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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Is it true that a quasiconvex, increasing and continous function, is convex?

Let $f:\mathbb R^n \to \mathbb R$ be a continuous and increasing function. Let $f$ be quasiconvex. Let $f(0)=0$. Can we say that $f(x)$ is convex ? If yes, how do we prove it ? Thank you very much ...
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463 views

Related Rates/ Optimization problem

I was having trouble figuring out this problem. A fisherman is in a boat 3 km from the nearest point A on the coast. The fisherman wishes to return to his camp C located 5 km from the point A. The ...
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30 views

Optimizing a ranch

"A rancher wants to fence in an area of 1000000 square feet in a rectangular field and then divide it in half with a fence down the middle, parallel to one side. What is the shortest length of fence ...
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51 views

What is the meaning of “girth” of a rectangular box?

Here's an optimization problem. A parcel delivery service will deliver a package only if the length plus the girth (distance around, taken perpendicular to the length) does not exceed 112 inches. ...
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Maximising a sum - closed form?

As a follow up to this question, I am wondering the following: Suppose $\sum_{i=1}^n x_i=0,\;\sum_{i=1}^n x_i^2=1$. Is it there a closed form for $\max \sum_{i=1}^n x_ix_{i+1}?$ ($x_{n+1}=x_1$) For ...
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Does every strongly convex function has a stationary point?

Say does every differentiable $\mu$-strongly convex function $f:\mathbb{R}^n\mapsto\mathbb{R}$, with $\mu>0$ have a point where its gradient is $0$? If not so which is the minimum you can impose ...
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46 views

Maximisation problem

I am trying the following question: If$$a+b+c+d=0,\;a^2+b^2+c^2+d^2=1$$ Then what is the maximum value of $ab+bc+cd+da?$ By the rearrangement inequality I can get $ab+bc+cd+da\leq 1$ but I am ...
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Example of a k-matroid

Let the set $K_i = (S, I_i)$ be a matroid for each $i \in \{1 \ldots k\}$. We define $K = (S, I) $ where $I = \{ X \subset S $ | $ X \in \bigcap_{i=1}^k I_i\}$ The claim now is that $K$ is a ...
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If a continuous function has one relative extremum, then it is actually an absolute extremum

The following statement makes sense intuitively, but is there a way to prove it mathematically? (This is something we make use of in applied optimization in calculus.) If $f$ is continuous on an ...
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499 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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Help with Gram-Schmidt problem

I'm supposed to show that the Gram-Schmidt process: $\textbf{a}_j = \left\{ \begin{array}{lr} \textbf{d}_j, \;\;\textbf{if} \;\;\lambda_j = 0\\ \sum_{i=j}^n \lambda_i\textbf{d}_i ...
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How can I use Mehrotra's predictor-corrector primal-dual interior point method to solve a problem that is not in the form of cTx?

I am not very familiar with optimization methods. I am studying the paper "Blind channel identification for speech dereverberation using l1-norm sparse learning" (here: http://linyq.com/NIPS2007.pdf). ...
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19 views

max and minimum qudratic function problem

A piece of wire $20$ metres long is cut into $2$ pieces and each piece is bent to form a square. Determine the length of the two pieces so that the sum of the areas of the two squares is a minimum. ...
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49 views

Optimization to minimize cost using the function C=Tq^(1/a)+F

I was given the function of $C=Tq^{1/a}+F$ where $C$ is total cost, $q$ is output, $a$ is a positive parametric constant, $F$ is the fixed cost, and $T$ measures the technology available (also a ...
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42 views

Optimization Problem (Linear Algebra)

I am not trying to cheat or anything, so any reference to online literature or MOOCs, that teach this stuff, will be highly appreciated. The problem is to prove that the following optimization ...
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Could someone explain the Lagrangian Method?

