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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Calculation of an expression ($\max_{U}\min_i \sum_j |U_{ij}|^2 |e_i^j|^2$)

There is an orthonormal basis $\{e_i\}(i=1,\ldots,n)$ in $\mathbb{C}^n$, each of them is represented in form of column vectors $$\begin{pmatrix} e_i^1\\ \vdots\\e_i^n\end{pmatrix}.$$ My purpose is to ...
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Methods to translate global constraints to local constraints

Are there any general methods for (global) optimisation which can translate a global optimisation problem to a "local" one? Or in other words, translate global constraints to local constraints. To ...
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Analysis and Optimization

Also, If we have f(x)=x^4-(2x^2)+1. What are the largest intervals on which the function is convex? Concave? Calculate the local minima for f; they are contained in intervals on which f is convex, ...
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How to argue that $f(x,y,z)=-x^2-xy-y^2+4yz-8z^2+2xz$ has a global maximum?

Let $f(x,y,z)=-x^2-xy-y^2+4yz-8z^2+2xz$. I know $f$ has a local maximum at $(0,0,0)$ but how do I argue that this is also the global maximum. The solution provided simply states it is a global ...
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optimize the set of nodes in a weighted graph

I have a set of weighted nodes (~100) some of which are connected (undirected) by weighted edges (~1000). I want to determine the set of nodes where the sum of the weights between the nodes in the ...
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Dictionary learning for sparse coding using ADMM

I'm trying to formulate an ADMM for performing dictionary learning (for sparse coding) on a set of data. Let's assume we have a data matrix of $X \in \mathbb{R}^{M \times N}$, a dictionary of $D \in ...
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How to optimizing a function that takes two different forms in two different regions

a,b,and P are non-negative constants. And $\theta$ is a random variable with distribution function $F(\theta)$ and density function $f(\theta)$. Denote $H(\theta)= {F(\theta)\over f(\theta)}$. No ...
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Moving die on cartesian plane so as to minimize sum of facing face

I have a problem that I have been working on for which I cannot find a solution. Problem: Assume you are on a cartesian plane, and you want to move a die to a specific point. You can move the die ...
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28 views

Code that minimizes the variance of sum of weighted random variables?

I'm trying to implement finding the weight that minimize the variance of the sum of random variables. I followed the formula from this question Minimizing the variance of weighted sum of two random ...
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1answer
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Is there an analytical result for this expression?

I have an expression $$ \max_{\{e_i\}(i=1,\ldots,n)}\sum_k \big|(e_k,\beta)\big|^4 + 2 \big|(e_1,\alpha)\big|^2, $$ where $\{e_i\}(i=1,\ldots,n)$ is an orthonormal basis in $\mathbb{C}^n$, ...
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How to optimizing a function that takes two different forms in two different regions [on hold]

H is a constant. I know that the positive part of function is non-differentiable, but I want to get the maximum this integral, namely, get f(z)* that make the objective function reach the maximum. ...
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27 views

What are “symmetry arguments” in the context of solving systems of equations?

What and how are the "symmetry arguments" used to solve a system of equations? My text makes extensive use of this argument but do not provide and explanation of how it works or the definition of ...
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is multiobjective optimization easier with fewer objectives?

it seems to me that optimization on nonconvex problems can usually reach better results with fewer criteria in the objective function. for example, in a given block of search time, if i am asking for ...
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1answer
32 views

Mixed Integer Linear Programming: Construction Rods

I have an interesting problem involving linear programming. The problem is the following, I have 4 different kinds of rods (rod sized found in the local market): 9m rod 11m rod 12m rod 15m rod ...
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24 views

Optimal choice of job based on multiple ranks

First of all I should state that I am a non-mathematics student but am pretty mathematically-inclined. I have a problem that I can't find a solution to on Google. Here is the hypothetical: I have ...
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1answer
31 views

The minimum value of $x^{-k} + (a - x)^{-k}$

The question I've been struggling with goes: 11. Given that $k > 0, a > 0,$ prove by considering the minimum value of the function $x^{-k} + (a - x)^{-k},$ that $\dfrac{1}{x^{k}} + ...
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Finding zeros of a piecewise function

Is there a general strategy for solving $$0 = \sum_i \left\{ \begin{array}{lr} f_i(x) \text{ if }p_i(x) \\ g_i(x) \text{ otherwise} \end{array} \right.$$ for $x$? To what ...
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1answer
33 views

