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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Dynamic Optimization - Transversality Condition for Infinite Horizon Case

When solving dynamic optimization problem such as $$ \max \int_0^\infty f(t,x,x')dt $$ $$ \ s.t. x(t_0)=x_0 $$ we can use the Euler equation to obtain a differential equation to solve. ...
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Find a point from the area tha is closest to the other point.

Given: $$A=\{\left(x,y,z \right)\in \mathbb{R}^3 : 2x-3y+z=1 \}$$ Find a point $\left(x,y,z\right)\in A$ that is closest to $\left(3,-2,1 \right)$. I do not know how to solve that problem, i need a ...
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variable transformation in optimization

I have an optimization problem with two sets of parameters, $x_i \in [0,1]$ and $y_k \in [-\frac{\pi}{2},\frac{\pi}{2}]$ where $i,k \in \{1...n\}$ are indices. One way to solve this problem is using ...
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3answers
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Finding the distance from the origin to the surface $xy^2 z^4 = 32$ using the method of Lagrange Multipliers

Problem: Find the distance from the origin to the surface $xy^2z^4 = 32$. Attempt: The Lagrange equation for this problem is $L(x,y,z, \lambda) = x^2 + y^2 + z^2 + \lambda (xy^2 z^4 - 32)$. Setting ...
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Calculating amount of cubes that fit in a sphere

I know that the problem of finding out how many spheres can fit in a cube is a commonly asked and well documented one, but I am struggling to find anything on the inverse of the problem, namely: ...
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2answers
528 views

linear least squares minimizing distance from points to rays - is it possible?

I'm writing a tool whose purpose is to process data from a sensor that provides the true bearing to a target, and combine measurements taken at various times into an estimate of the target's position ...
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1answer
77 views

Calculate the maximum area (maximum value)

TX farmer has 100 metres of fencing to use to make a rectangular enclosure for sheep as shown. He will use existing walls for two sides of the enclosure and leave an opening of 2 metres for a gate. ...
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Where to build a bridge to cross a river in the shape of an annulus

There is a river in the shape of an annulus. Outside the annulus there is town "A" and inside there is town "B". One must build a bridge towards the center of the annulus such that the path from A ...
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58 views

The maximum of a functional

Is the following statement true or false? $$ \max F \left( \theta \right)= \int_{\rho_{min}}^{\rho_{max}} g \left( \rho \right)\pi\left(\rho,\widehat{\theta \left( \rho \right)} \right)d\rho$$ ...
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Minimizing the sum of the $4^\text{th}$ power of a matrix entries.

Consider a real $n\times n$ matrix $X$. Suppose I would like to minimize the sum of the squares of its entries as a penalty term in some convex minimization. I can write the term using the Frobenius ...
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Intuition behind accelerated first-order methods

$\newcommand{\prox}{\operatorname{prox}}$ $\newcommand{\argmin}{\operatorname{argmin}}$ Suppose that we want to solve the following convex optimization problem: $\min_{x \in \mathbb{R}^n} g(x) + ...
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Show that inf $f(x)$ is achieved. Find $\inf f(x)$.

Let $$\Sigma = \{x\in R^3: x_1x_2 +x_1x_3 +x_2x_3=1 \}$$ and $$f(x) = x_1^2 + x_2^2 + \frac{9}{2} x_3^2$$ a) Show that $\Sigma$ is a smooth surface in $R^3$. b) Show that $\inf_{x\in\Sigma}$ f(x) ...
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1answer
30 views

Finding KKT conditions for nonlinear optimization problem.

I have an optimization like below: $\text{ minimize } \sum_k - \log_2 x_k $ $\text{subject to: } x_k \leq q , k =1,2, \cdots, N .$ I can form the Lagrange of the problem as below: $L(x, ...
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Scaling issue with Gradient descent methods

As is the common knowledge that gradient methods are affected by scaling issue of the variables. For example, If minimizing a function of say 2 variables $x_1$, $x_2$. Both variables have different ...
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1answer
459 views

How to see that K-means objective is convex?

I'm trying to proof that the objective of the K-means clustering algorithm is non-convex. The objective is given as $J(U,Z) = \|X-UZ\|_F^2$, with $X \in\mathbb{R}^{m\times n}, U\in ...
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1answer
52 views

Conjugacy relation in the primal and dual problem

The following is my derivation in the Conjugacy relation in the primal and dual problem. I am shaky in it; so hope for some advices. Consider the following problem, $f(x),g(x)$ are convex ...
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22 views

Some problems in finding conjugate function

Ask the following fundamental problems: How to derive the conjugate function of $g(y)$ if given "$\underset{y \geq 0}{\text{sup}}\{g(y)-y^Tx\}$"? My attempt is as following: \begin{align*} ...
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Maximizing the area of a transformed rectangle within some bounds

I need to solve this problem for a program I'm writing, but I'm struggling a bit with the maths behind it. Given a rectangle $R_{max\_layout}$ and a $3\times 3$ transformation matrix $M_{transform}$, ...
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2answers
35 views

The sum of the lengths of the hypotenuse and another side of a right angled triangle is given.

