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)

0
votes
0answers
38 views

How to find a function which maximizes a stochastic process containing sum?

Let $X=\lbrace X_t : t\geq 0\rbrace$ denote a Lévy process with initial value $X_0=0$. Let the process be sampled equally in time ($t_n-t_{n-1}=const.$). I am looking for the ...
1
vote
1answer
73 views

Argument to “linearize” an objective function

I have this optimization problem on the variables $\lambda_\ell^+, \lambda_\ell^-$ such that $ \lambda_\ell^+ \geq \lambda_\ell^-$ with $\ell=1,\ldots,n$ , and fixed $P\in [1/(n+1),1]$ \begin{align} ...
1
vote
2answers
24 views

Find the minimum possible order at a restaurant for a party of n people

I want to find an efficient algorithm for determining the minimum possible order total for a party of n people at a restaurant, assuming that the items in the order are unique, and they will each ...
0
votes
0answers
7 views

Ways to deal with Boolean constraint in optimization

For the optimization of $\text{min}_\alpha Q(\alpha)$, such that $\alpha_i \in \{0,1\}$, what will be popular way to deal with the Boolean constraint? Is there any methods to approximate the Boolean ...
0
votes
1answer
748 views

How do I construct the Jacobian for use in a Levenberg-Marquardt algorithm.

I am working on a 3D reconstruction system and I am looking to use a Levenberg-marquardt algorithm to do bundle adjustment. I am not too sure about how LM works and what it requires. The model I am ...
1
vote
3answers
38 views

Maximise the volume of an open triangular prism

An open container is to be constructed out of 200 square centimeters of cardboard. The two end pieces are equilateral triangles. The open top is a horizontal rectangle. Find the lengths of the sides ...
1
vote
1answer
28 views

Second order derivation optimization

Recently I am thinking about a problem that might be easy to answer but for me is a big challenge. Assume you have a function $f(x)$ that is second order derivative. So I am looking for a way to ...
0
votes
2answers
84 views

Spivak “min” notation confusion

Spivak uses a notation: min$(1, \frac{\epsilon}{2|a| + 1})$ What does he mean by this notation? especially by "min"??
1
vote
1answer
22 views

Check my solution for optimization problem

A piece of wire 40 units long is to be cut into two pieces. One piece will be bent to form a circle; the other will be bent to form a square. Find the minimum and maximum values of the area. I found ...
1
vote
0answers
11 views

Weierstrass conditions, what does strong mean, and are both conditions required?

I have the Weierstrass condition: In order that the extremal $\bf{C^*}: x = x^*(t)$ give a strong local minimum to $\bf{J[x]}$ it is sufficient that: # ...
0
votes
1answer
23 views

Dual simplex doubt (unrestricted)

I have this two problems and i only want to find the dual form: $\begin{gather} max\hspace{.1cm}z =5x_1+6x_2\\ s.t\hspace{.1cm}x_1+2x_2=5\\ -x_1+5x_2 \ge 3\\ x_2 \ge 0\\ x_1\hspace{.1cm} ...
0
votes
0answers
22 views

Grouping elements such that optimizing inter group communication cost

Suppose we have n elements. We need to group them into m group such that each group cannot take more than $x_1,x_2...x_m$ elements respectively. Sum of $x_i \geq n$. We need to optimize the inter ...
0
votes
1answer
34 views

Does maximizing an increasing function of two variables in more favorable conditions always increase both inputs?

Consider the problem of maximizing $\sqrt{x}y$ such that $x+y=10$. By basic calculus we can show that the maximum occurs at $x=10/3$, $y=20/3$. If we loosen the constraint to $x+y=12$ then the maximum ...
0
votes
1answer
26 views

how to differentiate this equation (contains absolute and norm)

how can I differentiate the following wrt $\mathbf{d}_i$? $\frac{|\mathbf{d}_i^T\mathbf{d}_j|}{\|\mathbf{d}_i\|_2\|\mathbf{d}_j\|_2}$ Thanks in advance.
0
votes
0answers
9 views

Algorithm to compare set of objects given a metric

Assume I have Objects $x\in X$ with an associated metric $d:X\times X\to\mathbb N_0$. I want to find a metric $d^*: \mathcal P(X) \times \mathcal P(X) \to\mathbb N_0$ wich compares sets of these ...
0
votes
1answer
165 views

Find a Square from n given Points.

Given set X of m Integeral Cords. I need to add minimum number of points to set X such that i get atleast one Square. For example: lets say X:{(0,0),(2,2),(3,3)}. Now i will have to add minimum 2 ...
0
votes
0answers
28 views

Finding the shape which gives the max area with constraint.

The problem is: Given $x$ feet of material with one side being $y$ long, what shape gives the maximum area that can be enclosed. My solution is having the $y$ side a straight line, and having ...
0
votes
2answers
53 views

Is a minimum a minimum in every norm? (for finite-dimensional vector spaces)

I have a question regarding the equivalence of the norms in finite-dimensional vector spaces. Basically the question is: if $\hat{x}$ is some minimum-norm solution in a subspace $\mathcal{K}$ under ...
1
vote
0answers
26 views

Is there any standard way to handle fractions of bilinear constraints in optimization?

