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 minmax over set of constrained continuous functions

I have this problem that I've been struggling with for a bit, and I don't know how to proceed. I'm looking for the solution to: $$ \min_f \max_s \frac{\int_s^t f(x) \;dx}{\int_s^\infty f(x)\; dx} $$ ...
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86 views

calculate malus value of a urn

In an urn, I can put 20 balls having a value from 1 to 7. So I got Ni, with i $\in$ [1,2,3,4,5,6,7], where Ni is the number of balls with a value i in the urn. And $\sum_{i=1}^7 Ni = 20$ The game ...
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28 views

Formulating an optimisation problem into a mixed-integer problem

I'm not sure if I understand this question and was wondering if anyone could provide any insight to an answer. The only thing I can think of adding is a constraint: "x2 = integer", so I'm clearly ...
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28 views

Is generalized mean convex / concave?

The generalized mean can be given using the following equation: $ M_p(x_1, \dots, x_n) = (\frac{1}{n}\sum_i x_i^p)^{1/p} $ Is it convex /concave when $p<1$ ?
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21 views

Interplanetary Optimisation using a simulator with PyGMO or SciPy

I am currently trying to use a N-body gravity simulator to model a spacecraft trajectory and using the simulator as a BlackBox to optimise the trajectory. I am thinking of using basin hopping/ ...
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1answer
24 views

Conjugate of difference of convex functions

I am reading through this tutorial on DC programming and the author makes a startling claim without proof: If $g$ and $h$ are two lower semi-continuous convex function, then the conjugate function of ...
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14 views

Minimize two-variable function

I wish to minimize a function of two variables $m$ and $L$ (both strictly positive). I have calculated the first two partial derivatives: $$\left[ \frac{-n}{2L} + \sum_{i=1}^n \frac{ (x_i - m)^2}{2m^2 ...
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1answer
25 views

Connection between complementarity problem and optimization problem?

I do not understand the connection between complementarity problems and optimization problems. I have tried to look at other definitions for complementarity problem to see if that would help me with ...
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22 views

Variational Inequalities - What excatly does the definition say? Why are they useful?

I am having issues understanding the definition of variational inequalities. We have the following definition: Given a set $X \subset \mathcal{R}^n$ and a mapping $F: X \rightarrow \mathcal{R}^n$ a ...
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12 views

Calculate best interval between peaks

I have a vector of values with zeros and some rare positive value (corresponding to the peaks in the hist) ...
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28 views

Newton conjugate gradient algorithm

In this video, the professor describes an algorithm that can be used to find the minimum value of the cost function for linear regression. Here, the cost function is $f$, the gradient is $g_k$ where $...
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30 views

Deducing MaxFlowMinCut from Menger

So the MaxFlowMinCut theorem with rational network capacities and (the edge-version of) Menger's theorem for undirected graphs are equivalent, both directions being not too hard. I gather that since ...
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22 views

How to programatically solve the optimal control problem?

I have to programatically (write a program) find a control function $u(\cdot)$ to minimize the following functional: $$ J(u,x) = \int_0^T { f_0(x(t), u(t), t)}dt + \Phi(x(0)) \rightarrow \min$$ ...
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15 views

How to find the value of lambda in following optimization problem?

Consider following optimization problem: $P_T = $minimize$(p_s + p_r)$ Subject to $p_s \ge p_{s,min}$ and $p_r \ge 0$. After solving the optimization problem, we get following equations for $p_s$ ...
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163 views

Least-squares problem with quadratic equality constraint

I want to find the solution of a Lagrange equation whose inputs are matrices. First I have the equation Ax=0. By decomposing $A$ into $A_3$ (columns 9 to 11 of A), $A_9$ (the rest of the columns), ...
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1answer
45 views

optimization equivalence

Given the functions $f_1(r,x)$ and $f_2(r,y)$: $[0,1]\times \Bbb R \to \Bbb R ^+$, solve the following problem $$\underset{r,x,y}{\text{argmin}}\; f_1(r,x)+f_2(r,y) \\ \text{subject to}\; x^2+y^...
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Enumerating (some) combinations of elements subject to a constraint

Consider this variant of the knapsack problem: I own an outdoor goods store, and hikers come from miles around because of my amazing variety of products for sale. There are 4 popular hikes in the ...
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1answer
26 views

How can I express the minimization of the p90th percentile mathematically?

