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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Optimal Strategy for Chosing Lottery Tickets

You have 2 types of lottery tickets: one that costs $c_1$ and has a probability of winning of $p_1$, and the other costs $c_2$ and has a probability of winning of $p_2$. The goal, as you might expect, ...
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Algorithm to find maximum possible value of the minimum expression in a list

Problem Let $x_1, x_2, \ldots, x_m \in \mathbb{R}$, and suppose I have a bunch of expressions which are linear combinations of $1, x_1, x_2, \ldots, x_m$. For example, I might have \begin{align*} ...
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Is it possible to reduce a lambda expression to it's smallest equivalent form?

In the Untyped Lambda Calculus, is it possible to reduce any arbitrary expression to it's smallest equivalent form? (defined as an expression with the smallest number of lambda terms) If so, is there ...
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Analytical Method to Minimise a Cost Function

I've been researching optimisation methods used to minimise the cost function of a neural network, such as: Back propagation, which calculates the error at each node in order to calculate the partial ...
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Mana Maximization (Hearthstone)

I recently started playing Hearthstone and a statistic / probability question came up my mind. Here's a quick breakdown: The game is a turn-based card game which involves "points" that you can used ...
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Find $\alpha$ such that the given point is critical for a implicitly defined funtion.

Can anyone check my solution for this exercise? Let $F:\mathbb{R}^3\rightarrow\mathbb{R}$ be given by $F(x,y,z) = \alpha xz + x\arctan(z) + z\sin(2x+y) -1.$ Prove that a function $z=f(x,y)$ ...
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Find the maximum value

Find the maximum value $$F(y)=\int_{0}^{y}\sqrt{x^4+(y-y^2)^2}dx$$ with $y\in [0;\: 1]$ This problem here
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Preconditioning of optimization problems

This question suggests that you can precondition an optimization problem by a simple multiplicative scaling of the variables in the objective function. However, when I look up literature on ...
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A question on minimizing $\| . \|_2^2$ vs $\| . \|_2$

Suppose we are in $\mathbb{R}^n$ Is the problem of $d(x,Y) = \inf\{ \| x - y\|^2 : y \in Y\}$ equivalent to $d(x,Y) = \inf\{ \| x - y\| : y \in Y\}$ Pardon me, let us keep it simple and just stick ...
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Maximum area under a curve by calculus of variations

I am asked to find the function that has the maximal area for a given length L when x runs from -a to a. I calculated the integral to be varied as follows: $$ \int_{-a}^{a}\ y + \lambda \sqrt{1 + ...
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Help deriving simple linear regression formulas

I'm reading the Wikipedia article Simple Linear Regression . In the article they write a function to be minimized by choosing $\alpha$ and $\beta$: $$Q(\alpha,\beta) = \sum_{i=0}^{n} (y_i - \alpha - ...
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Lagrange method for inequality

i have one question and any hint would be very helpful for me,we know how Lagrange multiplier works ,for example consider following problem Example 2 Find the maximum and minimum of subject to ...
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45 views

Extrema on a given set

Could you tell me if my approach to finding extrema on a set is good? Let's take a function $f(x,y,z)=x+y+z$ and a set $N= \{(x, y, z) \in \mathbb{R}^3 : x^2 + y^2 \le z \le 1 \} = \{(x, y, z) \in ...
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Find maximum of a double integral over a region

I have a region given by $$R = |{ax}|+|{by}| \le 1$$ and $$f(x,y) = \iint\limits_{R}{(ax-by)^2*(3ab^3+12a^3b-6a^3b^2)*\sin^2({\pi ax + \pi by}})dxdy$$ I need to find the values of $a$ and $b$ that ...
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34 views

Clarification about global extrema

Is everything in my statements $\textbf{(1)}$ and $\textbf{(2)}$ correct? A real valued function $f$ defined on a domain $X$ has a global maximum point at $x^{\bigstar}$ if $f(x^{\bigstar}) ...
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Maximal area in fixed perimeter [duplicate]

An old story I heard starts by two people that was arguing about how much land a man need. So they called to a young man, and said to him: You are stating in that point, and start running all the ...
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103 views

Definition of stability in the case of Levenberg-Marquardt optimization method

I've come across this guide: Fortunately, it inherits the speed advantage of the Gauss–Newton algorithm and the stability of the steepest descent method. What's a stability in this case? Does it ...
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Maximize a sum of sinusoids with comensurable periods

I'm writing a program that requires finding $$\text{argmax}_\theta\sum_{k=1}^na_k\cos(k\theta+b_k),$$ where $a_k$ and $b_k$ can be any real numbers. How can I do this efficiently?
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Optimization / personalization within clusters

I have the following optimization problem: I have a (random and very noisy) objective function f(A, P), where A is a vector of "observable" parameters of the input and P is the parameters that I can ...
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171 views

Optimization of Complex Functions, No Use?

