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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Minimize $\sum_i \arccos(v_i\cdot x)^2$ subject to the constraints $\|v_i\|=1$ and $\|x\|=1$?

Some background: (skip to the end for the actual question) Recently I have been trying to define some notion of an average of points on the surface of a sphere. My original idea was to ignore the ...
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85 views

Optimal Configuration for a Set of Points

Consider a set of $n$ points on the plane with positions $\mathbf{p}_1,\dots,\mathbf{p}_n$, such that each point $i$ has at least one neighbor $j$ at a distance of no more than $\lambda$ away from it ...
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120 views

A maximization problem in Sobolev space

For $k>0$, let $f_k$ be a sequence of positive functions in $H_N^1(0,1)$, where $H_N^1(0,1):=\{u\in H^1(0,1)|u^{'}(0)=0=u^{'}(1)\}$, $H^1(0,1)$ is the usual Sobolev space consisting of $L^2(0,1)$ ...
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59 views

What is an approach for optimizing the values of a matrix?

My apologies if I get some terminology wrong, I don't have a formal math background; half my problem is articulating what I'm trying to do and identifying the domain of math that deals with this kind ...
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500 views

Why is lasso not strictly convex

I know a nonmonotonic convex function which attains its minimum value at a unique point only is strictly convex. I didn't get how lasso is not strictly convex. For eg if I consider two dimensional ...
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74 views

Finding maximum of $ad-bc$ on $S^3$

One of my friends asked me to find the maximum of $ad-bc$ given that $a^2+b^2+c^2+d^2=1$ and $a, b, c, d \in \mathbb{R}$. I came up with the following. Can somebody please tell me if it is a ...
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160 views

Optimization for constrained problem

I'm reading about Lagrange multipliers from a Pattern recognition book appendix and on one point the following is stated: $\begin{align} ...
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167 views

Relationship between lagrange multiplier and constraint

I know there is one to one relationship between $\lambda$ and $t$ in the following two equivalent optimization formulation. But what is exact relationship? A) $$ \sum_i(y_i - \sum_k \beta_k ...
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62 views

Minimisation of Concave Function

$f\left(\mathbf{x}\right):\mathbb{R}_+^n\rightarrow\mathbb{R}_+$ is a concave monotonically increasing function to be minimised over the feasible region $\sum_{i=1}^n x_i=1$ and $x_i\geq ...
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1answer
496 views

finding maximum perimeter of a triangle

So, here we are given task to find maximum perimeter of a triangle with a given base 'a' and given vertical angle 'x' , now how should I proceed in given problem its confusing me Now supposing ...
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3answers
120 views

Solution to a scarce resources assignment game

I would like to tell you about this game, which can looks like very simple but there's a constraint which complicates it and prevents me from finding an analytical solution. Rules of the game: ...
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4k views

Solving a set of equations with Newton-Raphson

I want to solve this set of equations with Newton-Raphson. Can anybody help me? $$ \cos(x_1)+\cos(x_2)+\cos(x_3)= \frac{3}{5} $$ $$ \cos(3x_1)+\cos(3x_2)+\cos(3x_3)=0 $$ $$ ...
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621 views

How to justify the solution of this problem?

Assume $\mathbf{x} \in \mathbb R_+^N$ with support $P=\{p_1,p_2,\cdots,p_K\}$ ($P$ is unknown). We already know that $$f_1(\mathbf{x}) = f_2(\mathbf{x}) = \cdots = f_{N-1}(\mathbf{x})$$ where ...
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1answer
100 views

Showing that the function $f(x,y)=x+y-ye^x$ is non-negative in the region $x+y≤1,x≥0,y≥0$

ok, since it's been so long when I took Calculus, I just wanna make sure I'm not doing anything wrong here. Given $f:\mathbb{R}^2\rightarrow \mathbb{R}$ defined as $f(x,y)=x+y-ye^x$. I would like to ...
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334 views

Minimize Rosenbrock Function With Conjugate Gradient Method

I want to minimize $$ f(x,y) = (1-x)^2 + 100(y-x^2)^2 $$ For the conjugate gradient method I need the quadratic form $$ f(\mathbf{x}) = \frac{1}{2}\mathbf{x}^{\text{T}}\mathbf{A}\mathbf{x} - ...
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57 views

maximum of derivatives of lipschitz functions

Say $f$, and $g$ are nondecreasing functions on [a,b], differentiable with derivatives bounded by 1. can one infer that $$\int_a^b \min\left(\frac{d}{dx} f(x),\frac{d}{dx} g(x)\right)\,dx \leq ...
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258 views

