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.

learn more… | top users | synonyms (5)

1
vote
1answer
63 views

Minimization of Variational - Total Variation (TV) Deblurring

Under the Linear Blurring Model - $ f = H \ast u $. I'm trying to calculate the Euler Lagrange of with respect to $ u $ of the functional: $$ E \left( u \right) = {\left\| f - H \ast u ...
12
votes
4answers
4k views

Gradient Descent with constraints

In order to find the local minima of a scalar function $p(x), x\in \mathbb{R}^3$, I know we can use the gradient descent method: $$x_{k+1}=x_k-\alpha_k \nabla_xp(x)$$ where $\alpha_k$ is the step size ...
5
votes
0answers
538 views

Maximum and minimum of an integral under integral constraints.

Find the maximum and minimum of the following integral in terms of $f(x),a,C$: \begin{align}I=\int_{0}^{a} \frac{x}{f(x)}p(x)dx \end{align} s.t.: 1) $\int_{0}^{a} p(x)dx=1$ 2) $\int_{0}^{a} ...
4
votes
3answers
192 views

How do you prove ${n \choose k}$ is maximum when k is $ \lceil \frac n2 \rceil$ or $ \lfloor \frac n2\rfloor $?

How do you prove ${n \choose k}$ is maximum when k is $ \lceil \frac{n}{2} \rceil $ or $ \lfloor \frac{n}{2} \rfloor$ ? This link provides a proof of sorts but it is not satisfying. From what I ...
8
votes
6answers
4k views

Linear Programming Books

Do you know of a good book on linear programming? To be more specific, i am taking linear optimization class and my textbook sucks. Teacher is not too involved in this class so can't get too much help ...
14
votes
4answers
559 views

How find this inequality $\max{\left(\min{\left(|a-b|,|b-c|,|c-d|,|d-e|,|e-a|\right)}\right)}$

let $a,b,c,d,e\in R$,and such $$a^2+b^2+c^2+d^2+e^2=1$$ find this value $$A=\max{\left(\min{\left(|a-b|,|b-c|,|c-d|,|d-e|,|e-a|\right)}\right)}$$ I use computer have this $$A=\dfrac{2}{\sqrt{10}}$$ ...
6
votes
1answer
1k views

Trace minimization with constraints

For positive semi-definite matrices $A,B$, how can I find an $X$ that minimizes $\text{Trace}(AX^TBX$) under 'either' one of these constraints: a) Sum of squares of Euclidean-distances between pairs ...
13
votes
7answers
3k views

How to prove the sum of squares is minimum?

Given $n$ positive values. Their sum is $k$. $$ x_1 + x_2 + \cdots + x_n = k $$ The sum of their squares is defined as: $$ x_1^2 + x_2^2 + \cdots + x_n^2 $$ I think that the sum of squares is ...
10
votes
4answers
2k views

Looking to understand the rationale for money denomination

Money is typically denominated in a way that allows for a greedy algorithm when computing a given amount $s$ as a sum of denominations $d_i$ of coins or bills: $$ s = \sum_{i=1}^k n_i ...
4
votes
4answers
190 views

Not so easy optimization of variables?

What is the maximum value of $x^2+y^2$, where $(x,y)$ are solutions to $2x^2+5xy+3y^2=2$ and $6x^2+8xy+4y^2=3$. (calculus is not allowed). I tried everything I could but whenever I got for example ...
1
vote
0answers
73 views

optimization problem gaussian maximizes entropy

Let $X_1, X_2, Z_1$ be random variables and define $$Y=aX_1+bX_2+Z_1$$ I have the following optimization problem of difference of entropies, $$f=\max_{p(x_1x_2)} h(Y) - h(Y|X_2)= \max_{p(x_1,x_2)} ...
0
votes
2answers
206 views

Essential problem

I want to find one method or approach or idea which compute following statement: $$ \sup_{t \in [0,1]} \left( \inf_{X \in C^1([0,1])} \left\| \frac{dX(t)}{dt} - A(t)X(t) - F(t) \right\| \right) $$ ...
17
votes
5answers
6k views

How do Lagrange multipliers work to find the lowest value of a function subject to a constraint?

I have been using Lagrange multipliers in constrained optimization problems, but I don't see how they actually work to simultaneously satisfy the constraint and find the lowest possible value of an ...
5
votes
3answers
1k views

Maximal order of an element in a symmetric group

If we let $S_n$ denote the symmetric group on $n$ letters, then any element in $S_n$ can be written as the product of disjoint cycles, and for $k$ disjoint cycles, $\sigma_1,\sigma_2,\ldots,\sigma_k$, ...
4
votes
1answer
426 views

Duality. Is this the correct Dual to this Primal L.P.?

