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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Solve dual of linear program without simplex

I have a linear program and need to determine and solve the dual program. The primal program is $\begin{array}{lcl} \text{Maximize: }\\ f(x) := 6x_1+4x_2\\ \text{Subject to:}\\ -2x_1-4x_2 \leq ...
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Find the point on the ellipse where the cylinder intersects the plane furthest from the origin?

I'm confused about how I should set this problem up. It is a lagrange problem. The cylinder x^2 + y^2 = 1 intersects the plane x + z = 1 in an ellipse. Find the point on the ellipse furthest from ...
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minimize trace(AX) over X with a positive semidefinite X

I want to minimize trace(AX) over X, under the constraint that X is positive semidefinite. I guess the solution should be bounded only for a positive semidefinite A, and it's zero, or the solution ...
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Question about the constraint in Laplacian eigenmaps

When calculating Laplacian Eigenmaps, the original paper mentions about the constraint $$y^TDy=1$$ as "removes an arbitrary scaling factor in the embedding". My understanding is that it prevents $y$ ...
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32 views

Find the global extrema of $f(x,y)=\sin(xy)$ on $D=[(x,y)| x = [0,\pi], y=[0,1]]$

Find the absolute maximum and absolute minimum of the function: $$f(x,y) = \sin(xy) \text{ on } D=[(x,y)| x = [0,\pi], y=[0,1]]$$ I took the partial derivatives and got: $$\frac{df}{dx} = \cos(xy)y ...
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How does one rigorously prove that gradient descent indeed decreases the function in question locally i.e. show $f(x^{(t+1)}) \leq f(x^{(t)})$?

How does one prove that gradient descent indeed decreases the function in question locally? In other words if we take a step in the negative of the gradient as in: $$ x^{(t+1)} = x^{(t)} - \eta ...
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20 views

Differentiation minimization

This question taken from web based engineering mathematics online test. My answer to this question as below. This system says it is incorrect. Is there any mistake? Plz help.
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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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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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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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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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57 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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What is the optimal path between $2$ fixed points around an invisible obstructing wall?

Every day you walk from point A to point B, which are $3$ miles apart. There is a $50$% chance each walk that there is an invisible wall somewhere strictly between the two points (never at A or B). ...
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Optimal path around an invisible wall [duplicate]

The Problem On an infinite plane there are two points, $A$ and $B$, a unit distance apart. There is a $50\%$ probability that there is an invisible wall somewhere between the two points. The ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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1answer
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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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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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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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Values of parameter $\epsilon \in (0,1)$ that make a rational function decreasing

For $p \in (0,1]$, an integer $n \geq 2$ and $\epsilon \in (0,1)$, I want to show that $$\frac{p (1- \epsilon p)^{n-1}}{1- (1-p)^n}$$ is a decreasing function of $p$ for $\epsilon > g(n)$ for some ...
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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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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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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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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 ...
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generalised eigenvalue problem with absolute value

Problem: $\max_w |w^t A w|-|w^t B w|$ s.t $w^t C w=1$ If there was no absolute values, i.e. if the problem was $\max_w w^t A w-w^t B w$ s.t $w^t C w=1$ this would, by using the appropriate Lagrange ...
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Bipartite Matching with quadratic objective

I'm looking for the best way to formulate and solve the following bipartite matching problem: I have n nodes on the left hand side of the diagram, partitioned into ...