Tagged Questions
1
vote
1answer
28 views
What's wrong with this Kuhn-Tucker optimization?
The function $u(x,y,z) = xyz$ is to be maximized, under constraints: $ 0 \le x \le 1, y \ge 2, z \ge 0 $ and $ 4 - x - y - z \ge 0 $
Now I'm not quite sure how to translate the x-constraint into ...
1
vote
2answers
38 views
Solve the Lagrangian dual problem
Consider the (non-linear) optimization problem ($P$)
$$max \quad3x_1 + 4x_2$$
$$s.t. \quad x_1^2 + x_2^2 \leq 25$$
$$ \quad x_1,x_2 \geq 0$$
Solve the Lagrangian dual problem.
I ...
0
votes
2answers
29 views
Using Lagrange multiplier to find maximum value.
The maximum value of the function $f(x, y) = xy$, and subject to condition $x^2+y^2=1$:
So do I apply Lagrange's Multiplier method to find the maximum value?
I tried to find the numbers just by ...
0
votes
1answer
43 views
Prove or disprove that Y = AX-C
Let $A$ be an $m \times n$ matrix such that $\mathrm{rank}(A) = n \le m$. Prove, or disprove using a counter example:
Every $m\times n$ matrix $Y$ has a decomposition $Y = AX-C$, where $X$ and ...
-1
votes
0answers
35 views
A maximization problem [closed]
The problem is as follows:
max α ln x + (1 - α ) ln y subject to $p_x$x + $p_y$y $\leq$ I, where $0 < \alpha < 1$.
(Notice that ln $(x^{\alpha}y^{1-\alpha})$ = α ln x + (1 - α ) ln y)
1
vote
1answer
72 views
Cost minimization problem
The problem is as follows:
A firm uses $k$ units of capital and $l$ units of labor to produce
$(k^{\alpha}l^{1-\alpha})^{1/\beta}$
units of output, where $\alpha$ and $\beta$ satisfying $0 < ...
1
vote
0answers
36 views
Optimization problem (rectangle inscribed in a right triangle)
I am stuck on this optimization problem: http://imgur.com/lpKmxvh . I am supposed to find the largest rectangle that can be fit into the right triangle. I think I am having trouble with setting up the ...
1
vote
1answer
57 views
Maximum likelihood estimators, hypergeometric and binomial
I'm trying to solve a two part problem. The set up is as follows: consider a bag with $\theta$ red marbles and $7-\theta$ blue marbles, with $\theta$ being unknown. Let $x$ denote the number of red ...
1
vote
0answers
69 views
Home work on linear optimization problem
I'm sorry I could not upload my homework because I'm too new to post images. The image can be obtained here: http://i40.tinypic.com/2u5rl80.png.
I'm also sorry to ask such a big question. The ...
0
votes
1answer
28 views
Show using duality that exactly one of the following systems has a solution
(I) $Ax=b$ ; $0≤ x ≤e$
(II) $uA +v ≥0 ; ub + ve = -1 ; v ≥ 0$
1
vote
0answers
33 views
Prove mathematically
Q.1 Consider the dual simplex method applied to a standard form problem with linearly independent rows. Suppose we have a basis which is primal infeasible, but dual feasible, and let i be such that ...
-1
votes
2answers
48 views
Minimum ladder over wall optimization
A fence 6 feet tall runs parallel to a tall building at a distance of 2 feet from the building. We want to find the the length of the shortest ladder that will reach from the ground over the fence to ...
0
votes
1answer
20 views
Strict local minimiser
Let $\Omega$ be a convex subset of $R^n$ adm f is a real valued, twice differentiable function. Let $x^*$ to be a point in $\Omega$ and suppose that there exists $c \in R \, c >0$ s.t. for all ...
1
vote
1answer
65 views
Optimization: Minimize cost of pipeline
A small resort is situated on an island off a part of the coast of Mexico that has a perfectly straight north-south shoreline. The point P on the shoreline that is closest to the island is exactly 6 ...
0
votes
1answer
48 views
Optimization and distance (minimum time)
A small island is 5 miles from the nearest point P on the straight shoreline of a large lake. If a woman on the island can row a boat 2 miles per hour and can walk 3 miles per hour, where should the ...
2
votes
1answer
43 views
Minimum distance between $x = -y^2$ and $(0,-3)$
Find the minimum distance from the parabola $x + y^2 = 0$ (i.e. $x = -y^2$) to the point $(0,-3)$.
This is a homework question. When I try to use the derivative and substitute $-y^2$ for $x$, I ...
0
votes
2answers
44 views
Optimization and window area
A Norman window has the shape of a rectangle with a semi circle on top; diameter of the semicircle exactly matches the width of the rectangle. Find the dimensions of the Norman window whose perimeter ...
