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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Probability density function / maximum likelihood for correlating sequence

I have a stream that contains two consecutive identical sequences, each of length $N$. These sequences have a ideal autocorrelation property. So I want to have the probability density function over ...
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Maximize $xyz$ for $x,y,z>0$ satisfying $4xy+6yz+8zx=9$ [closed]

If $x,y,z$ are positive real numbers satisfying the equation $4xy+6yz+8zx=9$, then find the maximum possible value of $xyz$.
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Help with a homework problem involving $\textbf{H}$-conjugate vectors

My problem is the following: Let $\textbf{H}$ be a symmetric $n\times n$ matrix. Are the following claims true? Why? a) If the vectors $\textbf{d}_1$ and $\textbf{d}_2$ and vectors ...
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Proving the shortest function connecting two points is a straight line WITHOUT assuming the Euler-Lagrange equation

For learning purposes, I'm trying to prove that the shortest function passing through the two points $(x_1, y_1)$, $(x_2, y_2)$ is a straight line, without using the Euler-Lagrange equation. My ...
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2answers
51 views

Parameter optimization using a regression model.

I am working on an optimization problem. I build a regression model to understand the behavior of a system which depends on two variables which are functions of another two variables. My regression ...
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Maximum vertical distance between the line $y = x + 30$ and the parabola $y = x^2$ for $−5 ≤ x ≤ 6$

What is the maximum vertical distance between the line $y = x + 30$ and the parabola $y = x^2$ for $−5 ≤ x ≤ 6$? This is what I did but didn't work: Set $y_1=x+30$ and $y_2=x^2$, plugged ...
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1answer
14 views

Second Order Necessary Condition for Optimality

Question: [See context below.] What would be the analog of the Thm when $f$ is only defined on, say, a domain $D\subset\mathbb{R}^n$? In that case we can't take a general ...
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optimization with non smooth constraint

I am trying to maximize the profit of a power plant. I have a constraint which is that the power plant, when operating, has a minimum and maximum capacity. (So a power block either has an output of ...
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33 views

Can we ever have E(argmin(f)) = argmin(E(f))?

Consider a parametric real-valued function $f_{\boldsymbol{\alpha}}:\ \mathbb D^N \rightarrow\mathbb R$ whose parameters $\boldsymbol\alpha$ vary according to some distribution $\psi$, and $\mathbb D$ ...
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A textbook question from Fletcher's Practical Methods of Optimization

It is Q.2.19 on P.42: Given $q(x) = \dfrac{1}{2}x^{T}Gx+b^{T}x+c$ be a quadratic function, where $G$ is an $n \times n $ symmetric matrix and $b \in \mathbb{R}^{n}.$ (a) Show that a minimizer exists ...
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Optimization of a function

I need to optimize $$f(x,y,z)= x^2-y+e^{z}$$ with the restriction $$(x-2)^2+(y-3)^2+z^2=1$$ I've tried to substitute the restriction in $f(x,y,z)$ but it seems not to work. And when trying to use the ...
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Rank Minimization

I have a n*m matrix, the rank of matrix (r) is near to min(m,n) I want to minimize the rank by removing some of the rows or columns to get r << min(m,n) The goal is to achieve least rank ...
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Minimization of a weighted least-squares problem by Lagrange multiplier method

Problem: Let $Y = (y_1, y_2, \dots, y_m) \in \mathbb{R}^{m \times n}$ and $k \in \mathbb{R}^{m}$ satisfy $\sum_{i=1}^{m} k_i =1$ and $k \geq 0$. Show that $x=Yk$ is a minimizer for $h(x) = ...
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When can i solve simplex tableau

I saw exercises where they give an objective function ( without restrictions ) and a simplex tableau to be completed , if you can solve How do I know when it may solve the tableau ? What are the ...
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direct connection between gradient descent and follow the (perturbed) leader algorithm or weighted majority? [migrated]

Is there a direct conversion between gradient descent ([1], Alg 1 ) and any of the following algorithms? 1) Weighted Majority: http://onlineprediction.net/?n=Main.WeightedMajorityAlgorithm 2) ...
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Maximum area of a isosceles triangle in a circle with a radius r

As said in the title, I'm looking for the maximum area of a isosceles triangle in a circle with a radius $r$. I've split the isosceles triangle in two, and I solve for the area $A=\frac{bh}{2}$*. I ...
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1answer
61 views

Prove optimal solution to dual is not unique if optimal solution to the primal is degenerate

How do I prove an optimal solution to dual is not unique if an optimal solution to the primal is degenerate? I have no idea how to start this. Anyone know any books with these kinds of questions (and ...
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Optimization with Linear constraint $Ax=0$

