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
4 views

optimizing over a set of symmetric matrices

I need to minimize an objective function, $f\left(\Lambda\right)$ over a set of symmetric matrices, $S_{p}$ of dimension p, such that all the eigenvalues of $\Lambda \in \left[0,1\right]$. I can set ...
0
votes
0answers
10 views

Is there a name for this type of online optimization problem?

I have a sequence of items $1\leq i \leq n$ that arrive to me one at a time. Each item has a weight $w_j\geq 0$. If I pick up one item, I will not be allowed to pick up any of the next $k$ items ...
0
votes
0answers
4 views

Reliability/survival function raised to a power

Let $r(p)$ be the reliability function, and suppose that $r(p)=r(p,p,...,p)$ and that $r(p_0)=p_0$ for a certain $p_0$, $0\leq p,p_0\leq 1$. I'm asked to prove that $r(p)\geq p$ if $p\geq p_0$ and ...
5
votes
2answers
261 views

How to find the shift that minimizes the difference between two vectors?

I am looking for a efficient way to find the value of k that minimizes $\sum(s_t - b_{t+k})^2$ where $s$ and $b$ are N-dimensional vectors and the values are wrapped around like this: $b_{t+k} := ...
0
votes
0answers
7 views

multi-objective reduction of a given set

I have a set of arguments $v_k$. Each argument has a set of two different numeric values $x_{ak} \in [0,\infty]$ and $x_{bk} \in [0,\infty]$ associated to it. The set $V$ contains all $v_k$s. I’m now ...
1
vote
1answer
20 views

How come $Ax\le b$ and $c^Tx\ge \alpha +\epsilon$ has NO nonnegative solution.

Let $\alpha=c^Tx^*$ to be the optimum value of standard form of (LP)(= max $c^Tx$ subject to $Ax\le b$ and $x\ge0$ in $\mathbf{R^n}$) Then we know: $Ax\le b$ and $c^Tx\ge \alpha$ has a nonnegative ...
0
votes
0answers
12 views

How to optimize the repartition of samples in weighted channels?

This is more like an applied mathematics question, so my apologies if I am at the wrong place. Let S(n) be an infinite sequence of real numbers strictly growing from 0 to 1 (asymptotically). Let P be ...
0
votes
1answer
30 views

Is this function convex or non-convex? How do you decide?

The problem is: find $$\min⁡ \mathrm{P}\left[{\log(1+p||H^H \mathbf{w}||^2)\over 1+p||G^H \mathbf{w}||^2}<R\right]$$ constraint to: $||\mathbf{w}||^2=1$ where $H$ and $G$ are matrices of ...
0
votes
1answer
11 views

Optimize distributions for low mean, high variance

Assume a context with $N$ approximately normal distributions where a lower mean implies a 'better' distribution and a high variance or high standard deviation implies a 'better' distribution as well. ...
0
votes
0answers
12 views

Optimization problems with combinations of a finite set as the feasible area?

For example: Provided that $S\subset \Re$ is a known finite set ($n\leq |S| < \infty$), number $k$ is known, and $1 \leq k<n$ minimize $f(x_{1},\ldots, x_{n}) = \sin (\sum_{1\leq i\leq ...
1
vote
0answers
8 views

projection KKT optimal condition

Using the KKT optimality condition find the orthogonal projection of an arbitrary point $c \in$ to the closed convex set $C$ (non empty) defined by: (a) $C=\{x \in R^n : Ax\leq a\}$ where $A\in ...
1
vote
1answer
33 views

find the minimum value of this integral when $1>t>0$, $f(t)=\int\limits_0^1 x |e^{-x^2} - t|\ \mathrm{d}x = ?$

Is there someone who can show me How do i find the minimum value of this integral when $1>t>0$, \begin{align*}f(t)=\int\limits_0^1 x |e^{-x^2} - t|\ \mathrm{d}x &= \end{align*} Note : ...
1
vote
0answers
17 views

