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, complex quadratic function, Is my solutions correct?

I'm trying to compute $\max_{|z| \le 1} |(z+2)(z-1)|$. Here's how I do it: $\{z \in \mathbb{C} \ | \ |z| \le 1 \}$ is compact and $f(z) = (z+2)(z-1)$ is continuous, so it suffices to look for ...
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Gradient and hessian of square of quadratic form

I'm trying to differentiate a term of the form $(x^TA x)^2$ where $x$ is a vector and $A$ is a symmetric square matrix. Can anyone please tell me what the gradient and Hessian matrix of this term ...
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1answer
31 views

Optimization problem: rowing across a lake

A woman at a point A on the shore of a circular lake with radius $r=3$ wants to arrive at the point $C$ diametrically opposite $A$ on the other side of the lake in the shortest possible time. She ...
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1answer
19 views

Finding maximum of the basic Bernstein Polynomials

The basic Bernstein Polynomials $B_{n, k}$ are defined for all integers $n, k$ with $0 \leq k \leq n$ by $B_{n, k} = {n \choose k} x^k (1 - x)^{n-k}$ for $x \in [0,1]$. I want to prove that the ...
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Rearranging a sequence to minimize a series function on its subsequences

Given an arbitrary nonempty sequence of integers $Q$ with $p$ elements: Let $R$ be a rearrangement of $Q$. Let $A$ be the solution set of $1\leq n\lt m\leq p$, where $n,m\in \mathbb{Z}$. $F(n,m) = ...
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19 views

Newtons method for optimization

How can I solve this question? Use Newton's method for a system to write $x^2+y^2=25$ and $x^2-y=2$ in the form $J*\delta=-f$. Define the matrix $J$ and vectors delta, $f$. Dont perform iterations.
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Shortest distance from a point to vertices of a cube

A $d$ dimension cube has vertices $P_1,...,P_{2^d}$, where the coordinates of each vertex are either $0$ or $1$. To find which vertex of $P_1,...P_{2^d}$ is closest to a given point $P=(p_1, ...
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A question about the definition of Lipschitz continuity

Suppose $f(x,y)$ is a function on $R^2$. If $f$ is Lipschitz continuous with respect to $y$, then $|f(x,y_1)-f(x,y_2)|<C|y_1-y_2|$ for some constant $C$. But can anyone tell me whether the ...
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1answer
23 views

What optimization problem is this?

Minimize $$\sum_{i=1}^{m}w_i x_i$$ with $w_i \in \mathbb{Z}_{\ge0}$, and $x_i \in \{0, 1\}$ subject to a set of $n$ conditions of the form $$\sum_{i\in S_k} x_i \equiv c_k \pmod{2}$$ for $S_k ...
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10 views

Eigenvalues and Positive-Definiteness of the Hessian Matrix

Suppose we have a function $f \in C^{2}$ and the Hessian defined as follows: $Hf(x,y)(h) = \displaystyle\frac{1}{2} \begin{pmatrix} h_{1} & h_{2} \\ \end{pmatrix} \begin{pmatrix} f_{xx} ...
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1answer
34 views

What is the most efficient way of determining a date of birth using yes/no questions?

First question here on StackExchange! Sorry if this question is not quite of the correct style - please let me know if so. Anyway, here's the context. I'm trying to write a program to determine a ...
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1answer
13 views

Relation between minimum of a function and minimum of the sum of the same function and a linear term

I'd like to know if it's true that if given a function $f(x):X \mapsto \mathbb{R}$ and a vector $c \in X$, then if $$v = \arg\min_x f(x) + x^tc$$ one can say that $$v-c = \arg\min_x f(x)$$ Does this ...
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1answer
18 views

Optimization with changing objective function

Is there any theory about (convex) optimization where the objective function is allowed to change during the optimization process? I have a problem where the objective function depends on some ...
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7 views

Epi-convergence and normal cones

I have a series of lower semi continuous, eventually level bounded and proper functions $ f^\nu(p)$ that epi-converges to $f(p)$. In this context, it is known from e.g., [7.33, Variational analyis, ...
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15 views

Weighting with restrictions, but no clear objective function?

