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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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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1answer
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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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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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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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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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1answer
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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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4answers
42 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
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
48 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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1answer
23 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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26 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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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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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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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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26 views

A Question about Nested Maximizations

I am working on labor demand models where firms have to choose the optimal level of employment by maximizing profits. In particular, I have faced the following problem: Maximize with respect to $l$ ...
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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 ...
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1answer
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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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Minimum of function involving exponentials

I am trying to prove that this function involving exponentials: $g(x)=\frac{\sqrt{2 \pi } \left(1-2 e^{-2 \pi ^2 x ^2}\right) x }{2 e^{-\frac{1}{8 x ^2}}+\sqrt{2 \pi } x -1}$, when $x\geq1/2$ Is ...
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Homework help on eigenvalues function minimization

so I actually have two separate questions which are homework bonuses for my numerical methods course. Unfortunately, because of the time of the semester, our TAs are not available so I don't have many ...
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1answer
25 views

The functional take its maximal value for $y(t)=-t$

I want to show that the functional $J(y)=\int_0^1 [y'(t) \sin{(\pi y(t))-(t+y(t))^2}]dt$ ,where $y$ is a continuously differentiable function on $[0,1]$, takes its maximal value $\frac{2}{\pi}$ for ...
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1answer
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Bivariate probability distribution(s) over unit square, uniform marginals, midpoint is saddlepoint

Construct a bivariate probability distribution--or family of such distributions--over the unit square (corners $(0,0), (0,1), (1,1), (1,0)$) with uniform marginals and having a saddlepoint at ...
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1answer
44 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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number of maximizers

Suppose f(x) is continuous in x∈X, where X is compact. Let T(x):=argmaxf(x) be the set of maximum of f(x), where the maxf(x) is bounded. Then under what condition the set T(x) cannot contain ...
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1answer
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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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Smallest bound for convex combination of columns of non-negative matrix

The problem can be formulated as following linear program: $\min_{\mathbf{x},y}\;\;y$ subject to: $\mathbf{Ax}\le y\mathbf{1}$ $\mathbf{x}^T\mathbf{1}=1$ and $x_i \ge 0,\;\forall i$ Here, ...
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Find $\min_{y \in \mathcal{A}} J(y)$, if it exists.

Let $\mathcal{A}$ be the set of continuously differentiable functions at the interval $[a,b]$. Let $J$ be the functional $$J(y)=\int_a^b \sqrt{1+y'(x)^2}dx$$ Find $\min_{y \in \mathcal{A}} J(y)$, if ...
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Maximization of a function I came up while studying

So in a problem I am trying to solve, after calculations I came up with the following function: \begin{equation*} f(\overline{y},\theta)=\frac{e^{n\,min\{\overline{y},\theta)}-1}{n\theta} ...
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1answer
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Wouldn't this Greedy Algorithm achieve the highest possible of money in this situation?

I am doing a practice question from Midterm Dynamic Programming The Problem : Consider a row of n numbers a1, ..., an. The numbers are all positive, and n is even. We play a game against an ...
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How to find parameters from logistic equation

I have an function and assume that that is convex function. I want to use gradient decent to find parameters in that equation. Could you suggest to me the way to do it. Thanks. This is my function ...
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1answer
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Traveling salesman problem: why visit each city only once?

According to wikipedia, the Traveling Salesman Problem (TSP) is: Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city ...
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Converting a max-min problem to a max problem with a constraint

The objective is to find the greatest lower bound of the variable $\mu$. The lower bound is resulting from the positive-semidefinite (PSD) constraint $$\tilde{\mathbf{T}}:=\left( \begin{array}{ccc} ...
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Optimization of $e^{x^2 + y}$ on $x+y \leq 2$

Let $f(x,y) = e^{x^2 + y}$ and $M = {(x,y): x+y \leq 2}$. A. $f(x,y)$ on M is bounded above and not bounded below B. $f(x,y)$ on M achieves global minimum(a). C. (0,0) is point of ...
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1answer
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How can I solve the following exercise

Find the critical curves for the following functional : $$J[y(x),z(x)]=\int_{0}^{1}(y'^2+z'^2-xyz'-yz)dx$$ With the conditions : $$K[y(x),z(x)]=\int_{0}^{1}(y'^2-xy'-z'^2)dx=2$$ $$y(1)=z(1)=1$$ ...
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1answer
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How does the value of a functional change when you perturb the extremizing function?

