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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Devising a likelihood method for estimating disease prevalence in hunted deer populations

I am attempting to find the maximum likelihood estimate for disease prevalence in trapped mice by using data on the probability of being trapped each year and the number of mice actually trapped that ...
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8 views

Find $\min \sum_{1\le i\le n} x_i\mathbf{z}^T\mathbf{A}\mathbf{y}_i +\mathbf{b}^T\mathbf{x} +\cdots$

I have been stuck at this problem for a while :( Given $\mathbf{A}\in\mathbb{S}^{p\times p}, \mathbf{A}\ge 0,\mathbf{A} \text{ symmetric}, \mathbf{b}\in\mathbb{R}^n,\mathbf{c}_i\in\mathbb{R}^p\forall ...
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51 views

Assignment problem with multiple types, capacities and costs

I am trying to solve an optimization problem (variation of assignment problem). I'm stuck with how to represent this problem (as an LP or graph based). If it's formulated as a LP, I'm unsure of how to ...
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2answers
46 views

Max-min of a function on closed, bounded interval using EVT

I'm just having little bit of difficulty with the following question: Find the local maxima and minima of $f : [0, 1] \rightarrow \mathbb{R}$ defined by $$ f(x)=x^4(1-x)^6 $$ So we know the ...
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1answer
71 views

How to solve the coupled integer programming problem?

I have the following integer linear programming problem: $$\begin{equation*} \begin{aligned} & \underset{x}{\text{maximize}} && \sum_{k=1}^K\sum_{t=1}^Tx_{kt} \\ & \text{subject to} ...
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1answer
22 views

Is this reducible to a standard optimization problem?

There are $N$ agents who needs to be allocated $K$ discrete resources. There is a bottleneck threshold utility $R$ per agent. The $i$th agent has utility $r_{ij}$ if he is allocated $j$th resource. ...
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1answer
17 views

Motivated by Level Sets, how can I show that minimizing this functional is equivalent to this PDE?

I would like to show, that minimizing the functional $$F(g)=\alpha\int_\Omega |\nabla g(x)|^2dx+\mu \int_\Omega (g(x)-f(x))^2dx $$ is equivalent to solving the differential euqation $$-\alpha\nabla ...
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Multiple optimal solutions / LP

In the optimal primal simplex tableau, if we have a non-basic variable with a reduced cost of zero, can we say for sure the primal has multiple optimal solutions? Or can the same thing also happen ...
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2answers
26 views

optimization on two “max” function

Anyone knows how to use lagrange multiplier (or KKT conditions) to minimize an objective function such as $L(\beta,\beta_0)=\sum_{i=1}^n[a_i(1-y_if(x_i))_++b_i(1+y_if(x_i))_+$] where $a_i$, $b_i$ ...
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31 views

tangent cone to the set

I'm supposed to solve this problem: Let us consider the set $M=\{(x, \sin{x}):x\in\mathbb{R}\}\cup\{\big(\cos(x)-1,x\big):x\in\mathbb{R}\}$ The question is to find the tangent cone to the set $M$ in ...
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20 views

Derivative vector and Hessian for maximization

I'm having some troubles regarding maximization of approximated utility. I want to use the Newton method, but in order to do so I need the derivative vector and the Hessian matrix (I will be ...
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26 views

Karush-Kuhn-Tucker conditions for non-linear optimalization

I have the following problem: solve the local conditions (KKT) and find ALL optimal solutions: $$\min f(x,y)$$ subject to $$g(x,y)\le 0$$ $$x\geq0, y\in\mathbb{R}$$ I have some questions to this ...
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1answer
26 views

Rounding a distribution to minimize loss

This question deals with the problem of choosing cutoff points such that rounding a random variable down to the nearest cutoff point doesn't lose "too much" of its mean. Formally: Let $y$ be a ...
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14 views

Saturation Curve

I have a equation which is x/(x+40). I'm trying to find a point indicated in the graph. As you can see i drew 2 lines, one tangent to the region which it saturates, the other were it has max growth. ...
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24 views

Upperbound for a linear algebraic ratio?

Consider ($n\times 1$)-column vector $\mathbf{p} = (p_i)_{i=1}^n$ with $p_i > 0$ and a symmetric ($n\times n$)-matrix $\mathbf{A} = [a_{ij}]$ with $a_{ii} = 0$ and $a_{ij} \in [0,1]$ for $i \neq ...
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1answer
24 views

Minimizer of a quadratic form

Suppose I have a quadratic form of the form: $$q(x)=\frac{1}{2} x^T Q x$$ Now I want to find the minimum step length w.r.t the steepest descent. So I know the descent direction is $\nabla q(x)$. So ...
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55 views

What is the maximum value of the sum $\sum_{i=1}^L(\bar{x}-x_i)$, in this specific case.

