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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Maxima of $f(x)/e^x$ where $f(x)$ is an approximation of $e^x$ using Stirling's

Let $$f(x)=1+\sum_{n=1}^\infty\frac{x^n}{\sqrt{2\pi n}(n/e)^n}\tag1$$ and let $$g(x)=\frac{f(x)}{e^x}\tag2$$ If we plot $g(x)$ we get a graph that looks like this: Clearly there is a maximum at ...
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89 views

Find min/max $\|x\|_{1}$ subject to $Ax = b$, using the simplex method

Let $Ax = b$ be a linear system with $a_{i,j} \in \{0,1\}$ and $b_i \in \{0,1,2,3,4,5,6,7,8\}$. The constraints on $x$ are $x_i \in \{0,1\}$. We suppose that the system admits at least one solution....
3
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1answer
31 views

Where does the name “tracking type problem” come from?

In PDE-constrained optimization problems, the distributed control problem $$ \begin{array}{ll} \displaystyle \min_{y,u} & J(y,u) = \frac{1}{2}\|y-y_d\|_{L^2(\Omega)}^2 + \frac{\alpha}{2}\|u\|_{L^...
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84 views

Find $\theta$ and $\phi$ that maximize $\mid -2ia\sin\theta - 2ib\sin\phi + 2c(1-\cos\theta) +2d(1-\cos\phi)\mid$

How can you find for what values of the $\theta$ and $\phi$ angles the following modulus will assume its greatest possible value? $$\mid -2ia\sin(\theta) - 2ib\sin(\phi) + 2c(1-\cos(\theta)) +2d(1-\...
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1answer
35 views

Shortest possible distance to locate an unknown road

You are stranded in the middle of a large desert and the only way home is a through a straight road, which unfortunately you do not know the location of. If the perpendicular distance from you to ...
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0answers
15 views

Projected gradient descent with momentum

Can we apply momentum to projected gradient descent? If so, how should we do that? In the domain I'm working on, momentum greatly speeds up gradient descent. However, I want to do projected ...
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3answers
47 views

Find point on a line that is nearest to the origin

Can you help me with this exercise? Find the nearest point to the origin $(0,0,0)$ in the line given by the intersection of planes $x+y+z=2$ and $12x+3y+3z=12$. The intersection of the planes is ...
3
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1answer
69 views

Upper bound of $x_1x_2x_3+x_2x_3x_4+\dots+x_{n}x_1x_2$

Let $n\geq 3$ be a positive integer and let $x_i$'s be non-negative real numbers with $x_1+x_2+\dots+x_n=1$. What is the maximum of $x_1x_2x_3+x_2x_3x_4+\dots+x_{n}x_1x_2$? If the sum were symmetric ...
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0answers
14 views

Generate a class of matrices via optimization

I want to generate a matrix (using Matlab) with the following properties: (1) $A = (a_{ij}) \in \mathbb{R}^{n \times n}$; (2) $a_{ij} \in \{0,1\}$ and $a_{ii} = 0$ for all $i\in\{1,2,\cdots, n\}$; (...
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235 views

the continuity of argmin on convex funtion

Define $$x'=\text{argmin}_{x_1}f(x_1,\lambda),$$ where $f$ is a strictly convex function on $x_1$ and $\lambda$. I would like to ask if there is any theorem about the continuity of $x'$ w.r.t $\...
35
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1answer
785 views

A System of Matrix Equations (2 Riccati, 1 Lyapunov)

Setup: Let $\gamma \in(0,1)$, ${\bf F},{\bf Q} \in \mathbb R^{n\times n}$, ${\bf H}\in \mathbb R^{n\times r}$, and ${\bf R}\in \mathbb R^{r\times r}$ be given and suppose that ${\bf P}$,${\bf W}$,${\...
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17 views

Relationship between Newton's method in root finding and optimization

In both root finding and optimization, there are Newton's method. Wikipedia has 2 links here and here. Root finding is using first order derivative and optimization is using Hessian. What's the ...
3
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1answer
3k views

primal to dual solution conversion ??

i have an optimization problem $$\text{ maximize } z=3x+4y$$ $$\text{ such that: } x+y ≤ 450 \text{ and } 2x+y ≤ 600$$ the optimal solution to this problems comes to be $x=0$; $y=450$; $p=150$ (...
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21 views

How to cover a sphere with caps removed, with equidistant points?

I have a sphere with the caps removed, so: $$x^2 + y^2 + z^2 = R^2$$ for $|x|, |y| \leq R$ and for $|z| \leq R_z <R$. This creates a sphere with the top and bottom cap cut off. $R_z$ will be ~ $2R/...
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397 views

Multivariable optimization - how to parametrize a boundary?

