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, rectangle inscribed inside arch of the curve.

A rectangle is to be inscribed under the arch of the curve $y = 4\cos(0.5x)$ from $x = \pi$ to $x = -\pi$. What are the dimensions of the rectangle with largest area, and what is the largest area? ...
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21 views

minimum possible value of a linear function of n variables

Suppose $x_1,x_2,....x_n$ are unknowons satisfying the constraint $a_1x_1 + . . . + a_nx_n ≥ b$, where $a_1, . . . , a_n, b ≥ 0$. Then the minimum possible value of the expression $c_1x_1 + . . . + ...
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3answers
144 views

How do you find the maximum value of $|z^2 - 2iz+1|$ given that $|z|=3$, using triangle inequality?

Problem: How do you find the maximum value of $|z^2 - 2iz+1|$ given that $|z|=3$, using triangle inequality? My attempt: $$|z^2 - 2iz+1|\le|z|^2+2|i||z|+1$$ $$\implies |z^2 - 2iz+1|\le16$$ ...
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2answers
15 views

SVD with Sparisty Penalties

I am trying to solve a problem like $$\min_{u,v} \Vert X-uv^T \Vert_F^2 + \beta \Vert v \Vert_1 \mbox{ s.t. } \Vert u \Vert_2 \le 1$$ where $X \in \mathbb{R}^{N\times N}$, $u, v \in ...
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15 views

How to find local maximum of the following three variable function?

Find the local maximum $$ f(x,y,z) = (x^y{z \choose y})^{z+1} $$ where $$ (y+1)(z+1)\log_2 x=C $$ C is constant x,y,z are all positive real numbers
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1answer
23 views

Understand a compact Interval

For example , if I define a function of several variables $f : X \rightarrow Y$ , for example $f (a , b, c , x, y , z )$, then by putting the constraint $x + y + z = 2$ and $a + b + c = 3$ , for $ ...
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0answers
28 views

How to prove that the following function has a unique mode?

I am trying to prove that the function $$f(\alpha)=n\ln \alpha-n\ln\Big(\sum_{i=1}^{n}t_i^\alpha+\int_{a}^{b}x^{\alpha+\beta-1}e^{-\lambda x^\beta}\,dx\Big)+(\alpha-1)\sum_{i=1}^{n}\ln t_i,$$ where ...
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0answers
17 views

Show that any vertex $v$ of $P$ is half-integral.

Let $G$ be an undirected graph and define $$P=\{x \in R^{V}: x(u)+x(v) \leq 1 \:\:\text{for all edges}\:\: e=uv,\:\: x \geq 0\}$$ Show that any vertex $v$ of $P$ is half integral.
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51 views
+100

Interpolation and mapping between scattered vectors in two unequally dimensioned spaces

Imagine two spaces: An ‘input’ space with dimension $m$. An ‘output’ space with dimension $n$. $m \geq n$ There are points in each of these spaces defined such that some characteristic is ...
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+500

A System of Matrix Equations (2 Riccati, 1 Lyapunov)

Setup: Consider the following system of real matrices, \begin{align} {\bf P} &={\bf F}({\bf I}_{n}-{\bf K} {\bf H}^\top){\bf P}{\bf F}^\top+{\bf Q}, \;\;\;\;\;\;\;\;\text{where}\;\;\;\;{\bf ...
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0answers
10 views

Optimizing functions with a complex domain and a real codomain

In general I want to understand if it makes sense to optimize a function of the following form $f: \mathbb{C} → \mathbb{R}$ for my specific problem $f(z) = | z | ^{2} $ (wich I is not analytic since ...
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0answers
10 views

Removing variables from convex linear program

I am solving linear program (possibly non-convex). Then we know that dual is always convex. Then I noticed that depending on objective functional I can sometimes remove particular variables from this ...
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0answers
23 views

Understanding ADMM: how is it applied to this particular problem?

While reading Boyd's paper on ADMM I encountered an issue. Consider the following problem: Problem. Minimize $f(u) + g(v)$ subject to $Au + Bv = c$, where $f$ and $g$ are closed, proper, convex and ...
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1answer
532 views

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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0answers
61 views

Making a Matrix singular

During my research I came across the following problem. Intuitively this should be an easy one. However, the simplest version of it looks like this: Let $C \geq \frac{1}{2}$ be some fixed ...
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0answers
16 views

Negative eigenvalue of a hessian matrix entails a local decrease in function value?

