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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Max - min problem of a quotient of norms

For the $2\times2$ matrix $\begin{bmatrix}4&0\\-3&-5\end{bmatrix}$ Part 1 Find nonzero vectors $u$ and $w$ that maximize and minimize respectively the quotient $||Av|| / ||v||$. Part 2 ...
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Optimization problem with an added quadratic inequality constraint

Consider the following (non-convex) optimization problem on the real variables $\lambda_\ell^\pm$ with $\ell=1,\ldots,n$ \begin{align} \mbox{maximize}&\quad \lambda_{1}^+-\lambda_{1}^--2\sum_{\...
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How to solve sum of sines and cosines system of equations?

I have a set of equations to solve which in the following form: $ \cos(t_1 + t_2 + t_3 + t_4) + \sin(t_1 + t_2 - t_3 + t_4) + \cos(t_1 - t_4 + t_3 - t_5) + \sin(t_1 - t_2 + t_3 - t_5) + \cos(t_1 + ...
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190 views

Scale ellipsoid maximally within polyhedron

Given an ellipsoid around the origin with scaling parameter $e$ in the form $x^T E x \leq e$ and a polyhedron $P$ given by $A x \leq b$, how can we define an optimization problem that maximizes e such ...
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How to interpolate a function with a reproducing kernel

I am trying to interpolate a function that is noisy, but I know with a high amount of certainty about a third of the points in the series. I am trying to estimate the smooth mean of the signal via a ...
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123 views

Minimum Distance between a Triangle and a Distance Field 3D

I am looking for (possibly numerical) solution to this geometric problem: Given a filled 3D triangle $T = \text{conv}(p_1, p_2, p_3) \subseteq R^3$, and a distance field $D(x) : R^3 \to R$, what ...
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Counting the number of solutions to an equation.

In my work I am dealing with an equation that will have 1 or more solutions. Specifically I am trying to find local maximums. I am not interested in the solutions to the equation, but I am instead ...
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57 views

Combining multiple linear programming to minimize the sum

I have a math problem that looks like a bunch of linear programming problem combined where A matrix is shared. Here is the math definition of my problem Minimize \begin{align} & p_1 (x_{11} + ...
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34 views

Maximizing the sum of nonnegative functions.

I was trying to solve the problem A maximization problem when I ask myself if the general problem \begin{equation} \begin{array}{c} maximize\hspace{1cm} f(\mathbf{X})^p +g(\mathbf{X})^p \\ s.t. \...
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35 views

Does point's neighborhood have no local extremum?

I have polynomial of some limited degree: $f(x,y) = a_1 + a_2x + a_3y + a_4xy + a_5x^2 + a_6y^2 + a_7x^2y + \ldots$ There is a point $p_0=(x_0,y_0)$, which is NOT a local extreme NOR inflection ...
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Complexity of polynomial simplification into standard form

I am curious to know if any given $n$-variable polynomial in $\mathbb{R}[\mathbf{x}]$, not in standard form, can be simplified by an algorithm in polynomial time. The polynomial is $$ p(\mathbf{x}) = \...
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When is $D \max G = \max D G$?

All matrices are real. The operator $\max$ on matrices returns the largest value in each row. We are interested in characterizing the set of matrices $D$ of size $n \times m$, $m < n$ such that we ...
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56 views

Parts of the whole: Which base begets the largest percentage of fractions expressible as a terminating decimal?

Update: It appears the question I actually meant to ask was quite different. As Robert Israel explained in his answer I was calculating the wrong thing. After writing some ugly code (may take a sec to ...
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58 views

Optimization problem with integral

I have here quite a easy optimization problem, however I can't figure out how to solve it. Given a definite integral from a to b. I need to find values a and b such that the value of the integral is ...
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559 views

Optimisation problem - circle and square

A piece of wire of length $20$cm is cut into $2$ parts. the first part is bent into a circle of radius $r$ in cm, the second into a square of side length $s$ in cm. a) write down an expression for ...
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A maximization problem

I'm trying to find the maximum value of the function $f(x,y)=(ax+by)^p+x^p$ subject to the constraint $x^p+y^p=1$. Here, $a,b$ and $p$ are constants with $a,b>0$ and $p>1$, and $x,y>0$. I ...
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116 views

Given a set of 2D vertices, how to create a minimum-area polygon which contains all the given vertices?

