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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calculus of variations or optimize over function form

I have a question about optimizing the following quantity over function form . Given unknown function $f(\theta)$ such that $f(\theta)\geqslant 0$ and $\int f(\theta)d\theta\leq \infty$. And ...
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Finding an integral's max and min

I've been asked to find the max and min of the following: $F(x)=\displaystyle\int_0^{2x-x^2}\!\cos\dfrac{1}{1+t^2}\mathrm{d}t$ I tried applying the Fundamental Theorem of Calculus (taking the ...
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Dual program is wrong. Authors claim is right.

In a well respected book, I found the following. The authors claim that it is correct. But I think it is wrong. This is the primal Linear Problem: $$ \begin{array}{cccc} ...
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Optimization Problem Solving [closed]

I have some ambiguity about mathematical optimization problem modeling and solving because I don't have much more mathematical skills. Basically, I am from computer science background much more in the ...
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How far can the plane be tiled by congruent regular pentagons?

What is the limit, as the radius of the disk increases, of the greatest area, in proportion to the area of the disk, of the region covered by regular pentagons of the same fixed size, all lying within ...
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Finding the center of mass of a cylinder

Help finding center of mass of soda can? If you represent the soda can as a right-circular cylinder radius=$4$ cm height =$12$ cm We are told to neglect the mass of the can itself. When the can ...
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28 views

Identify if optimization problem is convex or non-convex?

I have formulated optimization problem for building, where cost concerns with energy consumption and constraints are related to hardware limits and model of building. To solve this formulation, I need ...
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35 views

Showing λu + (1 − λ)v is an optimal solution

$$\max \quad c \cdot x \\ \mathrm{s.t.} \ Ax \leq b\\ x\geq 0 \\$$ There are two optimal solutions to the LP $u$ and $v$. How do I show that for $\lambda \in [0,1]$, $\lambda u + (1-\lambda)v$ is ...
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Newton's Method in unconstrained optimization fails to converge

In order to show that Newton's method can produce a sequence of iterates that diverges, an example given in my book is apply Newton's Method to minimize $f(x)={2\over 3}|x|^{3\over 2}$. starting at ...
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16 views

Lagrange Optimization

I would like to ask for the optimization problem: $$\max_{x,y} g(x,y)$$ st. $x+y=1$ Would there be any difference if we formulate the problem as: $$g(x,y) + \lambda(1-x-y)$$ as opposed to ...
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Calculus optimisation with the speed formula

For a ship travelling at ${x}$ km/h the running cost in £ is ${(x^2 + {13500\over x})}$ per hour. Find the speed that minimises the cost of a 300km journey. The speed formula is ${speed = ...
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Maximization of Harmonic mean

Suppose x is a vector of size N with positive real elements sorted in decreasing order. Is it possible to find the analytical solution (no iterative solution) to the optimum value of M (1<= M ...
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31 views

How to describe $\lbrace \mathbf{x}\in \mathbb{R^n}: |x_j|\le1 $ for$ 1\le j\le n \rbrace $ in terms of $x_j=x_j^+-x_j^-$

How to describe the set $A$=$\lbrace \mathbf{x}\in \mathbb{R^n}: |x_j|\le1 $ for$ 1\le j\le n \rbrace $ in terms of $x_j=x_j^+-x_j^-$ where $x_j^+\ge0$ and $x_j^-\ge0$ The answer says: $B$=$\lbrace ...
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Optimization with trace and eigenvalues

Let $M \in \mathbb{R}^{n \times n}$ be a symmetric matrix with given eigenvalues $\lambda_1,\lambda_2,\cdots,\lambda_n$ with $\vert\lambda_1\vert > \vert ...
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46 views

A basic minimization problem

Does the following function has a global minimum $$f(x) = \frac{2x +1}{1-e^{-(1-\alpha) x}}$$ where $x$ is a positive real number. for $0< \alpha < 1$
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What is a good resource for a more intuitive/flexible understanding of optimization

Take the following example of optimization: $$cost = 10*x + 20*y$$ Where x = cans of soup, y = cans of juice It is easy to see in this scenario what we need to do in order to minimize cost. Just ...
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Optimization problem on graph with weights on nodes and edges

I am solving a problem where I have a complete undirected graph with weights on the nodes and on the edges. The weight on the node represents a profit that you obtain if you select that node. The ...
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Optimization: second order condition