Proof of a method to find the points of maximum slope

According to method described in a paper [1] if we want to find points of maximum slope in a signal $f(t)$, then one has to do following Convolve $f(t)$ with $g(t)$ where $g(t)=-cos(\omega ...
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1answer
144 views

The local minimum of the SQP (sequential quadratic programming) algorithm

Consider the constrained optimization problem \begin{eqnarray} goal~~&&\min f(x)\\ s.t.~~&&g_1(x)\leq0\\ &&g_2(x)\leq0\\ &&\cdots\\ &&g_n(x)\leq0 ...
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54 views

Is there an easy solution to this constrained discrete minimisation?

Given $\vec{a}$, $\vec{b}$, and $c$ I want to find a discrete combination $\vec{n}$ (i.e. a vector with non-negative integer elements) to $$\mathrm{minimise}\left(\vec{n}\cdot\vec{a}\right)$$ Under ...
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32 views

Minimizing an unknown system's output

Let we have an unknown system with two inputs and two outputs. inputs $x=[x_1 x_2]$ and outputs $y=[y_1 y_2 ]$ The system have the following properties $ y_1 = f_1(x_1,x_2)$ ; $y_2 = f_2(x_1,x_2) $ ...
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62 views

getting sign of LP solution variables

I have an LP where I'm only interested in the sign of some of the variables of an optimal solution. The value itself does not matter. Currently I'm using cplex to get an optimal solution and take the ...
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435 views

Linear programming simplex - can I have a constraint with a multiplication?

I'm not sure of this, can I have a constraint like this in a linear programming problem to be solved with simplex algorithm? $$n_1t_1 + n_2t_2 > 200$$ where $n_1$ and $t_1$, $n_2$ and $t_2$ are ...
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55 views

A basic question on unconstrained optimization

I am going through an introductory textbook on optimization where the following is said : "Optimization within a subspace or linear variety can often be reformulated as unconstrained optimization, ...
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71 views

On impulsive optimal control with functions of not bounded variation

I have the following optimal control problem $$ J=\int_0^TF(t,y_1(t),y_2(t))dt \to \min, $$ subject to \begin{align} &\dot y_1(t) = f(t,y_1(t),y_2(t)) + g(t)\nu(t),\\ &\dot y_2(t) = ...
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77 views

How would I answer this question? (Reworded)

George wants your help to work out how many of each type he should stock in order to maximise his profit. There are three types of Snackboxes: A, B and C. A and C both cost 5 to produce, and B cost 7 ...
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387 views

Making non-singular matrices singular

What is the minimum value of $k$ such that every non-singular $n\times n$ real matrices can be made singular by switching EXACTLY $k$ entries with ZERO ?
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1answer
167 views

Apostol's Calculus: optimize the perimeter of isosceles triangle inscribed in circle

As much as I hate to ask for a hint on this, I've gotta admit--I'm stuck. None of my previous attempts to solve this were successful, and I can't think of a fresh way to look at the problem. This is ...
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362 views

What is the global maximum of $x^{1/x}$

Let the following function be defined as such: $$F_x: \Bbb R \to \Bbb C, x \mapsto x^{1/x}, \forall x \ne 0$$ What I want to know is $$\max_{x<0}\Re\left(F_x\right)=\,?$$ and ...
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258 views

Minimize the sum of distance under maximum norm

Given a set of points (Xi, Yi). I need to find a point (doesn't have to be in the given set) that minimize the sum of distance to the other points. The tricky part is the distance is measured by ...
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63 views

A basic question on gradient

I am not understanding why gradient will show the direction in which the function value rises most quickly. It is just the vector of partial derivatives of the function. And why its magnitude will ...
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137 views

Maximum and minimum function $f(x,y)=x$ on $\left\{ (x,y): x^4+y^4 = 4xy \right\} $

Determine minimum and maximum of function $f(x,y)=x$ on $\left\{ (x,y): x^4+y^4 = 4xy \right\} $. I used Lagrange multiplier and received that $f(3^{3/8},3^{1/8}) = 3^{3/8}$ is maximum ...
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Linear programming with countably “infinite variables” and “finite constraints”!

