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 distinguish as Linear or Non-linear constraints

In optimization problem $$\arg\max_{\substack{l_1 \in [0, 1],\cdots,l_{M} \in [0,1]}} T$$ I have $m=1\cdots,M$ constraints such that $$\ln \left[a_m\right] + \ln \left[ \left(\frac{l_m}{l_m + ...
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1answer
32 views

Proportion in sets

We have $3$ sets of positive integers. $$A = \{x_1,y_2,z_3\},\quad{} B = \{x_2,y_2,z_2\}, \quad{} C= \{x,y,z\}$$ Which proportion do we use for adding $A$ and $B$ ($x_1+x_2$ and so on), so the ...
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Value of Lagrangian Multiplier

I have a two dimensional optimization problem of the form $$ v = \max_{x,y} f(x,y)+g(x,y) $$ Both $g,f$ are concave and continuously differentiable. Assume the solution can be reached by first order ...
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1answer
22 views

Transforming a $0$-$1$ knapsack problem into the standard form

I have the following $0$-$1$knapsack problem: $$\begin{align*} &\mathrm{Max} : \quad z= 3x_1 -4x_2+5x_3+7x_4-6x_5+x_6\\ & \mathrm{subject\ to}: -2x_1 +x_2 +10x_3 +3x_4 -5x_5+12x_6 \leq 4 ...
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4answers
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Compute the minimum value of $a^n + b^n + c^n$ subject to $a^2 + b^2 + c^2 = 1 $

Assume that $a,b,c$ are non-negative real numbers and $n$ is a natural number $n \ge 3$. What is $f(n)=$ the minimum value of $a^n + b^n + c^n$ ? I find ; $$f(3) = \frac{1}{\sqrt{3}}\qquad ...
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1answer
20 views

Asymmetric Least Squares Conversion from Equation to Matrix

In solving for asymmetric least squares baseline correction as defined in the article by Eilers and Boelens, the general equation is defined as: $$S = \displaystyle\sum_i w_i (y_i-z_i)^2 + \lambda ...
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Is there a reason that the maximal volumes of rectangular prisms with multiple open faces but constant surface area follows this pattern?

Suppose a rectangular prism has a surface area of $12 \text{ m}^2$. The optimal volume of this prism is well known. If the side lengths of the prism are $x$, $y$, and $z$, then the surface area is ...
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34 views

Finding extremas

Finding this quite tricky. $$e^{x}(\cos(x)-\sin(x))=0$$ Solve for $x$. This is an derivative of the original function: $$f(x)=e^x\cos(x)$$ And I am trying to find the extremas.
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KKT conditions with two inequality constrains

I need to minimize $f(x,y,z)=x^{2}+2y^{2}+3z^{2}$ subject to \begin{align*} &x-y-2z\leq 12\\ &x+2y-3z\leq 8. \end{align*} So I wrote the lagrangian of $f$. \begin{align*} ...
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460 views

Shortest distance between two lines in 3-dimensional space [closed]

Can someone explain to me how to solve this question? Find the shortest distance between the lines $L_1 = \left\{t \begin{bmatrix} 1\\ 1\\ 1\end{bmatrix} : t \in \mathbb{R}\right\}$ and $L_2 = ...
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52 views

Optimal control with state-dependnet solution

I'm trying to solve the following control problem $$ \begin{eqnarray*} \max & & \int_{0}^{T}\sum_{i=1}^{2}-c_{i}(x_{i}-u_{i})^{+}\\ s.t. & & ...
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14 views

What is the purpose of the 1/2 factor in SVM minimalisation equations?

The objective function for Support Vector Machines is in most sources formulated as: $\min\limits_{w,w_{0}} \frac{1}{2}||w||^2 + C\sum\limits_{i=1}^{N}\xi_{i}$ What is the signifance of the ...
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utility function question from my textbook

Suppose there are two goods with prices $ p₁ = 2, p₂ = 5, $ the income is $ M = 40 $ and the utility function is $ U (x₁, x₂) = (x₁)^⅓ . (x₂)^ ½, $ Find the optimum consumption plan. Attempt: I do ...
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12 views

SQP with non linear constraints

I am asked to solve a $SQP$ problem using Matlab, in the instructions for writing the code I am supposed to create two functions one for the function $f$ to minimize and one for the constraints $c_i$, ...
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1answer
27 views

Find the point on the plane xOy [closed]

Let $A(x_1; y_1)$, $B(x_2, y_2)$ and $C(x_3, y_3)$ be three points not lying on the same straight line. Find the point on the plane $xOy$ such that the sum of the distances from it to these points is ...
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18 views

Why do not we check second derivative test when solving constrained problems

when we are max/min an objective function z(x,y) subject to a constraint for example $$x^2+y^2\leq1$$ Then the solution will be in 2 steps : First : getting critical points inside this domain . ...
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16 views

Diagonal Newton's method for unconstrained optimisation.

Assume you are minimising a convex function $f$. The function is twice-differentiable. The well-known Newton's method consists in starting form some point $x_0$ and then using the iteration below. $$ ...
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2answers
54 views

Without Extreme Value Theorem, how do we find absolute extrema?

I have to find and classify the critical points of the following functions and then state which relative extrema are absolute extrema. $$f(x,y) = x^3 - y^3 - 2xy + 6$$ $$f(x,y) = xy + 2x - ...
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1answer
78 views

Find the Lagrange multipliers with one constraint: $f(x,y,z) = xyz$ and $g(x,y,z) = x^2+2y^2+3z^2 = 6$

Where $f(x,y,z) = xyz$ and the constraint is $g(x,y,z) = x^2+2y^2+3z^2 = 6$ I have tried this problem like three or four times and not gotten the solution, I even asked this question once and got the ...
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46 views

Utility maximization problem

I tried to search through the site and couldn't find a similar example. The task is to solve the utility maximization problem to identify the inverse demand for e(electricity). The utility function: ...
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20 views

A more general case of assignment problem

Recently I've learned hungarian algorithm for solving the assignment problem, Now I'm curious about how to solve more general problem: for given $n \times m$ table select several numbers, maximizing ...
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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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53 views

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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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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[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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1answer
32 views

Optimizing overlap between two reference frames

Let me share this little optimization problem with you: I have two orthonormal sets of vectors on $\mathbb{R}^3$, related by some Euler angles $(\alpha,\beta,\gamma)$ (corresponding to those of the ...
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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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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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Determine if fitted line is actually one line

I am trying to fit multiple lines through many data points in 3d space. My working method is sequential RANSAC, which now is fast enough and fits some lines, but produces some lines that don't fit one ...
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38 views

Minimum variance, fixed mean , discrete random variable

Consider the ordered set $\mathcal{S}$ $=$ $\{0,a_i,a_2,\ldots,a_n\}$, where $a_i$ are all stricly positive real numbers and $a_i< a_{i+1}$ forall indices i. What is the random variable $X$ which ...
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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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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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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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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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1answer
68 views

Method to find the extremal values of $xyz$ subject to $x^2+2y^2+3z^2=a$

This question has been asked before but I want to lay out my method and get feedback on reasoning and process this took me a long to put together as I am new to the formatting: Let the function $f$ ...
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1answer
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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12 views

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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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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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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1answer
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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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45 views

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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38 views

How to get explicit solution of the fractional minimization problem?

I need to minimize $f(x) = \frac{x^tQx}{a^tx-b}$, $Q$ is positive definite, $a,b$ are constant vector. But when I take gradient and set it equal to $0$, it's hard to get an explicit expression for ...
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1answer
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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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3answers
52 views

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 ...