I understand the method, technically, but what is actually going on? We set the gradient of the function equal to the gradient of the constraint (multiplied by a constant), and in doing so, we find ...
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How to approach this optimization problem with “sorted” constraint

I have formulated an optimization problem and I'm not sure how to go about solving it intelligently. I have three vectors $ a, p, n$, all with the same number of elements. I know what $p$ and $n$ are ...
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355 views

Convex Optimization of quadratic function with inequality constraints

How would I solve the following problem? $$\min_{x\in\mathbb{R}^n} x^T A x$$ subject to the constraints $$x_i\geq 1,\,i=1,\dots,n,$$ where A is positive semidefinite and symmetric. Is it possible to ...
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simple problem of calculus.

A company wishes to manufacture a box with a volume of $36ft^3$ that is open on top and twice as long as it is wide.Find the dimensions of the box produced from the minimum amount of material. My ...
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331 views

A variation of the Assignment Problem

In the following Wikipedia article about the Assignment Problem in the Example section, it says: Similar tricks can be played in order to allow more tasks than agents, tasks to which multiple ...
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927 views

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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Energy Function for Optimization with Time-Dependent Inputs?

I am working through a paper on energy functions for optimization and having some trouble understanding the notation. The author derives an E function for a neural network that is a function of both ...
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SOCP or SDP optimization problem

I am studying an optimization problem \begin{equation} \mathbf{w}^* = \text{argmax} \sum_{d=1}^D \log \bigg( \frac{|\mathbf{f}_d^H\mathbf{w}|^2+c_1}{|\mathbf{f}_d^H\mathbf{w}|^2+c_2} \bigg)\\ \\ ...
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maximization of a particular ratio

We are given a ratio: $$\frac{g(x)}{f(x)}$$ where: $$g(x) \in \mathbb{R}^{+}$$ $$f(x) \in \mathbb{N}\: \cap f(x)\ge 2$$ So $g(x)$ returns values in $[0,+\infty]$ while $f(x)$ returns values in ...
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How can the max-flow and min-cut problems, if dual to one another, both have unbounded optimal value?

The max-flow min-cut theorem states that the value of the maximum flow is equal to the minimum cut capacity. It is possible that the max-flow and min-cut is equal to $\infty$. However, reading ...
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Property of Newton step for self-concordant convex functions

Suppose $f(x)$ is a convex and self-concordant function minimized at $x^*$. I have two starting points $\tilde{x}_0$ and $\hat{x}_0$ such that $|\hat{x}_0-x^*| \le |\tilde{x}_0 - x^*|$. We also know ...
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24 views

Geometric interpretation of duality and Slater's condition

I am trying to study about optimization problems, Lagrange duality and related topics. I came across some presentation on the net, which claims to show the geometric interpretation of the duality and ...
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65 views

Optimization of a Cylinder In a Sphere WITHOUT Using Calculus

I have a quick question. I'm curious as to how to do an optimization question WITHOUT using calculus. Question: Determine the dimensions of the cylinder of maximum volume that can be inscribed in a ...
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48 views

Which is the better way to optimize a function with 3 variables

I have an optimization function depends on 3 parameters a, b, and c. Which is the better way to optimize it? ...
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a calculus optimization problem

Given points A(2,1) and B(5,4), find the point on the x-axis P(x,0) in the interval [2,5] that maximizes the angle APB. How can I devise an optimize equation and a constraint equation out of this?
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Newton's method for unconstrained optimization applied to a quartic function in R2

I am faced with the task of applying Newton's method to the following problem: $$ \text{min} ~~~~~ 8x_1x_2+\frac{1}{4}(x_1-x_2)^4 $$ where $x \in \mathbb{R}^2$. For clarification, the Newton method ...
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Optimization for minimum cost, with the total cost function $C=TQ^{1/a} + F$

I have the function $C=TQ^{1/a} + F$. Where C is total cost, Q is output, a is a positive parametric constant, F is fixed cost, and T measures the technology available to the firm (Parameter). We also ...
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Methods to select $m$ objects out of $m$ that minimize a function

I have a set of $n$ points $ x_i, i \in I = \{ 1, \ldots, n \}$ and I want to find $m \ll n$ points, $x_m, m \in M \subset I$, that minimize a cost function $ J = f(x_m) $. What is the name of the ...
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Simple Optimization Problem with linear Algebra