The sum of the lengths of the hypotenuse and another side of a right angled triangle is given.The area of the triangle will be maximum if the angle between them is: ...
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Difference between maximum and minimum?

If I have a problem such as this: We need to enclose a field with a fence. We have 500m of fencing material and a building is on one side of the field and so won’t need any fencing. Determine the ...
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1answer
41 views

Pontryagin's Maximum Principle as a sufficient condition?

It is know that Pontryagin's maximum principle provides in general only a necessary condition in the following sense: The ODE system which is known to be solved by the optimal control may have ...
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22 views

Minimum vertex cover of two edge disjoint perfect graphs

How well can the minimum vertex cover of the union of a perfect graph and bipartite graph (the two graphs are edge disjoint but not vertex disjoint) be approximated?
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20 views

Integer (Binary?) Optimisation of a problem

got a question regarding maximal optimisation of a problem. Refer to the table below: $$\begin{array}{c|c|c|} & \text{Area A} & \text{Area B} & \text{Area C} & \text{Area D} \\ ...
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Solving linear constrained optimization problem

So I have the following constrained optimization problem to optimize a circuit (electrical engineering) that I am working on: Minimize the following expression (power dissipation): $$I_{B1}(V - C_1) ...
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Find the critical curves for the following functional with subsidiary conditions

Find the critical curves for the following functional : $$J[y,z]=\int_{0}^{1}\left(y'^2+z'^2-xyz'-yz\right)dx$$ with subsidiary conditions : $$\int_{0}^{1}\left(y'^2-xy'-z'^2\right)dx=2$$ ...
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Solving Nonlinear system with logarithmic objective function

I have my objective function as : $\hspace{25mm} \text{Minimize} \sum_k- \alpha_k \log_2 W_k$ $\hspace{25mm} \text{subject to}: 0\leq W_k \leq q', 0 \leq \alpha_k \leq 1 $ $\hspace{25mm} k ...
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29 views

Placing Circles Onto Lines For Optimality

Suppose you have a yet to be determined number of vertical lines with length 50 on which you'd like to place as many circles as you can. Each circle is 10 units in diameter and its outside edge must ...
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21 views

Winning the relay race for your team

Relay race, members of a team of three take turns running from the point P to a point on the circle; To A for the first, B for the second, and C for the third, starting and returning to point P, ...
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1answer
64 views

Given matrices $B$ and $C$. What is the value of $L$ that minimizes the value $||L^T \times B \times L - C||_F$?

Where $L \in R^{m \times n}$ and $B \in R^{m \times m}$ and $C \in R^{n \times n}$ $B$ and $C$ are symmetric positive semi-definite. Where $\times$ denotes matrix multiplication and $||.||_F$ ...
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Minimum of a function given by integral and inequality type constraint

I need your help with the following problem I want to minimize $$2a + \int_0^1 tx(t) \, dt \to \min$$ s.t. $$1 - a - \int_t^1 x(s) \, ds \leq 0\text{ a.e. }t \in (0,1)$$ $$x(t) \geq 0 \text{ a.e. ...
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How can I solve a nonlinear optimization problem where constraint contains exponential term?

I have the optimization problem as below: $\hspace{10mm} \text{Maximize} \sum_{k} \alpha_k {R}_k $ $ \hspace{10mm}\text{subjcet to:} $ $ \hspace{10mm}\exp \left[ - (2^{{R}_k } -1) \left( ...
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Particular map from a square to a parallelogram

I would like to present you a problem I have to solve. I don't think its solution is elementary, so any hint you can give me is really welcomed. Let's consider $Q_1$ the square in $\mathbb{R}^2$ of ...
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Soviet Optimization books

I am aware of an answer on Soviet math books here: Soviet Russian Mathematical Books and the book by Boris Polyak on non linear optimization. I am also aware of a few books by Kantorovich which I do ...
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Optimization problem: solving one implies solving the reverse?

