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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Model linearly: What products to make, how much to make and in what plants to make them?

A company wants to make 3 new products for the upcoming week. We are given that: Each product can be made in 1 of 2 plants. At most 2 of the 3 new products should be chosen to be made. Only 1 of ...
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16 views

Minimizing sum of minimum

What are some theory/algorithm that talk about minimizing sums of minimums? For example, assuming y and z are discrete and the function is linear in x: $\min_{x} \sum_{y} \min_{z} f(x,y,z)$. I ...
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67 views

Calculus optimization problem leads to a quartic polynomial - is there a better way?

I am tutoring a student in first-semester Calculus. He needs to minimize the function $$f(x)=\frac{\sqrt{4+x^2}}{2}+\frac{\sqrt{1+(3-x)^2}}{4}$$ Taking the derivative and setting it equal to zero, we ...
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16 views

Normalizing eigenvectors when diagonalizing

Suppose your square matrix is symmetric and I want to diagonalize it. Why is it that at the end you normalize the eigenvectors to get your orthogonal matrix (actually orthonormal matrix)? Is this just ...
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29 views

Model linearly: Determine amount of units for production

A company produces 2 products in a week. Let $x_i$ denote the number of units of product $i$ to produce. Each product requires liters of Chemical X to make. Info is given below: \begin{array}{|c|c|} ...
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20 views

Maximize sum of squares

Lets say that I know that $n$ values $x_i$ sums up to $\mu$: $$ \mu=\sum_{i=1}^n x_i $$ I also now that $0\leq x_i\leq 1$ for all $i=1\cdots n$. I want to find an upper bound as tight as possible ...
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1answer
40 views

Relationship of radius of sphere to an inscribed right circular cylinder for max and min values

I cannot seem to find the correlation between having an interval of a radius of a sphere with finding the greatest lateral surface area of a right circular cylinder inscribed in it. The question goes ...
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14 views

Optimization PDE

I have an assignment where the question reads: $\min J(u) = 1/2 \int_0^1x^2u'(x)^2 dx - \int_0^1 u(x) dx$ with $u \in H_0^1(0,1)$ show $J(u) \geq -1/2 \forall u \in H_0^1$ So I try the usual ...
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30 views

$\max 2x_1 +x_2$ unbounded or unfeasible with the constraint $sx_1 +tx_2\le-1$

\begin{cases} \max & 2x_1 &{}+x_2\\ & sx_1 &{}+tx_2&\le-1\\ & x_1,x_2&&\ge 0 \end{cases} Find out when this program is not feasible, bounded Feasibility It ...
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26 views

Show convexity of $f(x,y,z)= x^2+y^2+z^2+xyz$

Let $f(x,y,z)= x^2+y^2+z^2+xyz$. Show that $f$ is convex on $\Omega=${$(x,y,z)\in R^3 : x^2+y^2+z^2<\frac{5}{2}$}. To prove it, I want to show that $\nabla^2f(x,y,z)$ is positive definite. I ...
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Does the existence of a Algebraic Riccati Equation implies the existence of an functional minimization?

Let $\forall k\ge 0. V_k(x)$ be the value function related to the recursive optimization problem $ J(x_0) = \underset{u}{\inf} \sum_{k=0}^{N-1} x_k^T Q x_k + u_k^T R u_k + x_{N}^T P_N x_N \\ s.t. ...
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1answer
41 views

How to rewrite/solve this differential equation

\begin{equation} \sin(\theta + d\theta) = \sqrt{1 + \frac{dy}{y}}\cdot{\sin(\theta)} \end{equation} I think this is a non-linear and non homogeneous first order equation. I found this whilst trying to ...
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2answers
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A differentiable function $f$ with maximum at $x=c\Rightarrow f''(c)<0$ true OR false

State true or false A differentiable function $f$ with maximum at $x=c\Rightarrow f''(c)<0$ I think this is a true statement but my book says this is a false statement.I do not understand why ...
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37 views

How many stationary points for this class of functions?

Let $f,g \in C^{\infty}[a,b]$ such that $f(a) = g(a)$ and $f(b) = g(b)$ and $f',g' \leq 0$ and $f'' > 0$ and $g'' < 0$. By the Rolle theorem I can say that there's at least one stationary point ...
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32 views

Weights in goal programming

I'm not quite convinced about assigning weights in goal programming. Here is an example formulation problem. What I tried: Let $x_j$ be the number of minutes for ad $j = R, T$ We want to ...
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37 views

$4$ or more type $2$ implies $3$ or less type $1$

I'm having difficulties with the logic with the last part of the reformulation part of the problem below. Let $x_i$ be the the number of ships of type $i$ to purchase. For $4a:$ (the ...
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22 views