During reading the appendix to an engineering text, I came across the following remark: "Complex cost functions are of no interest, because in the field of complex numbers no ordering (relations ...
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Maximization under constrains

I would like to maximize the function: $\frac{1}{2}\sum_{i=1}^{N}\lvert x_i-\frac{1}{N}\rvert$ under the constrains $\sum_{i=1}^{N}x_i=1$ and $0\le x_i \le 1$ $\forall i\in(1,...,N)$ I have done ...
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Determining domain interval for optimization problems

This example is from Paul's Online Notes for Calc I. You have $500$ feet of fencing material and you want to enclose a field with a fence. A building is on one side of the field (and so won't ...
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135 views

Bounded logarithmic function

I am trying to find any function that it grows logarithmically up to a certain point, and after that point it remains constant. Can anyone help me with that
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Propose suitable algorithm for min-max optimization problem

Consider: \begin{equation}\min_{x, y} \max_{\omega} | f(x, y, \omega) |\end{equation} where $(x , y)\in \mathbb{R}\times \mathbb{R} $ and $\omega \in (0, \infty)$. $f$ is the result of dividing ...
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How do I maximize $|t-e^z|$, for $z\in D$, the unit disk?

I guess this question doesn't have a closed form solution for all $t\in \Bbb C$, but I know one for $t=1$ provided by Daniel Fischer in a question I asked. $$\begin{align} \left\lvert ...
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Help solving an optimization problem involving inverse square roots

Does anyone know if the following optimization problem has an elegant solution? Let $A=\{a_1, a_2, \ldots, a_n\}$ be positive real numbers. Let $B=\{b_1, b_2, \ldots, b_n\}$ be unknown positive real ...
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305 views

Max/Min Problem using derivatives

Question: A professional basketball team plays in an arena that holds 20000 spectators. Average attendance at each game has been 14000. The average ticket price is 75 dollars. Market research shows ...
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Finding extrema of multivariable functions.

A problem asks me to find the absolute extrema of the function given by $f: \mathbb{R}^2 \rightarrow \mathbb{R} ,f(x,y)=(x^2+y^2)e^{-(x^2+y^2)}$. Now, how can I find the critical points?. As far as I ...
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268 views

Is the hessian negative semi-definite if we have an interior maximum?

Is it true that given a smooth scalar field f on a domain D , if f attains a maximum (minimum) on the interior of D then the hessian of f evaluated at this max (min) is negative (positive) ...
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Maximizing the area of rectangle inscribed in triangle

I'd like to ask if someone could help me out with this problem. Let's have a triangle with coordinates $[0,0],[4,0],[1,3]$. Inscribe a rectangle into this triangle, so its area is maximized The base ...
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Lagrange multipliers - finding maximum/minimum

I have solved the question, and obtained the critical points, but don't know how to show its a maximum or minimum of a function. I don't understand other answers because symbols confuse me so much and ...
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Help with local extrema of $f(x)=x^4-5x^2$

Find the coordinates of any local extreme points and inflection points of the function $f(x)=x^4-5x^2$ My try: Find critical points: $f^{\prime}(x)=4x^3-10x=0$ $f^{\prime}(x)=2x(2x^2-5)=0 ...
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Finding minimum of the trace of the matrix equals finding maximum of the trace of the inverse matrix?