This is the condition Where $L(x, \mu\,\lambda)$ is the Lagrangian function in a given point that satisfy the first order condition. Problem $ min (-4x -y)$ $ -x^2 -y^2 +1 <= 0 $ $ y- 1 ...
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Duality and the Positive Lagrange Multiplier

Suppose I have the following optimization problem: \begin{align} \min &f(x) \\ & f_1(x) \leq 0 \\ & \vdots \\ & f_k(x) \leq 0 \\ & g_(x) = 0 \\ ...
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Inapproximability of Combinatorial Optimization Problems

I've been reading the "Inapproximability of Combinatorial Optimization Problems" by Luca Trevisan (see: link). On pages 3-4 it mentions that a polynomial time algorithm for 3SAT would exist if there ...
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Optimization of the surface area of a open rectangular box to find the cost of materials

A rectangular storage container with an open top is to have a volume of 10 cubic meters. The length of the box is twice its width. Material for the base costs ten dollars per square meter and for the ...
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25 views

Conditions for a totally unimodular coefficient matrix of a Multi-Commodity-Minimum-Cost-Flow-Problem

I'm considering the following Multi-Commodity-minimum-Cost-Flow-Problem: This leads us to a coefficient matrix $A$ with $N$ donates the incidence matrix of a directed graph and $I$ is the ...
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44 views

Finding minimum of a two variable function

Let $D=\{(x,y)\in\Bbb R^2:1\le x\le1000,1\le y\le1000\}$. Define $$f(x,y)={xy\over2}+{500\over x}+{500\over y}$$ Then the minimum value of $f$ on $D$ is Finding $f_x=\frac y2-{1000\over ...
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Want to factorize one matrix into three, with L1 regularization, which optimization algorithm to choose?

I need to factorize one matrix $R$ into three component: $ R = P^TAQ $, in which I want to apply L1 regularization on $A$ to encourage sparsity, and apply L2 regularization on $P$ and $Q$ to prevent ...
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Optimization Problem Maximize $z= 60x_1+20x_2$

Restate the absolute value constraint as a combination of two linear constraints: I know how to find the optimal solution (std form, canonical form, simplex algorithm ...etc) I don't know how to put ...
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597 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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Optimization: determining maximum volume of a tube

I am unsure of how to go about solving this, the context is that there is a rectangular piece of paper with a perimeter of 100 cm that is to be rolled to form a cylindrical tube. The question wants to ...
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About Dual Simplex Method

I have a question about Dual Simplex Method (for minimization problem). While we are solving the method, when we obtain a non-negative $\bar b$, we stop the algortihm. But in addition to $\bar b ...
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51 views

Given $f(x)=4-e^{-cos(x-2)}$, find the maximum value of $f(x)$ in the range $[-2,0]$.

Given $f(x)=4-e^{-cos(x-2)}$ Find the maximum value of $f(x)$ in the range $[-2,0]$. $\forall a \in \mathbb R$, $e^a>0$ Hence, the maximum of $f(x)$ will occur when $e^{-cos(x-2)}$ is a minimum. ...
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How to show this integer program with irrational data has no optimal solutions.

I want to show the integer program with irrational data max$\{x_1-\sqrt{2}x_2:x_1\leq \sqrt{2}x_2,x_1\geq 1,x\in Z_+^2\}$ has no optimal solution, even though there exist feasible solutions with value ...
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Do standard gradient descent methods work on complex variables

I am currently whishing to optimize a function numerically $f(z)$ where $z \in \mathbb{C} $ ($f(z) \in \mathbb{R}$) . I am doing this via numerical packages (specifically scipy in python) and I have ...
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Optimization: Minimizing the cost of pipeline over land

I have the question "A Gas Outlet is one one side of a river 120 m wide. It is exactly 300 meters downstream and across the river from a cottage. A gas line is to be constructed to join the outlet to ...
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Deriving optimal time to change

I am working in economics and I am trying to build a model that take into account the fact that indivudal can take a decision once in their life time that changes the value of a parameter R. To be ...
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Maximum of $e^{-x} \sin(x)$, $x \geq 0$

What is the maximum of $e^{-x} \sin(x)$ for $x \geq 0$? Is there a closed-form solution? If not, what is a good approximation $y$ such that $\text{max}_{x\geq 0}e^{-x} \sin(x) \leq y$?
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Comparing a maximization to an integration with economics application

This seemingly simple question has interesting interpretation in economics, but I only state the mathematical problem here. Suppose $B(0)=C(0)=C'(0)=0$, $B'(\cdot)>0,\ B''(\cdot)\leq0,\ ...
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How do I construct the Jacobian for use in a Levenberg-Marquardt algorithm.

