ and $Y$ are known. But there is a modulus inside. I need to estimate $a$ and $...
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Maximizing this parametric expression with a certain range of integer inputs

Let $a,b$ be integers with $1 \le b < a \le n$ and $s,t$ be integers with $0 \le s < t \le m$ I would like to maximize the expression: $b^s (a^{t-s} - b^{t-s})$ My intuition says this should ...
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Can Extragradient method be expressed with proximal steps?

As we know, for solving saddle point problems, forward-backward algorithm is generally not guaranteed to converge. But extragradient method converge Structured Prediction via the Extragradient Method ...
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finding curve along which a function extremizes via theory of calculus of variations [closed]

Consider $$ I(y)= \int \limits _0 ^1 [y'(x)]^2dx \ +y(1)^2$$ with $y$ subsjected to the initial condition $y(0)=1$. Find the equation of curve along which $y$ extremizes.
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68 views

Weight Modification for Computationally-Efficient Nonlinear Least Squares Optimization

There was a time where I could figure this out for myself, but my math skills are rustier than I thought, so I have to humbly beg for help. Thank you in advance. I am solving a weighted nonlinear ...
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Using the Lagrange method to find max/min of $f(x,y) = \frac{x^3}3 + y$

Problem Use the Lagrange method to find max/min of $f(x,y) = \frac{x^3}3 + y$ Subject to the constraint $x^2 + y^2 = 1$ My attempt The constraint gives us $g(x,y) = x^2 + y^2 - 1$ $\displaystyle\...
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1answer
76 views

Find the maximum and minimum values of $\sin^2\theta+\sin^2\phi$ when $\theta+\phi=\alpha$

Find the maximum and minimum values of $\sin^2\theta+\sin^2\phi$ when $\theta+\phi=\alpha$(a constant). $\theta+\phi=\alpha\implies\phi=\alpha-\theta$ $\sin^2\theta+\sin^2\phi=\sin^2\theta+\sin^2(\...