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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Cutting a pie into n equal area pieces with the minimum distance of cuts. [duplicate]

Suppose we are to cut a unit circle into n equal area pieces. We can cut curves if we wish. What is the minimum distance we must cut? What strategy minimises this distance? Note: The shape of the ...
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1answer
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converting $\max\{\ldots,\ldots\}$ function to a set of $\min\{\ldots,\ldots\}$ functions

Suppose $\max\{A,B\} = A$ if $A\geq B$ and $\max\{A,B\} = B$ if $A <B$. Similarly, $\min\{\}$ is defined. We know that $\max\{A,B\} - A - B= - \min\{A,B\}$. Is it possible to write $\max\{A,B,C\} ...
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Non linear functional optimization under constraints

For some given positive functions $l(t)>0$ and $h(t)>0$, such that $h(t)>l(t)$, I want to solve this functional optimization problem on $a(t)$: $\min_a\int_0^T[l(t)\cdot\min(a(t),0) + h(t)\...
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9 views

Optimize linear function with integral as a constrain

How can I solve a linear optimization problem with an integral as a constrain? I am looking for a Matlab toolbox. My optimization problem is: $\max_{\alpha} \sum_{k=1}^K \alpha_k \cdot x $ such that ...
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Optimization problem with differential equations as constraints

I have formulated an optimization problem which I have to solve for a project but i do not have enough math skills to solve it. The problem is an optimization problem, whose constraints include both ...
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33 views

Does $\min\{x_1x_3+x_2^2 | x_1^2+x_2^2+x_3^4 = 4\}$ has an optimal solution?

Does $\min\{x_1x_3+x_2^2 | x_1^2+x_2^2+x_3^4 = 4\}$ has an optimal solution? I think continuous function over closed and bounded domain has an optimal solution but I am not sure. Can anyone give me ...
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1answer
24 views

Solving an optimization problem with a linear objective and quadratic constraint

The title is general, but what I am specifically interested in, is how to solve the following problem: $$\text{Maximize } c $$ $$\text{Subject to:}$$ $$a+b+c<0$$ $$b^2-4ac<0$$ $$a,b \in \...
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1answer
37 views

Can a quasiconvex function be made convex by composition with a diffeomorphism?

Assume we are given a continuous quasiconvex function $f: \mathbb{R}^n \to \mathbb{R}$. Intuitively I feel that quasiconvexity means that there should exist a diffeomorphism $h: \mathbb{R}^n \to \...
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How to solve a binary generalized assignment problem

I have the following generalized assignment problem: Z=max $\sum_{i=1}^{N}\sum_{j=1}^{M} x_{ij}R_{ij}$ such that $\quad 1)\quad \sum_{j=1}^{M} x_{ij}=1 \quad \forall i$ $\quad\quad\...
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50 views

What are some applications of vertex separators?

What are applications of finding a vertex separator that minimizes a cut in a graph. To clarify the problem I am talking about is is given a graph of n vertices and a partition $m_1,m_2,..,m_k $of ...
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2answers
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Let $Ax$ and $Ay$ minimize distance to $b \in \mathbb{R^m}.$ Show $x-y \in \ker(A).$

Let $A$ be an $m \times n$ real matrix and $b \in \mathbb{R^m}.$ Suppose $Ax$ and $Ay$ both minimize distance to $b,$ i.e. $||Ax-b|| = ||Ay-b||.$ Prove that $x-y \in \ker(A).$ Seems like there ...
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1answer
15 views

Maximum (edge)weight connected subgraph of an undirected graph.

Let G be a undirected graph with weighted edges. I want to find a connected subgraph which has at most L nodes(vertices) whose sum of edges is maximum. It sounds similar to MWCS or PCST but here only ...
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21 views

Formulating a linear transportation problem as a stochastic linear program

[Question provided in picture]http://i.imgur.com/avoARFG.jpg[/img] I am having trouble with part b of this question. For part a, I have the following: let xij = number of units produced by plant i ...
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52 views

Maximising $\int_{0}^{T} v(t)\, dt$, subject to constraints $|v(t)| \leq a; v(0)=0; v(T)=0$

Besides those constraints, we know nothing else about $v(t)$. Interpreting the integral as the distance travelled by a particle, a little geometry tells us that the answer should be $aT^{2}/4$ ...
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2answers
46 views

Coefficients($\sum 1$) of equation to get maximum output

Lets say we have $4$ variables: $$ x_1, x_2, x_3, x_4 $$ with coefficients: $a,b,c,d$ respectively, and output $y$ With different combinations of $a,b,c,d$, we have a blackbox/unknown function, that ...
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28 views

AI Parameters for Tetris-like Game

I am building an AI to play a variation of Tetris. The rules are changed in that there are 19 different types of pieces, rotation is not allowed, and the pieces can be placed anywhere in a 10X10 grid. ...
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1answer
26 views

Formulating deterministic and stochastic production models (not solving them) [Beginner's Operations Class]

Question provided in picture This question has been troubling me as I am not used to questions without numbers as it is hard for me to visualise. I also find stochastic problems hard in general. &...
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22 views

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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25 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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27 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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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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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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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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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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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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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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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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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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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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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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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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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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1answer
70 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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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
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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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46 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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1answer
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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44 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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45 views

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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17 views

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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46 views

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 ...