Let $x_i$ be a positive real variable, with $i=1,2,...,K$. We denote by $\bar{x}$ the average value of the values $x_1, x_2,...,x_K$. Let $a=\min_i x_i$ and $b=\max_i x_i$, then $x_i \in [a,b]$. My ...
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1answer
49 views

matrix gradient

I found the gradient of an optimization problem as $$ b*I + \rho\big(-A+diag(A)+X-2diag(X)\big) = 0 $$ But my problem is, I want to find the equation for $X$. From the above equation, because of the ...
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1answer
35 views

Constrained optimization of $f(x,y) = e^{-x^{2}-y} $

Let $f(x,y) = e^{-x^{2}-y} $ and the constraint set $M$ be $\{(x,y): y^2 = e^{-x^2}\}$. Then A. $f(x,y)$ is not bounded on $M$ B. $(0, -1)$ is point of local mimimum C. $(0,1)$ is ...
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21 views

How to divide items into nearly equal sized groups

I teach a class with about $N=290$ students who will be taking an exam next week. The exam consists of two sections ($A$ and $B$) each with 6 essay questions. Students must answer one essay question ...
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414 views

Can a dynamic programming problem be transformed into a linear algebra problem?

Here is a simple standard economic problem: Let Robin Crusoe have an endowment $w_0$ of lembas bread. She is immortal and discounts at a rate of $\beta$ per period. Each period (from $t=0$ on) she ...
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52 views

Proof that equation is non-convex function

I have a objective function as following $$E(\phi)=\int_{\Omega}(I(x)-m_1)^2H(\phi(x))dx+\int_{\Omega}(I(x)-m_2)^2(1-H(\phi(x)))dx+\int_{\Omega}|\nabla H(\phi(x)|dx$$ where $I$ is an image; $I: \Omega ...
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29 views

Constructing canonical tableau for a linear programming problem involving SVM

I have the following set of inequalities and equalites $$\begin{align}y_1x_1+\cdots +y_nx_n &= 0\\ x_1 &\geq 0\\\vdots\\x_n&\geq0 \\ x_1&\leq c\\\vdots \\x_n&\leq ...
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1answer
47 views

Sensitivity Analysis, RHS change in some constraints

I am going to first layout the problem, then I'll get to the thing that is troubling me. I am enrolled in a course called "Optimization I", and this exercise is from a chapter called "Sensitivity ...
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1answer
41 views

A weird Calculus of Variations problem

I became stuck with the following Calculus of Variations problem. The problem is related with something called as the "Nadaraya-Watson" model in statistics. We have $N$ inputs ${x_n}$ and each of ...
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1answer
52 views

How to project gradient vector to subspace defined by linear constraints

I have the following set of linear constraints: $$\begin {align}\textbf{y}^T\textbf {x} &= 0 \\ \textbf {0} &\leq\textbf {x} \leq C\cdot\textbf {1},\end {align}$$ where $\textbf {y} \in ...
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2answers
428 views

Prove that if all edge-costs are different, then there is only one cheapest tree.

Prove that if all edge-costs are different, then there is only one cheapest tree (minimum spanning tree). (Use contradiction and make sure to keep track of the costs of the different trees involved.) ...
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ZELDA Guardian Puzzle Part II - Shortest Path (Unsolved for new rules)

This question is in relation to the following previously asked question: Twilight Zelda Guardian Puzzle : Shortest Path (UPDATE: ADDED RULES) A 1-step-less solution was uncovered, but under an ...
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21 views

Lagrange Multiplier with Inequality Constraints

To maximize $f(x, y)$ subject to $g(x, y) \le b$, we define a Lagrangian $$L(x, y, λ) = f(x, y)−λg(x, y).$$ Then the conditions are: $$Lx = 0,\ Ly = 0,\ \lambda(g(x, y) − b) = 0,\ g(x, y) \le b$$ ...
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1answer
20 views

Perfect matching problem

We have a random graph G = (V,E). Two players are playing a game in which they are alternately selecting edges of graph so that in every moment all the selected edges are forming a simple path (path ...
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1answer
430 views

Using gradient descent and Newton's method combined

I have this function $f(\mathrm{X})$ where $\mathrm{X=A+B+C}$ where $\mathrm{A}$ is a diagonal element with variable $a$ on its diagonal. $\mathrm{B}$ is another diagonal matrix with variable $b$ on ...
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Must Number of equality constraints and decision variables be equal?

