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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Are maximize/minimize operators in optimization problem?

Note: I'm not sure if Math.SX is the best community to ask (TeX.SX might be also good), but I decided to post here because my question is about mathematical rule rather than about (La)TeX technique. ...
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31 views

Find the number of local extrema of that function without calculus.

I need to find the number of local extrema of that function without derivate or using calculus. I know that in $x = 1$ and $x = 3$ $f(x) = 0$ ... in which way I can affirm that this function has at ...
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12 views

Statistical metric for assessing optimality

In a stochastic computational model, I'm given a limited number of parameter sets and hope to identify the one set of input that is optimal, defined by the values of its numerical outputs, i.e., ...
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46 views

Minimal number of steps to construct $\cos(2 \pi /n)$

My question is related to this previous one. I was wondering what is the minimal number of steps $S(a)$ to construct a number $a \in \mathbb R$ that is constructible (as defined here). For instance, ...
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14 views

Quadratic Equality Constrained Quadratic Program and Convexity

There are a few questions on this topic already. However, none of them really answer my question. The most relevant are these: Quadratic optimisation with quadratic equality constraints Quadratic ...
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35 views

Is the logarithm of sum of multiple variables with the constraint on sum of them Concave?

I know that without any constraint $log \sum_{i=1:1:m} \alpha_i C_i $ is not Concave but I am wondering is this function Concave when we have the constraint that $ \sum_{i=1:1:m} \alpha_i =1 $ and ...
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Minimization with orthogonality constraints

I want to solve $$ {\rm arg min}_G \left|\left|A - XGB\right|\right|_F^2 \text{ subject to }G'G = I_u.$$ Here, $||\cdot||_F$ denotes the Frobenius norm and $A \in \mathbb{R}^{t \times g}$ and $G ...
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21 views

Optimization by KKT-method

I need to solve the following problem by KKT method. $$ \text{min} \ \ 2xy + 2yz + 2zx \\ \text{subject to} \ x^2 + y^2 ≤ 2, \ 2x + 2y + z = 0 $$ I have gotten as far as setting up the system of ...
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50 views

Integer programming model not working

I have to formulate an Integer programming model for the following using XPRESS; There are 10 items that need to assigned to 2 categories, A and B. Each item has a weight and 30 % of the weight is ...
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1answer
105 views

Simplex method - identity matrix

I want to solve the following linear programming problem: $$\min (5y_1-10y_2+7y_3-3y_4) \\ y_1+y_2+7y_3+2y_4=3 \\ -2y_1-y_2+3y_3+3y_4=2 \\ 2y_1+2y_2+8y_3+y_4=4 \\ y_i \geq 0, i \in \{ 1, \dots, 4 ...
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Conic formulation. Finding a point minimizing the maximum distance to a set of points.

I have to formulate (and I don't know) as a conic problem the next: Problem: Given a set of points $D=\{a_1,a_2, \dots, a_n\} \subset R^2$. Write like a conic problem the problem to find a point ...
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34 views

Refinery - Mathematical formulation of problem

In a refinery, two types of crude oil $T_1, T_2$ get mixed with two different procedures $R$ and $W$ and produce two types of petrol $P_1, P_2$ as shown at the following matrix: $\begin{matrix} ...
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optimal derivative position through optimization

So I have the following optimization problem: min. $-E^Q[u(h(x))]$ s.t $\int h(x)q(x)dx \leq \frac{V_0}{B_0}$ Where $Q$ is the subjective probability which then gives: $E^Q[u(h(x))]=\int ...
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21 views

Finding an optimal way to distribute government aid cases across multiple government offices

Note, this is for a friend who is an actual government employee, very real world application here, and the implications could be much more far reaching than you might think. Solving this problem could ...
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40 views

Break the connection between two edges by removing the minimum amount of edges

We know that a drug dealer is going to deliver from city A to city B. As the police, we want to avoid the delivery. Cities are connected to other cities by roads (undirected edges). We can place one ...
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37 views

is this function an ill-shape convex function?

I have a function with parameter $\vec{{\alpha}}$ where it is formulated by the formula: $$ f(D|\alpha)=n_1{\alpha}_{1}+...+n_m \alpha_m -Nlog \sum_{i=1:m} exp(\alpha_{i}+g_i(D)) $$ where $g_i{(D)}$ ...
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18 views

Mixed-integer (Linear) Programming (MILP) standard/canonical form

Is there a standard or canonical form for mixed-integer (linear) programming problems? For linear programms the standard form is sometimes given by: $$ \max_{\boldsymbol x} \boldsymbol c^T \boldsymbol ...
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13 views

Using sequence of least squares solutions to solve non-linear problems?

