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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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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Help optimizing payments on 3 loans - planning to use AMPL program but I have math problems first.

So the basis is that I have 3 loans with different interest rates and different principal amounts as well as different minimum monthly payments and different amortization (is that the right word? Time ...
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How do I include an integer constraint in Wolfram Alpha?

Related question: Tell Wolfram Alpha that a variable is a natural number I want to do the following in Wolfram Alpha: Minimise $$z = (y_1-x_1)+(x_2-y_2)+(1/2)(x_3)$$ s.t. $$0 \le ...
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Difference between minimizing and maximizing functions

Could someone please explain the difference between minimizing and maximizing functions or give me some links to explain the difference in very very very simple terms? I have searched online and I ...
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extremum under constraint question

I was tasked with finding the extremum of $z=xy$ under the constraint $x+y=1$, here is what I did: $$z=xy$$ $$x+y=1$$ from the second line we get $y=x-1$ and we substitute that back in the first ...
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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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Cookie Clicker Chocolate Egg strategy

Introduction Cookie Clicker is a silly Javascript based web game. Here is a brief description of what you do: (description taken from this question: Explain a surprisingly simple optimization result) ...
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[svm]A Problem of max(1/|w|) equal to min(1/2*|w|^2)

I've been search many SVM theory thesis for machine learning Those articles usually say max(1/|w|) equal to min(1/2*|w|^2) but they didn't write the detail of the mathematics process. I also read this ...
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Can an unfeasible solution be optimal in an LPP

In a linear programming problem, Is it possible to have an unfeasible solution that is optimal?
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Complex Least Squares With Magnitude Equality Constraints

For $\mathbf{x} \in \mathbb{C}^N$, I'd like to solve the following problem: $$ \mathbf{x}^\ast = \arg \min_{\mathbf{x}} \Vert \mathbf{Ax-b} \Vert_2 \,\,\,\,\,\, \mathrm{s.t.} \,\,\,\,\, \Vert x_i ...
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Second Principal Component Analysis Proof

I'm trying to prove that the 2 principal components are the 2 eigenvectors corresponding to biggest eigenvalues. So I'm in stage where I need to maximize: $$\sum_{i=1}^{i=n} \lambda_i\alpha_i^2 + ...
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Python - CVXOPT: What exactly should I check for G when "Rank(A) < p or Rank([G; A]) < n” exception is thrown?

I am new to using the CVXOPT module for Python and would definitely appreciate any illumination as to why the exception is thrown for my problem. (Also my first time posting a problem anywhere, so ...
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Extremum of a function under constraints

I have a function $f : E \subset R^n \to R$. $E$ is compact and $f$ is continuous so the extremums exist. But $E$ is not defined by an equation but an inequality, so i can't use the Lagrange method ...
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Minimizing the effort after toilet visit

We live together with 5 people (4 men and 1 woman) and the woman wants everyone to close the toilet after every turn (i.e. bring the seat and cover down, for smell reasons). To me this seems unfair. ...
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Find the extreme values of $f(x,y)=xy$ on $D=\{(x,y)|1 \leq x^2+y^2 \leq 4\}$

This would have to be done using conditional extremes(Lagrange method), and maybe some topological properties.I do not know how to do this, I have only done cases where the $D$ would be defined with ...
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11 views

Build a 4-regular, vertex-transitive, least diameter graph with v vertices

How to build a 4-regular, vertex-transitive, 'least diameter' graph with $v$ vertices? This implies to know what is the minimum diameter of a 4-regular vertex-transitive graph with $v$ vertices. ...
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Converges extremely slowly, using Douglas-Rachford splitting, how to improve?

my problem looks like this: $\min _{ E,A }{ { \lambda }_{ 1 }{ \left\| E \right\| }_{ 1 }+{ { \lambda }_{ 2 }\left\| A \right\| }_{ * }+{ \left\| D-ME-A \right\| }_{ 2 }^{ 2 } } $ the M is a ...
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Proving function is convex argmin

