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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33 views
Differentiating a complex equation in order to optimize for the parameter
$$E(TC)=C_{1}\left[\frac{\lambda[1-F(x)][1-G(y)]}{\mu\{1-\rho[1-F(x)][1-G(y)]\}}+\frac{\lambda[1-F(x)][1-G(y)]}{\mu\{1-\rho[1-F(x)][1-G(y)]\}^{2}}+\frac{\lambda}{\mu}\right]+C_{2}\mu$$
where ...
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2answers
58 views
How can I solve Lagrange multiplier equation with multi constraints?
This site is really awesome. :)
I hope that we can share our ideas through this site!
I have an equation as below,
$$ min \ \ w^HRw \ \ subject \ \ to \ \ w^HR_aw=J_a, \ w^HR_bw=J_b$$
If there is ...
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1answer
33 views
Determine conjugate function
Let $f:\mathbb{R}\to\mathbb{R}$ defined by $f(x)=e^x$. Determine $f^*(y)$.
I try to use some inequalities to get supremum but it is impossible. Seemingly, I must consider some cases of $x$.
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19 views
Smallest 3D Box that emcompasses N smaller Boxes??
I m trying to create an algorithmn that solves this optimisation problem. This means it needs to be coded and hopefully works for N boxes, lets not worry about calculation time.
So, if we have 3 ...
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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 ...
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0answers
39 views
Optimize fill factor by move objects between areas
I have a optimization problem which is about several small rectangles inside one outer rectangle. We have, let say, three outer rectangles which are in following order (similar to weeks).
Each ...
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1answer
49 views
Fastest Algorithm for NLP with linear constraints
I have an minimization problem of the following form:
(Im not a mathematician, i come from the programming side, so excuse me if i have not the perfect standard of writing the formulas)
$Z(x) = ...
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19 views
Constrained computational optimization of a functional of a vector valued function.
I am trying to increase the efficiency of a program I have written that must run in real time. I am asking this question in a broad sense, since I'm not sure what tools are available to me.
I am ...
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1answer
20 views
solve non-convex quadratic constrained quadratic programming
$\min_{\beta}\beta^{T} A \beta$
$s.t. \ \beta^{T} C \beta=1\ and\ \beta\geqslant 0$
Here $A,C\in \mathbb{R}^{M\times M}$, $\beta \in \mathbb{R}^{M}$
I saw in one paper saying that it could be ...
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2answers
61 views
Find min of $IA + IB + IC +ID$ in tetrahedron $ABCD$
Let the point $I$ in tetrahedron $ABCD$. Find $\min\{IA + IB + IC + ID\}$.
I can't solve this problem, even in the case ABCD regular. Please help
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1answer
36 views
Circle Packing: Unsolved Problem in Geometry?
Graham and Sloane minimize the second moment of the centres of a number discs in order to maximize their compactness. They use computational geometry techniques to find the optimal packings for ...
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19 views
Determine direction of minimum overlap of convex polygons
Given two convex polygons $P$ and $Q$ what is the minimum intersection polygon $A=P\cap Q'$ where $Q'$ is the polygon $Q$ offset by a vector $\overline r$ of fixed length?
Put another way, what is ...
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0answers
18 views
Transformation of binary data
I have a function that I try to optimize using Particle Swarm Optimization. Objective function gets a binary string. So these binary strings are candidate solutions of the subject function. I can ...
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0answers
19 views
Finding the smallest total sum of differrences
I wanted to find the $k$ so the sum of the differences of $k$ to all elements of a certain set $A$ with size $n$ is minimized. In other words:
$\underset{k\in \mathbb C}{\text{minimize }} ...
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60 views
A minimax problem
Let $\omega$ be a non zero real number, $(\lambda_i)_{i = 1}^n$ be a sequence of $n$ real numbers and
$$u_{i}:=1-(1-\lambda_{i})\,\omega.$$
Show that if we pick
...
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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 ...
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4answers
70 views
Given $x,y,z\geq0$ and $x^2+y^2+z^2+x+2y+3z=13/4$. Find the minimum of $x+y+z$.