I do know about the iteratively reweighted least-squares and have played around with it to some success finding non-linear solutions (like minimizing non-2-norms to achieve solutions which seem to be ...
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17 views

Percentage constraint for Integer programming model

Th question is as follows: There are 10 items that need to assigned to 2 categories, A and B. Each item has a weight and 30 % of the weight is allocated to A and the remaining 70% to B.The objective ...
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42 views

Computing optimal values of $\beta$ using log-likelihood optimisation

I have the following problem: Let us consider a log-likelihood function $L$ which is depended on the parameter $\beta = ( \beta_0, \beta_1, \beta_2)$ and has a form: $$L(\beta) = \beta_0 ...
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81 views

How many pawns, bishops, rooks or kings can be put on a $n \times n$ chessboard such that they don't threaten each other?

A friend of mine asked me this question and I know this is not easy to solve. I found some informations similar to this question here: https://en.wikipedia.org/wiki/Eight_queens_puzzle; First of all, ...
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60 views

Find the minimal value of $\frac{x(x-1)+y(y-1)}{2}-xy$

I need to find the minimal value of $f(x, y) = \frac{x(x-1)+y(y-1)}{2}-xy$ where $x$ and $y$ are both positive integers. I tried to find the minimum with the help of derivatives: 1) With respect to ...
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94 views

An optimal sequence of length 13

I'm looking for an optimal (or much better than I have now) increasing sequence $t_1, t_2, .., t_{13}: t_i \in N, 0 \le t_i<t_{i+1}$ where $3t_{12} + 4t_{13}$ is minimized, subject to the ...
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What is the moment matrix in converting polynomial optimization problem to a quadratic optimization problem

Happy new year, I have a function of the form below \begin{align} f(x,y,z)=\sum_i x_i y_i z_i + g(x)+h(z)\cr x,y,z \in R^n \end{align} where $h,g$ are quadratic functions. My difficulty lies in the ...
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32 views

Linear Optimization Problem with exponential variable

Hey Folks I've encountered an optimization problem which has a linear programming structure but it's coefficients are nonlinear function of another variable. here is the problem: $$\max ...
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1answer
33 views

How to write an absolute value expression in linear programming?

My objective function for the Xpress-IVE (Mosel lang) model is minimize |a-b| where a and b the number of elements in the decision variables which are arrays. Since there is no function to ...
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8 views

Subdifferentials of matrix norms

I am aware of the fact that, strongly unitary invariant matrix function $F:\mathbb{R}^{m\times n}\rightarrow \mathbb{R}^{q}$, where $q = \min(m,n)$, can be represented by the absolutely symmetric ...
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24 views

help with $\nabla$ and Lagrangian in optimization / portfolio theory?

So $\nabla$ as I know it from calculus means gradient. We have $\min \ \ \frac{1}{2}w^T\Sigma w$ $s.t. \ \ \ \ \ w^T1 = V_0, \ \ V_0 = 100$ where $w$ is weights in vector, $\Sigma$ is the ...
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36 views

Writing an objective function

I have this problem here: How to write a formula for the objective function? A powerhouse is located on one bank of a straight river that is $30$ feet wide. A factory is situated on the opposite bank ...
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28 views

Existence of maximum and minimum

Let $f:\mathbb{R}_+\rightarrow \mathbb{R}$ be continuous and such that $f(0)=1$ and $lim_{x\rightarrow+\infty}f(x) = 0$. Prove that $f$ must have a maximum in $\mathbb{R}_+$. What about the ...
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73 views

Minimize the sum of solution of linear equation

Let x(i,j) be a variable. All variables and constants can only have value of 0 or 1. Also, sum of two variables x(i,j) and x(k,l) is equal to (x(i,j)+x(k,l)) % 2 ...
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45 views

How to find the tangent cone to a set in a point?

Let $S\in R^{n}$ is a set and $x\in S$. We define tangent cone of $S$ in $x$ as: $$T_{S}(x)=\{z\in R^{n}:\exists (x_{k}), x_{k}\in S, x_{k}\rightarrow x, \exists (y_{k}), y_{k}>0, ...
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Linear programming and shortest path

Given the linear programming formulation of the shortest path problem: $$ \begin{align*} \min & \sum_{u,v \in A} c_{uv} x_{uv}\\ \text{s.t } & \sum_{v \in V^{+}(s)} x_{sv} - \sum_{v \in ...
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79 views

Shortest distance to a straight line

Find the coordinates of the point on the ellipse $4x^{2}+y^{2}=4$, which is closest to the straight line $x+y=10$. I could solve it by using Lagrange Multiplier. Is there any way to solve it ...
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43 views