I'm asked to find that the solution of $\displaystyle S(\mathbf{c})=\max_{\mathbf{c}}\frac{\mathbf{X' Z c}}{||\mathbf{X}||\cdot||\mathbf{Z c}||}$, where $\mathbf{X}$ is a $n\times1$ vector, and ...
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65 views

Maximizing total tax revenue with function Qs+-8+P and Qd=(80/3)-(1/3P)

The supply and demand equations of a good are given by Qs= -8+P Qd=(80/3) - (1/3)P P is measured in dollars. Suppose the government decides to impose a constant per unit tax of $t on the supplier. ...
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Distribute small number of points on a disc

Firstly I strongly know how many similar questions there are here. It's about sets of evenly distributed points inside a circle. If we need a big set of such points, good solutions are: Isocell ...
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32 views

Finding critical points of a multivariable function

Let $f(x,y)=e^{x^2-xy+y^2}$ (a) Find all the critical points of the following function. (b) Find the all the local maxima and local minima of the function if there is any. What i tried. I tried ...
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Duality in quadratically constrained quadratic program

I have been given the primal quadratic program with a single quadratic constraint as given below: $$ \text{min} ~~~~~~~~~~~~~~~~~~~~~~~~~ \frac{1}{2}x^{T}Qx $$ \begin{align*} \text{subject ...
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norm over differentiable functions computable from derivatives only

I'm running an optimization algorithm, minimizing the norm $||f-\hat f||$ of some objective function $f(x_1,x_2,x_3,y_1,y_2,y_3)$. The function $f$ cannot be computed directly, but its second ...
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39 views

Optimization to minimize cost function

I have the function $C=Tq^{\frac 1a }+F$. Where $C$ is total cost, $q$ is output, $a$ is a positive parametric constant, $F$ is fixed cost, and $T$ measures the technology available to the firm ...
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1answer
24 views

Optimization problem $L(R, PQ) \rightarrow \min$

Suppose we have some $n \times m$ matrix $R$ and we want to find non-negative decomposition on matrices $P$ of dimension $n \times d$ and $d \times m$-matrix $Q$. But since exact decomposition usually ...
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Finding max and min of function using a constraint equation.

I was presented with a problem in my linear algebra course but I haven't taken any calculus for awhile and can't seem to remember how to solve a problem like this. Here is the problem: Suppose T is a ...
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56 views

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 is ...
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Necessary condition of optimality for functionals

Let $C(a, b)$ denote the set of all surjective and continuously differentiable functions $\alpha:[a, b] \rightarrow [a, b]$. Consider the functional on $C(a, b)$ $$ F[\alpha(t)] = \int_a^b ...
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Faster gradient descent convergence by transforming the gradient?

If we modify the gradient descent update for a convex objective function $f(\boldsymbol{\theta})$ from $\boldsymbol{\theta}_{t+1} = \boldsymbol{\theta}_t - \nabla f(\boldsymbol{\theta}_t)$ to ...
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70 views

Equilibrium to maximize total tax revenue

The supply and demand equations of a good are given by Qs= -8+P Qd=(80/3) - (1/3)P P is measured in dollars. Suppose the government decides to impose a constant per unit tax of $t on the supplier. ...
0
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0answers
36 views

Optimization for minimizing average cost

I was given the function of $C=Tq^{\frac{1}{a}} + F$ where $C$ is total cost, $q$ is output, $a$ is a positive parametric constant, $F$ is the fixed cost, and $T$ measures the technology available ...
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mathematical model of an assignment/scheduling problem

I am solving a scheduling problem and I am able to abstract it into an assignment problem of assigning 45 machines to 42 jobs. the assignment problem was given as having 14 jobs, each with 3 tasks and ...
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Solving the problem of Affinity using Linear Programming

The affinity problem states that when we have a set of requested instances to be launched on a set of hosts, the placement of instances should be such that they must be close to each other. There can ...
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Why AM-GM gives us the lowest value?

I know, that by AM-GM we can sometimes find the lowest value (minimize) of some expressions. For example: Given that $x$, $y$ and $z$ are positive real numbers satisfying $xyz=32$, find the minimum ...