Here is the problem: I have 40 shares in an index and I want to weight them based on their market value, define the known value as $x_i$ In the traditional way, the weight of each share is ...
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4answers
129 views

Find absolute maximum and minimum with domain

Find absolute maximum and minimum of the function $f(x,y)=3-x^2+y^2$ on the region $R = \{(x,y):1≥x≥0, 2≥y≥0\}$ I found that the gradient is $∇f(x,y)=(2x,2y)$ and that the critical point inside ...
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Maximization over concave subset of variables

Let $f(x_1, \dots, x_N)$ be a concave function in $x_1, \dots, x_N$. For arbitray $n>1$, prove that the (constrained) truncated function defined by $$g(x_1, \dots, x_{n-1}) = \max_{x_n, \dots, x_N ...
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0answers
9 views

Line search Armijo, Wolfe, Strong Wolfe and Goldstein.

What are the articles (References) who proposed the line search of Armijo, Wolfe, Strong Wolfe, and Goldstein? Articles precursors of unidirectional searches?
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19 views

optimization terminology

For the function plotted below, x = 14.5/15 and x = 15.5/15 are two local maximizers. So gradient-based optimization methods could find the global minimum if the initial guess of x is in the range ...
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1answer
38 views

Algorithms: Maximizing $\Pi \, a^{b}$ (NOTE: Homework)

First I would like to say that this is a homework assignment, so I'm not looking for someone to give me a solution. Just a little guidance if what I have is wrong or inefficient: Given two sets $A$ ...
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1answer
31 views

Gradient descent for periodic function

Problem: minimize E=$\sum_{t=0}^T [ Y(t)-\sum_{k=0}^K(X(t+k)*cos(k*F+PHI)) ]^2$ where F, PHI - has to be optimized; Y(t), X(t) 1D arrays are given; t=0...T; T=1000; k=0..K; K=10; I can use FFT ...
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17 views

Least squares with three quadratic constraints (Ellipse fitting based on algebraic distance)

I would like to fit an ellipse to a given set of scattered data in $\mathcal{R}^2$. The fitting problem is in form least squares, minimizing the sum of squared algebraic distances \begin{equation} ...
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Max flow on undirected graph with constrained edges

I've been trying for a while to develop an algorithm that counts the maximum number of disjoint vertex paths in a graph, but with an addition of "forced paths". Forced paths are here marked with bold ...
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2answers
26 views

Why does gradient descent make sense?

Suppose I define two functions of $x$ in terms of a convex function $f$ with a unique minimum $x_0$: $$f_1(x) = 1 \times f(x)$$ $$f_2(x) = 2 \times f(x)$$ Suppose I wanted to minimize each of these ...
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28 views

Condition for trigonometric inequality

I want to prove the following statement: Suppose $\frac{1}{4}(\cos(\theta_1)+\cos(\theta_2))^2+\lambda^2(a\sin(\theta_1)+b\sin(\theta_2))^2\leq 1$ holds for all $\theta_1,\theta_2\in[-\pi,\pi]$, then ...
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1answer
30 views

Linear programming and the simplex method

I am trying to solve this system of equations. My approach would be to introduce slack variables and then somehow use the simplex algorithm to solve this. Can anyone show me how this is done?
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11 views

Need guidance on a Queuing problem

I can't really go into specifics, I'm more just looking for terms that I can research to get on the right track. Classes of model/processes etc. A close analogy to my problem: I need to optimally ...
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Optimization/Hamiltonian Robinson Crusoe model

Below is a practice problem for a midterm exam in Macroeconomic theory. I'm having a great deal of trouble getting this going as my background is Econ not math so the language transition has been ...
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1answer
50 views

A problem on positive semi-definite quadratic forms/matrices

Suppose $a+b+c=0$ and (without loss of generality) $a\leq b\leq 0\leq c$, $a^2+b^2+c^2=1$, is the following quadratic form positive semi-definite? Thank you very much. \begin{equation*} \begin{split} ...
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1answer
26 views

Finding the minimum distance between two lines

I really don't know how to tackle this optimization problem: We consider the two lines $$a(x) = x \begin{pmatrix}1\\2\\3\end{pmatrix}, b(y) = ...
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Projection onto a matrix where the diagonals are identity matrix

I'm trying the understand intersection of convex sets given in "Convex Optimization - Boyd" which I'm also trying to code in cvx. The two convex sets I'm trying to find the intersection are given ...
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1answer
18 views

Newton Raphson Method in optimiztion.

I understand that Newton Raphson Method is used to find the zero of a function. However when I'm trying to find the maximum or minimum point, why does diving the derivative by the second derivative ...
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Time-optimal control to the origin for two first order ODES - But wait, the node is unstable? Hard-mode active!

I want to find the time optimal control to the origin of the system: $$\dot{x}_1 = 3x_1+ x_2$$ $$\dot{x}_2 = 4x_1 + 3x_2 + u$$ where $|u|\leq 1$ I ran straight into the problem full strength, hit it ...
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5answers
65 views

Guessing on the SATs, is it ever better to leave it blank than to guess?

