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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Maximum Non-Crossing Connections (MNCC)

Playing around with some connections between nodes, I came up with a question that ask; "What is the maximum unique (non-crossing)connections that can be made to a structure of nodes?" So I set up a ...
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Finding optimal hyperplane

I have a set of vectors $\{V_i\}$ in $n$-dimensional space. There is a number corresponded to each vector $\alpha_i = f(V_i)$ ($\alpha_i$ can be negative). I want to find a hyperplane which would ...
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Matrix optimization

I'm trying to minimize over $U$ the objective $\|X^{\top}UU^{\top}UU^{\top}X\|_F^2 = \text{trace}(X^{\top}UU^{\top}UU^{\top}XX^{\top}UU^{\top}UU^{\top}X)$ subject to $U^{\top}U = I$, where $X \in ...
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Lagrangian method gives the wrong answer in a standard constrained optimization problem

I had this strange problem where the Lagrangian method gives the wrong answer in a constrained optimization problem. Here goes: The problem is $$\max_{c,n,q} \alpha\log(c)+(1-\alpha)\log(nq)$$ ...
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Is $argmin_{\mathbf{x}} f(\mathbf{x})=argmin_{\mathbf{x}} \log{f(\mathbf{x})}$ always true?

Assuming $\mathbf{x}\in \mathbb{R}^n$, $f(\mathbf{x})\gt0 \forall\mathbf{x}\in\mathbb{R}^n$, is $argmin_{\mathbf{x}} f(\mathbf{x})=argmin_{\mathbf{x}} \log{f(\mathbf{x})}$ always true? Why?
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Minimum of $|\det(X+iC)|$

Let $C$ be a fixed real $n\times n$ matrix, $X$ be an arbitrary real $n\times n$ matrix. Find the minimum value of: $$|\det(X+iC)|=\sqrt{\det(X+iC)\det(X-iC)}$$ When $n=1$ it's clear that the ...
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How to determine if a convex polytope is contained in a union of convex polytopes?

Given that we are in a Euclidean space of dimension d, that we have a bounded convex H-defined polytope P, and N possibly unbounded convex H-defined polytopes, I am looking for an "efficient" ...
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1answer
23 views

How to computer the Lagrange multipliers associated with an optimal solution

Suppose I have a solution $x^*\in\mathbb{R}^n$ to the following problem \begin{align*} \text{minimize}_{x}& \sum_{i=1}^n f_i(x)\\ \text{subject to}\quad &g_i(x) = 0\,\,i=1,\ldots,m\\ ...
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The equality of gradient between different calculations?

Suppose there is a problem $$\min\limits_v\max\limits_x E(v,x).$$ $E$ is a concave function w.r.t. $x$. But w.r.t. $v$, $E$ is a convex function plus a concave function. I can get ...
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19 views

Minimum in complex inner product vector space

I'm stuck at this problem, can someone give me a hint? Let $x_i$ and $y_i$ ($i=\overline{1,n}$) be vectors in an infinite dimensional vector space $V$ with inner product $(,)$ satisfy: ...
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Book recommendations for Binary Integer Linear Programming

I'm looking for a book on BILP, which focuses on algorithms / solutions methods. So far, I only found the following books on ILP "Integer and combinatorial optimization" by Nemhauser, George L. ...
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46 views

MaxMin: how much does the min “see”?

Consider the following quantity: $$ \max_{a \in \{-1,1\}}\left( \min_{b \in \{-1,0,1\}} ab\right).$$ Since the min is inside, we apply it first, but what value $b$ will be chosen? If the minimum ...
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30 views

Need help with optimization concepts.

In some optimization problems with inequality constraints some of the aforementioned constraints can be x>=0 , y>=0 and so on. I think these constraints are called non negativity constraints; they ...
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46 views

New variable in a convex optimization problem

Consider the convex optimization program $$ \min_{x \in X } x^\top P x + p^\top x \quad \text{ sub. to: } Ax = b $$ where $X \subset \mathbb{R}^n$ is compact, $P \succ 0$, $A \in \mathbb{R}^{m \times ...
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59 views

Finding maximum value of function [on hold]

If the composite function $(f_1 \circ f_2 \circ \cdots \circ f_n)(x) = f_1(f_2(f_3(...(f_n(x))))$ is an increasing function and if $r$ of the functions $f_i$ are decreasing functions while the rest ...
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Connected graph where edge costs depend on a parameter $t$. Find the $t^*$ which gives the minimum cost minimum spanning tree.