I was reading up on non-convex optimization, and I can across this sentence: "Since Hessian(f(w)) has a negative eigenvalue, there is always a point that is near w which has smaller function value" ...
2
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1answer
23 views

Win/Lose ratios and selection strategies

Imagine the following scenario: You're on a TCG tournament which allowed you to bring N decks with you. After each game, you might select another deck for your next game. You are allowed to keep ...
2
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1answer
69 views

Find and Evaluate Critical Points

I need to find al the critical points of the following function $f(x,y)=y^2-x^2y-3y+x^4-x^3$ Determine if they are local minima, local maxima, or saddle points, by looking at the Hessian matrices ...
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1answer
28 views

Inequality with three unknowns

Consider: \begin{equation} \Big(e^x-1\Big)\mathbb{1}_{(x\geq0)} \leq \lambda_1+\lambda_2e^x + \lambda_3x^2 \end{equation} where $\lambda_1$, $\lambda_2$, $\lambda_3$ are three unknown constants. By ...
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3answers
53 views

How to do a regression which includes reciprocals?

I'm trying to find an interpolating formula for a set of coefficients (I have $80$ at the moment). I tried first to find an interpolating polynomial, but that was not useful: using the first ...
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0answers
10 views

Gradient Descent with multiplication term

Say I have the objective: $\arg \min_{R, T} \|y - RTx\|^2_2$ where, R and T are matrices (not necessarily square) and y and x are known vectors. I wish to try and optimize R and T using Gradient ...
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2answers
1k views

Optimal Basic Feasible Solutions

In linear programming, is it true that you can only have at most 2 optimal basic feasible solutions? If so, why?
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Setting the right restriction in a simple linear optimization task

here is the task: One factory produces 3 types of cars: small, midsize and big. There are 6000 tons of steel and 60000 total time available. For each type of car produced, there must be 1000 cars of ...
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1answer
24 views

Continuity of Parameterized Optimal Solution

Suppose for every $y$, $f(x,y)$ is strictly convex in $x$. Further, $f(x,y)$ is continuous in $y$. Let $\mathcal X$ be compact (in my problem, $\mathcal X$ is an interval). Can anyone suggest any ...
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1answer
19 views

What is the maximum of the following function?

Let $f(x,y) = \frac{xy^\alpha}{x+y},\alpha\in(0,\infty)$. How to compute $$\sup_{(x,y)\in[a,b]\times [0,c]}\frac{xy^\alpha}{x+y},$$ with $b>a>0$?
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0answers
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One-dimensional deblurring

I just begun studying image deblurring on my own, and I have a question. Most books I found say that I can see the images as arrays, and that I can "vectorize" the arrays of the images by stacking the ...
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0answers
34 views

Basic optimization question

A teacher put this problem up the other day and I'm confused about how he got to the answer. Can you explain it to me? Job $X$ provides $20$ vacation days and $143,000$ euro annual salary. Job $Y$ ...
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0answers
251 views

Maximal minimum for a sum of two (or more) cosines

Please prove (or disprove, and give the correct answer): $$2 =\mathrm{argmax}_{r\geq 1}\min_{x\in \mathbb{R}}\left[\cos\left(x\right)+\cos\left(rx\right)\right] $$ In other words, find $r \geq 1$, ...
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1answer
15 views

Does optimal solution from primal problem follow from optimal solution to dual?

In a linear programming context, does the primal optimal solution yield an explicit way to find the primal dual solution? I vaguely remember something like this from an optimization class but can't ...
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2answers
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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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0answers
5 views

Biorthogonal (discrete wavelet) noise bases?

I am slightly interested in discrete wavelet transforms (DWT), but so far I have mostly used already-derived and existing well-known wavelets, such as Daubechies, Cohen-Daubechies-Faveau, Symlets and ...
2
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2answers
369 views

Multiple solutions for both primal and dual

If matrix $A$ in an LP (or $A^T$ in its dual) has full row (column- in dual) rank, is it possible that both primal and dual have multiple solutions?
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1answer
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Filling the area below a decreasing function by rectangles

Suppose $f:\mathbb{R}_+ \to \mathbb{R}_+$ is a strictly decreasing continuous function such that. Let $n$ be a natural number. I want to solve the following maximization problem $$ S_n = \max_{x_1 ...
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1answer
29 views

set up Linear programming problem

How do I set up this problem ? A product can be made in three sizes, large, medium, and small, which yield a net unit profit of $12, 10$ and $9$ respectively. The company has three centers where ...
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What the minimum of infinite tetration divided by $\sqrt{x}$?

