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 a function known to have a unique local and global minimum

Quasi-convex functions are a class of functions known to have a unique local and global minimum, which can minimized over convex sets using numerical methods with convergence guarantees. A function is ...
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Distinct Maximizers over a convex set

Let $u,u'\in\mathbb{R}^n$ be linearly independent and $B(x)$ be a smooth convex set (perhaps an $\epsilon$-ball) containing some point $x\in\mathbb{R}^n$. Under what conditions is it true that: ...
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Maximum of two skewed normal distributions

Does there exist a means to approximate the maximum of two skewed normal distributions in terms of another skewed normal distribution? To make it clearer, given 2 skewed normal distributions ...
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Convex optimization problem [closed]

Can anybody tell me how to convert this to quadratic programming format...?? objective is:- minimize {sum ( $(x(i)-x(j))^2 + (y(i)-y(j))^2$ ) for $j>i$; constraints:- $x(i)+r(i) \leq (1/2)w$; ...
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Variational optimization problem with several constraints

I am looking for solutions, approaches or hints to solve this variational optimization problem: Let $f:\mathbb{R}\rightarrow [0,\infty)$ be such that $\int f(x)\,dx=1$ and $\int x\,f(x)\,dx=0$ and ...
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Box Optimization: find best boxes for multiple products

I have a company that sells all kind of products and we have many issues with boxes. We are going to buy wholesale volumes, so we need to get the sizes right. We are trying to develop an algorythm ...
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17 views

Stuck on polynomial equation in optimization problem

I've been trying to solve an optimization problem, but I am completely stock on one step. I had the following Langrangian: $$\nabla\mathcal{L}(x,\lambda)= e\frac{\sum_{t\in I}e^t \Delta P(t)( x^t ...
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1answer
18 views

Explanation of strategies in infinite horizon dynamic programming problem

My question is regarding the Bellman equation regarding strategy $\sigma^{(1)}$ on the last 2 lines (I have attached pictures of the book below). If we know that all future states will have value of ...
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1answer
80 views

Is this dynamic optimization?

I would like to know what I should know to understand this IMF paper. What kind of optimization is used to maximize the utility function on page 9 (number 1) subject to constraints (2) and (3)? The ...
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1answer
27 views

lagrange method, linear constraints and unique global maximum

My book in linear programming states two things that I do not understand. We are working with the lagrange method with linear constraints.: From multivariate calculus we have that at a critical ...
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25 views

Finding the critical points of $f(x,y) = x y^2 - x^2 y + x y$

Trying to find the critical points of $f(x,y) = y^2x - yx^2 + xy$. I took partial derivative with respect to x, so $F_x = y^2 - 2xy + y$ $F_x = y(y - 2x + 1)$ Then with respect to y, $F_y = 2xy - ...
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1answer
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Find the largest range of values of the step size α for which the algorithm is globally convergent

Consider a fixed-step-size gradient algorithm applied to the following function f. $$ f(x)=1+2x_1+3(x_1^2+x_2^2 )+4x_1 x_2$$ Find the range of values of the step size α for which the fixed step ...
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Problem on EU commission

Consider the following problem. A collection of $n$ countries $C_1, \dots, C_n$ sit on an EU commission. Each country $C_i$ is assigned a voting weight $c_i$. A resolution passes if it has the ...
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1answer
12 views

Question about separability of convex envelopes

Given a function $f(\boldsymbol{x})$ defined on the hypercube $\boldsymbol{x} \in [0,1]^n$. Suppose $f(\boldsymbol{x})$ can be expressed as $f(\boldsymbol{x})=c(\boldsymbol{x})+g(\boldsymbol{x})$, ...
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1answer
29 views

Recovering the optimal primal solution from dual solution

I'm having trouble finding the optimal primal solution of a particular problem from its dual solution. Primal: $\texttt{Maximize} \ \ 10 x_1 + 24 x_2 + 20 x_3 + 20 x_4 + 25 x_5$ Subject to $x_1 + ...
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2answers
606 views

The meaning of $\lambda$ in Lagrange Multipliers

This is related to two previous questions which I asked about the history of Lagrange Multipliers and intuition behind the gradient giving the direction of steepest ascent. I am wondering if the ...
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The limit of an argmax function

Consider the function $$f(x,n)=x(A-cn)\frac{1-x^{n}}{1-x^{n+1}},$$ where $n\in\{1,2,...\}$, $x\in\mathbb{R}_{\geq0}$, and $A>c>0$. Let $n^{*}(x)=\mbox{argmax}_{n}f(x,n)$ denote the $n$ that ...
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Why are additional constraint and penalty term equivalent in ridge regression?

