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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Can I call my algorithm as EM algorithm? Can I claim convergence guaranty?

I am trying to minimize the following problem: $$\hat{x} = \underset{\bar{x}}{\operatorname{argmin}} \left( \left\|y- A_{MV}^{EPG} \bar{x}\right\|^2 + \|\mu_{T} \bar{x}\|^2 + \|D_{s} \bar{x}\|^2 \...
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1answer
44 views

Maxima of $f(x)/e^x$ where $f(x)$ is an approximation of $e^x$ using Stirling's

Let $$f(x)=1+\sum_{n=1}^\infty\frac{x^n}{\sqrt{2\pi n}(n/e)^n}\tag1$$ and let $$g(x)=\frac{f(x)}{e^x}\tag2$$ If we plot $g(x)$ we get a graph that looks like this: Clearly there is a maximum at ...
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16 views

How to find the maximal length of a system?

Let P be the set of $(a,b,c)^t \in \mathbb{R}$ which satisfies the following inequalities: $-2a+b+c \leq 4$ $a-2b + c \leq 1$ $2a + 2b-c \leq 5$ where $a \geq 1 $, $b \geq 2$, and $c \geq 3 $. ...
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1answer
26 views

Reducing KKT system

I was using CVXOPT library to solve one of my quadratic programming problem. I found that, CVXOPT library solves KKT system efficiently by reducing a 3x3 matrox into 2x2 blocks which has the following ...
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19 views

maximize a sum of unit fractions (without containing a subset of sum 1)

Let $ u \ge 2 $ be fixed. Then consider: $ S(u)=\max\left\lbrace \sum_{i=1}^{u+1} \frac{c_i}{t_i} \, \middle| \, 2 \le t_1 \le t_2-1 \le \ldots \le t_{u+1}-1, \, t_i \in \mathbb{N}, \, c_i \in \...
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1answer
24 views

Benders decomposition Master Problem

I am currently working on implementation of Bender's Decomposition for MIP. I am looking at the simplest model \begin{equation} \begin{split} \min_{x,y} &\; c^Tx + f(y)\\ s.t. & \; Ax + Dy \ge ...
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26 views

Canonical forms which have “minimal” Gershgórin discs, do they exist?

I'm wondering about if there is some way to define, uniquely or not a canonical form which has minimal radii for Gersgórin discs. To be more specific for a given matrix $\bf A$, find $\bf C$ and $\bf ...
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1answer
46 views

Are minimizing a function and root finding the same?

What is the relationship between minimizing a function and finding a root of an equation? Are the the same? I know in both problem we have similar algorithms, such as gradient decent, or newton's ...
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37 views

How can this be minimized?

I have the following function of $x_1$ and $x_2$: $$e(x_1,x_2)= (x_1^2+x_2^2)(a+n)+2a(-x_1+x_1x_2-x_2)+a^2$$ where $a$ and $n$ are real numbers. I want to find the values of $x_1$ and $x_2$ that ...
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113 views

How to maximize Std Dev given a range of possible values, a number of values, and a specific mean?

(I'm asking here and not stats.stackexchange because I'd like a mathematical proof of this) In this question: Prove how to maximize Standard Deviation given a certain mean $\bar{x}$ and set of values;...
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30 views

minimize quadratic form

In question Minimize Energy using Gauss-Seidel method with successive over- relaxation., when $$ E = \sum_i \|I_i - \mathbf N_i^T\mathbf L\|^2 + \lambda\sum_{i,j}\|\mathbf N_i - \mathbf N_j\|^2 = \...
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4answers
55 views

$\cos2\theta +\cos\theta +k = 0 $ - set of all values of $k$ for which there is a solution

The set of all values of $k$ (real), such that the equation $\cos2\theta +\cos\theta +k = 0 $ admits a solution for $\theta$ is? MY ATTEMPT: I substituted $\cos2\theta$ with $2\cos^2\theta - 1 $. On ...
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1answer
16 views

Formulating a problem involving sets with ILP

Consider set $\mathcal{G} = \{G_1, \ldots, G_K\}$. We are given $\mathcal{A}_i \subset \mathcal{G}$, $i \in \mathcal{N}= \{1,\ldots, N\}$ and for each $\mathcal{A}_i$, there is a corresponding cost ...
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17 views

Recommend a optimization book with more coding examples?