For some values of $x$ the limit of infinite tetration converges. For example when $x=\sqrt{2}$ this is fixed point $$\lim_{n\rightarrow \infty} \sqrt{2} \uparrow \uparrow n = ...
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1answer
35 views

Translate this problem to graph theory

Say I have a size $k$ set called $S_k$ with elements that are natural numbers (repetitions are allowed). For instance $\{2, 8, 6, 6, 1, 3\}$ is a valid set for $k = 6.$ I am trying to find the least ...
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1answer
22 views

Solve logistic problem with graph - fitting boxes

Suppose you have $n$ boxes, each of which falls into one of the $k$ sizes, and you want to nest smaller ones into larger ones, such that no two boxes $A$ and $B$ are nested inside the same box, if ...
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0answers
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Where does the name “tracking type problem” come from?

In PDE-constrained optimization problems, the distributed constrol problem $$ \begin{array}{ll} \displaystyle \min_{y,u} & J(y,u) = \frac{1}{2}\|y-y_d\|_{L^2(\Omega)}^2 + ...
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0answers
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Distance between point and ellipse - explanation of a paper

EDIT: I notice that the link is hidden, but this post is made with reference to THIS PAPER I'm trying to solve quite an old problem (once again) - to find the distance between a point (in 3d space) ...
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2answers
94 views

Solve $\min_{A \subset \mathbb{R}} \int_{A} (f(t)-g(t))dt$

Consider the functions $f:\mathbb{R}\rightarrow \mathbb{R}$ and $g:\mathbb{R}\rightarrow \mathbb{R}$ both integrable on any measurable set $A \subset \mathbb{R}$. Consider $$\min_{A \subset ...
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1answer
35 views

What is the difference between linear and integer programming?

Recently I tried to solve a maximization integer programming problem using linear programming by flooring the max point - but got the wrong answer. I'm wondering if someone can explain mathematically ...
2
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1answer
235 views

The sum of differences between two data sets is minimized when both are ordered the same way

Suppose we have two sets of data, $X$ and $Y$, each of which contains $10$ positive numbers. Now let us order the data sets $$X=\left\{ x_{1},\cdots,x_{10}\right\},\quad x_{1}\ge\cdots\ge ...
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Derivative of a variable across multiple functions

I'm just starting out learning about optimization. Not sure if this makes sense but... Given a variable $x$ that is passed to two separate functions, $f(x)$ and $g(x)$, why do we add the derivatives ...
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1answer
43 views

Project allocation optimization with tricky constraint

I have an allocation problem that should be straightforward, except that it has very specific constraints. I want to assign approximately 300 students to 170 projects in pairs - so that each project ...
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0answers
25 views

On a max-min problem from an exam.

I have asked a different question on the same exercise (from an exam) a couple weeks ago, I hope it is acceptable to have a different question on the same exercise, I searched the Meta and it seems ...
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2answers
13 views

MLE of a discrete random variable

For some reason I am having difficulty understand how to calculate the mle of a discrete rv. The pmf is: $$p(k;\theta) = \left\{\begin{array}{lcc}\frac{1-\theta}{3}&\mbox{if}&k=0\\ ...
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2answers
32 views

Newton's method for optimization

I have been reading about Newton's method and know that you can use it for optimization problems. However, does Newton's method only guarantee convergence to a local minimum or maximum, or can it be ...
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0answers
14 views

Find min and max of $\log n /\log an$ [on hold]

I would like to find min and max of $$\log n /\log an$$ where $a$ is a parameter such that $a>\log n/n$ and $n>e$. I think the min is $\sqrt{n}$, but i am not success to prove it. Sorry to ...
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1answer
23 views

True or false? “sum of an m-strongly convex and a convex function is m-strongly convex”

I would like to know if the following conjecture is true or false? If $f(x) = g(x) + h(x)$ where $g$ is m-strongly convex and $h$ is convex, then $f$ is m-strongly convex. NOTE: For a ...
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0answers
17 views

Is my proof correct? Convex optimization

There's a theorem that says that if $C \subset \mathbb R^n$ is a convex set, then $x^* \in C$ is the closest point in $C$ to $y \notin C$ if and only if $(y-x^*)\cdot(x-x^*)\leq 0$ for all $x \in C$. ...