A metal plate has the shape of the region $x^2 + y^2 \leq 1$. The plate is heated so that the temperature at any point $(x,y)$ on it is indicated by $T(x,y) = 2x^2 + y^2 - y + 3$. Find the ...
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1answer
105 views
+150

Two Approaches Two Different Solutions: Optimal Controls vs. Different Method

If I try to solve a problem two different ways, I get two different answers which generally means I am committing some horrible sin! Given the problem, \begin{align} \min_u\ S &= \int dt\ L(x, u) ...
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1answer
69 views

Find maximum value of a function [on hold]

$a$, $b$, and $c$ are real numbers, and $a+b+c=0$ and $a^2+b^2+c^2=2$. I need help finding the maximum value of: $$\big|a^2b^2(a-b)+b^2c^2(b-c)+c^2a^2(c-a)\big|$$ To be honest, I don't know where ...
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Finding solution to Calculus of Variation of linear functional whose domain consists of vector valued function

Problem Statement: Find $x^*$ such that it solves the optimization problem $$\max_{x \in \Omega} \quad f(x) = e_i^TAx$$ $$ \Omega = \{x: t \to \Delta^{n}|x \in C^1, x(0) = x_o\}$$ Where $\Delta^...
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2answers
41 views

Maximizing a convex quadratic function in CVX and Matlab

I understand that a convex function can not be maximized as there is no such value. However, consider the following function: $$\begin{array}{ll} \text{maximize} & 3x^2 + 5y^2\\ \text{subject to} ...
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633 views

Matlab optimization toolbox vs. CVX solver?

I would like to know what is the difference between the Matlab optimization toolbox and CVX solver which is a convex optimization toolbox? Can a convex optimization be solved in both?
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Error on optimization problem, maximize log determinant on CVX

$A$ is an $N \times N$ complex matrix $W$ is an $N \times N$ complex matrix $C$ is an $N \times N$ complex diagonal matrix $u$ is a scalar $V$ is an $N \times N$ complex matrix, whose diagonal elects ...
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2answers
59 views

For which point of $x+y+3z+k=10$, the expression $x^2+y^2+9z^2+4k^2$ is minimal? [closed]

For which $x,y,z,k \in \mathbb R$ of $x+y+3z+k=10$ is the expression $x^2+y^2+9z^2+4k^2$ minimal?
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13 views

When is this argmax-based function continuous?

Let $w: \mathbb{R}^+\to \mathbb{R}$ be a continuous, strictly-increasing and strictly-concave function. define the following function: $F: \mathbb{R}^+\times\mathbb{R}^+ \to \mathbb{R}$: $$F(s,t) = \...
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0answers
25 views

Scheduling grid optimization

I am trying to optimize the programming of multiple TV channels for a given week. For each show (a day, a time and a TV show) it is possible to forecast in advance the number of people that will watch ...
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2answers
849 views

How to maximize the volume of a cylinder with no top

A cylindrical can without a top is made using $A \text{ cm}^2$ of material. Find the dimensions that will maximize the volume of the can. What I have done was similar to the question: Optimization ...
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how to minimize $tr (X^T W - Y^T W_p)(X^T W -Y^TW_p)^T$ in closed form

Assume we are dealing with matrices. Then how to minimize $$ E(W,W_p) = tr (X^T W - Y^T W_p)(X^T W -Y^TW_p)^T $$ w.r.t both $W, W_p$ simultaneously? I can calculate the derivatives of $W$ and $W_p$ ...
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24 views

related to biconcave optimization

I have a bivariate function $f(x,y)$ both $x,y$ can assume values within closed interval i.e. $x_1\leq x\leq x_2$ and similarly $y_1 \leq y \leq y_2$. I know that for a fix value of $x$ the function ...
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1answer
29 views

Linear program with ceiling or floor functions

Is it possible to solve a linear program where constraints have ceiling or floor functions applied to variables (with maybe some constants)? For instance: $$\lceil (x_1 + a)/b \rceil + \lceil (x_2 + c)...
3
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1answer
56 views

Working with Lagrange multipliers, reducing gradients is okay, right?

I am employing the method of Lagrange multipliers to determine a maximum. As part of this, I arrive at the following equation involving two gradients and the parameter $\lambda$, as is common for ...
3
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2answers
33 views

L1 minimization problem with nested sums as LP problem

I've been trying to solve this problem but I have an issue with the fact that there is a sum under each absolute value. I'm trying to convert this minimization problem (with respect to $x, y_1, \dots,...
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1answer
36 views

Why isn't Linear Programming less convoluted? [Soft Question]

Just a quick question. So I'm taking a course in linear optimization, and one of the things that we're going over obviously is the simplex method. I just started the class so I may not be seeing the ...
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Convex optimization with $\ell_0$ “norm”