On most SAT questions, there are 5 answers of which exactly one is correct and exactly four are wrong. If one answers correctly you get $1$ point. If you answer incorrectly, you receive $-\frac14$ ...
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+50

Largest $n$-vertex polyhedron that fits into a unit sphere

In two dimensions, it is not hard to see that the $n$-vertex polygon of maximum area that fits into a unit circle is the regular $n$-gon whose vertices lie on the circle: For any other vertex ...
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1answer
25 views

Finding absolute extrema

The function is $f(x,y)=\sin x+\cos y+\sin (xy)$, which on {$(x,y)|0\le x\le 2\pi,~0\le y \le 2\pi$} I want to find the absolute extrema of function $f(x,y)$. I try to find gradient of function $f$, ...
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1answer
27 views

prove length-like function is convex

I'm trying to prove that $ F(u)= \int\limits_0^1 (1+(du/dx)^2)^{1/2}$ is a convex function of u ; however after squaring both side twice of $(1+(d(tu_1)/dx)^2)^{1/2}+(1+(d((1-t)u_2)/dx)^2)^{1/2} ...
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16 views

How to describe contours of a function

$$f(x) = 0.5x^TAx-b^Tx$$ $$A = 2I$$ $$b = (-1, -1, -1,...,-1)^T$$ How can I use this information (or in the general case, just $A$ and $b$) to describe the contours of $f$? I get the feeling that ...
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How to transform equivalent optimization problems

Can somebody either explain how to show the equivalence of the three alternative optimization problems in MPT (or point me to some literature)? I am looking for the necessary "algebraic steps" if at ...
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Regularization of equivalent optimization problems

In portfolio optimization three equivalent optimization problems exist. I am wondering if they are still equivalent when regularized, e.g by ridge regularization. E.g. are the following equivalent ...
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Exercice the converge $r$-superlinearly

Give $x_0 \in \mathbb{R}^*$. Show that $\{x_k\} \subset \mathbb{R}$ converge $r$-superlinearly for $x^∗=0$, where $x_k$ is defined by $x_{k+1}=(1−\beta_k)x_k$ and $\beta_k=1−2^{-k}$ if $k=i^2$ for ...
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Function Maximization

Good day. I have no idea how to solve the following question and will be grateful for any help. Suppose we are given $~n~$ real numbers $~y_1,...,y_n~$. Is there any simple way to minimize function ...
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Help in estimation of unknown parameter not present in the distribution

Question: The unknown parameter to be estimated is the volume of residuals points drawn from a process in multi dimensional space (embedded space after delay embedding) for the following situation : ...
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28 views

Help in finding the optimum value

There is a metric $H$ defined as = $\sum_{i=1, j \neq i}^{N} \min |u_i - u_j| * ..*|u_{i+d-1} - u_{j+d-1}|$ where $u$ is a multi dimensional vector of dimension $d$ and $u_i,u_j$ $\in \mathcal{R}^d$ ...
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1answer
28 views

How do I plot the following function? [closed]

I need to plot the following function, in order to get the intuition for optimization theory: $$ f(x) = \begin{cases} x , & \text{if $y=x^2$} \\ 0 , & \text{otherwise} \end{cases} $$ How do ...
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Help in deriving Maximum likelihood estimator

The problem is estimating a metric $\operatorname{Vol}(D)$ for the following situation : Given noisy observations the observed data are a random sample $Y_1,\ldots,Y_n$ where $Y_i \in \mathcal{R}^d$. ...
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The maximum volume of Tetrahedron

A optimization problem: Get the maximum volume of a tetrahedron its 4 vertices on the surface of cube whose edge length is 1 . From the geometrical intutition ,we can get : Selecting ...
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Show that $\{x_k\}$ converge $r$-superlinearly . [closed]

Show that $\{x_k\}$ converge $r$-superlinearly for $x^*=0$, where $x_k$ is defined by $x_{k+1}= (1-\beta_k)x_k$ and $ \beta_k = 1-2^{k}$ if $k=i^2$ for some integer $ i$, ande $ \beta_k=2$ ...
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gradient descent - cost reduces and then increases

I am optimizing a function using Gradient Descent. The learning rate is fixed. First for few iterations the cost decreases after that it starts increases. What is the reason for this?
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1answer
25 views

Optimization options to select multiple items with different features and values

I'm trying to identify which approach would work best to select a set of elements that have different features that minimise a certain value. To be more specific, I might have a group of elements with ...