The set-up: Let $G=(\,V,\,E\,)$ be a connected graph. Associated with every edge $e\in E$ is a cost/weight function $f_e(t) = a_e t^2 + b_e t + c_e $, where $a_e>0$. For a fixed $t$ we can define ...
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39 views

Need help with the graph of a function

In the optimization problem max: $$6x+2xy-2x^2-2y^2$$ subject to $x+2y\le2$ and $-x+y^2\le1$ I need to draw the graph of the feasible region in order to determine if the problem has global solutions, ...
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40 views

Local minimum and gradient [duplicate]

But the proof here below is specially elegant. Is there any function $f$ such that $f$ has a local minimum at $x$ but $\nabla f(x) \neq 0$? Only assumption on $f$ is that it has to be differentiable ...
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Sum of squares of series of boolean variables

I am going to simplify the following series: $$\sum^4_{v=1} \left(1 - \sum^4_{i=1} x_{v,i}\right)^2 + \sum^4_{i=1} \left(1 - \sum^4_{v=1} x_{v,i}\right)^2$$ Since $x_{i,j}$ is a boolean variable, ...
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26 views

How do I know if I have imaginary numbers when using Newton Raphson Method?

I am studying Newton-Raphson Method but I am facing questions in my head. As far as I know Newton Raphson Method works on real values, but what if Newton Raphson Method faces an imaginary number when ...
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Imaginary roots and Real values when using Newton-Raphson Values

I am studying Newton-Raphson Method but I am facing questions in my head. How do I know if I have an imaginary number or imaginary numbers? and What to do when I have them when using Newton Raphson ...
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34 views

Shortest path problem: dual formulation and proof of total unimodularity

The IP formulation of the shortest path problem looks as follows: \begin{align*} \min & \sum_{u,v \in A} c_{uv} x_{uv}\\ \text{s.t } & \sum_{v \in V^{+}(s)} x_{sv} - \sum_{v \in ...
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Sums of positive and negative distances to the least squares plane

Let $A_{1}, A_{2}, \ldots, A_{n}$ be points in $\mathbb{R}^{3}$ and $\pi_{*}$ be the least squares plane, i. e. $$ \sum \limits_{i = 1}^{n}\rho^{2}(A_{i}, \pi_{*}) = \min_{\pi}\sum \limits_{i = ...
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71 views

Optimize $x^2 + y^2 +2z^2 +z(x^2-y^2)$ subject to $x+y=2$

$$x^{2}+y^{2}+2z^{2}+zx^{2}-zy^{2}\overset{\left(x=2-y\right)}{\longrightarrow}4-4y+2y^{2}+2z^{2}+4z-4yz\rightarrow FOC: \; \begin{cases} -4+4y-4z=0\\ 4z+4-4y=0 \end{cases}\rightarrow y=1+z\rightarrow ...
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1answer
47 views

Find max and min subject to constraint ||x|| = 4

$Q(x,y)=7x^{2}+12xy+12y^{2}$ I only know how to do this is $\|(x,y)\|=1$ If $\|(x,y)\|=1$, the eigenvalues are $16$ and $3$. So obviously $\min=3,\max=16$. I don't know what to do if ...
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42 views

How do I know if a function has x roots on x-axis?

I am currently studying Newton Raphson Method. Now I am kind of having a question that how I know if the function ever has a x-root or roots on x-axis? Please let me hear your advice. I am sorry if I ...
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An error in least square optimization problem in Matlab

I am new to MATLAB and I want to formulate the following lease square expression in Matlab. I have some codes that I am typing here. But the optimization problem solution seems not to be correct. Does ...
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Minimization problem with amplitude constraint

I have the following minimization problem: $$\left\| \bf{A}x - y\right\|^2 \to min $$ $$s.t. \left|x_i\right| < 1, \forall i,$$ where $\bf{A}$ is the complex matrix with size of $(n\times m)$, ...
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optimization problem with integrals

There is a maximization problem of the following form \begin{equation} \max_{l(a)} \sum \int \bigg(U(c, 1-l(a)) \bigg) x(a,e) da \end{equation} where $$ c = a(1+ f(L)) + e G(L)l(a) - h $$ $$ L = ...
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How to find the global minimum or maximum of a data set

From some experiment, I am getting noisy data. I am interested in highest maximum value from data. Somehow data is periodic and I want to get the highest maximum value from first period. I am quite ...
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Help me out with this optimization problem