KKT Optimality Conditions

I am working with the following optimization problem: $$ \min_{\Delta} \boldsymbol{\theta}^T\boldsymbol{\Delta} \\ \text{Such that:} ~~~0 \leq \mu_i + \Delta_i \leq 1 ~~\forall~~ i\in\{1,2,\ldots, n\} ...
0
votes
1answer
32 views

Campbell's Source coding

In the usual Shannon's source coding problem one chooses code words that minimize $E[L]:=\sum_i p_il_i$ over all $L=(l_1,l_2, \dots), l_i\ge 0$ such that $\sum_i e^{-l_i}\le 1$ (Kraft inequality), ...
-1
votes
0answers
26 views

How do I maximize each value, while having them be as far apart as possible? [on hold]

I have three values V, S, and A. They sum to 1, and are all greater than 0. How do I maximize each value while having them be as far apart as possible? That is, I'd like V to be clearly greater than ...
0
votes
2answers
31 views

E-olymp: Cake. Giving Wrong Answer

Cake This is a e-olymp programming question mathematical optimization. In honor of the birth of an heir Tutti royal chef has prepared a huge cake, that was put on the table for Three Fat Man. ...
1
vote
1answer
33 views

Generating vectors in a non-orthogonal 3D lattice with increasing magnitude

I am trying to build an algorithm to generate a sequence of lattice vectors $\mathbf{v}_n$ in 3D such that: (a) the first vector $|\mathbf{v}_1|$ is the shortest vector of the lattice (b) for all $i ...
0
votes
0answers
13 views

A special case of GUBMKP

I am seeking a problem that resembles the Multidimensional Knapsack Problem with Generalized Upper Bound Constraints where the resources available are of equal sizes.I am only getting the case where ...
2
votes
2answers
41 views

Optimization of solve

Find the minimum value and the maximum value of the function $$y(x)=\frac{x^3}{x-3}$$ when $4\le x\le5$ I found that $f(x)$ is decreasing on the interval $[4,\frac{9}{2}]$ and increasing on ...
-1
votes
1answer
27 views

Image restoration in matlab via PDE toolbox

I want to remove a noise for an image using matlab, when the observed image is $$f=u+v$$ where $u$ is the restored image (is the image i want recovered) and $v$ is the gaussian noise. To restore $u$, ...
-4
votes
0answers
36 views

Journal related question. [on hold]

I want to know the quality of Optics and Photonics Journal(http://www.scirp.org/journal/opj). please let me know
0
votes
1answer
36 views

Local extrema in special directions

I am looking for the extrema of a function $G(y_1,y_2,y_3,y_4)$ subject to the constraint $y_1 = y_4 + y_2y_3.$ We know that $G$ is defined if $(y_2,y_3,y_4)$ is in the cylinder $\mathbb{D} \times ...
1
vote
1answer
35 views

How to resolve total variation $F(u)=f(u)+\lambda\sum_i^N\int_{\Omega}|\nabla u_i(x)|dx$

Given $u(x)=[u_1(x)..u_N(x)]$, $0 \le u_i(x) \le 1, \sum_i^N u_i(x)=1$ and the cost function is: $$F(u)=f(u(x))+\lambda\sum_i^N\int_{\Omega}|\nabla u_i(x)|dx$$ where $u_i(x)$ is a value that indicate ...
0
votes
1answer
15 views

Is it covex function?$J_{new}(u)=\int_{\Omega} \sum_{i=1}^{N} \lambda_if(x)u_i(x)dx$

I have a function such as $$J(u)=\int_{\Omega} \sum_{i=1}^{N} f(x)u_i(x)dx$$ where $f(x):\Omega \to R$, $0 \le u_i(x) \le 1,\sum_i u_i(x)=1$ Given that $J(u)$ is a convex function w.r.t $u$. Now I ...
0
votes
0answers
14 views

Adjacency matrices for wave length assignment [on hold]

I have some problem creating matrices with Path and wavelength assignment.I need to create a $P\times W$ matrix $A$, where $P$ is the number of path and $W$ is the wavelength assigned to that path. ...
0
votes
0answers
32 views

Derivation of energy function

Given the following energy function $E(d)$ (also found here on page 3): $$ E_d = \sum_{x,y \in \Omega} \left(d_{x,y} - \hat{d}_{x,y}\right)^2 + \lambda \sum_{x,y} \left( ...
0
votes
0answers
14 views

optimal control, semismooth newton, bounded norm

I'm solving an optimal control problem (Poisson's equation with dirichlet BVP) $F(y,u) :=\frac{1}{2}\int_{\Omega} (y-y_d)^2 dx + \frac{\lambda}{2} \int_{\Omega} u^2 dx$ with finite element method. ...
1
vote
2answers
55 views

Do we really need the constraint qualification?

