Tagged Questions

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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8 views

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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9 views

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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16 views

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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2answers
17 views

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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0answers
5 views

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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1answer
22 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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0answers
38 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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0answers
11 views

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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1answer
25 views

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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26 views

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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0answers
21 views

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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12 views

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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0answers
30 views

Minimizing average cost through optimization [on hold]

A reasonably realistic model of a Firms cost is given by the short-run Cobb-Douglas cost curve: C=T(q^1/a)+F where C is total cost, q is output, a is positive parametric constant, F is the fixed cost, ...
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0answers
27 views

Optimization of competitive scenario

Suppose we have a function $f(x_1,x_2)$ with the following properties: Let $x^*=\arg \max_{x_1} f(x_1,x_2=x^*)$ and $x^*=\arg \min_{x_2}f(x_1=x^*,x_2)$. $f(x_1,x_2)$ is concave in $x_1$. ...
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1answer
58 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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1answer
26 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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1answer
26 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 parameter). ...
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0answers
12 views

slaters condition - Duality - KKT condition [on hold]

Can someone give a more intuitive idea to Slater's conditions and how it is related to KKT condition and duality ?
4
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2answers
38 views

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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0answers
39 views

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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1answer
28 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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0answers
35 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
23 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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0answers
23 views

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 ...
0
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1answer
19 views

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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1answer
29 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 ...
2
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1answer
33 views

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 ...
4
votes
1answer
51 views

Smallest possible triangle to contain a square

I was looking at this stack exchange question* and started thinking about the case of a polygon with 4 sides: a square. The question asks for a program that can take a polygon of N sides and return ...
0
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1answer
41 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
votes
1answer
23 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 ...
2
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0answers
18 views

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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0answers
10 views

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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0answers
28 views

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 ...
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0answers
42 views

Convergence au sens des compacts et au sens de Hausdorff / Compact and Hausdorff convergence

Montrer que, si $(\Omega_n)$ est une suite d'ouverts qui converge au sens des compacts vers un ouvert $\Omega$ et au sens de Hausdorff vers $\Omega '$, alors $\Omega \subset \Omega ' \subset ...
3
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1answer
51 views

Finding lowest possible value

please help me with this problem: Find the lowest possible value of $$ x+y^3 $$ where both x and y are positive and x*y=1. I know how to solve this one using my method, but I was suggested to use ...
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1answer
14 views

How to tell of extrema lies on the boundary or interior of a function? (Lagrange Multiplier)

For example: Q: Find the extreme values of f(x,y,z) = x + yz on the solid ellipsoid x^2+2y^2+8z^2 <= 32. The solution manual does: " f_x = 1 not equal 0, f has no critical points. -> all ...
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1answer
26 views

How to maximize the volume of a cylinder with no top

A cylindrical can without a top is made using $A \text{ cm}^2$ of material. Find the dimensions that will maximize the volume of the can. What I have done was similar to the question: Optimization ...
0
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1answer
18 views

Minimizing an open box (Calc I)

A rectangular container with an open top is to have a volume of $12 \;\text{m}^3$. The length of its base is twice the width. Material for the base costs (in dollars) 10/$\text{m}^2$. Material for ...
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0answers
12 views

Determine whether it's min or max of functional.

so I have such functional: $$\phi(y)=\int\limits_0^1 (y^2+2y'^2+y''^2)dx, \ \ y(0)=y(1)=0, \ y'(0)=1, \ y'(1) = -\sinh1.$$ By using Euler-Lagrange formula, I get $$y^{IV} - 2y'' + y = 0$$ After ...
3
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1answer
28 views

Lagrange multipliers: More than one constraint

I have more or less understood the underlying theory of the Lagrange multiplier method (by using the Implicit Function Theorem). Now, I try to extend this understanding to the general case, where we ...
0
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0answers
9 views

Necessary and sufficient conditions for a feasible Linear Programme

I am trying to solve the following problem. I have set up the dual, and drawn a graph of the dual. I know solutions must be in the first quadrant as $ x\ge0$ but I don't know how to complete the ...
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4answers
41 views

Optimization problem - Trapezoid under a parabola

recently I've been working on a problem from a textbook about Optimization. The result that I get is $k = 8$, even thought the answer from the textbook is $k = \frac{32}{3}$ The problem follows: -- ...
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0answers
22 views

Stuck with an optimization problem with 2 constraints (Lagrangian multiplier method)

I am really stuck with a certain minimization task. I thought I would understand the Lagrangian multiplier method (at least I could solve simple 2-variable optimization problems with 1 constraint). ...
2
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1answer
24 views

Linear Programming Problem - Looking for an Explicit Solution

How can I solve a linear program of the form: $$\min c^Tx\\ \mathrm{s.t.}\ Ax=b\\ x\geq0\\$$ where $c$ is fixed. In the specific case I am looking at, $$x \in R^n$$ $A$ is an $m\times n$ ...
1
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1answer
36 views

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 ...
1
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0answers
28 views

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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0answers
15 views
20
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3answers
3k views
+50

The Stupid Computer Problem : can every polynomial be written with only one $x$?

When I was a child, I wanted to be a mathematician so I asked my parents to buy me a computer to make super complex calculations. Of course, they were not crazy enough to buy an expensive super ...
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0answers
15 views

Spectral methods with linear programming

Is it possible to model and solve some fundamental spectral methods (say Singular-Value Decomposition) with (Integer?) Linear Programming? Update: say you want to do SVD. Can you model it as a ...
1
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1answer
11 views

Optimization Question Regarding proving Maximum Area of Window

A window has the shape of a rectangle of height $h$ surmounted by a semi-circle of radius $r$. The area of the window is given by $A = 2rh + \frac{1}{2}\pi r^{2}$ NOT drawn to scale. If the ...