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 the shortest path length on a curved surface(hyperboloid)

I wish to find the minimum path length between two points $P_1(\sqrt2,0,-1)$ and $P_2(0,\sqrt2,1)$ on a hyperbolic surface $S =\{(x,y,z)\in R^3\ |\ x^2+y^2-z^2=1\}$ I faintly recall studying ...
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158 views

Minimum distance to points in plane

Someone told me that the the following problem is elementary. Given three points $a=(-5,0)$, $b=(0,5)$ and $c=(5,0)$ in $\mathbb R^2$ with Euclidean norm: $$\mbox{minimize}\;\; \; f(x)=\|x-a\| + ...
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KKT point of a constrained optimization problem

Min$_{x}~x$ Subject to $x \geq 0$ For this problem, is $(x^{*}, \lambda^{*})=$$(0,0)$ a KKT point ? My try : I formulated corresponding Lagrangian and tried to find out the KKT point(s). ...
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42 views

Chord Maximisation

I am currently going back through all the "Challenge" questions in preparation for exams, and for this I do not know where or how to start, any hints would be appreciated. Now the way I have ...
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2answers
22 views

Local minimum of the function

We have a function $$\sum_{i=1}^{n}x_{i}^{2}=min,\quad\mbox{ subject to }\sum_{i=1}^{n}x_{i}=c,$$ which should have the minimizer $$\frac{c^{2}}{n}.$$ However, from the Lagrangian ...
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116 views

Minimizing Height of a Table

This optimization question popped into my mind while working with latex tables: Suppose we have a table with $m$ rows and $n$ columns, and for each $1\le i\le m,1\le j\le n$ we are given $T(i,j)$ ...
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48 views

Expanding variance

Could someone please expand on line 2 and 3 of: Thank you.
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990 views

Closest distance between two quadratic curves

I'm having trouble with the following problem : "find the closest distance between $x^2+4y^2=4$ and $xy=4$" I tried to solve using the properties of ellipse and hyperbola, but the relatively tilted ...
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201 views

Maximizing the Magnitude of the Resultant Vector

Given a set of $n$ two-dimensional unit vectors: $\left\{ \mathbf{v}_1, \dots, \mathbf{v}_n \right\}$, I want to find the coefficients $\left\{ \alpha_1, \dots, \alpha_n \right\}$, $0 \leq \alpha_i ...
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92 views

How to minimize $\max(x_1, x_2)$ and $x_1^2 + 9x_2^2$ subject to constraints?

My textbook came up with a solution without explanation. I'm looking for a systematic way of solving the following optimization problems and similar ones (by hand), because I'm drawing a blank: ...
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52 views

Find min and maxima

Find local min and maxima of $ \sin(x^3)$ on the interval $]-2,2[$. I take the derivative and get: $$3x^2 \cdot \cos (x^3)$$ I set this equal to zero and get $$x^3 = \cos^{-1}(0)$$ $$ \Rightarrow ...
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Why is that a risk averse consumer buys the optimum insurance when there is actuarially fair insurance?

I've asked the same question at the Quantitative Finance StackExchange. Consider the following example: "As a risk-averse consumer, you would want to choose a value of x so as to maximize expected ...
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Minimizing distance between 2 arrays (or points)

I would like to get a solution or receive guidance on how I can solve the optimisation problem below. Let's say I have two arrays of length N , say A and B, and I want to find 2 coefficients $k_1$ ...
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39 views

Finding coordinates of nodes in a graph

I have a complete graph in which the edges represent the euclidean distance between the nodes which is known. Assuming a node to be (0,0), I want to find (approximately) the coordinates of other ...
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32 views

$\max_x \max_y f(x,y) = \max_y \max_x f(x,y)?$

Just come across a question regarding sequential maximization and simultaneous maximization, and I do not recall whether there are any established conditions for the equivalence. Anyone has some idea? ...
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75 views

Lagrange multiplier - Find maximum on surface

I need someone to walk me through a 3 variable lagrange problem, since I haven't been able to find a reliable source to teach me, please. Here it is: Find the maximum of the function $F(x,y,z) = ...
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76 views

Finding minimum value of trigonometric function

Find the minimum value of $$\displaystyle \frac{2\cos^{-1}(x)}{\pi(1 - x)} , x \in [-1,+1) $$
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Optimization question about box

An open box is to be made from a rectangular 30cm x 18cm cardboard piece by cutting out squares of equal size from the four corners and bending up the sides. Find the dimensions of the resulting box ...
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Unique critical point and psd implies pd and hence strict relative maximum