I can't keep my fingers off Nocedal/Wright's Numerical Optimization (1999,1E) and I apologize. But maybe YOU can shed light on the question: Why does a point $x \in \mathbb{R}^n$ need to satisfy the ...
1
vote
0answers
14 views

Convergence results for block coordinate descent methods

I am trying to solve the problem minimize $f(x)$ subject to $x_1 \in C_1, x_2\in C_2, ... x_m\in C_m$ where $x_1, ..., x_m$ are block subvectors of $x$, and $C_i$ are each closed convex sets (not ...
3
votes
1answer
40 views

Usefulness of prime numbers as Threading Timeouts in programming [closed]

I am a .NET programmer, founded in math. I am having an argument with a fellow programmer. When I add a Threaded Timer to the program, the interval in milliseconds I use is always a prime number. ...
0
votes
1answer
35 views

Optimization Example with some constraints !? [closed]

We want to optimize the following function: $$ f(x,y)=x^2+3y^2+2xy+2 $$ with constraints $-2 \leq x< 2$, $-2 <y<2$, and $3y^2+x \leq 10$. Who can help me for the above example from ...
0
votes
1answer
12 views

Maximization: KKT on unbounded region

Solve the following NLP: $$\left\{\begin{matrix} \min & -3x+y-z^2\\ s.t& g(x,y,z)=x+y+z \leq 0\\ & h(x,y,z)=-x+2y+z^2z=0 \end{matrix}\right.$$ My attempt Using kkt conditions, we ...
0
votes
1answer
47 views
+50

Modeling, Measuring, and Maximizing “Mixedness”

Background: My class has $10$ students and $3$ tables; naturally, the students are distributed with $3, 3,$ and $4$ seated at the individual tables. On the second day of class, students sat in the ...
0
votes
0answers
22 views

non-linearity and non-convexity

I am taking a course on linear regression online and it talks about the sum of square difference cost function and one of the points it makes is that the cost function is always convex i.e. it has ...
1
vote
1answer
30 views

Matrix norm in the objective of an optimization problem

I am stuck with the following optimization problem from research. The optimization problem have the following objective function: $\|Q-H\|_\infty$. Here $Q$ is a PSD matrix and $H$ is a symmetric ...
1
vote
1answer
43 views

Minimizing the function with a log determinant and trace function?

I am trying to minimize the following argument, which is unbounded in case one of the eigenvalues of $A$ is equal to zero. $\arg min_{S} \log|S^H A S| - tr\{ \Sigma^{-1}S^HAS\}$ Let $A > 0$, ...
0
votes
2answers
62 views

An exam`s points dilemma

On July 2 I have an exam, in this exam will be 40 questions in test with 5 variants of answer for each question. For each correct answer will be given +1 point. For each incorrect answer will be ...
0
votes
0answers
29 views

Compressive sensing for complex matrix

I'm fairly new to compressive sensing, and I have been looking for a MATLAB implementation of the problem $$ A x = b $$ where $A$ is non square, $x$ is kind of sparse and all the numbers involved are ...
0
votes
1answer
18 views

Eliminate cases before calculting all KKT conditions

I have the following non linear programming to solve: $$\left\{\begin{matrix} \min & (x-3)^2 + (y-2)^2 \\ s.t. & x^2 +y^2 \leq 5 \\ & x+y\leq 3 \\ & x \geq 0\\ & y\geq 0 ...
0
votes
0answers
7 views