I am interested in continuous optimization problems. However, I feel it is very difficult for me to understand the classic books such as Convex Optimization or Numerical Optmization. My problem with ...
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35 views

Finding a maximum with some constraints

I would like to maximize the term $ l_1b_1+l_2b_2+l_3b_3-2 $ such that the following conditions hold: $ 1>l_1>l_2>l_3>0 $, $ l_1,l_2,l_3 \in \mathbb{Q} $, $ b_1,b_2,b_3 \in \mathbb{N} $...
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31 views

In Nonsmooth Optimization, whats the point of $x \in G$ being a local minimum results in $0 \in \delta\ f(x)+N_G(x)$

I'm reading about nonsmooth optimization and specially the book 'Nonsmooth Optimization: Analysis and Algorithms with Applications to Optimal Control' by Marko M. Mäkelä, Pekka Neittaanmäki. At page ...
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1answer
42 views

Prove that $-4\leq5\cos\theta+3\cos(\theta+\frac{\pi}{3})+3\leq10$

Prove that $$\color\red{-4}\leq5\cos\theta+3\cos(\theta+\frac{\pi}{3})+3\leq\color\red{10}$$ My attempt:- I simplified the equation to $$\begin{align} &\;\;\phantom{=} 5\cos\theta+3\cos(\...
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27 views

Minimum norm of analytic function may not be achieved on the boundary of its domain

I need to show that the minimum modulus of an analytic function may not be achieved on the boundary of its domain. I'm stuck with this question, would appreciate if someone could help me with it. I ...
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3 views

Revenue Leakage optimization/Minimize Revenue Leakage

I'm a beginner in Operation research field.I need to do an optimization of Revenue leakage.Can you please suggest me some good reading links where I can get idea about revenue leakage optimization ...
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1answer
29 views

argmin as projection in the dual averaging algorithm

I am struggling to understand the dual averaging algorithm as presented in this paper. More precisely the update of the parameters given as $$\Pi^\psi_\chi (z,\alpha) := \operatorname{argmin}_{x \in \...
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Acyclic orientation of a mixed graph with minimization of the critical path

I already asked this question as a guest but I was not able to edit it or add comments after I registered with my e-mail address. A apologize for asking the same question again. A mixed graph is a ...
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23 views

maximize 3-variable linear function [version 2.0]

This problem came up when I was trying to solve a bigger, probabilistic problem. So at the end it boils down to this: how can we maximize the function $f(x_2,x_3,x_4) = \frac{18}{100}\frac{x_2}{6}...
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1answer
46 views

maximize 3-variable linear function [version 1.0]

This problem came up when I was trying to solve a bigger, probabilistic problem. So at the end it boils down to this: how can we maximize the function $f(x_2,x_3,x_4) = \frac{18}{100}x_2 + \frac{...
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27 views

Sum/product of two functions of two variables are to be minimized

I have two functions $f(x,y)$ and $g(x,y)$ whose sum/product (whichever is possible) is to be minimized. The values of $x,y$ can vary in the interval $0<x,y<1$ (hence none of them can have a ...
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21 views

Skew-Symmetric Parts of Stochastic Matrices

It's easy to see that the set $\{W - W^T : W \in \mathbb{R}^{n \times n}\}$ is precisely the set of real skew-symmetric matrices. This continues to be the case if we restrict to (entry-wise) non-...
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29 views

Multiplying quaternions vs multiplying rotation matrices

It's a trivial question, but one I'm not 100% clear about. Given two matrices $$P_{\{1,2\}} = \left[ \begin{array}{cc}R & t \\ \textbf{0} & 1 \end{array}\right]$$ where $R$ is a 3x3 ...
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Find minimum and maximum on range

$f(x,y)=x^{4}-x^{2}+y^{2}$ $B={(x,y)\in \mathbb R, x^{2}+y^{2}\leq 1 }$ I should find minimum and maximum of this function on the range B. I tried it with Lagrange Multiplier and I got these points ...
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1answer
31 views

Least Squares Sensitivity to data

Let ($x_1$,$y_1$),...,($x_n$,$y_n$) be my data set. I have a function $f(x,{\bf c})$ where ${\bf c}=(c_1,...,c_m)$ is a vector of $m$ parameters. I want to fit to the data using non-linear least ...
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Orient edges in a mixed graph to minimize the critical path

3 down vote favorite A mixed graph is a graph that has directed and undirected edges. Is there an efficient algorithm that allows the orientation of undirected edges in a mixed graph in such a way ...
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Solve $ \max U = [\sum\limits_{i = 1}^2 {a_i^{{1 \over \sigma }} } \cdot X_i^{{{\sigma - 1} \over \sigma }} ]^{{\sigma \over {\sigma - 1}}} $

The problem is $$ \eqalign{ & \max U = \left[\sum\limits_{i = 1}^2 {a_i^{{1 \over \sigma }} } \cdot X_i^{{{\sigma - 1} \over \sigma }} \right]^{{\sigma \over {\sigma - 1}}} \cr & \...
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3answers
95 views

A convex optimisation problem involving the Euclidean norm

Any ideas on how to approach the following optimisation problem? $$\begin{array}{ll} \text{maximize} & \|Ax\|_2^2+\|Bx\|_2^2+\|Cx\|_2^2 \\ \text{subject to} & \|x\|_2 = 1\end{array}$$
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1answer
111 views

Minimum of a function in $(0,1) \times (0,+\infty)$

I would like to minimize the function $$ (\alpha,\theta) \mapsto F(\alpha,\theta) := -\theta x^\alpha + \sum_{k=1}^N \ln(1+p_k(e^{\theta \ell_k^\alpha}-1)) $$ where $\theta \in (0,+\infty)$, $\alpha \...
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1answer
120 views

