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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Minimizing trace with equality constraints

I would like to solve the following trace-minimization under equality constraints optimization problem: $$W^* =\arg\min \operatorname{Tr}[WCW^T] \text{ s.t. } A=B^TW^TWB$$ where $W,C\in\mathbb{R}^{...
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3answers
653 views

How to prove Lagrange multiplier theorem in a rigorous but intuitive way?

Following some text books, the Lagrange multiplier theorem can be described as follows. Let $U \subset \mathbb{R}^n$ be an open set and let $f:U\rightarrow \mathbb{R}, g:U\rightarrow \mathbb{R}$ be $...
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1answer
77 views

Inverse Vectorization Vec^-1

Hope that you will find this post in good health. I am Mr. Adnan from Pakistan with research background in Control systems. I am working on one problem in which Hadamard weights are using. During ...
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5answers
204 views

Maximum value of $\sin A+\sin B+\sin C$?

What is the maximum value of $\sin A+\sin B+\sin C$ in a triangle $ABC$. My book says its $3\sqrt3/2$ but I have no idea how to prove it. I can see that if $A=B=C=\frac\pi3$ then I get $\sin A+\sin ...
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196 views

Prove that $\sin A+\sin B+\sin C\leq \frac{3}{2}\sqrt3$ [duplicate]

If $A,B,C$ are the angles of a triangle, prove that $\sin A+\sin B+\sin C\leq \frac{3}{2}\sqrt3$ I want to prove this inequality without Jensen's inequality, as Jensen is not in my syllabus. Let $\...
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1answer
93 views

Maximizing area of a pentagon

Suppose $a,b,c,d,e$ are pairwise distinct positive integers. Consider a pentagon with sides $a,b,c,d,e$ and with angles maximizing its area (we assume that a pentagon with such sides exists). It is ...
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Failure of second derivative test of two variable function where the all the partial derivatives are equal

Actually I am looking to find the local minimum of the following function : $$F(x,y)=\frac{\Gamma(x+y+1)\Gamma(n-x-y+1)}{\Gamma(n+1)}$$ The partial derivatives of this function are: $\begin{align} ...
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2answers
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Upper bound for $\gcd(a,b)$ if $\frac{a+1}{b}+\frac{b+1}{a}\in\Bbb{N}$

Suppose that $a,b$ are two positive integers so that $\frac{a+1}{b}+\frac{b+1}{a}$ is also a positive integer.Find the best upper bound for $\gcd(a,b)$. My work: $\frac{a+1}{b}+\frac{b+1}{a}=\frac{...
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1answer
18 views

Individually checking constraints for convexity in Optimisation problem valid?

I have a quadratic minimisation problem where both the objective fn and constraints have some quadratic terms. (Such as a throttle variable (continous) * On/Off (integer variable)). My question is: ...
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1answer
28 views

Minimization of vector sum through rotation

I'm curious if there's an algorithmic way to find the minimal vector sum of $N$ vector magnitudes by applying a rotation $\Phi$ to each individual vector. As an example in two dimensions, if I have ...
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234 views

Minimize $-\sum\limits_{i=1}^n \ln(\alpha_i +x_i)$

While solving PhD entrance exams I have faced the following problem: Minimize the function $f(x)=- \sum_{i=1}^n \ln(\alpha_i +x_i)$ for fixed $\alpha_i >0$ under the conditions: $\sum_{i=1}^n ...
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0answers
51 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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1answer
27 views

Robusness of median

If we let $X$ be a set of pints in $\mathrm{R}^2$, and let $g(X) = \arg \min_{y \in \mathrm{R}^2} \sum_{x_i \in X} \parallel x_i -y \parallel_2$ (geometric median of $X$). If $X$ and $X'$ are ...
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2answers
1k views

How to calculate the hessian of a matrix function

I am estimating a model minimizing the following objective function, $ M(\theta) = (Z'G(\theta))'W(Z'G(\theta))$ $Z$ is an $N \times L$ matrix of data, and $W$ is an $L\times L$ weight matrix, ...
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3answers
61 views

Maximum und Minimum [closed]

I need help for c) also I solved a and b, but c) I could not. Let $f~:~\Bbb R^2\to \Bbb R$ be defined as $f(x,y)=e^{-x^2-y^2+2x-2y}$ a) Determine all local extrema for $f$. b) Determine all ...
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14 views

Optimization of function on integer hypertetrahedron

I have the following optimization problem: Let $k,n \in \mathbb{N}$ with $k < n$. Let $N:=\{1, \dots,n\}$ and $D := \{^{t}(x_1, \dots, x_k) \in N^k \vert \sum_{i=1}^k x_i = N\}$ (hypertetrahedron)...
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23 views