Given a problem: Find the dual: $$ Primal =\begin{Bmatrix} max \ \ \ \ 5x_1 - 6x_2 \\ s.t. \ \ \ \ 2x_1 -x_2 = 1\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x_1 +3x_2 \leq9\\ ...
1
vote
2answers
2k views

Derivation of soft thresholding operator

I was going through the derivation of soft threholding at http://dl.dropboxusercontent.com/u/22893361/papers/Soft%20Threshold%20Proof.pdf. It says the three unique solutions for $\operatorname{arg ...
9
votes
1answer
192 views

Minimum of $|az_x-bz_y|$

I am trying to minimize the following function: \begin{align} &f(z_x,z_y)=|az_x-bz_y| \\ &\text{ s.t. } z_x,z_y \in \mathbb{Z},1 \le z_x \le N_x \text{ and } 1 \le z_y \le N_y \text{ and } ...
5
votes
1answer
228 views

On minimizing the area of an enclosing surface subject to nonnegative Gaussian curvature

This is inspired by this previous question on physical processes that might give rise to convex hulls. Consider the problem of gift-wrapping a three-dimensional object using an inextensible material, ...
4
votes
2answers
2k views

Find the maximum of $f(x)=x^{1/x}$

Find the maximum of the function $$f(x)=x^{1/x}$$ and the value of $x$ which gives the maximum value?
3
votes
2answers
629 views

Finding the widest angle to shoot a soccer ball from the sideline using optimization

I'm trying to do an independent project for my Math class, but I was stuck and couldn't figure out how to use optimization to find position along the sideline that gives the widest angle to shoot. ...
3
votes
1answer
109 views

minimizing $\sum_{i=1}^n \max(|x_i - x|, |y_i - y|)$

Suppose there are $n$ points $(x_i, y_i)$ for $i = 1,\ldots,n$. Please find another point $(x, y)$ to minimize function: $$\sum_{i=1}^n \max(|x_i - x|, |y_i - y|)$$
2
votes
2answers
3k views

Maximizing volume of a rectangular solid, given surface area

Maximize the volume of a rectangular solid, given that the sum of the areas of the six faces is $6a^2$ for a constant $a$. So basically they tell you it's a rectangle with 6 sides. 2 sides are ...
1
vote
2answers
136 views

Summation notation problem

Any help is greatly appreciated! Outline: Hermione has been thinking about the imminent return of the Dark Lord, so she has been busy packing her bag with all the items required for her survival. ...
0
votes
1answer
69 views

Trust region sub-problem with Jacobi Condition

Consider the $2 \times 2$ trust region sub-problem. Given $Q$ symmetric $2 \times 2$, vector $\mathbf b$ and $\Delta > 0$, find $\mathbf x$ that minimizes $f(x)=\frac {1}{2} \mathbf x^T Q \mathbf x ...
0
votes
1answer
203 views

Surf Rescue - word problem - pythagoras

Angela works at the local beach as a part of a surf patrol unit. During her patrol, she notices that an old lady is having difficulty in the surf. She estimates that the distance is approxiamtely 220 ...
39
votes
4answers
1k views

AM-GM-HM Triplets

I want to understand what values can be simultaneously attained as the arithmetic (AM), geometric (GM), and harmonic (HM) means of finite sequences of positive real numbers. Precisely, for what points ...
11
votes
3answers
7k views

Lagrange Multipliers with Inequality Constraints

I do not have much experience with constrained optimization, but I am hoping that you can help. My current problem involves a more complex function, but the constraints are similar to the ones below. ...
9
votes
2answers
1k views

Euler-Lagrange, Gradient Descent, Heat Equation and Image Denoising

For an image denoising problem, the author has a functional $E$ defined $$E(u) = \iint_\Omega F \;\mathrm d\Omega$$ which he wants to minimize. $F$ is defined as $$F = \|\nabla u \|^2 = u_x^2 + ...
15
votes
2answers
456 views

Does a polynomial that's bounded below have a global minimum?

Must a polynomial function $f \in \mathbb{R}[x_1, \ldots, x_n]$ that's lower bounded by some $\lambda \in \mathbb{R}$ have a global minimum over $\mathbb{R}^n$?
9
votes
3answers
2k views

Understanding the concept behind the Lagrangian multiplier

I've been trying to understand the principles behind the Lagrangian multipliers and I think I've got a rough understanding of it. Would appreciate it if you guys could help me answer a few questions! ...
10
votes
2answers
429 views

Explain a surprisingly simple optimization result

The following optimization problem came to my attention as an idealization of the silly browser game Cookie Clicker, but is representative of a range of strategy games: You have an initial ...
3
votes
2answers
158 views

Find maximum of $P$

Let $$P = \frac{{{x^2}}}{{{x^2} + yz + x + 1}} + \frac{{y + z}}{{x + y + z + 1}} - \frac{{1 + yz}}{9}.$$ Find maximum of $P$ where $x, y,z$ are nonnegative real numbers such that ${x^2} + {y^2} + ...
5
votes
5answers
441 views

How to find the minimum of $a+b+\sqrt{a^2+b^2}$

let $a,b>0$, and such $$\dfrac{2}{a}+\dfrac{1}{b}=1$$ Find this minimum $$a+b+\sqrt{a^2+b^2}$$ My try: since $$2b+a=ab$$ so ...
5
votes
3answers
4k views

Primal- degenerate optimal, Dual - unique optimal

Simple question- Is it possible for a linear programming optimization problem possible to have a degenerate optimal solution whereas the dual has a unique optimal solution? I can't find a scenario ...
3
votes
2answers
176 views

How is $L_{2}$ Minkowski norm different from $L^{2}$ norm?