By fractional bilinear constraints, I mean this form: $$\frac{a_1 + a_2 + a_3 + \cdots}{b_1 + b_2 + b_3 + \cdots} \frac{c_1 + c_2 + c_3 + \cdots}{d_1 + d_2 + d_3 + \cdots}$$ Here, $a,b,c,d$ are ...
0
votes
1answer
14 views

Simplex method - infeasible basic variables

I am working on an optimization problem right now, and I am using the simplex method on the initial tableau. At first, the basic variables are all non-negative and are equal to the slack variables. ...
0
votes
0answers
22 views

Sharing between 3 people

I have a problem on what im thinking since 2 weeks but i got no idea how to solve it. There are gifts with given values and we need to share these gifts between people fairly. Fair sharing in this ...
3
votes
4answers
70 views

Find the maximum of $\frac{1}{1+|x|}+\frac{1}{1+|x-a|}$

Let $a>0$. Show that the maximum value of the function $$f(x)= \frac{1}{1+|x|}+\frac{1}{1+|x-a|}$$ is $$\frac{2+a}{1+a}.$$ really need some help with this thing
1
vote
1answer
41 views

Strongly minimizing curve optimisation with Weierstrass condition

No idea where to start on this one: Find the strongly minimizing curve and value of $J_{min}$ for cases: $$\int_1^2 (t^2\dot{x}^2 + 2x^2) dt$$ where $x(1)=0,x(2)=7$ Using the Weierstrass ...
0
votes
1answer
27 views

Linear Programming : Is there any other way to solve than graphs?

In my highschool curriculum there's a a chapter on Linear Programming Problems. In the chapter there are bunch of unproved statements and mechanical ways to solve linear problem. But my question is- ...
2
votes
2answers
73 views

Research Area Choice: PDE vs Optimization

I am on track to starting in applied math PhD this coming Fall. My area of interest is PDE where I have a strong background and have even published a paper. It seems the particular area of PDE I am ...
0
votes
0answers
32 views

is $R^N_{ ++}$ a convex set?

Is $R^N_{ ++}$ a convex set? I'm working on some optimization hw problems that have some functions of the type: $f:\mathbb{R}^2_{++} \rightarrow \mathbb{R}$ And it seems like in general whenever ...
1
vote
1answer
28 views

By minimizing the function $\phi(s,t) = \frac{1}{2} \mid\mid \textbf{b} - (s\textbf{a}_1 + t\textbf{a}_2) \mid\mid^2$, find a for

Suppose $\textbf{a}_1$ and $\textbf{a}_2$ are linearly independent vectors, $L = \text{span} \ \{{\textbf{a}_1, \textbf{a}_2}\}$, and $\textbf{b}$ is a vector not in $L$. By minimizing the function ...
2
votes
1answer
24 views

Maximize $ax + by + cz$ given $x^2 + y^2 + z^2 = k^2$. Write answer as Schwartz inequality for dot products.

Maximize $ax + by + cz$ given $x^2 + y^2 + z^2 = k^2$. Write the answer as the Schwartz inequality for dot products $(a, b, c) \cdot (x, y, z) \le \_\_\_\_\_\_\_\_ \ k$. I'm stuck on this problem. I ...
2
votes
0answers
24 views

Maximize and minimize a function using Lagrange multipliers.

I want to maximize and minimize $$h(a,b) = a + b$$ given the constraint $$g(a,b) = a^{\frac{1}{3}}b^{\frac{2}{3}} = l$$ I'm trying to use Lagrange multipliers. Here's what I did: \begin{align} ...
0
votes
1answer
64 views

Give example of a set which has No Extreme Point !!..

Give example of a set in R^2 , which has no extreme point ?? We were given this question for assignment !!..I thought of a simple line but doing some research i stumbled upon this solution which ...
0
votes
1answer
19 views

Rate of growth of $a$ in $f(x) = \frac{\ln(ax)}{x}$ causing movement of extrema

Problem description Look at the function $f(x) = \frac{\ln(ax)}{x}$ on a cartesian system with steps of 1cm on both axes. a) Show that $f$ has a local maximum for $x = \frac ea$. b) When $a$ ...
0
votes
0answers
24 views

Constrained non-linear optimization problem

For some background, this comes from a sample size allocation problem in statistics. I am trying to minimize the following function (a sum of three variances), and could use some help with direction ...
1
vote
1answer
355 views

Find the regular $n$ -side polygon of A Constant Area that can contain Most Number of Circles