Extrema on (compact) vinculum

My textbook ask to find the extrema of $f(x,y) = 2x^2+y^2$ on $x^4-x^2+y^2-5=0$. It uses the lagrangian multipliers to find critic points.. Then it computes the function on these points then says "...
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24 views

IF statement as Linear Constraint

I am writing a linear program, but I am currently having troubles writing a certain constraint, which is basically an IF-statment. I will try to explain it as detailed as possible: IF: $x_{it'}(t' +...
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1answer
479 views

Find pressure in a sinusoidal function

Tiffany is a model rocket enthusiast. She has been working on a pressurized rocket filled with laughing gas. According to her design, if the atmospheric pressure exerted on the rocket is less than 10 ...
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Optimized placing of same-size squares into rectangles

Suppose that we have several squares of the same size. We want to draw n rectangles (red and yellow rectangles here) to contain these squares. The goal is to have ...
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8 views

Is it possible to convert a loss function minimization problem into an eigenvector problem?

I have a very vague feeling that the problem of optimizing a loss function is related to the problem of finding the smallest/biggest eigenvector. However, I don't have the expertise to see this ...
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46 views

Cutting a pie into 2 unequal peices with a single cut, minimising its length. [closed]

Suppose we have a circle with an area of 1, which we are to cut into two pieces, of area (x) and (1-x) respectively. Let x<0.5. How should we make the cut, to minimise its length? What is the ...
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21 views

Solving Equations involving max operation

I would like to know how to solve this set of equations for v*(h) and also v*(l) Assume all other variables are known..concentrate on the 3rd equality in case of v*(h)..the first 2 are not needed. I ...
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1answer
78 views

Linear program with two equality constraints

Compute the minimal value of $$x_1 + 2x_2 + 3x_3$$ when $x_1$, $x_2$, $x_3$ satisfy $$x_1 − 2x_2 + x_3 = 4$$ $$−x_1 + 3x_2 = 5$$ and $$x_1 \ge 0, \qquad x_2 \ge 0, \qquad ...
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32 views

Minimizing a function including max functions

Consider the following problem. Let $\mathcal{N} = \{1,2,\ldots,N\}$ and $\mathcal{N}^i = \mathcal{N}\setminus \{i\} $. For each $i \in \mathcal{N}$ and for each $S \subset \mathcal{N}^i$, we have a ...
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modelling a composite objective function (max + argmax) as an (integer) linear program

Suppose $\mathbf{x} = [x_1, x_2, \ldots, x_n]$, where $x_i \in \{0, 1\}$ are binary variables. We know for a fixed $\mathbf{w}$ the following problem is an Integer Linear Program: $$ \arg\max_{\...
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Second-order stochastic optimisation - proof of convergence.

Consider the following scheme. We want to minimise some function $f(\theta) : \mathbb{R}^n \rightarrow \mathbb{R}$. We have access to noisy estimates of the first order derivative $g_t$ such that $E[...
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1answer
798 views

lagrangian minimisation problem and Karush-Kuhn-Tucker conditions

A rectangular box without a lid is to be made from 50m² of cardboard. Find the maximum volume of such a box.( i know how to solve this in the conventional way, i am trying to figure out how to do it ...
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How find the least value of the expression: $M = \cot^2 A + \cot^2 B + \cot^2 C + 2(\cot A - \cot B)(\cot B - \cot C)(\cot C - \cot A)$?

Consider all triangles $ABC$ where $A < B < C \leq \frac{\pi}{2}$. How find the least value of the expression: $M = \cot^2 A + \cot^2 B + \cot^2 C + 2(\cot A - \cot B)(\cot B - \cot C)(\cot C - ...
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1answer
57 views

In a $\triangle ABC,$ Evaluation of minimum value of $\cot^2 A+\cot^2 B+\cot^2 C$ [duplicate]

In a $\triangle ABC,$ Evaluation of minimum value of $\cot^2 A+\cot^2 B+\cot^2 C$, Given $A+B+C = \pi$ $\bf{My\; Try::}$ Using $\bf{A.M\geq G.M}$ $$\frac{\cot^2 A+\cot^2 B}{2}\geq \cot A\cdot \...
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1answer
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Is the EM-algorithm the same thing that variational inference in LDA?

I am new in the probabilistic topic modeling, and I need to understand deeply the LDA process, I understand what want to do the inference process in LDA, and I understand too that there is 2 "types" ...
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29 views

Properties of polyhedron solving constrained max problem

This is a question for people who don't have trouble to think in more than two dimensions. Don't hesitate to ask clarifying questions! Let us suppose we have $n$ random variables $X_i$ that are iid ...
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What is meant by a function being linear in two variables?