I confront with this problem: $$\min_{x \in \mathbb{R}^{n}} \dfrac{1}{2} \left\| x- a \right\|_{2}^{2}$$ subject to $$Ax=0.$$ My tactic is to use Lagrange multiplier method that: $$\mathcal{L}(x, ...
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27 views

Selection of the mean of random variables to optimize the expected value of objective function

Here is the objective function to be maximized: $$ E_{v}(\log(1+v^{\mathsf T} \Lambda v) ) $$ where $v$ is a Gaussian distributed random variable vector $v ∼ \mathrm{CN}(M,I)$ with its mean vector ...
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Smallest value taken by a quadratic polynomial in two variables.

Let $p$ be a degree $2$ polynomial with integer coefficients, say $$p(x,y) = Ax^2 + By^2 + Cxy + Dx + Ey + F.$$ I would like to find an algorithm which solves the following: Problem 1: Given ...
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Transform a minimization problem to LP

This is a past examination question. I was asked (Q.1) to find an equivalent linear programming problem of: $$\min_{x \geq 0} \left \|Ax-a \right\|_{1} + \left\|Bx-b \right\|_{\infty}$$ where $A$ ...
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Will this optimisation give the global maximum?

My book optimises a type of functions using the lagrange method. From calculus I remember that we had to check the boundary when using lagrange, because it only gave local max, but it is not mentioned ...
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Find the Maximum and Minimum of the Given Function on the Given Plane Region

I've been good with most of the max/min finding in different regions, but this one's really messing with me. Can anyone lend a hand? Thanks. z = 2xy Region is the circular disk $x^2 + y^2 =< 1 $
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minimal sum of product of triples

I have a bunch of positive integers $a_1, a_2, ... a_{2n}$. I split these numbers into groups of 2, calculate the product of each pair, und sum over those. E.g. $Sum = a_1a_2 + a_3a_6 + a_4a_5$. I ...
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Polar cones' property [duplicate]

I am trying to prove: $A \subseteq B \implies B^\circ \subseteq A^\circ$ where $A^\circ$ is polar cone of $A$ ($A$ convex cone) and $B^\circ$ is polar cone of $B$ ($B$ convex cone)
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Slice an ellipsoid into equally thick slices for maximal surface

After seeing a colleague slicing a nearly ellipsoid piece of ginger for his cup of tea into almost equally thick slices to get more surface area (so the tea would suck out the ginger taste better), i ...
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266 views

De Jong's Fifth Function's Minimum?

What is the minimum solution to De Jong's fifth function, in the range $-65.536\leqslant x_1\leqslant 65.536, -65.536\leqslant x_2\leqslant 65.536$?
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56 views

The minimum value of $\frac{a(x+a)^2}{\sqrt{x^2-a^2}}$

The problem is to find the minimum of $A$, which I attempted and got a different answer than my book: $$A=\frac{a(x+a)^2}{\sqrt{x^2-a^2}}$$ where $a$ is a constant ...
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Intuition behind gradient VS curvature

In Newton's method, one computes the gradient of a cost function, (the 'slope') as well as its hessian matrix, (ie, second derivative of the cost function, or 'curvature'). I understand the intuition, ...
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What does 'the level set is bounded' exactly want to tell?

'The level set is bounded.' occurs in many theorems and other places. I think I can understand the definition of 'level set' but I don't know what does 'it's bounded' want to tell me exactly in ...
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Solving for f(t) in presence of f'(t)

Here's the situation: I have a function $$e(t) = \frac{a~d(t)}{b + d(t)}$$ with first derivative $$e'(t) = \frac{a~b~d'(t)}{[b+d(t)]^2}$$ where $a$ and $b$ are constants. For a given constant $K$ I ...
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Maximize area of a corral

See problem 7 and figure 9 in picture What I've done so far: Not sure if $P=2l+2w$ or just $l+2w$ (dashed line makes me think the latter) $600=\pi r+l+2w$ $600=\pi r+2r+2w$ $w=\frac{600-\pi ...
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Shannon Entropy Continuity Constraint

I have the following problem: I want to find the probability density $p$ which maximizes the Shannon entropy \begin{equation} S := - \int_{x_b}^{x_c} dx ~ p(x) \log (p(x)) \end{equation} under the ...
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What is this sort of optimisation called?