Given $x,y,z\geq0$ and $x^2+y^2+z^2+x+2y+3z=13/4$. Find the minimum of $x+y+z$.
I tried many method, such as AM-GM, but all of them failed.
Thank you.
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0answers
37 views
how to solve this optimization problem?
$maximize\frac{l_{1}+l_{2}+\cdots+l_{n}}{max(\frac{l_{1}}{r_{1}},\frac{l_{2}}{r_{2}},\cdots,\frac{l_{n}}{r_{n}})}$
Where $1\leq l_{i}\leq17$are optimization variables and $r_ {i} $s are postivie ...
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0answers
59 views
minimization of function $F(a) = \int_0^1 (G(x) - P_a(x))^2\,dx$?
I have the following questions referring to this link to a previous question on this site : Approximate a function over the interval $[0, 1]$ by a polynomial of degree $n$ (or less).
a) Explain why ...
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26 views
Optimization problem - maximizing number of satisfied linear inequalities subject to quadratic constraint
I am wondering if anything is known about optimization problems of the following type.
Our control $x$ is a unit vector in $\mathbb{R}^n$. We are given a finite number of linear inequalities
$$A z ...
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42 views
Projection Methods
I found out that in fact two classes of numerical algorithms are called "Projection Methods". Projection methods á la Yousef Saad to solve a linear system Ax=b (f.e. krylov subspace methods) and ...
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1answer
64 views
How to minimize studying, using mathematics?
A friend asked me this question earlier today, and it made me wonder how to come up with a general solution (where each variable is an integer):
I have a vocabulary test tomorrow at school. On it, ...
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1answer
35 views
Solving exponential equations for maxent with opensource applications or coding
I am working on maximum entropy. And I have a problem solving some equations.
Look at these 2 equations:
$\frac{(2*e^{-a*2-b*2^2}) +(6*e^{-a*6-b*6^2}) +(10*e^{-a*10-b*10^2}) ...
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2answers
44 views
Find the minimum of $|a+\frac 2 {a-1}|$ where $|a|\leq2$.
Find the minimum of $|a+\frac 2 {a-1}|$ where $|a|\leq2$.
I tried using differentiation, but the absolute makes things troublesome...
Please help. Thank you.
2
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1answer
38 views
Minimizing Mean Squared Error for Exponential Function
I have a function that I'm trying to model using an exponential function and I'm trying to determine the constants for the exponential. I know I could optimize it using trial-and-error in R or another ...
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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 ...
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0answers
27 views
Is duality theory in optimization as useful as it seems?
I have been reading a lot about nonlinear optimization and duality, and it seems that duality theory is extremely useful. I feel that I am missing some of the negative aspects/difficulties associated ...
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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 ...
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28 views
maxima of the sum of unimodal functions .
I have a set of unimodal functions. Each function has real roots. All roots of each function lie outside a certain limit points. These limit points are the same for each function. Each function is in ...
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2answers
36 views
Linear programming problem neither max nor min
Heres the actual question:
television provider broadcasts two movie channels, A and B. Channel A broadcasts 1 romantic
movie, 3 action movies and 3 comedies per month at a cost of 50 Euro. ...
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2answers
35 views
Determine the points where $f$ is has a local minimum/maximum. Multivariable calculus question.
This is no homework, but it is in my book and I find it hard to solve:
Determine the points where $f$ is has a local minimum/maximum.
Determine if it strong/weak and absolute/relative and ...
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0answers
33 views
Suggestions for a reference-level text on optimization theory?
I'd be interested in knowing if anybody has suggestions on an advanced but still self-contained reference on optimization theory, centered around linear and convex problems. The key feature of my ...
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1answer
52 views
Local and global extremes
I Wrote problems and solutions, I need just few explanations.
1.Let
$$J(x)=\int_{0}^{1}x'^{2}dt,\quad x(0)=0, x(1)=1. $$
Find the extrema value for $J$.
I'm doing this using Euler equation ...
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0answers
55 views
Minimizing distance variance between points on a sphere
If we uniformly distribute N points on a sphere, and must visit each point exactly once in succession such as to form a fully covering path, is there an approach to selecting this order such as to ...