Must Number of equality constraints and decision variables in an optimization problem be equal ? If not, how can I solve the equality constraint equations with a solver e.g. ...
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Find an example of critical point

Find an example have the following property: Let $\Omega $ be open in $\mathbb{R}^{n}$, $f, g : \Omega \rightarrow \mathbb {R}$ be $\mathcal{C}^{1}(\Omega)$ and $S=\begin{Bmatrix} ...
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find mean of matrices $A_i, A_j$ given $d_{A_{ji}}=\ln{\left|\left| A_{ji} \right|\right| \left|\left| A_{ji}^{-1} \right|\right|}$

Given a finite set $\mathbb{A}$ of $k$ like-shaped, square, non-singular matrices $A_i\in\mathbb{R}^{n\times n}$, let's define $A_{ji}=A_j A_i^{-1}$, then the distance of the two matrices $A_i, A_j$ ...
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27 views

Finding global maximum

I have a program which must quickly find $x$ and $y$ where $x,y\in\mathbb{N_0}$ which correspond with maximum value of a function: $$f(x,y)=\frac{\sum_{i=0}^{|b|-1}{|b_i ...
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1answer
681 views

Linearization of a product of two decision variables

I am trying to solve a problem that involves constraints in which products of two decision variables appear. So far, I read that such products can be reformulated to a difference of two quadratic ...
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35 views

Optimization of sum of logs

I have an optimization problem of the form $$\operatorname*{argmax}_{\mathbf{w}} \sum_i \log(1 + \mathbf{w} \cdot \mathbf{k_i})$$ given some set of vectors, $\mathbf{ \{k_i\} }$. I have tried both ...
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4answers
43 views

Monotonicity of $f(x)= \frac{ e^{\alpha x}-1}{\alpha x}$ where $\alpha \geq1 $ and $x >0$.

There is this function I encountered when I was solving a problem and I am trying to study its monotonicity. So the function is $f(x)= \frac{ e^{\alpha x}-1}{\alpha x}$ where $\alpha \geq1 $ and $x ...
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1answer
16 views

Dealing with free variables in Linear Programming

I have a free variable in my formulation. In the objective function, this free variable has a cost, and another cost coefficient which is only incurred when the free variable is negative. I used the ...
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1answer
30 views

proving that the shortest line conntecting a point and a line will be perpendicular to that line

So I have a problem for my final math project that I've been fiddling with for hours without success. I have to use calculus to prove that the shortest line connecting a point to a line will always be ...
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1answer
25 views

How to solve this kind of Lagrangian function?

Suppose $\mathbf{a} = (a_{0}, \dots, a_{N-1})$ and $\mathbf{b} = (b_{0}, \dots, b_{N-1})$ with $a_{i}\geq0$, $b_{i}\geq 0$. I would like to minimize $$-\sum_{i=0}^{N-1}a_{i}b_{i}$$ subject to ...
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1answer
21 views

Find a critical point satisfied the Lagrange condition is not local extremum

We know that Lagrange Multiplier gives necessary conditions for an extremum.It locates all possible condidates.But not all such points need be extrma. I want to find an example of the point is ...
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2answers
784 views

How to calculate the hessian of a matrix function

I am estimating a model minimizing the following objective function, $ M(\theta) = (Z'G(\theta))'W(Z'G(\theta))$ $Z$ is an $N \times L$ matrix of data, and $W$ is an $L\times L$ weight matrix, ...
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1answer
463 views

Finding the center of mass of a cylinder

Help finding center of mass of soda can? If you represent the soda can as a right-circular cylinder radius=$4$ cm height =$12$ cm We are told to neglect the mass of the can itself. When the can ...
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2answers
35 views

Line of Best Fit Optimization Problem (Stewart's Early Transcendentals, 14.7, #55)

I know posting pictures is kind of frowned upon here, but I didn't want to type the whole problem out, diagram and all. I'm feeling pretty lost on this one. We've been learning about absolute ...
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27 views

what kind of optimization is this?

I have an optimization problem that looks like this: \begin{array}{cc} min & x'\varSigma^{2}x+k^{2}e'e-2ke'\varSigma x\\ s.t. & x'\varSigma x=ke'x\\ & ...
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3answers
385 views

Problem on EU commission

Consider the following problem. A collection of $n$ countries $C_1, \dots, C_n$ sit on an EU commission. Each country $C_i$ is assigned a voting weight $c_i$. A resolution passes if it has the ...
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1answer
34 views

How can I mathematically model the combinatory problem?

I have the following problem, and I would like to model it using a mathematical formula, for a purpose of optimization problem: let's say that I have two tasks $[T_1, T_2]$, and $3$ resources ...
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1answer
26 views

Find local and global extrema for $f(x,y) = y^4 -3xy^2 +x^3$

above you find a function and some questions I have to answer. I'll give you a more or less detailed input of what I did. I'll be glad if you could help me with the questions I inserted with "->". ...
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2answers
42 views

Simple Lagrange Multiplyers Problem

Can anyone please help me with the following: Find the stationary values of $u=x^2+y^2$ subject to the constraint $t(x,y) = 4x^2 + 5xy + 3y^2 = 9$. The answer is given as $u = 9$ and $x = \pm ...