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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Duality - linear programming

I have to find a respective dual programme for the given LP $ max \ 2 x_1 + 2x_2$ s.t. $ -x_1 - x_2 \ge -5$ $x_1,x_2 \ge 0$ I got this: $min \ 5y_1$ s.t. $y_1 \ge 2 $ $y_1 \ge 0$ But I'm not ...
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Plackett-Burman designs for screening experiments

i am looking for a way to investigate the relations between deferent factors in a experiment. basically i have 4 deferent lysis buffers, for each buffer i use deferent variables high/low amount of ...
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Prove convexity of function over space of positive definite matrices

I want to show that the function $f(X) = -log \ det(X)$ is convex on the space $S$ of positive definite matrices. What I have done: It seems like this problem could be tackled by considering the ...
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Newton's method: Is the change of parameter values between consecutive steps always decreasing?

Assume that I have a twice differentiable function $f(x)$ which I try to maximize with respect to $x$ (let's say $x$ is $k$-dimensional vector). When performing optimization via Newton algorithm, ...
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Lattice fitting to points

I have a set of points (shown as little black circles) which ideally form a hexagonal lattice shape, each point having an equal distance to all of its neighboring points. (Sorry for my drawing, some ...
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Express this linear optimization problem subject to a circular disk as a semidefinite problem.

I have to express following problem as a semidefinite problem: $ min \, F(x,y) = x + y +1$ subject to (1) $(x,y) \in \mathbb{R}^2 : (x-1)^2+y^2\leq 1$ Only affin equality conditions should be used. ...
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Are there any algorithms for simultaneous optimization of multiple objective functions?

I would like to minimize a set of similar objective functions $$f_\boldsymbol{s}(\boldsymbol{x}),$$ where $\boldsymbol{x} \in A \subseteq \mathbb{R}^M$ and the parameterization $\boldsymbol{s} \in S ...
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Why $f^{+}(v)-f^-(v) =val(f)$ if $v$ is the source?

I'm reading Bondy/Murthy's Graph Theory: He defines $x$ as the source and $y$ as the sink, reading a bit later in the chapter, he presents this definitions: $$ f^{+}(v)-f^-(v) = \left\{ ...
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Arithmetic mean is less than geometric mean (Spivak Calculus 3rd Chapter 2 Problem 22)

If $a_1, \ldots, a_n \ge 0$, the arithmetic mean $$A_n={a_1 + \cdots + a_n \over n}$$ and the geometric mean $$G_n = \sqrt[n]{a_1 \cdots a_n}$$ satisfy $G_n \le A_n$. As a first step to prove this ...
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Optimization using Karush-Kuhn-Tucker conditions

min $y^Tx$ subject to $\|x\|^2 \le 1$ where y is a nonzero vector in $\mathbb R^n$ I rearrange the constraints so that the RHS is $0$. New constraint: $x_1^2 + \cdots + x_n^2 - 1 = \|x\|^2 - 1 \le ...
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Why does $\int f(x)(y-r(x))\;dP(y,x) = 0$?

My question is, why does: $$\int f(x)(y-r(x))\;dP(y,x) = 0,$$ where $r(x) = \int y \;dP(y|x)$ and $P$ is a probability distribution function. It was also given (in my book) that: $$\int ...
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Thomson problem vs. maximizing sum of distance

Given $N$ equally charged points lying on the unit sphere ("electrons"), the Thomson problem consists of finding the configuration of these points such that the electrostatic potential energy $$ ...
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Linear Programming: Three variable graphical solution

A small bank offers three type of loans: housing loans at $8.50$% interest, education loans at $13.75$% interest rates, and loans to senior citizens at $12.25$% interest. Further, it needs to ...
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Partial derivatives of multivariable function with maximum

I have a question regarding the derivative of a maximum. I found the answer below which suggests evaluating the function at several intervals: Derivative of max function However, I have a function of ...
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Feasible solutions in semidefinite programming - beyond Slater's theorem

Suppose a semidefinite program is given by: \begin{align} \text{Maximize:}&\quad \text{Tr}\left[AX\right],\\ \text{subject to:}&\quad \Phi\left(X\right)=B,\\ &\quad X\geq 0. \end{align} ...
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Formulate the Marriage Problem into a Maximum-flow problem (Graph theory)

Suppose I have $M=\{1,\ldots, n\}$ men and $W = \{1, \ldots, n\}$ women and $B =\{1, \ldots, m\}$ brokers, such that each broker knows a subset of $M \times W$ and for each pair in this subset a ...
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Type of convex function?