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0answers
40 views
What is that type of TSP
I'm searching for the name of the TSP-like problem.
The basic principal is like it follows:
When a city is visited by the salesman, he will came in one point and exit the city in another. The ...
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1answer
16 views
Minimizing deviations from threshold value from a given group of numbers
Given a set of numbers $a_n$, a threshold level $t$, how do I find the combination of numbers that will sum to at least the threshold with minimum deviation? Added: That is, they must always exceed ...
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0answers
52 views
Divergence of Gradient Method
Is there any example of a continuous differentiable function out there, in which the gradient method with Armijo's stepsize-rule doesn't converge?
I found it pretty hard to create one myself because ...
3
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1answer
60 views
Taking derivative of $L_0$-norm, $L_1$-norm, $L_2$-norm
I am a little confused about taking derivatives w.r.t. the norms.
$L_0$-norm: $L_0$ means number of non-zero elements in a vector. Say, I am interested in an $x_i$.
...
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1answer
38 views
Partial linear relaxation yields an integer solution
Consider a binary integer program
\begin{align}
\min \quad &\sum _{j \in J}f_j x_j +\sum _{i \in I} c_i y_i \notag \\
\mbox{s.t.} \quad &\sum _{j \in N_i} x_j \ge 1-y_i, \quad \forall i\in I ...
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0answers
32 views
How to solve this type of equation with posynomial form?
I have an equation with the following form where the goal is to find $x$:
$$ \sum_k c_k x^{\gamma_k} = 1$$
where $c_k, \gamma_k \in \Re^+$ and $\gamma_k > 1$
Alternatively using $y = \log(x)$ I can ...
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1answer
32 views
How to maximize $n!\sum^n_{k = 0}\frac{a^k(1+(-1)^{n-k})}{k!(n-k)!} \pmod{a^2}$ for a given $a$?
Short Version of the Question:
How do I maximize the value of $n!\sum^n_{k = 0}\frac{a^k(1+(-1)^{n-k})}{k!(n-k)!} \pmod{a^2}$ for a given $a$?
Long Version of the Question:
I'm currently attempting ...
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1answer
15 views
Notation minimum of a column vector
I'd like to know the notation to express the minimum of a column vector.
Is this notation correct?
\begin{equation}
\min
\left[\matrix{
\left|b_{n}-b_{n+1}\right| \cr
...
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0answers
22 views
L1 penalty can serve as a convex surrogate for an L0 penalty. Why?
I have heard machine learning practitioners say that the $L_1$ penalty is a (or can serve as) convex surrogate for an $L_0$ penalty (in the context of optimization and statistical fitting).
What do ...
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0answers
34 views
Vector Minimization Problem
Let $ m > n$, $A \in \Bbb R^{m\times n}$, and $\mu \in \Bbb R$. Is it possible to determine $0\ne u \in \Bbb R^n$ so that $u$ minimizes $$\|Au\|_2 ^2 + \mu \|u\|_1$$ ?
An algorithm or general ...
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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)
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17 views
Simplex with edges of length at least s having smallest circumradius
Is it true that of all $k$-simplices with edge lengths greater than or equal to some parameter $s$, the regular simplex with edge lengths $s$ has the smallest circumradius?
Please supply a proof or ...
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0answers
25 views
KKT conditions of this convex optimization problem
Consider a continuous function $f: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}$ such that $\forall y \in \mathbb{R}^m$ $f(\cdot,y)$ is convex, and $\forall x \in \mathbb{R}^n$ $f(x,\cdot)$ ...
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1answer
71 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 < ...
6
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0answers
49 views
Do balls optimize the boundary area for a fixed volume?
I recently saw this question, asking if the circle is the planar figure which has the least perimeter given the area. As far as I know, this is a classical problem, and the answer is affirmative. I am ...
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0answers
30 views
non-degenerate basic feasible of Polyhydron
I couldn't show this problem. Can somebody help me by this question?
Consider a polyhedron $\{X \in \mathbb{R}^n | AX \leq b, X \geq 0 \}$ and a non-degenerate basic feasible solution $X^*$. We ...