Minimization problem with infinite variables and linear constraints

How can this minimization problem be solved? $$ \left\{\begin{matrix} \begin {aligned} &\sum_{i=1}^{\infty}P_i^3 \rightarrow min \\&\sum_{i=1}^{\infty}P_i=1 \\ &P_i\geqslant 0 \:for\: ...
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Time independent vs. time dependent lagrange multiplier

What are the differences between these two in applications? For example: $$max\sum_{t=0}^{\infty} \beta^t u(c_t)$$$$s.t.f(c_t,c_{t+1},x_t,x_{t+1})=0$$ What are the differences between: ...
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Where does the duality comes from in linear programing and can we get the optimal basis from it?

$$\begin{cases} \max & c^Tx\\ & Ax\le b\\ & x\ge 0 \end{cases}\Leftrightarrow \begin{cases} \min & y^Tb\\ & y^TA\ge c^T\\ & y^T\ge 0 \end{cases}$$ Then we come to the ...
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45 views

Nonlinear constraints replaced by parameters and estimated iteratively

I have an optimization problem with nonlinear constraints in the following form: $x + y + 0.5(x+y)^2-z = 0$ $s+(x+y)*t\ge M$ I linearize these constraint by replacing the nonlinear terms by ...
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Regularization as an alternating objective or combined objective

I have a "primary" task loss function $L=L1$ which I want to minimize. Adding a regularization term via $L=L1+\lambda L2$ can be thought of as "forcing" the optimal solution to be meaningful for a ...
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32 views

Linearize non-linear constraint [closed]

I have a problem which may be defined as: $$\max 5 x_{11} + 6 x_{12} + 2 x_{21} + 3 x_{22} \\ x_{ij}\in \{0,1\} \\ x_{11} + x_{12} = 1 \\ x_{21} + x_{22} = 1 \\ t_1,t_2 \text { integer} \\ (t_1 - ...
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Finding the lower bound of a linear program with the duality method

The issue I have some difficulties understanding the lower bound of a program when applying the duality method. It seems that it comes from $$c^T\underbrace{\le}_{x\ge 0\\y^TA\ge c^T} ...
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15 views

Value of Taylor Approximation

I'm studying Trust Region Methods and there's one simple thing I'm finding it hard to wrap my head around. The premise of the method is to approximate $f(x)$ as a quadratic model: ...
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26 views

Maximize $f(\textbf{x},\textbf{y}) = f_1(\textbf{x}) + f_2(\textbf{x},\textbf{y})$

(Sorry if I'm not formal enough) Let $$ f(\textbf{x},\textbf{y}) = f_1(\textbf{x}) + f_2(\textbf{x},\textbf{y}) $$ a real function of vector variable $\textbf{z} = (\textbf{x},\textbf{y})$ , that ...
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23 views

Quadratic approximation of function of two variables near critical point

I know that to second-order, the Taylor polynomial of a function $f(x,y)$ at a critical point $(a,b)$ which gives a negative minimum is $$f(x,y)\sim f(a,b)+c_1(x-a)^2+c_2(y-b)^2+c_3(x-a)(y-b)$$ for ...
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Is there a name for smoothed maximum value function?

I have several arrays that look something like this: Spectrum Plot. Think gaussian curves, but shorter and with lots of noise. I've been comparing values for the sake of peak detection. Through ...
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58 views

Maximum and minimum value

$f(x)=2x+7$ has absolute max value at $x=3$, true or false? The question is wrong, right? Because he should mention an interval?
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Solve convex optimization problem with objective function having power \alpha >1

I am not able to figure out how to get the explicit solution of the following minimization problem: $$\min_{\mathbf{w}\in \mathbb{R}^n} = \sum_{i = ...
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Can the constrained optimization problem (1) be transformed into the unconstrained form (2)

(1) \begin{equation}\label{constrained} \begin{array}{cl} \arg \min \limits_{\mathcal{C}_k} & \text{rank}(\mathcal{C}_k)\\ \mathrm{s.t.} & \mathcal{E}(\phi_{j}^{k})\le \epsilon ...
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general framework for proof of convexity in least square problem

Let $f(x_i,\theta) : X \times \Theta \rightarrow Z$ be a parametric function with parameter $\theta$ that we wish to fit to set of samples $S =\{(x_i,z_i)\}_{i=1}^N$, where $(x_i, z_i) \in X \times ...
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58 views

Fourier-Motzkin - exercise

I'm going to solve this exercise but I'm stuck. Use Fourier-Motzkin elimination to compute the minimal value of $$x_1 + 2x_2 + 3x_3,$$ when $x_1$, $x_2$, $x_3$ satisfy $$x_1 − 2x_2 + x_3 = 4,$$ ...
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Where do we use the fact that we chose b > =0 in P of the general primal-dual algorithm?