Tikhonov regularization (or ridge regression) adds a constraint that $\|\beta\|^2$, the $L^2$-norm of the parameter vector, is not greater than a given value (say $c$). Equivalently, it may solve ...
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optima; value of a function

Suppose we have the following function $$Err(f) = \frac{1}{2}E|Y-f(X)| = P(Y=1,f(X)=-1) + P(Y=-1,f(X)=1),$$ where $Y, f(X) \in \{-1, 1\}$. How can find the optimal value of the above function, Err? I ...
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2answers
42 views

Minimize a non-convex function subject to linear dynamics constraint

I want to solve the following problem: $$\min\limits_{\bf u} \frac{\bf c^T {\bf x} (T_f)}{\| \bf c\|\|{\bf x} (T_f)\|}$$ subject to $$\dot{\bf x} (t) = A {\bf x}(t) + B {\bf u}(t)$$ $$x(0) = x_0$$ ...
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2answers
99 views

Optimization of parameter for recursive Cauchy sequence

I have the following recursive sequence I'm analyzing: $$V_0 = 50, V_1 = (1-10k)V_0,$$ $$V_{n+1} = (1-10k)V_n - 5kV_{n-1}$$ where $k > 0$ is a parameter that I'm investigating by running ...
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Calculating minimum optimization of wall panel leftovers in excel

6,4,2,8,8,3,8,9 Group the following values based on the constraints listed below. Each value can only be used once. There is no limit to the amount of values in a group. The sum of each group must ...
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1answer
52 views

linear approximation with respect to L1 norm

I am trying to solve this problem: Find the best $L^1$ linear approximation of $e^x$ on [0,1] i.e. minimize $\int_0^1|e^x-\alpha-\beta x| dx$ any hints how to proceed
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Longest chord in the intersection n disks (circle areas)

Given n disks that intersect, there is a shape in the space where they intersect. Given that, what is the longest chord, more generally longest line, that can be drawn in this space? For n=1, this is ...
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6answers
229 views

I am trying to maximize an exponential function [closed]

I am looking for the value of $x$ that will maximize $y$ in the following equation $$ y=e^{-(x-a)^2/b} $$ where $a$ and $b$ are constants. Any help is appreciated
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52 views

If g(x) is the maximum value of f(t)

Let f be continuous on [a,b] and define a function g(x) on [a,b] as follows g(a)=f(a) and for a $\lt\ $x $\le\ $b then g(x) be the maximum value of f(t) on [a,x]. Prove that g(x) is continuous of ...
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55 views

Solving non-linear optimization using generalized reduced gradient (GRG) method

Consider the following elementary maximization problem: \begin{align} f{=}\mathrm{argmax}_{y_{l,c}, p_{l,c}}~\sum_{l=1}^{L}\sum_{c=1}^{C} y_{l,c}\text{log}_2\left(1+\frac{p_{l,c}}{I_{l,c}}\right) ...
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41 views

The meaning of 'worst case'

When giving bound on convergence rate, complexity and so on, people sometimes will specify it by 'worst case'. What is the meaning of 'worst case'?
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Can Three Equilateral Triangles with Sidelength $s$ Cover A Unit Square?

A previous question on the site asked for a short proof of the fact that three equilateral triangles with unit side length cannot be arranged to cover a square with unit side lengths. Given the truth ...
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How to optimize this types of problems?

Given that $min [ t_{f} - t_{0} ]$ such that $x(x'(t_{0}),y'(t_{0}),z'(t_{0}),t_{0}) = 0$ $y(x'(t_{0}),y'(t_{0}),z'(t_{0}),t_{0}) = 0$ $z(x'(t_{0}),y'(t_{0}),z'(t_{0}),t_{0}) = 0$ $x(t_{f}) = ...
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Optimization problem with variables in the subscript

I want to solve a optimization problem, which mimics the actions between a seller and several buyers. A seller has several goods, 1, 2, ... J, with prices $p_j$ and quantity $q_j$. A buyer can only ...
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1answer
33 views

The Euler-Poisson equation

$$\int_{0}^\pi (x''^2+4x^2) dt$$ $$ x(0)=x'(0)=0; x(\pi)=0;x'(\pi)=sinh(\pi)$$ This is The Euler-Poisson equation, i found: $$\frac {\partial f}{\partial x}-\frac {d}{dt} \frac{\partial f}{\partial ...
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1answer
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Rate of Convergence of complicated sequence with interactions

I have been working on a problem where the sequence turns out to be so complex that i am unable to find its convergence rate with necessary and sufficient conditions on the parameters.After working ...
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2answers
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How to maximize shipping box volume

Earlier last week I realized I needed to ship a large volume of things domestically. Of course, I decided that I wanted to do so as cheaply as possible. I first looked at USPS standard post rates. I ...
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Can a dynamic programming problem be transformed into a linear algebra problem?