I want a convex function $f:\mathbb{R} \to \mathbb{R}$ with the following property: given points $x,d \in \mathbb{R}$, and $\alpha \in (0,1)$, we have $$f(x + \alpha d) \geq \alpha f(x + d).$$ Is ...
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Minimization of the integral with respect to a parameter

Intro Let $f$ be a a real-valued function parametrized by a parameter $\alpha \in \mathbb{R}$ and let $J\colon \mathbb{R} \to \mathbb{R}$ be a functional defined as follows: $$J(\alpha) = ...
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Finding maximum value of a 3-variable function using inequality.

Let $a, b, c$ be positive real numbers satisfying $a^2 +b^2+c^2=14$. Find the maximum value of $f(a,b,c)=\frac{4(a+c)}{a^2+3c^2+28}+\frac{4a}{a^2+bc+7}+\frac{5}{(a+b)^2}-\frac{3}{a(b+c)}$
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Introduction to Linear Optimization: Driving the artificial variables out of the basis (case: no entries in the $j$-row are nonzero)

Reading the book Introduction to Linear Optimization by Bertsimas and Tsiklisis, I've come across the following subject: Driving the artificial variables out of the basis. The description is as ...
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Is my idea of incoming/outgoing arcs correct?

I'm reading Jungnickel's Graphs, Networks and Algorithms. I've met the following lemma: I know that $e^{-}$ are the incoming vertices and $e^{+}$ are the outgoing vertices. Then I've tried to ...
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Analytical solution to the first PCA direction

It is known that the first PCA direction for a dataset of $n$ points is the unit vector with max variance after projecting the points onto this vector. I wonder whether there are some analytical ...
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Finite difference method

I wanted to ask something regarding the finite difference approximation. I used the finite difference to calculate the numerical derivatives of my function. The finite difference is given by the ...
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0 to the power of 0 follow up question: Finding the “optimum curve” such that one can just avoid limiting into 1 or 0

Previously in $0^0$ shape of discontinuity we investigated the limiting behavior and shape of the singularity of the function $x^y$ as we approach the origin We knew that from the answers given in ...
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Minimum of $x+y+z$ on $\{(x,y,z) \in \mathbb{R}^3 | z \le x^2+2y^2+3, z\ge 3x+2y\}$

Find the minimum of $x+y+z$ on $$\{(x,y,z) \in \mathbb{R}^3 | z \le x^2+2y^2+3, z\ge 3x+2y\}.$$ My first naive thoughts would be to consider setting up a nasty triple integral and evaluate it or ...
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Solving for stationary points for questions of the following type

How do you solve questions like $f(x,y) = x^2y + y^3x -xy$ for stationary points? A link to an educational resource that goes over this would be very helpful as well, as I don't even know what ...
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Maximum volume of an open box with a square base?

A box with a square base and an open top is to be made. You have 1200cm^2 of material to make it. What is the maximum volume the box could have? Here's what I did: 1200 = x^2+4xz; where x=length of ...
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when do we have polynomial local minimum = to global minimum

When does a multivariate polynimial has only one stationary point so that local minimum is global minimum?
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Optimization problem?

Hi I was having trouble figuring out this question. Find the point on the circle $x^2 + y^2 = 1$ in the first quadrant where the tangent line to the circle encloses with the coordinate axes a ...
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Optimization of competitive scenario

Suppose we have a function $f(x_1,x_2)$ with the following properties: Let $x^*=\arg \max_{x_1} f(x_1,x_2=x^*)$ and $x^*=\arg \min_{x_2}f(x_1=x^*,x_2)$. $f(x_1,x_2)$ is concave in $x_1$. ...
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*SOLVED* Largest lower bound that covers p percent of the data

Solved: Rahul wrote the solution to the problem in the comment to the question. Thank you Rahul. The original question is below: Suppose that you have a finite set $X\subseteq \mathbb R$, and you ...
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An example of a continuous function on $\mathbb R^2$ with two critical points, both of them minima

Knowing you can not use the minimum bound, there exists a function $f ( x , y )$ continuous in $\mathbb R ^ 2$ that has exactly two critical points which are (both) the minimum? Can you give me an ...
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Dual cone of a L1 norm cone?