In deriving the Euler equation for etremizing a functional \begin{equation*} J[y] = \int_a^b F(x,y,y')\,dx, \end{equation*} we look at: \begin{equation*} J[y+h]-J[y] = \int_a^b(F_yh+F_{y'}h')\,dx + ...
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15 views

Binary depending on the sign of another variable

I'm writing a mixed integer linear problem, where I have an indicator function in the objective function counting the instances of negative values of a decision variable. I thought of defining a ...
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1answer
37 views

A maximization problem within the simplex

Let $\lambda_i$ be an ordered list of $N$ positive numbers, $\lambda_1<\lambda_2<\dots<\lambda_N$. I'm looking for the extrema of the function $$ f=\left(\sum_{i=1}^N p_i \lambda_i ...
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How to determine the optimal step size in a quadratic function optimization

I have the following optimization problem: $$\underset{\alpha\in\mathbb{R}}{\text{min}}:\;\;f(\textbf{x}+\alpha\textbf{d})$$ $$\text{subject to}:\;\;0\leq\alpha\leq \alpha_{max},$$ where ...
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1answer
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Quadratic programming with constrained number of free variables

I started with a (positive-definite) quadratic programming problem subject only to a single equality constraint. i.e. $$ f(x)=x^{T}Qx+c^{T}x $$ $$ s.t. x_1+x_2+x_3+...+x_n=y $$ I now have to find ...
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1answer
70 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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How can we maximize the following functional?

$\max_{} \; \int_0^1 \left( -\frac{1}{2} \left( \lambda_1(1-t) - \int_t^1 \lambda_2(s) ds \right)^2 - 1.25 \lambda_2(t) \right)dt + \lambda_1$ s.t $\lambda_1\geq0$, and $\lambda_2(t) \geq 0$ for ...
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1answer
21 views

Trace minimization-Revised

The problem is as follows: $\displaystyle\min_{V}$ trace($V^TH^T\Phi HV$)$\\$ s.t. $V^TV=I_d$ in the case when $H$ is not known. When $H$ is known, the solution is given by the eigenvectors ...
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Find the edges of a polyhedron P.

Given the polyhedron $P = \{v \in \mathbb R^2 \mid Av \le b\}$ with $A = \begin{bmatrix} -1 & -1 \\ 2 & -1 \\ -1 & 2 \\ 1 & 2 \end{bmatrix}$ and $b = \begin{bmatrix} 0 \\ 1 \\ 1 \\ 2 ...
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1answer
29 views

Solving an optimization problem with KKT-conditions

I've been studying about KKT-conditions and now I would like to test them in a generated example. My task is to solve the following problem: $$\text{minimize}:\;\;f(x,y)=z=x^2+y^2$$ ...
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Say optimal solution to the primal is degenerate. Does it hold that optimal solution to dual not unique?

I think it's supposed to be that existence of a degenerate and unique solution of the primal implies multiple solutions to the dual, according to this book (pages 141-145, proof of Theorem 4.5). In ...
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$v^TAv\ge \left\|A\right\|\cdot\left\|v\right\|_2$ for all positive definite $A\in\mathbb{R}^{n\times n}$

Let $A\in\mathbb{R}^{n\times n}$ be positive definite and $v\in\mathbb{R}^n$. Let $\left\|\cdot\right\|_2$ be the Euclidean norm. Can we prove $$v^TAv\ge \left\|A\right\|\cdot\left\|v\right\|_2$$ for ...
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2answers
30 views

Find maximum and minimum of funсtion on set

I have the task: find maximum an minimum of $$f(x) = x_1(\pi - x_1)\sin x_2 + x_2 \cos x_1$$ on X where $$X = \{x\in R^2\ |\ x_1\in [0, \pi], x_2 \ge 0\}.$$ First thing i did was system : ...