0
votes
2answers
29 views
Optimization and Rent
The manager of a large apartment complex knows from experience that 110 units will be occupied if the rent is 342 dollars per month. A market survey suggests that, on the average, one additional unit ...
1
vote
1answer
22 views
Optimization and fence size
A fence is to be built to enclose a rectangular area of 250 square feet. The fence along three sides is to be made of material that costs 6 dollars per foot, and the material for the fourth side costs ...
1
vote
0answers
42 views
Formulation of a problem as semidefinite programming
I would appreciate some help with this problem:
$R$ is a positive semidefinite matrix $\in{R}^{n\times n}$, $A \in{R}^{n\times m}$.
I need to formulate this optimization problem as semidefinite ...
8
votes
2answers
85 views
How to find the minimum of f(x)?
I need to find the minimum of $f(x)$ with
$$f(x)=(\sin(x)+\cos(x)+\tan(x)+\cot(x)+\sec(x)+\csc(x))^2$$
Could you help me with some clues?
2
votes
1answer
75 views
Finding the max. of an integral
I have a question which asks:
Let $g\in C[-1,1]$ and the usual inner product $\langle f,g\rangle = \int_{-1}^{1} f(x)g(x)dx$.
Find the max value of $\int_{-1}^{1}g(x)x^3dx$ where $g$ is subject to ...
2
votes
1answer
42 views
Determining the Existence of Global Minimum/Maximum
Determine whether the function defined as $$f(x,y,z)=x+y+z$$ has a maximum or a minimum value on the set $xy+yz=1$, $xz+yz=4$, $x>0$, $y>0$, $z>0$.
It is clear to me that it does have a ...
3
votes
2answers
62 views
In Search of a More Elegant Solution
I was asked to determine the maximum and minimum value of $$f(x,y,z)=(3x+4y+5z^{2})e^{-x^{2}-y^{2}-z^{2}}$$ on $\mathbb{R}^{3}$.
Now, I employed the usually strategy; in other words calculating the ...
1
vote
1answer
59 views
Tricky algebra for minimization
Find the local minimum for $f(x, y) = 2x^4 + y^2 - 4xy + 5y,\:x,y \in \mathbb{R}$ find the local minimum.
Okay this seems easy enough, the necessary condition dictates that candidates are of the form ...
3
votes
2answers
93 views
Optimisation problem choose x to minimize y
I have stumbled upon a sample maths question during my revision, and I have no idea how to solve it. Can anyone help or guide me along?
Given a piece of rectangular paper of 11 cm by 8.5 cm. The ...
0
votes
1answer
79 views
Linear programming problem
Some additional information:
In the next season the harvesting amount is estimated at 900 for farm A, 1200, 1500, 1800 for farm B,C and D respectively.
In this scenario I'm asked to minimize the ...
1
vote
1answer
46 views
minima of $\frac{(1-k)x\log(x^2-x)}{(1-k')(x-1)\log x^2}$
Can anyone help me in finding minima of $\frac{(1-k)x\log(x^2-x)}{(1-k')(x-1)\log x^2}$ where $k$ and $k'$, are constants. I found the differential but it was too big to be equated.
...
7
votes
1answer
194 views
Graph theory problem (edge-disjoint matchings)
Find the smallest number $x$ so that if an $n$-vertex simple graph has at least $x$ edges then it contains $k$ pairwise edge-disjoint perfect matchings* ($k$ is a positive integer, $n$ is an even ...
0
votes
0answers
48 views
0
votes
0answers
18 views
two or more negative cases for solving integer programming with Gröbner basis
Below are maple code, i have tried several cases for two more negatives, still failed.
how to treat two or more negative coefficient in order to solve with Gröbner basis
Two negative cases
...
1
vote
0answers
59 views
Minimum value of given expression?
You have two lists of $N$ integers $x_{i}$ and $y_{i}$ where $i\in\{ 0,1,\dots,N-1\}$. We know the value of $x_{0}$ and $y_{0}$, remaining $N-1$ values is calculated using formula given below:
...
1
vote
1answer
66 views
Optimization, two numbers
The sum of two nonnegative numbers is 36. Find the numbers if
A) the difference of their square roots is to be as large as possible.
B) the sum of their square roots is to be as large as possible.
...
1
vote
1answer
72 views
Properties of shortest walks and simple paths during optimization
Let $G=(V,E)$ denote a digraph, $s,t\in V$ two different vertices in $G$ and $w:E\to\mathbb R$ the weighting function for all edges. Moreover $\mathcal K$ denotes the set of all walks, $\mathcal E$ ...