Here is a simple standard economic problem: Let Robin Crusoe have an endowment $w_0$ of lembas bread. She is immortal and discounts at a rate of $\beta$ per period. Each period (from $t=0$ on) she ...
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Help Required in eigenvectors for sparse matrix?

I have a large sparse matrix A(~400000,~400000) . If I randomly remove few rows from the matrix will there be considerable change in the eigenvalues and the eigenvector's compared to eigenvector's of ...
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354 views

Optimization, rectangle inscribed inside arch of the curve.

A rectangle is to be inscribed under the arch of the curve $y = 4\cos(0.5x)$ from $x = \pi$ to $x = -\pi$. What are the dimensions of the rectangle with largest area, and what is the largest area? ...
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Maximum Volume of a rectangular box in ellipsoid

This is the problem I am working on: Find the maximum volume of a rectangular box that can be inscribed in the ellipsoid: $x^2/25 + y^2/4 + z^2/49 = 1$ with sides parallel to the coordinate axis I ...
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Radio factory linear program

I need a help with this exercise. I’m supposed to write a liner program for the problem below and then solve it using simplex method, but I’ don’t know how to include all the factors into variables. ...
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Hammersley–Chapman–Robbins bound for Rice distribution

I am trying to evaluate the Hammersley–Chapman–Robbins bound for the variance of an unbiased estimate $\hat{\alpha}$ of $\alpha$ (for a given $\sigma$) for the Rice distribution: $$p(x|\alpha,\sigma) ...
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418 views

Dual cone of a L1 norm cone?

I am listening to convex optimization lectures and I hear that dual cone of a $L1$ norm cone is a $L-\infty$ norm cone. Can anybody please explain how? I understand that every point in the dual cone ...
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How to Solve this maximization Problem?

You are given two s: N and K. Lun the dog is interested in strings that satisfy the following conditions: The string has exactly N characters, each of which is either 'A' or 'B'. The string s ...
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What is the derivative of this?

I have a function of the following form: $J = \|W^TW-I\|_F^2$ Where, $W$ is a matrix and $F$ is the Frobenius Norm. How can I find the derivative of $\frac{\partial J}{\partial W}$ ?
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Mana Maximization (Hearthstone)

I recently started playing Hearthstone and a statistic / probability question came up my mind. Here's a quick breakdown: The game is a turn-based card game which involves "points" that you can used ...
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1answer
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Finding Maximum and Minimum for f(x,y)

The problem I am working on is: Find the maximum and minimum values of the function: $f(x,y) = -3x^2 - 14xy - 3y^2 -8$ on the disk: $x^2 + y^2 \leq 4$ The $-14xy$ term is severely throwing me for a ...
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How to minimize $x^2+4xy+5y^2-4x-6y+7$ without using calculus

I would like to find the smallest possible value of the function $$f(x,y)=x^2+4xy+5y^2-4x-6y+7$$ without taking any derivatives. My thoughts were to complete the square on both $x$ and $y$ and ...
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Maximizing the area of a triangle with its vertices on a parabola.

So, here's the question: I have the parabola $y=x^2$. Take the points $A=(-1.5, 2.25)$ and $B=(3, 9)$, and connect them with a straight line. Now, I am trying find out how to take a third point on ...
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How to Solve this Linear Programming Problem?

$$\max[Z(x,y)=x+y]$$ $$-x+y\le 1$$ $$x\ge 0$$ $$y\ge 0$$ What i have done so far ? I tried simplex method , but i can't stop iterating . It really seems like a live lock . So how can i solve ...
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measurable selection for almost-minimizers of an irregular functional

I'm faced with the following problem: I have a functional $F$ defined on $H^1$ curves $[0,1] \rightarrow \Omega \subset \mathbb{R}^n$ where $\Omega$ is either a compact subset or the whole ...