Let $f(x)$ be of class $C^{(2)}$ on an open set A, $x_0\in A\subseteq R^n$ a critical point. In addition, the hessian matrix of f(x) at $x_0$, $H(x_0)=\{f_{ij}\}|_{x=x_0}$, is positive semi-definite. ...
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Regularization for unordered vectors

Let suppose we have two vectors u, v $\in \mathbb{R}^n$ and we want a function that returns $0$ if the ordering of the elements of both vectors are the same or a positive number otherwise, where the ...
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184 views

Minimum of a quadratic form

If $\bf{A}$ is a real symmetric matrix, we know that it has orthogonal eigenvectors. Now, say we want to find a unit vector $\bf{n}$ that minimizes the form: $${\bf{n}}^T{\bf{A}}\ {\bf{n}}$$ How can ...
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94 views

Convert SOCP from quadratic form to generalized inequality form

I have formulated a Second-order Cone Problem (SOCP) in “quadratic” form with a norm inequality constraint. To use a certain solver (ECOS, to be precise), I need to rewrite it to a form that makes use ...
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35 views

Calculus-based proof that $ x_1^{p_1}\cdots x_n^{p_n}\le p_1x_1+\dots+p_nx_n$ when $\sum p_i=1$

Let $$g(x_1...x_n)=x_1^{p_1}\cdot...x_n^{p_n}$$ $$u(x_1...x_n)=p_1x_1+...p_nx_n$$ Where $\sum p_i = 1$. I have to show that $f(x)=g(x)-u(x)$ is always negative or $0$ over $\Bbb R_+^n$. I've ...
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Examples of non trivial problems in this structure.

I'm looking for examples of non trivial problems that match with the follow structure. Let the function $$g: U \times V \rightarrow \mathbb{R}$$, where $U$ and $V$ are complex vetorial spaces of ...
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98 views

How do I properly set up this optimization equation?

So I've been the given the task to fully optimize any packaging. I chose a DS game box. So first I took the measurements of the cartridge itself ($3.5 \text{ cm} \times 3.3 \text{ cm} \times 0.38 ...
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49 views

How can I find four colors with maximum equal difference?

I need to find four colors, expressed as triple $(r_i, g_i, b_i)$ where $0 \le r_i,g_i,b_i \le 1$, $0 \le i \le 3$. Define color difference as $D_{i,j}=\sqrt{(r_i-r_j)^2+(g_i-g_j)^2+(r_i-r_j)^2}$. ...
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100 views

Minimize the minimum - Linear programming

Consider an optimization problem with variables $x_1, x_2, \dots, x_n \in \mathbb{R}$ (maybe subject to some linear constraints), and linear functions $\{f_i(x_1, \dots, x_n)\}_{1\leq i\leq m}$. We ...
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Optimization that involves inverse operation.

$\newcommand{\diag}{\operatorname{diag}}$ I have the following optimization problem: \begin{align} \mathop{\arg\min}_\beta & \frac{1}{2} a' [ M + \diag( \beta ) \otimes I_d ]^{-1} a + ...
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47 views

Convex functions -> quasi-convex functions -> … can we weaken the assumptions?

First of all let me say that I'm new to optimization. I realized that quasi-convex functions share with convex functions some nice properties, so I wonder if we can push the weakening a little ...
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Optimization with a constrained function

Okay so I understand how to find points of extrema when for example, We have $3x^2 + 2y^2 + 6z^2$ subject to the constaint $x+y+z=1$. I followed the method of the Lagrange multiplier and resulted in ...
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steepest descent with quadratic form converge in 1 iteration

Well I'm stuck on an exercise given: The steepest descent method is applied to the quadratic form $$Q(\mathbf{x}) = \tfrac{1}{2}\mathbf{x}^TA\mathbf{x} - \mathbf{b}^T\mathbf{x} + c$$ where $A$, ...
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Maximum area of quadrilateral of given perimeter.

Let $0\lt a\lt b$ (i) Show that among the triangles with base $a$ and perimeter $a + b$, the maximum area is obtained when the other two sides have equal length $b/2$. (ii) Using the ...
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Maximizing the product of first Eigenvalues of rank-1 hermitian matrices

Suppose we have $L$ complex vectors $\mathbf{a}_{l}$ with dimension $N\times 1$ I want to solve this optimization problem $\mathbf{x}_{\mathrm{opt}}=\arg ...
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What is the largest circle that fits in $\sin(x)?$

Imagine dropping a circle into the trough of $\sin(x)$. Would it reach the bottom or get wedged between two points on the curve? Depends on the size of the circle. So, what is the radius of the ...
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How many routes possible in the traveling salesman problem with $n$ cities? And more…

SO the general answer I come across on the internet is $(n-1)!/2$. But it would seem to be $n!$, or at least $(n-1)!$. Which one is it? If you have 2 cities, you would have 1 path. So $(n-1)!/2$ ...
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Can the search space of a solvable linear optimization problem be discontinuous?