Proximal mapping of $f(U) = -\log \det(U)$

This is an assignment problem which I failed to solve in a couple of days. Denote the set of all $n \times n$ symmetric matrices and the set of all $n \times n$ symmetric positive definite matrices ...
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19 views

Global optimization algorithm for up to 20 parameters [on hold]

I have cost functions $f:[0,1]^n\to[0,\infty)$ with the following properties The dimension $n$ is at most $20$. The function $f$ is not continuous (there are punishing terms that prohibit ...
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27 views

Optimization of utility function with Lagrange multiplier

Let u: ${\mathbf R^n_+ \rightarrow \mathbf R}$ be a utility function of n goods which you buy in quantities $x_1,…,x_n$ to the prices $p_1,…,p_n$ under the budget K. So maximize $u(x_1,…,x_n)$ subject ...
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Is there a difference between inequality and equality sign when using Lagrange multiplier?

For example, find the extreme values of z=xy subject to the condition x+y=1 This is quite simple example of finding extreme using Lagrange multiplier When the constrain is changed from x+y=1 to ...
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Definition of CVaR of the gain profile

The CVaR for a loss profile is defined as: $F(x,\beta) = \beta + \frac{1}{1-\alpha} E[f(x,y) - \beta]^+$ where $[t]^+$ is $\max(0,t)$. I have an extremely naive question, how this definition will ...
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NP-hard optimization problems for which approximations would be useless?

The optimization problems which I am familiar with (e.g. the Traveling Salesman Problem) are such that approximate solutions to these problems are still quite useful. I'm wondering, however, if there ...
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You construct a rectangular Box with volume K cm^3

Prove that a cube uses the least amount of material to construct the box
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112 views

Closest Matrix with Specific Eigenvector

Consider a vector ${\bf x}$ and a matrix $A_0$ with $A_0(i,j)\ge0$. What is the best way of getting matrix $A$ s.t. $$A = \arg \min |A-A_0|$$ subject to $$A{\bf x} = \lambda {\bf x} \hspace{2mm} ...
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Nonlinear Programming examples

I'm an optimization newbie. I am looking for a simple nonlinear optimization problem that I can work through in Excel. For linear optimization I used the "Giappeto Inc" problem and I wonder if there ...
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Minimum of the difference of two logarithms

I am trying to find an analytical expression of the minimum of $$ f_n(x) = \frac{2x}{n^2+n}\log(x) - \frac{2x+2}{n^2+3n+2}\log(x+1) $$ when $x\in [1;n]$ I used to think from graphing it that this ...
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2answers
38 views

Stationary points of general function

When studying the stationary point(s) of the following $$ Q=\frac{K(x)}{x} $$ I find $$ \frac{dQ}{dx}=\frac{x\frac{dK(x)}{dx}-K(x)}{x^2}=0 $$ and hence $$ \frac{K(x)}{x}=\frac{dK(x)}{dx} $$ I'm ...
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Residual error between two transformations

Suppose I have two unknown 3D transformations matrices in homogeneous coordinates: X and Y. I want to calculate both of these by ...
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Many-objective optimization

I have a set of functions $f_i(x) = \phi(x) \cdot \theta_i$ where $i=1,\dots,n$ and $\phi = (\phi_1, \dots, \phi_k)$ is a vector of radial-basis functions with different centers each and $\theta_i$ is ...
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Minimize $f(X)=trace\left(X^{T}\begin{bmatrix} 1&0\\0&16\\ \end{bmatrix}\right)$

Minimize $$f(X)=trace{\left(X^{T}\begin{bmatrix} 1&0\\0&16\\ \end{bmatrix}X\right)}$$ subject to the condition $g(X)=det(X)=1$. Then for taking $X=\begin{bmatrix} ...
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Making a least material rectangular box [on hold]

Show that the shape that uses the least amount of material to construct a box with volume $ K cm^3 $ is a cube.
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Linear Optimization [on hold]

In advance thank you for helping me, I have the following optimization problem that I have to solve unsuccessfully max(30-5*$x_{1}$-5*$x_{2}$-10*$x_{3}$-4*$x_{4}$) under the following constraints ...
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Finding minimal distance between column sums by permuting row values

I'm sure the title is not very clear, that's because I don't know the proper name for this problem - if there is any. Consider a matrix with integers. The task is to find such a permutation of the ...
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1answer
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Optimization problem [on hold]

Find Minimum value of $f(x_1,x_2)=x_1+x_2$, subject to $g(x_1,x_2)=x_1^2+x_2^2-4x_1-4x_2+7\leq0$.
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1answer
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Optimising Over a Parameter in an Integral

