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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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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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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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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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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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 ...
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How can I use Mehrotra's predictor-corrector primal-dual interior point method to solve a problem that is not in the form of cTx?

I am not very familiar with optimization methods. I am studying the paper "Blind channel identification for speech dereverberation using l1-norm sparse learning" (here: http://linyq.com/NIPS2007.pdf). ...
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max and minimum qudratic function problem

A piece of wire $20$ metres long is cut into $2$ pieces and each piece is bent to form a square. Determine the length of the two pieces so that the sum of the areas of the two squares is a minimum. ...
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Optimization to minimize cost using the function C=Tq^(1/a)+F

I was given the function of $C=Tq^{1/a}+F$ where $C$ is total cost, $q$ is output, $a$ is a positive parametric constant, $F$ is the fixed cost, and $T$ measures the technology available (also a ...
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Optimization Problem (Linear Algebra)

I am not trying to cheat or anything, so any reference to online literature or MOOCs, that teach this stuff, will be highly appreciated. The problem is to prove that the following optimization ...
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Could someone explain the Lagrangian Method?

I understand the method, technically, but what is actually going on? We set the gradient of the function equal to the gradient of the constraint (multiplied by a constant), and in doing so, we find ...
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How to approach this optimization problem with “sorted” constraint

I have formulated an optimization problem and I'm not sure how to go about solving it intelligently. I have three vectors $ a, p, n$, all with the same number of elements. I know what $p$ and $n$ are ...
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Convex Optimization of quadratic function with inequality constraints

How would I solve the following problem? $$\min_{x\in\mathbb{R}^n} x^T A x$$ subject to the constraints $$x_i\geq 1,\,i=1,\dots,n,$$ where A is positive semidefinite and symmetric. Is it possible to ...