How can I show that the following function is convex in which Z is a random variable? $$\rho(Z)=\frac{2}{3}argmin_{t}\{t+10\mathbf{E}[Z-t]_{+}\}+\frac{1}{3}argmin_{t}\{t+5\mathbf{E}[Z-t]_{+}\}$$ I ...
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56 views

Setting up an LP problem on producing linear board in jumbo reels

I have to set up a linear programming problem corresponding to the following scenario: What I tried: I think we have 8 templates for 1 $68 \times l$ reel (or whatever): $22,22,22$ (66) ...
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2k views

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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Convex solver returns disordered dual variables, how to re-order?

I have the following convex optimization problem: $$ \begin{align*} \min_{x} &f(x) \\ \text{subject to} \\ x_{k+1} &= x_k+\cdots\qquad k=0,1,2,\ldots,N-1 \end{align*} $$ I managed to solve ...
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Total support in matlab

I need help writing an algorithm in Matlab telling me if a radom matrix has total support or not. I'm trying to use the Linprog formula, but I don't understand it.
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Limitations of Gaussian Response Surface Methodology for Optimization

I recently went through some material to learn about Gaussian Response Surface Methodology in the field of optimisation. However, I couldn't find the limitations or applications where gaussian ...
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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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How to derive 2D equation representing minimums of constrained 3d equation?

I have a 3D (multivariate) function f(x,y) which can be represented as a surface with constraints. When the surface is viewed from the side (as below), such that the Y axis is not visible, there is ...
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Use Levenberg marquardt to compute model parameters of cosine equation [closed]

my measurement device measures pairs of values $(r_i, p_i)$. These values are supposed to stem from the following model: $r_i =\alpha (y_I - y_S) \cos(p_i) + \alpha(x_I -x_S) \sin(p_i)$ For each ...
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Minimize or maximize the powers

I came up with this problem and I could not find a proof. Basically the problem is, suppose positive numbers $a_i$, $i=1,2,\ldots,N$ satisfy $$\sum_{i=1}^Na_i=1$$ then for $p>0$ when the expression ...
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Can 'Frobenius product method' be used to get analytic expression for **vector derivative**?

this objective function is shown as follow: $$\min_{u*, i*}\sum_{ui}c_{ui}(p_{ui}-x_{u}^Ty_i)^2 + \lambda(\sum_u\|x_u\|^2 + \sum_i\|y_i\|^2) + \lambda_f(\|x_u-\frac{1}{|N(u)|}\sum_{f \in ...
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Relationship between two-equation constrained optimization and one-equation version

I am learning about the Lagrange multiplier. Here's what I understand so far. Suppose a point $P$ is a minimizer of $f(x)$ subject to $g(x)=0$. Then any movement along that level-curve of $g$ must ...
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Discrete Approximation to Dynamic Lagrangians

Suppose I have the following dynamic optimization problem, where I want to maximize the function $u(c,h)$ over time that's differentiable in both $c$ and $h$. I'm going to assume that the function $u$ ...
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Why can't this be done? Or can it?

I was writing an answer to this question here From AM-HM $$\frac1{1+x}+\frac1{1+y}+\frac1{1+z}=2$$ $$\frac{1+x+1+y+1+z}{3}\ge \frac{3}{\frac1{1+x}+\frac1{1+y}+\frac1{1+z}}$$ $$\implies ...
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the optimal solution of $min$ $-x-y+(1/2)(x+2y-3)$

the optimal solution of $min$ $-x-y+(1/2)(x+2y-3)$ s.t. $x,y=0,1,2, or $ $ 3$ Attempt: if we tale the gradient of the objective function we have $[-1/2,0]^T$. This means that y could take any ...
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Optimize over measure on function space

I'm an absolute newbie in analysis, so this might be a dumb question. Let $S$ the space of non-negative, monotone functions from R to R. Is the following optimization problem well-defined? ...
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Minimize a particular function in one variable