Here is my task: Maximise over $a>0$ the following function: $$ f(a) = a e^{-a\sqrt{2r}} \int_0^\infty \exp\left(-{1 \over 2}x^2 - {{\lambda (a - S_0)^2} \over {\sigma^2}} {1 \over x^2} ...
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1answer
23 views

Schatten p norm p>1

The Schatten p norm is differentiable away from the origin for p> 1. Does a stronger condition of Lipschitz continuity of the gradient also hold?
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Portfolio Optimization Problem: Variance Co-variance matrix

I have a set of daily returns and using these daily returns I calculated the average annual return for each asset and also by using the daily returns I calculated the var-cov matrix. To get optimize ...
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1answer
41 views

Maximize $a_1^{a_2^{\ldots^{a_n}}}$, where $(a_1,a_2,\ldots,a_n)$ is a permutation of $(b_1,b_2,\ldots,b_n)$

You are given a tuple of integers $B=(b_1,b_2,\ldots,b_n)$. Find $(a_1,a_2,\ldots,a_n)$ - a permutation of $(b_1,b_2,\ldots,b_n)$ - that maximizes $a_1^{a_2^{\ldots^{a_n}}}$. For example - If ...
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21 views

Indicator Variable, Mixed Integer Linear Programming

Assume $x$ is a real variable, and $0\leq x \leq1$. Besides, $y$ is a binary random variable. I need a linear program that: if $y$ is $1$: $x>0$, if $y$ is $0$: $x=0$ I know the following ...
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1answer
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A Variant of Gradient Descent

Suppose I have some objective function $f(\beta)$ which I would like to minimize for $\beta$. A standard gradient descent would be $\beta^{(t+1)}=\beta^{(t)}-\alpha \nabla f(\beta^{(t)})$, where ...
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is this a convex optimization problem?

Can someone clarify is this a convex optimization problem or not. $min \| X-UV\|_{F}\quad $ s.t $ \quad U \geq ,V\geq0$ . If not , what makes the problem non-convex?
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1answer
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A fast algorithm for a simple multi-objective minimization?

I have a set $S$ consists of n (arbitrary) integer numbers which I want to partition into $k$ subsets $S_i$ each of size $\frac{n}{k}$ (assume $k$ divides $n$). Let $A$ be the arithmetic mean of ...
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2answers
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How to interpret max(min(expression))?

How to interpret and evaluate an expression such as this one? $$\max_{x \in (-3,5)} \min_{y \in \{-1,1\}} \frac{x-2}{y}$$ I found such expressions in Step 2 of section 3 of this paper. What do ...
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1answer
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Is it possible to find an analytical solution for “x” in this equation?

In my research I have come across the equation $$\prod_i^n \left( \frac{a_i}{x} \right)^\frac{b_i}{x} = \prod_i^n (1-d_i)^{(1-b_i)c}$$ Is it possible to obtain $x$ from this analytically, or do I ...
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Another naive question: Objective in (maximization) optimization increases and decreases during iterations!

I have following optimization problem: $\max_{u,z} z$ $s.t \quad z - J_i(u,\theta_i) \leq 0 \quad\forall i$ $z$ is a scalar, $J$ is a nonconvex,non linear function of $u$ and $\theta_i$ is just ...
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The $\alpha$ estimation for the model $x_i = \xi_i \cdot \alpha$

We have $n$ sensors $X_i$ which estimate the scalar value $\alpha$ with different relative accuracies $\delta_i \ll 1$: $$ x_i = X_i(\alpha) = \xi_i \cdot \alpha, \ \ \ \xi_i \sim N(1, \delta_i) $$ ...
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How to determine the gradient of this cost? [duplicate]

I have asked a similar question before, but I guess I haven't provided clear information. The cost of my function $f:\mathbb{R}^5\rightarrow\mathbb{R}$ is $$f(\vec{\alpha}) = ...
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1answer
29 views

Maximization: Volume of paraboloid within cone?

Given a right circular cone with the line of symmetry along $x=0$, and the base along $y=0$, how can I find the maximum volume paraboloid (parabola revolved around the y-axis) inscribed within the ...
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1answer
45 views

Exchanging max and limit

Suppose I have sequence of function $f_n$ that converge to $f$. Suppose I want to find maximum of $f$ over some set $S$ that is \begin{align*} x^*={\rm arg} \max_{x \in S} f(x) \end{align*} ...