Let $K$ be a positive definite, symmetric matrix. Let $C$ be a nondegenerate matrix of the same order. Elements of $K$ and $C$ depend on some parameter $a.$ Is it true to say that $$ ...
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Simplifying This Boolean Expression? (A Little Rusty)

I have the Boolean expression: F = A'B'C'D + A'BC'D' + ABC + AB'C'D' + ABCD'. Note that the ' indicates the negation of the variable by my convention. I am trying to show that F = BC + A'C' + B'D' is ...
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Strictly convex self-concordant function

Some definitions: A function $f:R^n\rightarrow R$ is convex[strictly convex] if for every $\lambda\in[0,1]$ [$\lambda\in(0,1)$] and for every $x,y$ [$x\neq y$] in $R^n$ we have $f(\lambda ...
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L1-norm minimization

This is undoubtedly a trivial question but might as well ask: Why is the L1 norm minimization a heuristic for finding the sparsest vector? What I mean is that if the L1 norm sums the elements of a ...
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Direction of steepest descent and minimization?

I have the following linear function: $min$ 1/2 $<x, x>$ + $r^Tx$ for every x belonging to $R^n$, $r^Tx$ belongs to $R^n$ Now, = $x^TAy$ and A is symmetric positive definite. = $x^TAy$ is ...
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Sum of weighted squared distances is minimized by the weighted average?

Let $x_1, \ldots, x_n \in \mathbb{R}^d$ denote $n$ points in $d$-dimensional Euclidean space, and $w_1, \ldots, w_n \in \mathbb{R}_{\geq 0}$ any non-negative weights. In some paper I came across the ...
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Effect of approximating a non-differentiable function on optimisation of minimisation

I am looking at a problem of constrained minimization, where the function to be minimized contains the Heaviside function, and as such is not twice continuously differentiable. My question is what ...
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Find $x, y$ such that $\left | \frac ab -\frac xy \right |$ is minimal

Given positive integers $a, b, D$. How to find $x, y \in \mathbb{Z^+}$ such that $$M =\left | \frac ab -\frac xy \right |$$ is minimal and $x + y \le D$? For a solution, I can get it by ...
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52 views

Range Of Quartic Polynomial Of Two Variables

$a$,$b$ are real numbers such that $~3\leq a^{2}+ab+b^{2}\leq 6$. I would like to find the range of $~a^{4}+b^{4}$. Is it possible to find it with (well-known) AM-GM, CS, etc...?
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Convex hull of a 0/1 set in $ \mathbb{R}^d$

Reading in my textbook, I found the following example: $$ $$Let S $ \subseteq $ {0,1}$^d$ be an arbitrary 0/1 set in $ \mathbb{R}^d$ and the Polyhedron Q = conv(S). It can be shown easily that the set ...
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Minimise Total $x$, Maximum $x$ or $|x|$ in integer/linear programming

Suppose we have a linear program (it may be integer, it probably doesn't matter). Suppose the variables $t_j$ give the tardiness for job $j$, and we want to minimise something to do with this ...
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Maximize $ax + by + c$

Working on a problem of comparative advantage of the economist David Ricardo, I've gone into solving a more general case of that study in which I stumbled over this question : how do we maximize the ...
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Representing a 2D function as a sum of rectangles of arbitrary shape and orientation

Suppose I am given a non-negative function $f(x,y)$ defined for $x \in [0,1]$ and $y \in [0,1]$. I'd like to represent this function as a weighted sum $w_i$ of a small number of rectangular apertures. ...
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Measure minimization for a combination of overlapping sets

This problem may have been worked out before but I don't know where to start looking so I hope one of you can help me. The problem is as follows: There are $N$ variable-sized finite sets ...
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Zisserman Lecture and $x_{MLE}$

In the Zisserman Lecture below http://www.robots.ox.ac.uk/~az/lectures/est/lect34.pdf page 36, he derives $x_{MLE}$ for Gaussian sensor fusion. There are two noisy measurements $z_1$ and $z_2$ ...
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Need (solid) proof for finding a maximum value

Can anyone please verify my logic to find maximum value of a function? here is my work: Goal is to find $x,y$ which maximize $f(x,y)$ ($f(x,y)$ is a function developed by myself) only $y$ has a ...
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First order necessary conditions for $\max_{x_1}f(x_1,g(x_1)).$

$$\max_{x_1}f(x_1,g(x_1)).$$ And, let $f$ attends max at $x_1^*$, so first order necessary conditions imply that $$\dfrac{\partial f(x_1^*,g(x_1^*))}{\partial x_1}+\dfrac{\partial ...
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596 views

Finding global minimizer and maximizer

Let $f(x,y)=x^4-8x^2+y^4-18y^2$ Find the set of global minimizers of f? Does f have a global maximizer?Justify? I first calculated the gradient of f and then let partial derivative of x and y to ...