Is it possible to do a linear programming with countably "infinite variables" and "finite constraints"? If not, what do you purpose? (Example Link): Maximum and minimum of an integral under integral ...
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90 views

Summation notation [duplicate]

Does anyone have any suggestions as to how I would be able to formulate this problem using summation notation for those of you who are familiar with it? Hermione has been busy packing her bag with ...
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1answer
443 views

Maximum of product of numbers when the sum is fixed

This is the problem I'm working on. $$\begin{array}{rl} \text{maximize} & (n+\ell+x_1)\cdots (n+\ell+x_{k-1})(\ell + x_k) \\ \text{subject to} & 0 \leq x_1, ..., x_k \leq n \\ & x_1 + ...
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1k views

“Box With No Top” Optimization

I am having some trouble with this problem, A box with no top is to be constructed from a piece of cardboard of dimensions $A$ by $B$ by cutting out squares of length $h$ from the corners and ...
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112 views

Operations research - summation notation [duplicate]

Outline: Hermione has been thinking about the imminent return of the Dark Lord, so she has been busy packing her bag with all the items required for her survival. Because she has so many different ...
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1answer
405 views

How to see that K-means objective is convex?

I'm trying to proof that the objective of the K-means clustering algorithm is non-convex. The objective is given as $J(U,Z) = \|X-UZ\|_F^2$, with $X \in\mathbb{R}^{m\times n}, U\in ...
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226 views

Optimization of three variables with a constraint

I have a question puzzling me for a while. I tried using Lagrange multipliers, however it began to get messy as well as the fact that i am new to the method of Lagrange multipliers! The question is ...
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402 views

Is Positive Semidefinite matrix Same as Positive Number in Convex Optimisation?

Consider the optimisation problem expressed in a crude form $\max_{\mathbf{Q}}\sum w_ir_i$ where $w_i$ are constants, $r_i$ are concave functions of positive semidefinite matrix $\mathbf{Q}$ ...
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509 views

Find the point on graph of $xy=12$ that is closest to the point $(5,0)$

This is from a Derivatives chapter in the section on Optimization. Find the point on graph of $xy=12$ that is closest to the point $(5,0)$ I believe I have to use the distance formula. So, so ...
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Joint discrete and continuous optimization problem

What is an optimization problem which involves joint discrete and continuous variables called? I hope here be a good place to ask this. If it is not, please tell me where to ask it.
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Chemical reactions and solutions of a constrained optimisation problem

I have to find a solution for this problem: given $N$ materials of density $\rho_k$, find the mixture of them giving a compound of density $\rho$. From a mathematical point of view, we have to find ...
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optimal road layout problem - how to convert to maths and see the shapes it makes?

I have this puzzle going round my head about optimal road layouts, but I'm a programmer not a mathematician and I don't really know how to specify it as a maths problem. Once it's well-specified I can ...
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extremal points on a manifold intrinsiclly

I am wondering if there is a geometric object for real analytic manifolds that characterizes extremal points of the manifold intrinsically. For instance, suppose I live in the manifold, can I ...
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67 views

Correlated Equilibrium

I have a question about the definition of the correlated equilibrium. I see that some authors define it as "expected payoff of playing the recommended strategy is no less than playing another ...
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31 views

Optimization problem interpretation

I posted a question in http://math.stackexchange.com/ and got a solution. But the solution is a bit hard for me to understand. The actual question is here : minimizing $\sum_{i=1}^n \max(|x_i - x|, ...
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Minimization values for a function [duplicate]

I have got function - non linear(I thk), and a set of variable S=[(x1,y1),(x2,y2)...]. The objective is to find the value for ...
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1answer
52 views

Approaches to fitting noisy oscillatory data?

I have observations $\hat{f}$ from data at points $\mathbf{x}=\{x_1,\ldots,x_N\}$, that is modeled as a known oscillatory form $f(k\ x)$ (for example, the sinc function), where $k$ controls the ...
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Which optimization algorithm converges faster?

everyone. I'm having a large scale unconstrained optimization problem. If I treat the unconstrained problem as a constrained problem with infinity constraints, I should be able to use both the ...
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46 views

Directive on Dimensionality Reduction

I have a data set (24 data records) which is in $\mathbb{R}^{13}$ and I need to project it to a lower dimension (at least to $\mathbb{R}^{3}$). My objective of the dimensionality reduction is to ...