I'm trying to understand the Mangasarian condition in the context of dynamic optimization (see here p 8.12) and am not sure what exactly is meant by a function $f(x,u)$ being linear in $x$ and $u$. If ...
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34 views

Construct set of basis vectors with most zeros

I have a set of basis vectors that describes a linear subspace. The following is an example of three 16-dimensional vectors: $$\quad\quad\!\! \left( \begin{array}{c} b_1 \\ b_2 \\ b_3 \end{array} \...
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linearize average success probability constraint

In my optimization problem, I've a constraint to calculate the average success probability of a path. $x_{i,j}$ is binary variable defined as: $$ \begin{align} \label{eq3:1} x_{i,j} = \begin{cases} ...
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convexity of 3 variables minimax problem

Here is the function $f(x)=\underset{y\,\in \,C}{min}\quad \underset{z\,\in \,\mathcal{K(x,y)}}{max} \quad z^TQz$ in which $\mathcal{K(x,y)}:=\{z\,|\,x=y+z^a-z^b\}$ $x,y,z \in C$ $C \subseteq R^...
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Tuning schemes for additional valves on a brass instrument

So this question is very much about music, but it's entirely about a mathematical part of music, and that part is tuning and how it relates to pipe length in a brass instrument. Background: I am ...
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0answers
18 views

What is the difference between these two Bellman equation?

Bellman equation: $V(x) = max \{F(x,y)+ \beta V(y)\}$ $V(x) = max \{F(x,y), \beta V(y)\}$ When to use the plus and when to use the comma? Do they get the same form of sequence problem? Would you ...
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1answer
593 views

Cookie Clicker Chocolate Egg strategy

Introduction Cookie Clicker is a silly Javascript based web game. Here is a brief description of what you do: (description taken from this question: Explain a surprisingly simple optimization result) ...
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Potentially A Double Optimization Problem

I am trying to figure out how to handle this optimization problem, but I am having difficulty in processing the calculation. Consider $E$ to be the set $E = \{(x,y) \in \mathbb{R}^2 | x^2 + y^6 = 2\}.$...
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Orthogonal Projection onto the $ {\ell}_{\infty} $ Ball

What is the Ortohogonal Projection onto the $ {\ell}_{\infty} $ Ball? Namely, given $ x \in {\mathbb{R}}^{n} $ what would be: $$ {\mathcal{P}}_{ { \left\| \cdot \right\| }_{\infty} \leq 1 } \left( x ...
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1answer
34 views

Convex or non-convex function

I want to minimize the following function $$\frac{a}{bxy+cd}e^{\frac{a}{bxy+cd}}H+2-\Gamma(1,\frac{eaf}{b(1-x)},\frac{eagf}{bx(1-y)})$$ where $a,b,c,d,e,f,g,H$ are constants and greater than $0$. $\...
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1answer
37 views

System of equations with multiple answers, but only natural numbers (and other constraints)

Okay so basically I want the solution with the smallest sum when you add up the integer variable solutions that give a solution to: $975a + 880b + 790c + 585d + 487e + 440f + 292g + 260h + 530i + ...
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1answer
22 views

Algorithm for scheduling event observers

I'm reviewing different algorithms to solve a scheduling problem and was hoping someone with a better breath in the area might help me focus on the right class of algorithms. Basically the problem is ...
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35 views

Maximum of Contour Line

Consider the potential $U(x,y)=ay^{2}+b(e^{x-y}-1)^{2}+c(e^{x+y}-1)^{2}$ where $a$, $b$ and $c$ are known constants. I want to move through a contour line of this potential $U(x,y)=k$, say $y=g(x)$. ...
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1answer
115 views

Expressing an inequality constraint as a linear matrix inequality (LMI)

I am trying to formulate an optimization problem as a semidefinite program (SDP). My optimization variable is $\bf x = [x_1,\dots,x_N]'$, where $\bf x$ is an $N \times 1$ vector, and one of my ...
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2answers
1k views

Time complexity of a convex quadratically constrained quadratic program (QCQP)

Could someone tell me the time complexity of a convex quadratically constrained quadratic program (QCQP) problem? And any references? Thank you very much.
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SDP relaxation of non-convex QCQP and duality gap

Short version Is there a duality gap between a QCQP problem and the SDP problem obtained through Lagrangian relaxation? A paper I'm studying is using this fact, but I cannot achieve the authors' ...