1
vote
0answers
112 views
Find local and global minimizers
Consider the problem:
minimize $f(x)=x_1$
subject to $x_1^2+x_2^2≤4$, $x_1^2≥1$.
Find all local minimizers for the problem, and determine which of those are also global minimizers.
My answer: By ...
6
votes
6answers
188 views
Optimal ladder position to maximize height reached
Suppose we have a ladder (with a unit length) and we want to position it to reach the highest point possible on a vertical wall. The terrain is sloped and can be described with a variable k. What is ...
4
votes
1answer
112 views
Kuhn-Tucker condition is not satisfied
Show that the solution to finding minimum of
$f(x)=-x_{1}$
With conditions
$-\sin(x_{1})+x_{2} \leq 0$
$x_{1}-x_{2} \leq 0$
is point $(0,0)$, but the Kuhn-Tucker condition is not satisfied in this ...
2
votes
1answer
63 views
Approximating a function with a convex function
Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a continuous, differentiable function. Is there a known algorithm that fits $f$ with $g$, which is an order-$n$ polynomial that is convex, in the least ...
8
votes
2answers
299 views
Optimization of the area of a cross inscribed in a circle
I've really been scratching my head over this optimization problem. "Consider a symmetric cross inscribed in a circle of radius $r$." The length from the center of the cross to the middle of one of ...
1
vote
0answers
123 views
Linear programming: writing a problem with artificial variables?
Use artificial variables to write a linear programming problem in canonical form with non-negative resource vector whose solution will determine whether there exists (and if so, find) non-negative ...
1
vote
1answer
108 views
Using Lagrange multipliers for restricted extrema
Consider the function $f(x,y) = x^2 + xy + y^2$ defined on the unit disc $D = \{(x,y) \mid x^2 + y^2 \leq 1\}$.
I can not simplify the equations to the point where I find a constant for the lagrange ...
0
votes
0answers
33 views
Constrained Extrema: Questions about derived solution.
I have $f(x,y,z) = x + y + 2z$. I'm constraining it with $E = \{(x,y,z) \in \mathbb{R}| x>0, y>0, z>0, x*y*z =5 \}$ I find the gradient of $f$ to be $<1,1,2>$ and then the gradient of ...
0
votes
0answers
30 views
checking whether or not a function is concave
Given $t$ as a random variable with a smooth distribution $f(t)$ which is independent of $y_{i}$ and a function
$f(x)=g(y₁)-αE[max(y₁,y₂)]-βE[max(y₁+t-x,0)]$
with the following properties:
...
1
vote
1answer
139 views
A sufficient condition for a unique maximum of the product of two concave functions
Given two concave functions $f(x)$ and $g(x)$, what conditions in terms of these functions can ensure that
$h(x)=f(x)g(x)$
have a unique maximizer on an interval $[a,b]$ for $a<b$?
1
vote
0answers
99 views
sufficient condition for KKT problems
For the Karush-Kuhn Tucker optimsation problem, Wikipedia notes that:
"The necessary conditions are sufficient for optimality if the objective function f and the inequality constraints g_j are ...
2
votes
2answers
81 views
Is this question erroneous? (Stationary points)
Using the second partial derivative test, I have found (-1,1) to be a saddle point but this option is not available in the MCQ. Have I made a mistake?
The person who set the question insists ...
2
votes
1answer
141 views
Minimise the entropy of a probability vector using Lagrange multipliers
Problem statement:
The entropy of a probability vector $ p = (p_1, ... , p_n)^T $ is defined as $ H(p)= - \sum\limits_{i=1}^{n} p_i \log{p_i} $,
subject to $ \sum\limits_{i=1}^{n} p_i = 1 \mbox{ ...
3
votes
0answers
79 views
Solving Linear Inequalities for Optimization
I want the max of:
$100-(2x_1+3x_2+4x_3+5x_4+6x_5+7x_6)$
I am given 5 inequalities:
$x_1+x_4\le6$
$x_2+x_5\le8$
$x_3+x_6\le7$
$x_1+x_2+x_3\le9$
$x_4+x_5+x_6\le11$
and
...
0
votes
1answer
399 views
proving kantorovich inequality
Nevermind! I worked it out. This question seems really silly now.
I am working through the proof of the kantorovich inequality on pages 6 and 7 of the following lecture notes: u.mich optimisation ...
0
votes
1answer
95 views
Prove that an optimal solution $x^*$ of the problem 1 $\min f(x)$ s.t $x\in \mathbb{R}^n$ and..
Prove that an optimal solution $x^*$ of the problem 1 $\min f(x)$ s.t $x\in \mathbb{R}^n$ and an optimal solution $(\bar{x},\bar{z})$ of the problem 2 $\min z $ s.t $z\ge f(x)\,, x\in \mathbb{R}^n$ ...