I would like to minimize the 90th percentile of a function with a normally distributed variable. If I wanted to minimize the expected value, I would do it something like this: $$ min_s \ z = E(f(X,s)...
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30 views

Is this the correct way of using Variational Principle (Minimization Principle)?

I am constructing a smooth function $f(x)\equiv f(u(x),v(x))$, such that $u(x)$ and $v(x)$ are some trial parameters. I have the following integral $$G=\int_{x_i}^{x_f} f(u(x),v(x)) \mathrm{d}x.$$ My ...
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2answers
37 views

Trigonometry minimum value

If $\alpha , \beta$ and $\gamma$ are angles of triangle How can we prove that $$ \cot^2(\alpha) + \cot^2(\beta) + \cot^2(\gamma) $$ has a minimum value of $1$. I actually used the AM-GM inequality ...
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Minimizing the Frobenius norm with linear inequality constraints

How to solve the following system for $\mathbf{C}$ and $\mathbf{a}$: $\min\|\mathbf{X-XC} \,\mbox{diag} (\mathbf{a})\|_F^2$ subject to $\mathbf{c}_{ik}\geq 0$, $1^T \mathbf{c}_k = 1$ and $1-\delta\...
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73 views

Equation of the form $\mathbf{\Phi}'(t)=\mathbf A(t)\mathbf{\Phi}(t)$.

Let $\mathbf{\Phi}(t)$ and $\mathbf A(t)$ be matrices satisfying the differential equation $$ \mathbf{\Phi}'(t)=\mathbf A(t)\mathbf{\Phi}(t)\ . $$ If I am not mistaken, if $\mathbf A$ and its integral ...
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Solving Optimization Problem (Orthogonal Projection) Using Projected Sub Gradient / Dual Projected Subgradient

Given the following optimization problem (Orthogonal Projection): $$ {\mathcal{P}}_{\mathcal{T}} \left( x \right) = \arg \min _{y \in \mathcal{T} } \left\{ \frac{1}{2} {\left\| x - y \right\|}^{2} \...
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44 views

Why do we minimize the squared norm instead of the norm in this optimization problem?

When reading about the optimization problem for Support Vector Machine in Bishop's book (Pattern Recognition and Machine Learning) he wrote that: The optimization problem then simply requires that ...
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1answer
16 views

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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47 views

Optimizing a problem using Lagrange multipliers

$\newcommand{\norm}[1]{\|#1\|}$ I have the following problem: $$ \min_{w,\theta}\frac{1}{2}\norm{w-w_t}^2+\frac{1}{2}(\theta-\theta_t)^2 \text{ s.t. } w^\top(z(n-\theta)-\hat z(\hat n - \theta)) \ge 1 ...
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42 views

Is Lagrangian Multiplier Equivalent to Brute Force for binary decision variables

I have a set of variables $x_{i} \in \{1,k\} $ in a non linear optimization problem. As this variable has only two possibilities I have encoded this into a constraint. I assumed having equality ...
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45 views

How can I solve this optimization problem?

How do I solve this optimisation problem? $$W = \left(\frac{n(X-Y-Z)p}{Zq}\right)^{1/a},\, a>0$$ $\operatorname{Max}\{ W\}$, subject to $0\leq n \leq 1$, $0\leq Y \leq X$ and $Z \leq Z_{max}$ ...
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Minimal lateral surface of a cylinder

Inscribe in a given sphere a cylinder such that its lateral surface (without the bases) shall be maximal. So lateral surface is = $2 \pi rh $ $ \Rightarrow 4 \pi x \sqrt{r^2-x^2}$ Now take ...
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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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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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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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3answers
36 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 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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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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61 views

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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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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1answer
104 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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195 views

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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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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45 views

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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1answer
26 views

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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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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27 views

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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How to minimize a linear function over a halfspace efficiently and intuitively

Consider the following fundamental problem: Two methods: By duality: ($\lambda, b \in R$) $L(x,\lambda)=c^Tx+\lambda(a^Tx-b)=x^T(c+\lambda a)-\lambda b \ \ $. Therefore, $g(\lambda)=-\...
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27 views

'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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1answer
37 views

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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185 views

Extending the ordered sequence of 'three-number means' beyond AM, GM and HM

I want to create an ordered sequence of various 'three-number means' with as many different elements in it as possible. So far I've got $12$ ($8$ unusual ones are highlighted): $$\sqrt{\frac{x^2+y^2+...
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23 views

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{...