Background Say you have a traditional linear-optimization problem, there is a linear cost function, $\vec{c}\cdot\vec{x}$ and a set of linear constraints, $A_1\vec{x} \geq b_1 $ $A_2\vec{x} \leq ...
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Minimize ||A.x-b||

A is a n x m matrix with known real elements and b is a known real n-dimensional vector. I would like to find all x such that ||A.x-b|| is a minimum. Is there a theorem that deals with it ? Update ...
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Microprocessor pinout problem. How many pins do I need for given N buttons?

At the begginning I'd like to sorry for the long intro, but I think it helps in fully understanding the problem. Of course the math problem is much shorter, so if you are not interested in the full ...
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least-square optimization with linearly depend solution $x$

What is the exact solution $x_{n \times 1}$ of the following constrained optimization problem \begin{align*} &\min \|A x - b\|^2 \\ s.t.& C x = 0 \end{align*} where $A$ is full column rank $m ...
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187 views

How to calc $\min ||J\Delta\tau + D||_*$

How to calculate $$ \min_{\tau} ||J_1 \tau_1 + \cdots + J_p \tau_p + D ||_* $$ where $\tau_1, \cdots, \tau_p \in \mathbb{R}$ $J_1, \cdots, J_p, D \in \mathbb{R}^{m \times n}$ $||\cdot||_*$ is sum ...
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72 views

Critical points of absolute value function

For this question, if I divide this function into two parts, which are $x \ge 0$ and $x<0$, then the part that doesn't include "0" will have no critical point, and I also have no idea of how to ...
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Find closest vector to a given vector from a particular set of vector

Let $x=\left(x_t\right)_{t=1}^n$ be a vector such that $$ x_t = \prod_{i=1}^t u_i, \tag{1} $$ where each parameters $u_i$ can take any of two value $$ u_i \in \left\{a,b \right\} = \left\{ 1.3, 0.8 ...
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Finding all the minima in a unconstrained minimization problem

I've just read a textbook stating that the first order condition $\frac{df}{dx} = 0$ and second order sufficient condition $\frac{d^{2}f}{dx^2} \gt 0$ of unconstrained minimization will find the all ...
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Classification of critical point

The critical point of this function are $(0,0),(-1/3,-2/9),(-1/3,2/9)$. And for $(0,0)$, I get the difference that $△f=f(0+a,0+b)-f(0,0)=a^4 + (3a+1)b^2$ will always greater than zero where a,b are ...
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62 views

What does the adjoint operator do? Is this Frechet derivative correct?

Problem statement Let $x \in l^2$ and $J(x) = \sum_{n = 1}^{+\infty} x_{2n - 1}^2$ Find first and second Frechet derivatives. Attempted solution Let's note that $J(x) = \sum_{n = ...
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1answer
72 views

Frechet derivative of double integral.

Problem statement Let $u(t) \in L^{2}(0, 1)$ and $J(u) = \int_0^1 tu(t) \int_0^t u(s)dsdt$ Compute first and second Frechet derivatives. Attempted solution $$ \begin{split} J(u + h) - J(u) &= ...
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What trick to calculate this Frechet derivative?

Let $u(t) \in L^{2}(0, 1)$. I need to calculate the first and second Frechet derivatives of $$J(u) = \int_0^1 \left(\int_0^{t^3}u(s)ds\right)^2dt$$ I am completely at a loss here: I know several ...
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57 views

Minimize the trace of a combination of PSD matrices analytically

I have the following problem: Define $H$ and $R_k$ for $k=1\dots N$, to be $M\times M$ positive definite matrices. The problem is to find optimal weights $p_k$that solves the following problem ...
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How do you prove ${n \choose k}$ is maximum when k is $ \lceil \frac n2 \rceil$ or $ \lfloor \frac n2\rfloor $?

How do you prove ${n \choose k}$ is maximum when k is $ \lceil \frac{n}{2} \rceil $ or $ \lfloor \frac{n}{2} \rfloor$ ? This link provides a proof of sorts but it is not satisfying. From what I ...
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48 views

Find max and min on some region

Find maximum and minimum values of $f(x,y,z)=x^2yz$ on the region $x^2+y^2\leq1,$ $0\leq z\leq1.$ First, I get $\nabla f= (2xyz, x^{2}z, x^{2}y) = (0,0,0) \implies x = y = z = 0$, so the critical ...