This excercise has been taken from an exam. In the following problem: opt:x+y^2-2 subject to y^2<=x and x<=2-y and y>=0 I've found the green area to be the feasible region. (Sorry for the ...
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Example of a math problem that is not an optimization problem [closed]

Is there a problem in mathematics that cannot be formulated as an optimization problem? If so, any example? Thanks Edit: To clarify the point and spirit of the question. The question is meant to ask ...
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Norman Window Optimization

A Norman window has the shape of a rectangle surmounted by a semicircle. Find the dimensions of a Norman window of perimeter 24 ft that will admit the greatest possible amount of light. I know that I ...
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Two-way matrix optimization

I have run into a problem like this. Looks a bit unusual, but I think should be doable. Find $U$ achieving $$\min_U \left( \| A - UW \|_2^2 + \| RU - H \|_2^2 \right)$$ $A,U,W,R,H$ are all ...
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explicit function between transformation matrix and vertex in polyhedron

recently I am stuck in solving a geometric problem. I hope someone could give me some tips, thanks for all in advance!!! Question 1: given a constant polygon $M1$ with 4 vertices: ...
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1answer
41 views

Need help with second derivative test

In an optimization problem with restrictions, when I have already found the critical points of a function and I have to classify those points (they can either be maxima or minima or saddle points), do ...
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First and Second Order Necessary Conditions [closed]

1)Let Ω = {x belong to R : Ax=b} and min cT x .Find all points satisfying in the First and Second Order Necessary Conditions. 2)Let f(x) = cT x. Show that if c ≠ 0, then we cannot have an optimal ...
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Quadratic Optimization Problem with Box Constraints

I want to solve a problem of form $$\min_x x'Ax + b'x \;\;\mbox{ s.t. } l\leq x \leq u$$ where $A$ is a positive semidefinite matrix, thus the function I'm optimizing should be convex. However the ...
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Differentiation of cost function in adaptive CFO estimator

I'me trying to simulate the steepest descent algorithm for CFO estimation using null subcarriers (OFDM wireless). And some mathematic difficulties have arised. In the core of algorithm lies cost ...
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Needing help with this problem

can anybody help me out with this? opt: $x^2+y^2$ subject to $(x-1)^2-y^2=0$ I couldn't even find the critical points.
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Optimization Calculus Problem- Flight

If exactly 230 people sign up for a charter flight, the operators of a charter airline charge Dollars 330 for a round-trip ticket. However, if more than 230 people sign up for the flight, then fare is ...
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Rectangular Box Optimization Problem

A rectangular box is to have a square base and a volume of 40 ft3. If the material for the base costs \$0.31 per square foot, the material for the sides costs $0.05 per square foot, and the material ...
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Optimizing the distance [duplicate]

A painting is mounted on a wall. The bottom of the painting is 5 feet above eye level, and the top of the painting is 14 feet above eye level. If you stand directly underneath the painting, you cannot ...
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1answer
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Convex hulls for a finite amount of points

I'm trying to understand what a convex hull intuitively is, and say given for a set of points $(x,y)\in\mathbb{R}^2$ how is it generated from these points? I tried reading the wikipedia article and ...
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Minimizing Question

A closed box constructed from a tin sheet has a square base and a volume of $343 \text{in}^3$. Find the dimensions of the box, assuming the minimum amount of material was used in its construction. ...
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23 views

Select machines to minimise latencies between them

I am working in an optimisation problem. I am still trying to model it and solve it. The problem is: There is a number of different types of virtual machines. Each type has different hourly cost ...
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Optimize volume of an open cardboard box made from flat square of cardboard…

Consider the following problem: A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the sides. ...
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113 views

Find maximum of $P$

Let $$P = \frac{{{x^2}}}{{{x^2} + yz + x + 1}} + \frac{{y + z}}{{x + y + z + 1}} - \frac{{1 + yz}}{9}.$$ Find maximum of $P$ where $x, y,z$ are nonnegative real numbers such that ${x^2} + {y^2} + ...
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50 views

Finding the maximum and minimum

Can't understand how to find the maximum and minimum with the given definitions (with both x and y).. can someone explain step by step?
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A simple optimization problem of reciprocal function

Can someone tell me the answer to this question? I cannot seem to figure it out The function $y=\frac{2}{x}$ is decreasing in?? a.$(0,\infty)$ b.$(-\infty,0)$ c.$(0,2)$ d,$(-\infty,\infty)$ I ...