I am listening to convex optimization lectures and I hear that dual cone of a $L1$ norm cone is a $L-\infty$ norm cone. Can anybody please explain how? I understand that every point in the dual cone ...
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The Stupid Computer Problem : can every polynomial be written with only one $x$?

When I was a child, I wanted to be a mathematician so I asked my parents to buy me a computer to make super complex calculations. Of course, they were not crazy enough to buy an expensive super ...
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Weighted Norm Minimization

I have a minimization problem of the form $min (w_1 \|x\|+w_2\|y\|)$ subject to constraints $A_1x=b_1$, $A_2y=b_2$, $0< l_1 \leq x \leq u_1$ and $0< l_2 \leq y \leq u_2$ where ...
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optimization help calculus [on hold]

A police department purchases new patrol cars for $22,000. The department estimates that average capital cost and average operating cost are a function of x, the number of miles the car is driven. The ...
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Does the maximum cut implies the minimum flow?

Is it possible to reverse the result of the min-cut max-flow theorem and obtain the result that if you have the maximum cut, then you have the minimum flow? I've been thinking about it, but I have no ...
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Optimization question on graph

I was having trouble with this question. A triangle has one side parallel to the x-axis, two vertices on the part of the parabola $$y =3 − {x^2\over 12}$$ above the x-axis and the third vertex at ...
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Symmetric matrix and its Conjugate

Let H be a symmetric matrix. Show that if u and v are eigenvectors of H corresponding to distinct eigenvalues, then u and v are H-conjugate. Need some help, what and how should I approach this? :/ ...
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Is it true that a quasiconvex, increasing and continous function, is convex?

Let $f:\mathbb R^n \to \mathbb R$ be a continuous and increasing function. Let $f$ be quasiconvex. Let $f(0)=0$. Can we say that $f(x)$ is convex ? If yes, how do we prove it ? Thank you very much ...
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Related Rates/ Optimization problem

I was having trouble figuring out this problem. A fisherman is in a boat 3 km from the nearest point A on the coast. The fisherman wishes to return to his camp C located 5 km from the point A. The ...
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Optimizing a ranch

"A rancher wants to fence in an area of 1000000 square feet in a rectangular field and then divide it in half with a fence down the middle, parallel to one side. What is the shortest length of fence ...
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What is the meaning of “girth” of a rectangular box?

Here's an optimization problem. A parcel delivery service will deliver a package only if the length plus the girth (distance around, taken perpendicular to the length) does not exceed 112 inches. ...
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Maximising a sum - closed form?

As a follow up to this question, I am wondering the following: Suppose $\sum_{i=1}^n x_i=0,\;\sum_{i=1}^n x_i^2=1$. Is it there a closed form for $\max \sum_{i=1}^n x_ix_{i+1}?$ ($x_{n+1}=x_1$) For ...
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Does every strongly convex function has a stationary point?

Say does every differentiable $\mu$-strongly convex function $f:\mathbb{R}^n\mapsto\mathbb{R}$, with $\mu>0$ have a point where its gradient is $0$? If not so which is the minimum you can impose ...
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Maximisation problem

I am trying the following question: If$$a+b+c+d=0,\;a^2+b^2+c^2+d^2=1$$ Then what is the maximum value of $ab+bc+cd+da?$ By the rearrangement inequality I can get $ab+bc+cd+da\leq 1$ but I am ...
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Example of a k-matroid

Let the set $K_i = (S, I_i)$ be a matroid for each $i \in \{1 \ldots k\}$. We define $K = (S, I) $ where $I = \{ X \subset S $ | $ X \in \bigcap_{i=1}^k I_i\}$ The claim now is that $K$ is a ...
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If a continuous function has one relative extremum, then it is actually an absolute extremum

The following statement makes sense intuitively, but is there a way to prove it mathematically? (This is something we make use of in applied optimization in calculus.) If $f$ is continuous on an ...
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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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Help with Gram-Schmidt problem

I'm supposed to show that the Gram-Schmidt process: $\textbf{a}_j = \left\{ \begin{array}{lr} \textbf{d}_j, \;\;\textbf{if} \;\;\lambda_j = 0\\ \sum_{i=j}^n \lambda_i\textbf{d}_i ...