I am reading the book Multidimensional Particle Swarm Optimization for Machine Learning and Pattern Recognition. They use $L_{2}$ Minkowski norm (Euclidean) as the distance metric in the feature space ...
2
votes
1answer
134 views

Maximum of the sum of cube

(1) $-2\leq a_{i} \leq 2$ $~(i=1,2,3,4,5)$ (2) $\displaystyle\sum_{cyclic}a_{i}=0$ then, find the maximum value of $\displaystyle\sum_{cyclic}a_{i}^{3}$ also, can it be generalized as for ...
2
votes
4answers
158 views

Three Variables-Inequality with $a+b+c=abc$

$a$,$b$,$c$ are positive numbers such that $~a+b+c=abc$ Find the maximum value of $~\dfrac{1}{\sqrt{1+a^{2}}}+\dfrac{1}{\sqrt{1+b^{2}}}+\dfrac{1}{\sqrt{1+c^{2}}}$
2
votes
2answers
127 views

Maxima and minima of multivariable function $f(x,y)=6x^3y^2-x^4y^2-x^3y^3$

$$f(x,y)=6x^3y^2-x^4y^2-x^3y^3$$ $$\frac{\delta f}{\delta x}=18x^2y^2-4x^3y^2-3x^2y^3$$ $$\frac{\delta f}{\delta y}=12x^3y-2x^4y-3x^3y^2$$ Points, in which partial derivatives ar equal to 0 are: ...
2
votes
1answer
84 views

Cubes, squares and minimal sums

I have trouble solving the following task: i need to find positive integers a and b such that 1) $a \neq b$ 2) $ \exists c \in \mathbb{N} : ~ a^2 + b^2 = c^3$ 3) $\exists d \in \mathbb{N}: ~ a^3 + ...
2
votes
2answers
610 views

satisfy the Euler-Lagrange equation

Two circles of unit radius, each normal to the line through their centers are a distance d apart. A soap film is formed between themas shown below; energetic considerations require the filem to ...
2
votes
1answer
219 views

Multiple solutions for both primal and dual

If matrix $A$ in an LP (or $A^T$ in its dual) has full row (column- in dual) rank, is it possible that both primal and dual have multiple solutions?
1
vote
3answers
3k views

How to plot a phase portrait for this system of differential equations?

I beg your help.. I'd like the phase portrait for this system. I don't know how to use Mathematica/Matlab ... :( If anyone can make this portrait and post a print screen here, I would thank you ...
1
vote
1answer
94 views

Positive semidefinite cone is generated by all rank one matrices.

The positive semidefinite cone is generated by all rank one matrices $xx^T$ . They form the extreme raysof the cone. The positive definite matrices lie in the interior of the cone. The positive ...
1
vote
0answers
181 views

Duality for Support Vector Machines

SVM classifier for two linearly separable classes is based on the following convex optimization problem: \begin{equation*} \frac{1}{2}\sum_{k=1}^{n}w_k^2 \rightarrow \min \end{equation*} ...
0
votes
2answers
2k views

Solving a set of equations with Newton-Raphson

I want to solve this set of equations with Newton-Raphson. Can anybody help me? $$ \cos(x_1)+\cos(x_2)+\cos(x_3)= \frac{3}{5} $$ $$ \cos(3x_1)+\cos(3x_2)+\cos(3x_3)=0 $$ $$ ...
0
votes
2answers
57 views

minximum and maximum of $P=x+y+z$

Let $x,y,z \in R;x \ge 1,y \ge 2,z \ge 3$ and $$\sum\limits_{\large{\text{cyc}}} {\frac{{{x^2} - x + 1}}{{x + \sqrt {x - 1} }} = 12} $$ Search minximum and maximum of $P=x+y+z$
0
votes
2answers
88 views

Explain this statement $\bar 0 \in \partial f(x^*)$ where $\partial f(x^*)$ is subgradient

I haven't understood this theorem "$x^*$ is global minimum iff $\bar 0\in \partial f(x^*)$". What does it mean? Visually? P.s. Studying Nonlinear-optimization -course, 2.3139.
10
votes
2answers
1k views

Augmented Reality Transformation Matrix Optimization

i am a software developer, i'm working on an Augmented Reality system. I'd like to receive some advice in order to optimize my math model. My program has to be slim and fast. Here's the situation: ...
8
votes
1answer
309 views

Simple question: the double supremum

Let $f:A\times B\to \mathbb R$. Is it always true that $$ f^* = \sup\limits_{a\in A,b\in B}f(a,b) = \sup\limits_{a\in A}\sup\limits_{b\in B}f(a,b). $$ I proved it by the $\varepsilon$-$\delta$ ...
6
votes
1answer
740 views

Why is Dantzig's solution to the knapsack problem only approximate

For a bunch of items with values $v_i$ and weights $w_i$, and with a total weight $W$ that our bag can carry, how do we achieve maximum total value without breaking the bag? Dantzig proposed that we ...