Find $\alpha$ which makes the problem have an optimal solution or none at all + dual problem LP

min $x_1 + \alpha x_2 $ subject to $4x_1+3x_2\leq29$ $x_1+x_2\geq4$ $x_1\leq5$ $x_2\leq7$ Find for which $\alpha$ the given problem has an optimal solution or no solution at all. Provide the ...
0
votes
1answer
17 views

multi objective optimization

Suppose we want to maximize two positive bounded objectives. A usual approach for this aim is to maximize a weighted sum of these two objectives. Now, my question is why not to maximize the product ...
2
votes
0answers
41 views

How does one evaluate the derivative of a matrix with a tensor $\frac{\partial \operatorname{Tr}[A(\mathrm{Id}\otimes w)]}{\partial w}$?

I am stuck on the following: $$\frac{\partial \operatorname{Tr}[A(\mathrm{Id}\otimes w)]}{\partial w}=\text{ ?}$$ with $A$ a $d\times d^2$ matrix, $\mathrm{Id}$ the identity matrix of $d\times d$ ...
1
vote
1answer
36 views

binary quadratic optimization problem

I am trying to solve the following binary quadratic program. $$ \min_{\Delta} \Delta^T H \Delta + c^T\Delta \\ \text{Such that:} ~~~\Delta\in \{0,1\}^n ~~\text{and}~~ \sum_{i=1}^n \Delta_i \leq \Gamma ...
5
votes
3answers
75 views

Optimal approximation of quadratic form

Let $\mathbf{x}\in\Bbb{R}^n$ and $A\in\Bbb{S}_{++}^n$, where $\Bbb{S}_{++}^n$ denotes the space of symmetric positive definite $n\times n$ real matrices. Also, let $Q\colon\Bbb{R}^n\to\Bbb{R}_{+}$ be ...
2
votes
1answer
33 views

If $H$ is positive definite and $s^Ty>0$, then $s^THs-\frac{s^Tyy^Ts}{s^Ty+y^TH^{-1}y}\ne -1$

Let $H\in\mathbb{R}^{n\times n}$ be symmetric and positive definite $s,y\in\mathbb{R}^n$ with $s^Ty>0$ How can we show, that $$s^THs-\frac{s^Tyy^Ts}{s^Ty+y^TH^{-1}y}\ne -1\;?\tag{1}$$ ...
3
votes
1answer
50 views

Minimizing Area by Approximation

Suppose I have an increasing step function $E_c$ given by $$E_c(\phi) = \sum_{i=1}^n E_i \theta(\phi - \phi_i),$$ where $\theta$ is the Heaviside step function and $E_i$, $\phi$, and $\phi_i$ are all ...
0
votes
1answer
17 views

Maximization of quadratic form over complex unit cube

I am trying to find the maximum of a hermitian positive definite quadratic form $xQx^H$ (where $Q=Q^H$ and all eigenvalues of $Q$ are non-negative) over the complex unit cube $|x_i|\leq 1$, ...
1
vote
0answers
23 views

Coin distribution problem to optimize

There are $N$ users, with each user having a money request. There are $T$ coins, these coins are to be assigned to the user in such a way that its request is fulfilled. Assume each coin may have ...
0
votes
0answers
10 views

Prove that if $e \in \left ( S\to \overline S \right )$ when $\left ( S, \overline S \right )$ is a min-cut, then $f(e) = c(e)$

Given a min-cut $\left( S, \overline S \right )$, we define $\left ( S\to \overline S\right ) =\{\left (u\to v\right )|u \in S, v\in \overline S\}$ and $\left ( \overline S \to S \right )$ similarly. ...
0
votes
0answers
13 views

For which values of $c_1, c_2$ and $c_3$ is (1, 2, -2) a local minimum

Consider the problem $$\left\{\begin{matrix} \min & x^2 -2xy + 2xz +y^2 + 4yz + z^2 + c_1x + c_2y + c_3z \\ s.t & g(x,y,z)=-x^2 -4xy - 4xz -2y^2 -4yz - 2z^2 + x -y+z+4 =0 \\ \; & ...