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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Is $f(x)=-3x_1+x_2-x_3^2$ pseudoconvex at $\bar x$?

Is $f(x)=-3x_1+x_2-x_3^2$ pseudoconvex at $\bar x=[-115/588, -95/588, 5/14]^T$? Pseudoconvexity: If $\nabla f(\bar x)^T(x-\bar x)\ge0$, then $f(x)\ge f(\bar x)$ for any $x\in \mathbb{R^3}$ (in this ...
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Finding the range of a $y=-x^2(x+5)(x-3)$ without calculus?

I was helping a precalculus student with this question. The graph wasn't given. My only idea was to find the inverse and try to find its domain. When trying to find the inverse, I arrived at ...
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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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1answer
31 views

how to find mininimum $f(x)$ using $\int_{-\infty}^{\infty} f(x)g(x)dx$?

I would like to know the $f(x)$ which minimizes the $\displaystyle\int_{-\infty}^{\infty} f(x)g(x)\,dx$. Actually, this question start from the MMSE (Minimize Mean square error) ...
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How find this maximum $S_{\Delta ABC}$

in $\Delta ABC$,and $\angle ABC=60$,such that $PA=10,PB=6,PC=7$, find the maximum $S_{\Delta ABC}$. My try:let $AB=c,BC=a,AC=b$, then $$b^2=a^2+c^2-2ac\cos{\angle ABC}=a^2+c^2-2ac$$ then ...
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How to form a dual problem in convex optimization (in a broad view)

After reading some papers, this problem confuses me. There are different forms of dual problem to the primal problem: $$\underset{x}\min \ \ f(x)$$ where $f(x)$ is a convex function. By ...
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Maximizing the Nullity of a Symbolic Gram Matrix

I have a symbolic gram matrix, that is, a matrix $AA^T$ with some entries being variables. I would like to find a solution for my variables which maximizes the nullity of this matrix, or equivalently, ...
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534 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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Travelling salesman - organising a tour of any European destination based on the cheapest flights available.

I apologise if this has only a tenuous link to a mathematics forum I'm sure everyone is familiar with the £10 one-way flights by Ryanair and similar airlines in Europe. I was wondering whether there ...
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58 views

the minimum and maximum values of the determinant of order $3\times3$ matrix with entries $\{0,1,2,3\}$

The minimum and maximum values of the determinant of order $3\times3$ matrix with entries $\{0,1,2,3\}$.
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Unbounded variables and dual of a linear program

I have to find the dual of \begin{cases} \max & -x_1 &-2x_2+x_3\\ & -3x_1 &+x_2&\le-1\\ & x_1 &-x_2&\ge 1\\ & -2x_1 &+7x_2&\le6\\ & -5x_1 & ...
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26 views

Minimize Energy function

I got the following equation: $$V = \frac{1}{2}x_2(t)^2 + \gamma(x_1(t),x_2(t))x_1(t)$$ Now the goal is to decrease this "energy" function in as little time as possible, as much as possible. ...
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28 views

How to determine constant $C$ in $p(x) = Cx^{-D}$?

Given a distribution obeying the power-law (fractal) relation, such as the cumulative distribution function $L_{cf}(> X) = CR^{-D}$, if $X$ is given, how does one find the constant $C$ from a given ...
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18 views

Linear integer programming

I am trying to find the optimal solution for the following linear integer programming: \begin{eqnarray} &&\underset{x_i, \forall i} {\text{maximize}} ~ \sum_{i=1}^N x_i a_i \\ && ...
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The number with minimum sum of differences

Let $a_1,a_2,...,a_n\in\mathbb{R}$. I wonder how to find the number $x$ with $$|x-a_1|+...+|x-a_n|=\mbox{min}\{|a-a_1|+...+|a-a_n|\mid a\in\mathbb{R}\},$$ namely the sum of the differences with ...
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Expected number of times a set of 10 integers (selected from 1-100) is selected before all 100 are seen