I have an optimization problem of the form $$\begin{align*}\text{minimize }\;&f(x)\\ \text{subject to }\;&||x||_0 \le t,\end{align*}$$ where $t$ is a given constant and $f:\mathbb{R}^d \to \...
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21 views

Finding dual of certain problem

Could anyone help me finding lagrangian function and lagrangian dual of the following problem: \begin{equation} \begin{split} \max_{X}\quad & \operatorname{trace}(H X H^T)\\ \text{s.t} \quad &...
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1answer
23 views

Prove this property of the Hessian

I have been reading about the hessian for a scholar work about optimization and I find this property: Let be $H_{P_0}$ the determinant of the hessian matrix for the Lagrangian function $\mathscr{L}(x,...
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2answers
70 views

How to covert min min problem to linear programming problem?

I have the following problem: set $P=\{1,2,3...,n\}$ for index $i$, set $K=\{1,2,3,...,m\}$ for index $k$. Value $B_i^k$ is indexed by both $i$ and $k$, while value $l_i$ is indexed by only $i$. Here ...
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6answers
7k views

The median minimizes the sum of absolute deviations

Suppose we have a set $S$ of real numbers. Show that $$\sum_{s\in S}|s-x| $$ is minimal if $x$ is equal to the median. This is a sample exam question of one of the exams that I need to take and ...
2
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1answer
449 views

Gradient-descent and Hidden Markov Models

I would like to use gradient-descent to fit the parameters of a simple 2-state HMM. This paper Levinson, S. E., Rabiner, L. R. and Sondhi, M. M. (1983), An Introduction to the Application of the ...
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Economics maximization problem linear activity

Consider the vectors: $a_1 = \begin{pmatrix} 0 \\ -1 \\ 1 \\0 \end{pmatrix}, a_2 = \begin{pmatrix} 0 \\ 0 \\ -1 \\1 \end{pmatrix}, a_3 = \begin{pmatrix} 2 \\ 0 \\ 0 \\ 1\end{pmatrix}$ Find a single ...
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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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1answer
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Advantage of multi-objective optimization over single objective

What are the advantages of multi-objective optimization over single objective? I am specifically thinking about MO and SO in Genetic Algorithm. I have surfed the net and found many articles talking ...
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1answer
480 views

Find pressure in a sinusoidal function

Tiffany is a model rocket enthusiast. She has been working on a pressurized rocket filled with laughing gas. According to her design, if the atmospheric pressure exerted on the rocket is less than 10 ...
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1answer
34 views

What is the easiest way to optimize the weighted sum of L2 norms?

I have the following cost function (solving for $M$ - the $x_i$s are known): minimize $\sum_i\sum_j(w_{ij} \cdot (x_i-x_j)^T\cdot M\cdot(x_i-x_j))$ ($w_{ij} \in [-1,1] $) subject to: $M \succeq 0$ (...
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1answer
15 views

Proof: $\underset{\|q\|=1}{\max} q^TAq = \lambda_{\max}$ with $q$ the corresponding eigenvector ($A$ symmetric)

This problem is quite old and there should be similar problems. I know the following technique: \begin{equation} \begin{aligned} q^TAq=q^TU\Lambda U^Tq=(U^Tq)^T\Lambda (U^Tq) \end{aligned} \...
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631 views

Absolute extrema

Find the absolute extrema of the function $f(x,y)=2xy-x-y$ over the region of the xy-plane bounded by the parabola $y=x^2$ and the line $y=4$ I was wondering if I needed to use Lagrange multipliers to ...
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1answer
41 views

Linear programming with a product term in the objective function

The title might sound a little weird. I actually want to ask if this problem can be solved as a LP. And if so, how to convert the product term? set $P=\{1,2,3,\ldots,n\}$ for index $i$. Variables $...
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1answer
67 views

Ideal shape for underwater habitat

Is there an analytic solution to this problem or do I need to compute a discrete approximation using a relaxation procedure - or something similar? I want to find the shape of a roughly spherical ...
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1answer
4k views

Linear programming vs. Integer programming

I was trying to solve a problem where I want to choose which items to choose where each item has a number b_i associated with it and a reward r_i associated with it. I need to choose items that ...
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0answers
36 views

Showing the integrality property for an Integer Linear Program

I am trying to figure out why solving a relaxed Integer Linear Program (ILP) always give an integral solution. The ILP can be summarized as: $$\min \sum_{t\in T} \sum_{s \in S} c_s k_s^t $$ subject to:...
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1answer
92 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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131 views

Is 0-1 integer programming always NP-hard?

I have the following problem. Maximize $\sum\limits_{m=1}^M\sum\limits_{n=1}^N x_{mn}$ subject to: $\sum\limits_{\substack{m^\prime=1\\ m^\prime \neq m}}^M\sum\limits_{\substack{n^\prime=1\\ n^\...