Not sure whether this question belongs here or mathoverflow. You can assume that all the vertices are unique. The given vertices can be the vertices of the polygon, thus they do NOT have to be inside ...
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66 views

Maximizing expected value when distribution is binomial

Consider the following problem: $$\max_{n\in\mathbb N}\;f(n)= \frac12 \left[v_0 \sum_{i=\lceil k_n \rceil}^n \binom{n}{i}p^i (1-p)^{n-i} + v_1\sum_{i=1}^{\lfloor k_n \rfloor}\binom{n}{i}q^i(1-q)^{n-i}...
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49 views

Why is $\max\limits_b \|b\|_2 = \max\limits_b \|b\|_2^2$?

I'm currently reading up some optimization theory. I've come across this equation (subject to some constraints): $$\max_b \|b\|_2 = \max_b \|b\|_2^2$$ In this case, it is ensured that $\|b\| > ...
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27 views

Does the terms 'LP primal' and 'LP dual' usually refer to any primal/dual, or just the optimal primal/dual pair

As the title says, I'm wondering whether the terms LP primal and LP dual usually refers to any primal/dual pair of an LP (feasible or not), or just the optimal primal/dual pair. The reason that I'm ...
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178 views

Is $\text{Trace}(e^{XA+A^TX})$ a convex function of X?

Is $\text{Trace}(e^{XA+A^TX})$ a convex function of $X$? $X$ is diagonal and positive definite, $A$ is symmetric negative definite definite. And by the way, what is the best way to solve a problem of ...
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Finding Extremas of $|x|$.

I'm trying to find the extrema of$\mod(x)$ but I'm not being able to do so. My attempt: $f(x, y) = |x|$ $f_{xx} = 0, f_{yy} = 0, f_{xy} = 0.$ So, $D(x, y) = 0$. And second derivative test isn't ...
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Partial derivative with matrices

I have reforumulated my problem of computing some quantities $\mathbf{a}\in R^{m}$ from $\mathbf{b}\in R^{n}$ in a matricial form: $$\mathbf{b} = (C\odot(\mathbf{1}_{n}\cdot \mathbf{a}^{T}))\cdot \...
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Equality case in elementary form of Holder's Inequality

A well known elementary formulation of Holder's Inequality can be stated as follows: Let $a_{ij}$ for $i = 1, 2, \dots, k; j = 1, 2, \dots, n$ be positive real numbers, and let $p_1, p_2, \dots, p_k$ ...
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61 views

How do you compare carsharing plans to calculate the cheapest?

Call hourly rate = HR. Assume that I can guess my monthly usage in hours, which I call $g$. Beware that the fixed fees are presented in different units of time, so first convert everything into months,...
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Optimization 101 for electrical engineers…Where to start from?

I have never taken any optimization class. From an electrical engineering point of view, how should I approach learning this field? What kind of information I should be looking at in my problem to ...
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linear programming slater condition

I am wondering if anyone could help to come up with a such example: ...
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Get the minimum value for multivariable

I need a way to calculate the very minimum of each variable for a grade average. Each grade variable have a weighing percentage (I don't know if it's the right term) The weighing sum must be the ...
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The cylinder of maximum volume that can be drilled from a sphere of given radius [closed]

Find the dimensions of the cylinder whose maximum volume can be drilled from a steel sphere of radius 8. As Amanda Kelius said, my problem is how to setup the expression for the volume. I know that ...
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Trouble with formulation of objective function (constraint optimization)

I am new to optimization and I will try to state my question as clear as I can. I need to solve this constraint optimization problem. I want to find real vectors $\mathbf{f}$ and $\mathbf{g}$ that ...
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Nested optimization problems solving using mixed integer linear programming

Let us have two vectors of decision variables, $\mathbf{x}$ and $\mathbf{y}$, two linear objective functions, $F \left( \mathbf{x}, \mathbf{y} \right)$ and $f \left( \mathbf{x} \right)$, and two sets ...
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Conditional inequalities

Let a,b,c be positive real numbers such that $abc=1$. Prove that $$\frac 1{a^3(b+c)}+\frac 1{b^3(c+a)}+\frac 1{c^3(a+b)} \ge \frac 32$$ We can derive the following inequalities from the given equality ...
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Gauss-Newton versus gradient descent

I would like to ask first if the second order gradient descent method is the same as the Gauss-Newton method. There is something I didn't understand. I read that with the Newton's method the step we ...
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Add vectors from a set to reach the goal vector, using the minimum possible cost

I am trying to solve a problem in an optimal way. The problem is as follows: We have an n-dimensional space In this space, we have a "finish" point with n coordinates, all non-negative We have a set ...
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Given 500 parts and a list of orders, pick 50 parts to maximize the number of fulfillable orders