It's in in the general primal-dual algorithm and I don't know why we choose b>=0 in P. I guess it may be related to the RP problem but I am not sure because i don't not have a deep understanding of ...
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maximizing a ratio of sums

Consider 3 n-uplets of real positive values : $a_i$ such that $\forall i , \frac{1}{X} \leq a_i \leq X$ where $X>1$ $b_i$ such that $\forall i , \frac{1}{Y} \leq b_i \leq Y$ where $Y>1$ and ...
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Equality constraints into inequalities constraints through elimination

I read here in Section 10.1.2 of this text that a way to eliminate linear equality constraints of the type $$Ax = b$$ in convex optimization problems is to parameterize the related affine space as a ...
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Overconstrained linear system. Minimizing error for “quasi-solutions”.

This is a practice-motivated problem, and I know very little about optimization, so I come here for help. Consider the system $$A\mathbf x=\mathbf b$$ Where $\mathbf x, \,\mathbf b$ are ...
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Are the constrained optimization problem equal to the unconstrained one?

(1) \begin{equation}\label{constrained} \begin{array}{cl} \arg \min \limits_{x} & \|Ax-b\|_2\\ \mathrm{s.t.} & \|x\|_1\le \epsilon \end{array} \end{equation} (2) ...
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22 views

Showing how a dot product is less than 0

Considering a function $f\in C^1(\mathbb{R}^N)$ with $g=\nabla f(x)$, I want to prove a result about optimizing it. However the part I am struggling with I think can be reduced to simple matrix dot ...
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38 views

Calculus 1: Optimization Word Problem - Right Triangle

Find the maximal area of a right triangle with hypotenuse of length 6. I've labeled my triangle with Z being the hypotenuse and the two sides X and Y. I know $$A = BH/2 = XY/2$$ Using the ...
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A ratio of two convex functions with different minima cannot be monotone. Proof?

Let $\lambda(x)=\frac{f(x)}{g(x)}$ where $f(x)$ is a differentiable function minimized at $x=x_1$ and $g(x)$ is a differentiable function minimized at $x=x_2\neq x_1$. How can I show that $\lambda(x)$ ...
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Show that every extrema of $\phi_a(x)+\phi_b(x)$, sum of angular diameters, is a linear combination of $a,b$

Where $a,b\in \Bbb{R}^n$, $|a|=5,|b|=10$, $a,b$ are linearly independent, define $\phi_a(x)$ to output the angular diameter of a point $x$ with respect to the sphere $S_1(a)=\{x\in ...
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Global minimum of the function 'maximum of absolute values of linear fuctions in two variables'

Let $a=(a_{1},a_{2},\ldots,a_{n})\in\mathbb{R}^{n}$, $b=(b_{1},b_{2},\ldots,b_{n})\in\mathbb{R}^{n}$, $c=(c_{1},c_{2},\ldots,c_{n}) \in\mathbb{R}^{n}$ be given. Let us firstly consider the problem of ...
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How to solve a multiple knapsack problem?

I have the following binary LP max $\sum_{l=1}^{L}\sum_{f=1}^{F}[S_{f} \sum_{k=1}^{K}a_{kl}b_{kf}]x_{lf}$ s.t $\quad 1)\quad \sum_{f=1}^{F}x_{lf}S_{f}\leq C_{l} \quad \forall l$ $\quad 2)\quad ...
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Finding minimum energy graph, subject to constraints

I imagine there's a known algorithm for this, but am not totally sure what to search for, and so my search didn't turn up much. Basically, I have have a set of $N$ nodes $\hat x_i $ in a graph $\hat ...
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67 views

Are there any global extrema in this Lagrange Multiplier problem?

I'm trying to find the max and mins of the equation $f(x,y,z) = xy + 3xz + 2yz$ on the constraint, $g(x,y,z)=5x+9y+z-10$. So according to the Lagrange Multiplier procedure, I take the partial ...
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Minimization over a specific set

Suppose we want to find the argmin of a function over a specific set: \begin{equation} \tilde z= \text{arg}\min_{z \in I} \Phi(z), \end{equation} where $z$ is a vector (say, $z \in \mathbb{R}^n$) and ...
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48 views

Property of Rayleigh Quotient

I want to know on how do I prove this following statement of the Rayleigh Quotient. If A is symmetric, the optimization values (I) and (II) below have the same optimal value. If A has at least one ...
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70 views

What programs or websites solve linear integer or goal programming problems?

I don't think I can use Excel. My solver doesn't work so I can't even use Excel for regular linear programming. Something like this but for integer or goal programming. This seems to allow integer ...
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21 views

computational-expensive signal reconstruct - a combination problem

My problem is: I have a time series signal (vibration signal), use BSS algorithm (Blind Source Separation, we can regard it as a black box), separate the source signal into 100 components. Now I ...
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25 views

Egyptian fraction with least possible sum

Suppose that $~a~$ and $~b~$ are coprime positive integers. Then there exists representation of $~\frac{a}{b}~$ as egyptian fraction: $$~\frac{a}{b} = \frac{1}{d_1} + \cdots + \frac{1}{d_s} ~$$ There ...