I am working on a 3D reconstruction system and I am looking to use a Levenberg-marquardt algorithm to do bundle adjustment. I am not too sure about how LM works and what it requires. The model I am ...
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315 views

Why is Newton's method faster than gradient descent?

Can you provide some intuition as to why Newton's method is faster than gradient descent? Often we are in a scenario where we want to minimize a function f(x) where x is a vector of parameters. To do ...
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What is the range of $y$ if $x+y+z=4$ and $xy+yz+xz=5$ for $x, y, z \in\mathbb{R}_+$

What is the range of $y$ if $x+y+z=4$ and $xy+yz+xz=5$ for $x, y, z \in\mathbb{R}_+$ How to explain the following method? Let $x=z$ then: $$2x+y=4\quad;\quad 2xy+x^{2}=5$$ $$\implies \left( ...
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Newton's optimization: why wouldn't the results be good if we take $x_0$ between $(-\frac{1}{\sqrt 3},\frac{1}{\sqrt 3})$ for $f(x)= x^3- x-1\;?$

I was reading about the numerical method of Newton for finding the roots of $f(x)$ in Thomas' Calculus ; the author presented an example Find the x-coordinate of the point where the curve $f(x)= ...
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2answers
64 views

What is the maximum number of boxes that can fit in a rectangular container

I'm looking for an algorithm for the following question: What is the maximum number of boxes with sides a,b,c that can fit in a rectangular container with sides $x$,$y$,$z$. For example, the ...
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Is closed convex set with finite number of extreme points convex polyhedron

I have this simple question related to convex set and convex polyhedron. As the content in the title, it's basically my question: Is closed convex set with finite number of extreme points convex ...
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Matlab: need help with optimization

I am trying to minimize the objective function over [x(1),x(2)]: exp(x(1))*(4*x(1)^2+2*x(2)^2+4*x(1)*x(2)+2*x(2)+1)+b subject to constraint ...
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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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Multivariable optimization for time to build a ship in a game, and maybe some possible application in “everyday” life

I precise first that english is not my monther tongue and I may will not be as clear as I would like, just ask me question if you need, thank you. I am playing a game (Galaxy Empire) for a while, ...
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34 views

Weakly lower semicontinuous functional on a bounded closed and convex set

Let $J$ be a sequentially weakly lower semicontinuous functional on $C$ with values on the real line. Moreover let $C$ be a bounded, closed and convex subset of a Hilbert space $H$. Is it true that ...
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Explanation of formula

Suppose that we have $M$ production stations $A_1, \dots, A_M$ of a product and $N$ destination stations $B_1, \dots, B_N$ of the product. We suppose that $x_{ij}$ units of the product are ...
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Maximise volume given inequality constraint on its dimensions without using Lagrange, KKT or Linear Programming

The problem (from Calculus for Business, Economics, Life Sciences and Social Sciences 12e): I found this and that, but they use Lagrange/KKT. What I tried: Girth $= 2w + 2h$ Maximise ...
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Find the smallest possible value of $a^4+b^4+c^4-136abc$

Let $a$, $b$, and $c$ be real numbers such that $a+b+c=-68$ and $ab+bc+ca=1156$. The smallest possible value of $a^4+b^4+c^4-136abc$ is $k$. Find the remainder when $k$ is divided by $1000$. I ...
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How to describe minimization of L1 norm error using linear programming?

Given a set of $n$ pair points $(x_1, y_1), ..., (x_n, y_n)$ in the plane, I need to find a line $ax + by = c$ that fits the points of the L1 norm error points as closely as possible. I need a linear ...
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Maximum and minimum of the function $xy+z^2$

Find the maximum and minimum values of the function $f(x,y,z)=xy+z^2$ in the circumference obtained by intersections between the sphere $x^2+y^2+z^2=4$ and the plane $y-x=0$. I did Lagrange and found ...