I have a constant area $A$ , and I can mold that area into a regular $n$ -side polygon, where $n\geq 3$. The issue now is how to find the $n$ such that it can contain the most number of circles, ...
0
votes
0answers
33 views

hint for investigating positivity/negativity of a function

I'm investigating if and how the positivity or negativity of a multivariable function can be proved. Consider $y_{1},y_{2},y_{3}\in\mathbb{R}$ and the following function ...
0
votes
0answers
27 views

Saddle point problem (KKT) with block-diagonal matrix

Consider the following saddle point problem originating from an interior-point method algorithm: $$ \begin{bmatrix}\mathbf{H} & \mathbf{A}^{T}\\ \mathbf{A} & \mathbf{0} ...
0
votes
0answers
10 views

Convert the problem into an equivalent stalndard lpp

Consider the optimization problem, Minimize, $C_{1}|x_{1}|+C_{2}|x_{2}+C_{3}|x_{3}|+...+C_{n}|x_{n}|$ subject to, $AX = b$ , $C_{i} \neq 0$ Convert the above problem into an equivalent standard ...
1
vote
2answers
47 views

Minimize and maximize the function $g(a,b) = a + b$ given the constraint $h(a,b) = \frac{1}{a} + \frac{1}{b} = 1$.

I want to minimize and maximize the function $g(a,b) = a + b$ given the constraint $h(a,b) = \frac{1}{a} + \frac{1}{b} = 1$, and I want to find the values of $a, b,$ and $\lambda$. This is what I've ...
1
vote
0answers
39 views

Minimizing a function using a direct approach (no Lagrange multipliers).

I want to minimize $g = x^2 + y^2$. My constraint is $h = 2x +y = l$. I know that using Lagrange multipliers is unnecessary here. I solved the constraint to get $y = l - 2x$. I then substituted this ...
0
votes
2answers
30 views

Finding minimum on graph for given domain

So I want to find what is the minimum value of a graph on a certain domain. For example, for $y=x^2+1$ between $x=-3$ and $2$, the minimum value is at 1 at x=0. I think I know how to find minimums ...
0
votes
0answers
25 views

how to derive this equation?

How can I derive this? $\min_{d_m} \|Y - DX\|_2^2 = \min_{d_m} x_m^Tx_md_m^Td_m - 2R_mx_m$ where $R_m = Y - \sum_{i \neq m } d_ix_i^T$ $x_m $ is a vector represents a row in $X$
0
votes
0answers
22 views

What is a good optimization textbook for the theoretically inclined student looking for a rigorous and concise proof-based book?

I'm looking for an optimization book that is more like a classic pure math textbook without requiring any actual prior pure math courses. A book that puts focus on the theoretical aspects of ...
1
vote
0answers
24 views

Prove circle packing solution is optimal

Background: This is a follow on from this question of how to maximise the area of two non overlapping circles of arbitrary radii packed into a rectangle of arbitrary width and height. I proposed a ...
1
vote
1answer
35 views

How to approach a minmax problem?

Starting with a certain geometric problem, I have reached this function: $$R(s,t,u,v)=\max(s-u,s+u,t-v,t+v,sX+tY+u, tX-sY+v)$$ where $X\geq0$ and $Y\geq0$ are parameters. I have to find the minimum ...
0
votes
1answer
75 views

Optimization Calculus.. a box/shelter with sides missing..

I'm solving a problem involving calculus optimization. The problem is the following: "We plan to build a boxshaped shelter with no floor and one side open. (Hence we need a roof and three sides). The ...
0
votes
0answers
10 views

BFGS update formula

In this pdf : http://www.ing.unitn.it/~bertolaz/2-teaching/2011-2012/AA-2011-2012-OPTIM/lezioni/slides-mQN.pdf, in slide 46, the BFGS updata rule is given and is simplified to a second form. How did ...
0
votes
1answer
25 views

How to apply Sherman Morrison formula for rank 2 update?

For obtaining the inverse update in BFGS, Sherman-Morrison needs to be applied twice since it is a rank 2 update. But what does it mean to apply it twice?
0
votes
5answers
47 views

Minimum perimeter of a three-sided rectangular fence with given enclosed area

A three-sided fence is to be built next to a straight section of a river, which forms the fourth side of a rectangular region. The enclosed area is equal to 1800 ft^2. Find the minimum perimeter and ...
1
vote
1answer
22 views

Modelling problem

i have this problem and i have to model it in a boolean formula. Assuming that variables can have value 0 or 1 and V is OR and ∧ is AND. I have n boolean variables x1,x2......xn. i want a formula ...
0
votes
2answers
24 views

When is $A^t+B^{1-t}$ for $t \in[0,1]$ maximized?

When is $A^t+B^{1-t}$ for $t \in[0,1]$ maximized. Suppose that $A,B \in \mathbb{R}^{+}$. This is very similar to convex combination but only in exponents.
0
votes
0answers
22 views

Linear Programing : get the maximum constant associated to a non null variable

Does anyone know a way get the maximum constant associated to a non null variable using only linear programing ? For example, supposed that there is 3 linear variable x, y and z. x being associated ...