Suppose I have a set of 100 integers. I randomly choose 10 of those, make a note of which ones I selected, and repeat the process. What is the expected number of times this process must be repeated ...
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465 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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38 views

Optimization question related to calculus. [closed]

Suppose we have arbitrary real numbers $a,b$. We want to maximize $a^2 + b^2$ subject to $a + b = c$, for some constant $c$. How would one do this?
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Utility maximization of n goods

I have a question that involves finding the optimal demand of $n$ goods for a consumer. However, I haven't anything like this before and I'm not sure how to proceed. The consumer has a utility ...
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Find the Min of P(x,y)

Find the Minimum of the following function : $$P(x,y) = \frac{(x-y)}{(x^4+y^4+6)}.$$ This is a math problem I found in an internet math competition but it is really complex to me !!!
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Gradient descent method for real function of complex matrix

Suppose $\mathrm{a}$ is $N\times 1$ known complex vector, and we need to solve this following optimization problem with the gradient descent method: ...
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Maximize $|Ax|/N$ for binary $x$

Is there a systematic way of going about solving \begin{align} {\text{maximize}} &\hspace{3mm}& \frac{|\mathbf{Ax}|}{N} \\ \text{subject to} &\hspace{3mm}& \mathbf{x} = [0,1] ...
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Maximizing the volume of a rectangular prism

A rectangular prism has a surface area of $300$ square inches. What whole number dimensions give the prism the greatest volume? This is a math olympiad problem. It involves the volume and surface ...
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Optimal positioning of tokens on a unit disk

Suppose we have $N$ tokens, labeled $x_1,x_2,\ldots,x_n,\ldots,x_N$. Our goal is to place these tokens optimally (defined below) on a unit disk. Formally (and please let us know if our notation is ...
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26 views

Determining the most appropriate set of eigenmodes for a modal decomposition of an experimental data set

I have a complex vector of the transverse amplitude and phase distribution of a laser beam, derived from experimental data. When modelling these field distributions, ordinarily the eigenmodes of the ...
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Quadratization of $5 x_1 x_2 − 7 x_1 x_2 x_3 x_4 + 2 x_1 x_2 x_3 x_5$ using Rosenberg's algorithm

In section 4.4 of Pseudo-Boolean Optimization by Boros et al., the authors have reproduced Rosenberg's quadratization algorithm as follows. Then they have given an example of implementing the ...
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30 views

Showing that the unique function satisfying $f''(x)=\alpha f(x)$ is $e^{\sqrt{\alpha}x}$, $\alpha>0$

I am trying to avoid doing some tedious case work on an optimization problem and I think this is true but I am struggling to prove it. I have seen the proof of $\begin{equation*} f'=af ...
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Is the gamma function a solution to this problem?

A real differentiable function on $\mathbb{R}^+$ satisfying $f(x) = x!$ for $x\in\mathbb{N}$ and having minimal derivative $\left|\frac{\partial f}{\partial x}\right|$ everywhere, say in the sense ...
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Dependence of the derivative of a pseudo-Boolean function on its variables

I am going through Pseudo-Boolean optimization by Boros et al. In the section 2, the paper introduces the idea of derivative and residual of a peudo-Boolean function. It is claimed that both ...
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22 views

Limit of absolute difference

So let's say I know that I have 2 real functions $a_n(x)$ and $a_0(x)$, and both functions are greater than zero for any $x$. $a_0(x)$ represents an optimal solution and $a_n(x)$ represents a solution ...
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Inverse Vectorization Vec^-1

Hope that you will find this post in good health. I am Mr.Adnan from Pakistan with research background in Control systems. I am working on one problem in which Hadamard weights are using. During ...
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Issue with optimization problem

Consider the following configuration: Now, we can minimize the length $L=2\ell_1+\ell_2$. Let the top left angle be $\theta$ so that $\ell_1=\frac{a}{2}\sec\theta$ and ...
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How to maximise the minimum distance of a lattice using orthogonal matrices?