I am reading a book in mathematical finance. There is something about constrained optimisation. They have specialised it for the financial market, but I am wondering what the general name for this ...
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Strong duality and its relation with perturbation functions

From the strong duality wiki page Strong duality holds if $F^{**}=F$ where $F$ is the perturbation function relating the primal and dual problems and $F^{**}$ is the biconjugate of $F$. I ...
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Optimizing a functional with a differential equation as a constraint

I am working on solving the following optimization problem. I think it is well-poised but, if not, please give me some pointers that could make the question make more sense. We have a parametric ...
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1answer
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Given an optimal solution to the LP, show how it can be used to construct a directed cycle with minimal directed cycle mean cost.

Let $\mathcal G = (\mathcal V, \mathcal A)$ be directed graph with associated edge costs $c_{i,j}$ that has at least one directed cycle. Define the directed cycle mean cost to be $\frac {\{\text {sum ...
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1answer
74 views

A interesting max min problem

Let $\mathcal{S}\subset\mathbb{R}^2$ be a bounded, closed, compact, convex set which contains origin in its interior. Define \begin{align} c_1^{\star}=\min_{{(x_1,0)\in\mathcal{S}}}~&x_1 ...
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Split a vector into three

Say we have a vector of length n<100, $v(w_1,w_2,\ldots,w_{n})$. My problem is to divide the vector $v$ into groups of $3$, eg $u_m =(w_i, w_k, w_k)$ with as close weight as possible. Eg to ...
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1answer
878 views

How to describe minimization of L1 norm error using linear programming?

Given a set of $n$ pair points $(x_1, y_1), ..., (x_n, y_n)$ in the plane, I need to find a line $ax + by = c$ that fits the points of the L1 norm error points as closely as possible. I need a linear ...
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Binary Linear Programm: Check for feasability and multiple solutions

Assuming, I have binary integer program, e.g. given by: $ \arg\min_x \quad 0\\ \text{such that}\quad A_\text{eq} x = b_\text{eq}, x_i \in \{0,1\} $ Where also $[A_\text{eq}]_{ij} \in \{0,1\} $ and ...
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Property of monotone operator (Positive definite)

I would like to prove this statement: "$F$ is monotone if and only if $\nabla F$ is positive semidefinte." I only know $F$ is monotone with respect to $\Omega$ if and only if ...
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why in Phase I of the simplex method, if artificial variable become nonbasic, it never become basic?

Does anybody has idea how to solve this problem ? "Show that in Phase I of the simplex method, if an articial variable becomes nonbasic, it need never again become basic. Thus, when an articial ...
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1answer
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How to show these two problems have equivalent solutions

I have two problems, where $A$ is positive definite: $$\inf \{ x^TAx + b^Tx : 1-x^Tx \ge 0,\ x \in \mathbb R^n\} \ (1)$$ and $$ max_\lambda \ q(\lambda) = -0.25b^T(A+\lambda I)^{-1}b - \lambda : ...
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1answer
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How can I show that these two problems have the same optimal solution?

How can I show that these two problems have the same optimal solution: $$\inf \{ x^TAx + b^Tx : 1-x^Tx \ge 0,\ x \in \mathbb R^n\}$$ $$\inf \{ x^TAx + b^Tx : 1-x^Tx = 0,\ x \in \mathbb R^n\}$$ when ...
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1answer
526 views

Notation: what is “arg min”

there is a function that says j=arg_min{ f(x),g(y) } What does that mean? As noted by the comment, arg_min(f(x)) is the x that gives smalles f. But what happens when arg_min takes two function, ...
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Discretization of a convolution integral for constrained optimization problem

I'm working on a constrained optimization problem in which an unknown forcing function, $u(\eta)$, is in the integrand of a convolution integral. To find an optimal shape for $u(\eta)$, the integral ...
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Optimization problem with embedded absolute values (how to turn to LP)

say I have a problem of the form $$\begin{align*}\min&\sum_i{c_i\,|x_i-|y_i-z_i|+|s_i|\,|}\\[0.3cm] \text{s.t. }&0\le x\le 1\\[0.2cm] &0\le y \le 1\\[0.2cm] &0\le z\le 1\\[0.2cm] ...
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478 views

How to find the minimum/maximum distance of a point from elipse

I have the point $(1,-1)$ and the ellipse $$x^2/9 + y^2/5 = 1 $$ How to find the minimum and maximum distance of the point from the ellipse ? from exploring the ellipse I know that $$a = 3$$ , $$b ...
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Given a fixed volume for a solid cylinder, is it possible to find the minimum or maximum curved surface area?

Given a fixed volume for a solid cylinder, is it possible to find the minimum or maximum curved surface area $ 2 \pi r h $ of this cylinder?