I'm going to start with a proclamation that this kind of optimization is new to me, so don't fault me for setting up the problem in a weird way. Please let me know if this is unclear. In a ...
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Job scheduling to minimise squared completion times using mixed 0-1 quadratic program

I have come across an Optimization question as follows: There are $n$ jobs that have to be processed on a machine. The machine can process only one job at a time. The time taken to process job $i$...
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Maximum time-to-exit of random walk in R^n

I am trying to solve the following problem : Given a set $A$ in $\mathbb{R}^n$ and a point $p$ , I want to find a convex subset of $A$, call it $C$, such that $p$ is in $C$ and random walk starting at ...
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399 views

IF a cone is inscribed in a larger cone,then what will be the radius of the small cone if it has the maximum volume?

If a smaller cone is inscribed in a larger cone as shown, then what will be the radius of the smaller cone if it has the maximum volume? Attempt I know that the volume of a cone =$\dfrac{1}{3}\pi ...
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Non-decreasing Convex function?

When my textbook states, "Non Decreasing Convex Function", does it mean that the function is convex and increases in y for every x from its minimum? That is if f(x) = y is convex. Please explain if ...
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Strong duality for nonconvex quadratic program (with multiple constraints)

Consider the following optimization \begin{eqnarray} P_1: \quad &\underset{x\in\mathbb{C}^N}{\mathrm{minimize}}&\; f_0(x) \\ &\mathrm{subject\;to}&\; f_i(x) \leq 0, i=1,\ldots,m \\ &...
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solving LP problem : no optimal solution exists?

$$\max[Z(x,y)=3x+2y]$$ $$-x+y\le 1$$ $$-x+2y\le4$$ When I tried to solve the above maximization LP problem using the simplex method, from the first iteration, all basic variables became negative. ...
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SOS relaxations for polynomial optimization

I do not understand how SOS (Sum-Of-Squares) relaxation for polynomial optimization works in some cases. For instance, consider the polynomial optimization problem: \begin{equation} \begin{array}{c}...
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Structural / design / meta optimization - is there mathematical theory. Optimization over categories?

There is huge branch of mathematical optimization theory, but it mostly considers the finding optimal parameter values for the predefined structures. There are variational calculus and optimal control ...
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How to understand ' Let $\mathcal{H}$ be a Hilbert space of functions $f$ : $ \mathcal{X} \rightarrow R$, denoted on a non-empty set $\mathcal{X}$.'

I am a beginner. By asking this question, I means that, to construct a Hilbert space, should $\mathcal{X}$ satisfy some properties? Furthermore, in some papers especially on machine learning, ...
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Global optimality of a convex but non-smooth function

I have a question. The answer may be too obvious but I cannot be sure about the right answer. Let say that we have a convex but non-smooth function which is defined as $f : \mathbb R^2 → \mathbb R$. ...
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How to reconstruct a sparsely sampled multiperiodic function?

I have $m$ oscillators, where $m$ is unknown, with periods $\vec p = p_1, p_2, \ldots, p_{m}$. Each of the oscillators $j$ has associated with it a vector of sine coefficients $\vec A_j$ and angle ...
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How to solve an optimization problem where the size of the solution is part of the objective

I want to find the smallest vector $\vec p$ such that some constraints are satisfied, so something like: $$\hat p = \underset{\vec p}{\arg \min} \; |\vec p| \\ s.t. \; F(x_i, \vec p) \leq \epsilon_i ...
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Maximum of $f(x) = (45-2x)\cdot (24-2x)\cdot (2x)\;,$ Where $0<x < 12$

How Can I Maximise $f(x) = (45-2x)\cdot (24-2x)\cdot (2x)\;,$ Where $0<x < 12$ Using Inequality $\bf{My\; Try::}$ In $0<x<12\;,$ The value of $(45-2x)\;,(24-2x)\;,2x>0$ and we can ...
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Maximize ratio of logarithms

How can one maximize the ratio of two logarithms $ \frac{\log{f(x)}}{\log{g(x)}}$ where the argument to each logarithm is the (positive) ratio of two first-degree polynomials? I have tried ...
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Dynamic programming problem for discrete linear time varying system

I'm working on a linear time varying discrete(LTV) multi input multi output(mimo) system. I formulate the problem description in the following way $$x_i(k+1) = x_i(k)\cdot A_i(k) + B_i(k)\cdot u_i(k)$...