Given an $n$-dimensional real lattice $\Lambda$ with generator matrix ${\bf L}_{n\times n}$ (basis vectors are columns of ${\bf L}$). What is the solution to the following optimisation problem? ...
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Inequality from IMO 2000 problem 4 question $\Pi_{cyc}\left(a-1+\frac{1}{b}\right)\leq 1$ $abc=1$

I know the problem is repeated but my question is somehow different. I want to know whether my proof is correct because I have troubles with the last part. Since $abc=1$ we can homogenize the ...
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Max-Min values of $f(x,y) = x^3+y^3-6x^2-y-1$

I am asked to find the extrema of the function $$f(x,y) = x^3+y^3-6x^2-y-1$$ I understand that we have to equal the partial derivatives to zero, which means $$ f_x = 3x^2-12x = 0\\ f_y = 3y^2-1 = ...
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Integer programming with linear constraint

I am trying to find the optimal solution for the following problem \begin{eqnarray} &&\underset{x_i, ~y_i ~\forall i} {\text{maximize}} ~ \sum_{i=1}^N x_i f_i(y_i) \\ && ...
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Questions about simplex algorithm

I'm trying to understand how simplex algorithm works, and here are my questions: 1. Why we choose the entering variable as that with the most negative entry in the last row? My understanding is that ...
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Local minimum for a polynomial

Can someone please help me to answer this question: we consider $V \in E_r =\lbrace P \in \mathbb{R} [X_1,X_2,..,X_d] \mid \deg P \le r \rbrace$. If $\exp^{-V(x)}\in L^2(\mathbb{R}^d)$ then $V$ ...
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How to solve a linear program with additional equality constraints?

The following optimization problem $$\max_{\substack{x \ge 0,\\Ax^T+b^T\ge 0}} c x^T$$ where $x \in \mathbb{R}^n$, $A \in \mathbb{R}^{m \times n}$, $b\in\mathbb{R}^m$, and $c \in \mathbb{R}^n$ is ...
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Distributed control problem which involves the p-Laplacian operator

Someone could help me to deduce the optimality system for the optimal control problem: \begin{align} &\min_{u\in L^{2}(\Omega)} ...
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Minimum number of $m \times m$ matrices needed to recover a single large matrix

This problem was motivated by the need to efficiently train a neural net on a dataset in which the labels represent dependencies between examples, but nothing about it is machine-learning specific so ...
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Different ways of solving $\underset{\mathbf{s}}{\text{min}}\;\|F\mathbf{s}-\mathbf{x}\|_{l_2}^2 + \|W\mathbf{s}\|_{l_2}^2$ least square problem?

The problem that I am going to describe arises from compressed sensing technique and after using weighted least squares it can be transformed into the following least squares problem: ...
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Drift management optimization

I have a problem in which I am having trouble formulating the optimization. A portfolio value is $10M I have a vector of current weights [.10,.15,.15,.10,.05,.10,.20,.15] and another vector of ...
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how I can minimize this equation using derivation

I'm a software engineer and have not much mathematical knowledge. Now, I'm facing with a problem in my research. I have a system of equations as below: $$P_1 = \alpha V_p + \beta I_c^2 $$ $$P_2 = ...
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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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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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757 views

Optimization, rectangle inscribed inside arch of the curve.

A rectangle is to be inscribed under the arch of the curve $y = 4\cos(0.5x)$ from $x = \pi$ to $x = -\pi$. What are the dimensions of the rectangle with largest area, and what is the largest area? ...
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Maximize function symbolically

I have the following expression: $$ \sum_{i,j=1}^n\rho_{ij}^2-\frac{2}{n}\sum_{i=1}^n\left(\sum_{j=1}^n\rho_{ij}\right)^2 +\frac{1}{n^2}\left(\sum_{i,j=1}^n\rho_{ij}\right)^2 $$ My goal is to ...
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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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Optimization Cost of candy

You have decided to buy candy for the trick-or-treaters and have estimated there will be 200 children coming to your door, and plan to give each children three pieces of candy. You have decided to ...