Normal lines to a parabola, and areas bounded by them

This is the question: What I have done: (a) Show that the equation of the normal to the parabola at a point $(x_0,y_0)$ is $y = {-1\over 2kx_0} + kx_0^2 + {1\over 2k}$ $$ f(x) = kx^2 $$ $$ ...
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35 views

ways to change a function locally for my poker texas holdem bot

I apologize for the lack of math format and possible ambiguity. If I had more maths knowledge, this could be a two-sentence question, but I don't have that. I want to find a local optimum for my ...
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61 views

Maximizing the sum of the products of endpoints of edges in a graph

Let $G$ be a graph with vertex set $V=\{v_1,v_2\dots v_n\}$ and edge set $E$. Let $f:V\rightarrow \mathbb [0,\infty)$ be a real valued function such that $\sum\limits_{i=1}^n f(v_i)=A$. What is the ...
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27 views

Is there a name for this discrete version of Jensen, specifically when applied to binomial coefficients?

We have $2k$ integers greater than or equal to $j\geq0$ $a_1+a_2+\dots + a_k=n$ and $b_1+b_2+\dots + b_k=n$. If for all $1\leq i\leq k$ we have $|n/k-a_i|\leq|n/k-b_i|$. Then ...
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39 views

Understanding the solution to a problem on Lagrange Multipliers

This is a problem from Riley-Hobson "Mathematical Methods For Physics And Engineering". QUESTION: Two horizontal corridors, $0\le x \le a$ with $y\ge 0$ and $0\le y \le b$ with $x\ge 0$, meet at ...
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61 views

How to efficently solve a convex optimization problem with positive semi-definite Hessian matrix?

Consider the following optimization: $$ f(x)= \min \sum_{i=1}^n \left(x_i-\sum_{j=1}^n x_j\right)^2 $$ Let $g_i(x)=x_i-\sum_{j=1}^n x_j$ , then $$ f(x)= \min \sum_{i=1}^n g_i(x)^2 $$ The ...
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1answer
44 views

M/M/1 or M/M/n?

In a queuing systems with a single queue that receives $n$ poisson arrival streams with arrival rates $\lambda_1, \lambda_2, ...,\lambda_n$, and exponential service rates of $\mu_1,..., \mu_n$, we can ...
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18 views

Softmax for Continuous Functions?

The softmax $\log \sum_{i=1}^n \exp(f_i)$ of vector $f$ is a smooth upper bound on $\max_i f_i$. However, the same cannot be said of $\log \int_{X} \exp(f(x))dx$ in relation to $\max_{x \in X} f(x)$ ...
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1answer
48 views

Positive-Semi-Definite form of Variance?

first thing: I'm an informatics student and know some algebra. However, this seems to be a bit over my head, so please be gentle with me. ;) I have multiple sets of real variables. Let these sets be ...
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1answer
37 views

Linear vs non-linear Least Squares

I am trying to understand the difference between linear and non-linear Least Squares. In the book I have it says: "If the parameters enter the model linearly then one obtains a linear LSP." "If the ...
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57 views

Minimizing a function (with boundary conditions)

I tried the following thing just for fun (so it does not have a deeper sense), but unfortunately i failed to solve it. Assume this function is given: $$ ...
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27 views

Polynomial time solvable cases of the knapsack problem.

Is there some restricted version of the knapsack problem, which is not $Np$-complete and there is a polynomial time algorithm? In my cases the weights are all power of $2$, so $(1, 2, 4, 8, 16, ...
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178 views

Monotonic Function Optimization on Convex Constraint Region

So I have the following function, which I want to maximize: $$f(x_1,...,x_n) = \sum_{i=1}^n\alpha_i\sqrt{x_i}$$ (where all $\alpha_i$ are positive), subjected to the following equality and inequality ...
3
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1answer
29 views

First order condition in constrained optimization: Alternative characterization via normal cones

Consider the following constrained optimization: Min $f(x)$, $x\in C\subset R^n$, where C is convex. We know that one characterization of a local minimum (necessary condition) is the following: ...
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115 views

What are common Mathematical Programming Languages out there?

I've seen the term used Mathematical Programming to describe a superset of: Linear programming Quadratic programming Nonlinear programming Mixed-integer programming Mixed-integer nonlinear ...
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13 views

minimizing function with exponential term (no exponential constraints)

$\frac{((a-1) j+a ){e^t}^a}{(a (2 a-1))}-\frac{k {e^{a-1}}^ t)}{(a-1)}+t (i+j+k-1) $ with j=-1 i+j+k=1 a-1 >= 0 t >= 0 How can this expression be simplified so it can be used by any solver like ...