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

Minimizing average cost through optimization

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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12 views

Optimization of competitive scenario

Suppose we have 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 $x_1$. $0 \le ...
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2answers
33 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
18 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
17 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
10 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 ?
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2answers
20 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
32 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
22 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
31 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
18 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 ...
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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
24 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
30 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 ...
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1answer
46 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 ...
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1answer
32 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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1answer
21 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
12 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
9 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
27 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
36 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 ...
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1answer
49 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 ...
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1answer
17 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
10 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
27 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
38 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 $m\times n$ ...
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0answers
29 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 ...
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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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2answers
222 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
14 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 ...
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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 ...
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0answers
15 views

Projection of n-Simplex into k-Simplex

I try to find properties of orthogonal projections such that a standard n-Simplex $S_n$ is projected into a k-Simplex $S_k (k\leq n)$. Literature provides work on "smallest projections" in this ...
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0answers
6 views

Optimization Formula for Specific Gravity Modification [on hold]

I have 200 gallons of apple juice with a specific gravity of 1.04. I would like to remove through freezing an amount of water (SG=1.000) such that the final SG of the juice equals y. I am assuming ...
0
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0answers
20 views

Minimize function

I am struggling since couple of days with a function that, according to my supervisor, is nice and easy. Now, I have to show this function is always bigger or equal to one, but I am far to the ...
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0answers
18 views

Simplex Algorithm go wrong [on hold]

When can Simplex Algorithm go wrong? Is there any other way to solve Simplex other than the traditional way of pivoting?
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0answers
40 views

Classify the degenerate point;

Given $$f(x,y)=y^4+y(x-1)^2-8y^2$$ Find the three critical points, use Hessian method to classify the two non degenerate points. Then ; By considering the value of $f$ along the curve ...
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2answers
69 views

Show this function has infinitely many critical points and classify;

Show that $$f(x,y)=x^2-x+\cos(xy)$$has inifinitely many critical points and classify; Partial Derivative w.r.t. $x$ $$f_{x}=2x-1-y\sin(xy)=0$$ Partial Derivative w.r.t. $y$ ...
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1answer
18 views

Tiling a rectangle with $n$ rectangles to maximize product of areas

Consider a rectangle $R$ with integer width and integer height. We want to tile $R$ using exactly $n$ rectangles with integer dimensions. Now we carefully want to choose such rectangles so that ...
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0answers
16 views

Optimizing Sum of Maximum value of Matrix Rows

I have somewhat of a unique optimization problem. I have a AxBx2 matrix. Think of the rows of AxB as the time axis, and the columns of AxB as actions to take. Every element of AxBx1 is the value ...
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1answer
31 views

maximize function with two variables

I would like to maximize the function: $f(x,y) = c[x\log(2y) + (1-x)\log(2(1-y))]$ subject to constraint that $x,y \in (0,1)$ to find a relationship between $x$ and $y$ that maximizes $f(x,y)$. My ...
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2answers
45 views

Calculus of optimization help ):

If I need to sell 400 chairs. The price per chair is 90 dollars up to and including 300 chairs. Above 300, the price will be reduced by 0.25$ (on the whole order) for every additional chair over 300 ...
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1answer
30 views

Finding if a function is Concave or Convex

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 ...
0
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1answer
11 views

Is there a way to analytically find a stationary point along an arbitrary line in a multivariable quadratic function?

Let's say I'm working with a quadratic function with an equation of $f(\mathbf{x}) = \frac{1}{2}\mathbf{x}^TA\mathbf{x} - b^T\mathbf{x}$. Now, let's take a direction $\mathbf{p}$ and transform the ...