Largest rotated ellipse inscribed in a rectangle

Let's say I have a parametrized ellipse $$x (t) = a \cos(t) \cos(r) - b \sin(t) \sin(r)$$ $$y (t) = a \cos(t) \sin(r) + b \sin(t) \cos(r)$$ Where $r$ is the rotation around the axis and $t \in [0,2\...
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26 views

The gas cloud covering problem

I'm faced with problem described below. My goal in posting this here is having you guys lead me in the right direction. Maybe there is a scientific article that treats a similar problem? Maybe a ...
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17 views

Genetic algorithm optimize and minimize

I'm using a Genetic Algorithm to increase a certain value and decrease another. I'm trying to find the best parameters for a trading strategy. There are 2 values important for me. The netto profit ...
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2answers
53 views

Lagrange multipliers with trigonometric functions. Stucked figuring out x and y values.

I want to find the maximum of the function $f(x,y) = \cos^2(x) + \cos^2(y)$ with the constraint $x-y = \pi/4$. Here are my partial derivatives: $$f_x = -2\cos(x)\cdot\sin(x)$$ $$f_y = -2\cos(y)\cdot\...
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Confused about solution to the piecewise constant regression model

I am confused about the solution to the following solution to fitting piecewise constants: Specifically, are we minimising the sum of squares, that is, finding the vector $\beta = (\beta_1,\beta_2, ...
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22 views

Distance between a point and a conic curve

I have a point $r=(100,0)$ and want to find the closest point to it from this set: $$k = \{(a,b) : b^2=1+a/4\}$$ where $a$ belongs to $[-4,0]$. I thought about defining function $h(x)=|r-x|$, and ...
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1answer
44 views

Optimize wrt a partial matrix?

I have a common optimization problem $$\arg\min_A \text{tr}( A^TWA),$$ where $W$ is a positive semi-definite matrix, and $A$ is the matrix to be optimized. If $A$ is completely unknown, with some ...
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17 views

How write one optimization formula.

In this game I start with a Galleon with capacity 400. I can upgrade the harbor to get more Galleons, or upgrade the technolgy to increase the Galleon base cargo size by 10%. Right now have 8 ...
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151 views

A graph problem

Consider the following graph problem. We are given a set of vertices $A_i$, $B_i$, and $C_i$ where $i \in \{1,2,3 \}$. For each vertex, there is a corresponding weight where the weight of vertex $A_i$ ...
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1answer
36 views

What are spurious local optima?

I keep seeing that word "spurious" (when used in the context of optimization), but I'm having trouble finding a good reference on what the definition of the term is.
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64 views

Can't find minimum using Lagrange multipliers

I want to find the minimum of the function $f(x,y) = x + y^2$ with the constraint $2x^2 +y^2 = 1$. Here are my partial derivatives: $$f_x = 1$$ $$f_y = 2y$$ $$g_x = 4x$$ $$g_y = 2y$$ I have the ...
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30 views

Lasso with non-linear objective

I have a non-linear objective function that I want to minimize considering some constraints in order to obtain a sparse solution (lasso type). min f($\theta$) s.t. $\sum_i|\theta_i|\leq t$ $\theta_i ...
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48 views

Why does the “printing neatly” algorithm use cubes rather than squares?

In Introduction to Algorithms, 2nd ed. (Cormen, Leiserson, Rivest, and Stein), ch. 15, Dynamic Programming, problem 15-2 Printing neatly (a copy of which is here), the official solution given in ...
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1answer
21 views

floor/ceiling/round functions in the constraints of an optimization?

I have a constrained optimization problem in which I have to impose a "floor" or "ceiling" constraint to the solution. In fact I decided to use these nonlinear rounding functions because I needed to ...
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1answer
48 views

What is the name of this problem? linear Matrix equation optimization?!

I have almost no knowledge in linear algebra but I need to understand the process of solving a problem. In fact I'm looking for some keywords or hints to know what exactly should I be Googling! So any ...
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2answers
46 views

Logistic regression for football results - Estimating coefficient through maximum likelihood

Consider two football teams $V$ and $L$ with strengths $W_V$ and $W_L$, respectively. Let's assume that the draw probability $\mathbb{P}(Draw)$ is known. Then this model is supposed to give estimates ...
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0answers
14 views

Multi-objective optimization or single objective optimization?

I have this function: A(x)= P(x) / B(x) Firstly I thought about doing an multi-objective optimization, maximizing A(x) and minimizing B(x) because this two values are very important. But if I just ...
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4answers
82 views

Minimize the function $f(y_1,y_2)=3 y_1^2+8y_2^2$ [closed]

I would like to minimize $f(y_1,y_2)=3 y_1^2+8y_2^2$ with the constraints $g(y_1,y_2)=y_1^2+y_2^2=1$. I thought I could use the Lagrange multipliers, but it is not work. Is there anyone could show me ...