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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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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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
25 views

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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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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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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39 views

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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33 views

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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40 views

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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15 views

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
79 views

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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30 views

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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36 views

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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24 views

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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13 views

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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29 views

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
29 views

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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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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51 views

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
23 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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21 views

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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33 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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32 views

$v^TAv\ge \left\|A\right\|\cdot\left\|v\right\|_2$ 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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31 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 : ...
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47 views

Definition functions, integrals on $\mathbb R^{|N|}, \mathbb R^{\mathbb R}$

Is there a standard/reasonable way of defining functions on the sets $\mathbb R^{|\mathbb N|}, \mathbb R^{\mathbb R} $. How about defining integrals over these sets? I guess a function on $\mathbb ...
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calculus optimization: solution two equations

I am to do this maximization problem. I simplified it down to this triangle. I am given no dimensions, instead I am given that G is fixed and H is fixed. The rest are variables. We are to maximize ...
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Quadratic program with complementarity/modular constraints is NP?

Is the following program NP/NP-hard? Any neat way to prove it, or a helpful reference? $\min x^TMx$, subject to $\|x\|_1=1,e^Tx=0$ Here $M$ is a real, symmetric and semidefinite positive matrix, ...
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calculus optimization problem: give answer in equations

I have a weird question as part of a homework assignment I am having trouble with. I think I am getting tripped up using only letters and no numbers. The question goes: Fred and Sally are adding a ...
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34 views

Shortest smooth paper Möbius Strip

I want to make a familiar Möbius strip of width 1 unit satisfying the physical properties of paper. Assume paper is a ruled surface, and the strip has to be smooth and non-self-intersecting. What ...
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Derivatives - optimization (minimum of a function)

For which points of $x^2 + y^2 = 25$ the sum of the distances to $(2, 0)$ and $(-2, 0)$ is minimum? Initially, I did $d = \sqrt{(x-2)^2 + y^2} + \sqrt{(x+2)^2 + y^2}$, and, by replacing $y^2 = 25 - ...
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29 views

How to do optimization

My teacher gave me a very complicated explanation on how to solve an optimization problem so I just wanted clarification. To do so I have laid out what I think is the simplest way to solve it. Take ...
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43 views

ADMM formalization

I found lots of examples of ADMM formalization of equality constraint problems (all with single constraint). I am wondering how to generalize it for multiple constraints with mix of equality and ...
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Lipschitz Number in Gradient Descent

During gradient descent, if an objective function's value is greater than the previous iteration, would use of an orthogonal vector to the update vector be advantageous? Regarding trust regions, the ...
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2answers
83 views

An easy question about NP-hard

Consider an optimization problem includes two variables. If we fix the value of one variable, then the optimization problem over the other variable is NP-hard. Can it be concluded that the original ...
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optimization for the area of a garden

I have been working this problem for awhile and cannot seem to solve it even though its probably easier than I think... There is a rectangular garden that needs fencing. For one side of the fencing ...
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Maximum of norm

Given a matrix $A$ with $N$ rows and $d$ columns, I would like to prove (or disprove) the following. Let $q(f)=\|(\begin{pmatrix} ...
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Gluing two strongly convex function

Definition: We call $f:\mathbb{R}^n\rightarrow \mathbb{R}$ a $\lambda$-strongly convex function iff for every $x,y\in \mathbb{R}^n$ and $t\in[0,1]$ it follows $$f(tx+(1−t)y)\leq ...
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Present and future value: selling now vs selling for a higher price later

A wine dealer contemplates whether to sell his bottle of wine for $\$30$ today, or wait to sell it in the future. If he sells it in the future, then he can sell it for a higher price. However, the ...
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1answer
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How to model a multiobjective problem with a large dataset?

I have a large dataset of businesses (around 5k venues with distance from a predefined point, average price and service quality rating) and I need to create the objective functions to minimize the ...
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39 views

Effect on Minimizer of Tightening Constraints

The Statement of the Problem: Consider the minimization problem $f(x,y)=14x+20y$ under the constraints $x+2y \ge 4 $, $7x+6y \ge 20$, and $x,y \ge 0$. Don't use the simplex method! (i) Draw the ...
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19 views

Do quasi-Newton methods check the second-order optimality condition?

I have a practical question about quasi-Newton methods. In quasi-Newton methods, Hessian matrix is approximated. It seems to be impossible for them to check the second optimality condition. In ...
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38 views

Find Min: $A= \frac{bc}{a(b+2c)} +2 [ \frac{ac}{b(c+a)} + \frac {ab}{c(2a+b)}]$

Given $a,b,c>0$ such that: $ \frac{4a}{b} (1+ \frac{2c}{b}) + \frac{b}{a} (1+ \frac{c}{a})=6$ Find Min: $A= \frac{bc}{a(b+2c)} +2 [ \frac{ac}{b(c+a)} + \frac {ab}{c(2a+b)}]$ My try: Let: ...
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37 views

Weighted least squares with nuclear norm minimizaiton, how to optimize?

Nuclear norm minimization is very popularization and formulation is least squares term with nuclear norm term as following, $$\min\limits_{X} ...
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Reference Request for Penalty Method for Optimal Control?

Is there a good book or review article to read about the methods like penalty method, method of duality and method of relaxation in problems of calculus of variations and their relations to optimal ...
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18 views

Grouping constrained optimization

I am looking for an efficient solution to solve the following problem. Can anybody help? S is a finite set of elements $k_i$ V is a subset of S, e.g. $v_4$={$k_1$,$k_3$} E is a finite ensemble of ...