I am looking to solve an optimazation problem $Maximize_{x} [A(x)]$ s.t. $B(x)\geq B_0$, where $B_0$ is a constant. If I solve this problem (i.e, finding the optimal $x^*$ that optmize while ...
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Classifying stationary points without the Hessian

Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be $\mathcal{C}^{\infty}$ in $\mathbb{R}^n$. I can calculate the gradient which results in an expression of the form $ \nabla_{\mathbf{a}} ...
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Maximum occurring at all points in a set

Is there any term for "sets where maximum of a mathematical expression in attained"? I just want to know if the set has any specific name. The set is infinite (do not consider discrete points). The ...
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Who knows Krotov's Method in Optimal Control Theory

I'm finishing my PhD thesis about applications of optimal control theory in the field of energy harvesting. In the course of my PhD I dealt with different ways to compute optimal controls, and I found ...
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1answer
36 views

Existence of minimum in bounded but open set

According to the Extreme Value Theorem, a continuous function achieves at least one minimum and one maximum whenever the set is bounded and closed (i.e. compact). In my case, I have a bounded and ...
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2answers
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If $x,y,z>0$ and $xyz=32,$ Then the minimum of $x^2+4xy+4y^2+4z^2$ is

If $x,y,z$ are positive real no. and $xyz= 32\;,$ Then Minimum value of $$x^2+4xy+4y^2+4z^2$$ is $\bf{My\; Try::}$ Here I have Used $\bf{A.M\geq G.M}$ Inequality So $$\displaystyle ...
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2answers
69 views

Lagrange's multiplier not working

Given the function $f(x,y):=xy+x-y$. Let $D:=\{(x,y)\in\mathbb{R}^2:x^2+y^2\leq25\wedge x \geq 0\}$. Find the absolute maximum and minimum of $f$ on $D$. My working is as follows: $\begin{array} ...
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1answer
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What is $H^\infty$ norm and why is it used in control theory?

Can anyone knowledgable elaborate on what exactly is a $H^\infty$ norm and why it is used in control theory instead of some other norms? Thanks!
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Finding maximum of a function represented by a back-propagation neural network

First, I train a standard feed-forward neural network over a training set of data points. I get an approximate function, say $F(x)$, represented implicitly by that neural network. Now I want to find ...
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Global maxima/minima of $f(x,y,z) = x+y+z$ in $A$

Find the global maxima/minima of $f(x,y,z) = x+y+z$ for points inside of $A = \{ (x,y,z) \in \mathbb{R}^3: x^2-y^2 = 1 \wedge 2x+z = 1 \}$ I renamed the conditions of $A$ to a function $g(x,y,z) ...
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Maximization of a nasty Gaussian likelihood

I have a Gaussian likelihood function, $$p(y|x) = \mathcal{N}(y; Ax, (x^\top V x + \lambda) \otimes I)$$ where $A,V,\lambda$ is known, and $\otimes$ is the Kronecker product. (the notation indicates ...
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$\left\{x^k\right\}$ converges to $x^*$ superlinearly iff $\left\|\nabla^2f(x^k)^{-1}\nabla f(x^k)+x^{k+1}-x^*\right\|=o(\left\|x^{k+1}-x^*\right\|)$

Let $(x^k)_{k\in\mathbb N}\subseteq\mathbb R^n$ be convergent to $x^*$. We say, that the convergence is superlinear iff $$\left\|x^{k+1}-x^*\right\|=o\left(\left\|x^k-x^*\right\|\right)\tag{1}\;.$$ ...
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“Optimaly” reordering the vertices of a hypergraph.

I am not even sure of how to search for an answer to this, or how to approach the problem myself, so I thought I would try to ask it here. Consider an n-vertex hypergraph where the vertices are ...
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Condition for product of increasing and decreasing functions to be quasiconcave?

Is there any condition for product of increasing and decreasing functions to be quasiconcave? More specifically, I am having in mind a condition for $F(x)\cdot(1-G(x))$ to be quasi concave where $F$ ...
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91 views

Trace minimization subject to diagonal constraints

Problem Revisited - Edited for conciseness: We are given two set of data points X [$p \times n$] and Y [$q \times n$]. Let us assume $X = \hat{X} + \tilde{X}$ and $Y = \hat{Y} + \tilde{Y}$ I am ...
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constrained optimization including sum of two upper incomplete gamma function in both fitness function and constraint

i'm trying to solve this constrained optimization problem the constraint is $$\zeta=\frac {\Gamma \left(M,\frac {\lambda}{\left |\sum_{i=1}^Nw_i*h_{ei} \right|^2 + \sum_{i=1}^N\left |w_i \right|^2} ...
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Classifying Critical Points of $f(x,y)=xy-x+2x^3-yx^3$

I am classifying the critical point(s) of $ f(x,y)=xy-x+2x^3-yx^3 $: I first found the critical points by solving for $ f_x=f_y=0 $: $f_x= y-1+6x^2-3yx^2=0 $ $f_y= x-x^3=0$ Hence $x=0$ and ...