For given $a,b$, what is the minimum value of the following expression? $$ \frac{a}{x^2+b}+x,\qquad x>=0, a>0,b>0 $$ Differentiating the above gives a messy polynomial. I tried plugging ...
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Probable mistake in calculation of maxima

QUESTION: Given function is $$E=\frac{1}{4}\cdot \frac{F^2}{m}\cdot \frac{\omega_0^2+\omega^2}{(\omega_0^2-\omega^2)^2+4\alpha^2\omega^2}$$ We have to maximise $E$ with respect to $\omega$. MY ...
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585 views

Maximize profit with dynamic programming

I have 3 tables… $$\begin{array}{rrr} \text{quantity} & \text{expense} & \text{profit}\\ \hline 0 & 0 & 0 \\ 1 & 100 & 200 \\ 2 & 200 & 450 \\ 3 & 300 & 700 ...
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Has $\max\{cx:Ax\le b\}$ an optimal solution $x_1=\sqrt 2$ with $A\in \{-1,0,1\}^{m*n}$ with exactly one $1$ and one $-1$?

Let be $A\in \{-1,0,1\}^{m*n}$ with exactly one $1$ and one $-1$ and zeroes at each line. $c\in\mathbb{Z}^n$ such that $\sum\limits_{j=1}^{j=n}c_j=0$. $b\in\mathbb{Z}^m$ positive. How to start to ...
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Do corner points optimise a linear function over a bounded convex region?

This proof says if $Z_P \ne Z_Q$... ...then $Z$ is maximised (or minimised, I guess) at one of the $\color{red}{\text{endpoints}}$ -- of what exactly? $\overline{PQ}$? So the maximum value of ...
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50 views

Develop a model for determining the optimal production schedule in a manufacturing facility

I have to formulate (linearly) the following problem mathematically: What I tried: 1. Variables Let $x_{ijk} = 1$ if, in month k, product i should be made in production line j, where ...
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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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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: ...
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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 ...
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If the primal is unbounded, then the dual is infeasible.

In the context of duality in linear programming, prove that If the primal is unbounded, then the dual is infeasible. My book says that this is a corollary to complementary slackness. What's ...
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Why does a distance and its square reach their minimum at the same point?

There is a question in my calculus textbook that asks to find a point on the parabola $y^2 = 2x$ that is closest to point $(1,4)$. They want us to first use the distance formula, but then proceeded ...
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Why is Maximizing Marginal Log-Likelihood Difficult?

I was reading this tutorial on expectation maximization and in section 3 the author mentions that a marginalization inside a log function is difficult (impossible?) to take the derivative of. I am ...
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Minimize and maximize the sum of dot products at the same time

this is the problem. I have a set of numerical positive vectors of equal length. For each pair of vectors $(\mathrm{i}, \mathrm{j})$ I define the vector $\mathrm{ij}=\mathrm{i} - \mathrm{j}$. I also ...
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Matrix-free conjugate gradient

In the conjugate gradient method for solving $Ax = b$, to update the search direction $p$ you would need to evaluate the matrix-vector product $Ap$, i.e. making sure that each search direction are ...
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Why this two problems are equivalent?

I was reading about Support Vector Machines and I found that it's equivalent to solve the problem of maximize this number: $\frac{1}{\left \| w \right \|}$ with to minimize this number: ...
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Construct a matrix of polynomials to optimize condition-like score

I'm a physicist currently working on my PhD. Within my studies, my colleagues & I encountered a (strictly mathematical) problem that baffles us (and anyone else we've talked to so far) and is also ...
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Why the dual and the canonical dual may have the same optimal solution?

For instance let be \begin{cases} \max & 3x_1 &+2x_2 &+4x_3 & -x_4\\ &x_1 &+x_2 &-2x_3&&\le4\\ &2x_1&+3x_3&&-4x_4&\ge 5\\ ...