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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2
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2answers
20 views

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 ...
0
votes
1answer
30 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 ...
0
votes
0answers
4 views

Project a function on a space?

The problem I'm solving is $\begin{cases} & \dot{x}_{1} = -u \\ & \dot{x}_{2} = 4x_{1} \end{cases}$ $x_{1}(0) = x_{2}(0)=0, |u| \leq 3, t \in [0;2], u^{0}(t)\equiv0, J[u] = -2x_{1}(2) + ...
0
votes
1answer
20 views

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 ...
0
votes
0answers
18 views

0 to the power of 0 follow up question: Finding the “optimum curve” such that one can just avoid limiting into 1 or 0

Previously in $0^0$ shape of discontinuity we investigated the limiting behavior and shape of the singularity of the function $x^y$ as we approach the origin Now I am interested in trying to find the ...
0
votes
1answer
23 views

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 ...
0
votes
0answers
12 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 ...
0
votes
0answers
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 ...
0
votes
0answers
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)\\ \\ ...
1
vote
2answers
18 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 ...
0
votes
0answers
7 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 ...
0
votes
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 ...
0
votes
0answers
40 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? ...
0
votes
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). ...
0
votes
1answer
27 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?
0
votes
0answers
27 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 ...
1
vote
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 ...
0
votes
0answers
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 ...
0
votes
0answers
31 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, ...
1
vote
0answers
28 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 $x_1$. ...
4
votes
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 ...
1
vote
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 ...
0
votes
1answer
31 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 ...
0
votes
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
votes
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 ...
2
votes
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 ...
0
votes
1answer
30 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. ...
0
votes
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 ...
0
votes
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 ...
0
votes
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
votes
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 ...
0
votes
1answer
31 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
votes
1answer
35 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
votes
1answer
42 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
votes
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 ...
0
votes
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 ...
0
votes
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 ...
-1
votes
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
votes
1answer
52 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 ...
0
votes
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 ...
0
votes
1answer
28 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
votes
1answer
19 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 ...
1
vote
0answers
14 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
votes
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
votes
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 ...
1
vote
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: -- ...
0
votes
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
votes
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 $m\times n$ ...