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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Is this function convex?

I have a model - function of two vectors $A$ and $B$. I have data that I want to fit to the model and find the model's parameters. The function needs to be convex to find the parameters using ...
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244 views

Portfolio Optimization Problem Without Correlation Info

I received this interesting problem from a friend today: Assume that you are a portfolio manager with $10 million to allocate to hedge funds. The due diligence team has identified the following ...
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65 views

Optimisation Problem

I'm given a lattice with particles having charges which have known magnitude but unknown signs. The primary aim is to stabilize the lattice (or decrease the force acting on the system) by assigning ...
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146 views

Maximum in a cell intersected with a sphere

I have a rectangular cuboid-shaped 3D "cell" with scalar values at each vertex $(v_1,\ldots,v_8)$. Within this cuboid I do tri-linear interpolation. What I want is the maximum value of that function ...
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1k views

Differences behind different methods of fminunc in MATLAB?

Assume I have some .m file with a function (and it's gradient) to be used by fminunc() in MATLAB for some unconstrained optimization problem. To solve the problem ...
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245 views

Underlying assumption in a Primal/Dual table

I just read in one of the questions answered by @MikeSpivey that the following table is provided in Sierksma's Linear and Integer Programming: Theory and Practice, Volume 1, page 144. ...
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Basic Optimization Problem

I sat for an exam a few days ago. I managed to answer every question except for question $1$c in the calculus paper. Provided that I got question $2$d correct (my answer was $m=0.5$), the absence of ...
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How do I set up the following problem to arrive at the answer?

A warehouse has 10 unlabelled rows of pallets. Each row of pallets contains thousands of cell phones destined for different countries. Each 100 gram cell phone is exactly the same except for those in ...
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how to compute this optimization problem

Given $A,B$ are positive semidefinite matrices, how to compute $\max_{0\leq P\leq I}\|APBPA\|$, where the norm is spectral norm, i.e. the largest singular value.
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Equivalent optimization problems

I am reading about optimization and I am having difficulty in understanding the following: If a matrix A is $n\times n$ Hermitian, then $\max_{x^{*}x=1} x^{*}Ax$ is solution equivalent to ...
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42 views

Relation of $\max_P \; x^T \cdot Py$ and $\min_P \; \|x-Py\|_2$

Consider a permutation matrix $P$ and two vectors $x$, $v$ with 2-norm = 1 and all positive entries. Are the optimal solutions $P^\ast$ of $\max_P \; (x^T \cdot Py)$ and $\min_P \; \|x-Py\|_2$ the ...
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667 views

Finding minimum of multidimensional function

My calculus knowledge is pretty limited, but unfortunately I need to solve a problem of the following kind: I'm given a 2 dimensional function $f(x,y)$ from $\mathbb{R}^2$ to $\mathbb{R}$ and I want ...
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Optimization - Get value of Lagrangian

We know that $f(x) \to \min$ subject to $g(x) = t$ and $h(x) \leq m$ can be written as $f(x) + \lambda g(x)\to\min$ subject to $h(x) \leq m$. How do we get value of lambda so that the two problems ...
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Positive semidefinite vector $\bar{x}$ as $\bar{x}>0 :=\bar{x} \lambda \bar{x}^{T}>0$?

$A \lambda A^{T} $ (quadratic form?) is used with matrices to check definiteness. What about with vectors? If I see conditions such as $\bar{x} > 0$, how can I know whether it means $\bar{x}_{i} ...
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164 views

Show $\nabla \bar{x}^{T} M \bar{x} = \lambda ( \nabla \bar{x}^{T} \bar{x} )$

I am trying to prove the sentence ...
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151 views

Finding the “best” way to map set of points to another set

I've got a set of points (currently 4, but I can increase the number for better accuracy), and I want to find the optimal transformation so that they can be mapped to another set of points. For ...
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Lagrangian dual in continuous domain

The continuous max flow problem is posed as follows : sup $\int_\Omega p_s(x)dx$ subject to : $|p(x)| \le C(x); \forall x \in \Omega $ $p_s(x) \le C_s(x); \forall x \in \Omega $ $p_t(x) \le ...
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479 views

Optimization problem with two-step discontinuous function

imagine my function as a staircase with two steps. This function is to be fitted to some empirical data and I'm searching for an algorithm which minimizes the Root Mean Squared Error between this ...
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174 views

Relaxation of a linear constraint in a quadratic programming problem

the problem i have is like following: $x'Qx + f'x \rightarrow \min_x$ subject to $Ax \le 0$. $Q \ge 0$, so there's nothing wrong there, usual QP with a linear constraint. Is there a way to ...
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Dimensional Consistency in Grids used in Optimization

I am working on an optimization problem in the research I am doing and my partner and I have found that in order to quickly converge on a solution using a specific PSO (the firefly algorithm - it's ...
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Given a satisfactory real number = [any integer]/(2b) where a and b are integers, how would one find the minimum value of b?

For instance, 0.625 = 5/(2*4). Given 0.625, how would one find 4? 0.75 = 1/(2*2). Given 0.75, how would one find 2? I should ...
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compute a certain maximum in MATLAB

let $ C \in \mathbb{N} $ and $ c_1>c_2>\ldots>c_k \in \mathbb{N} $ with $ C>c_1 $ and $ c=(c_1,c_2,\ldots,c_m)^\top \in \mathbb{N}^m $, where $ \mathbb{N} $ are the natural numbers without ...
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Find scaling factor that minimizes f(x) - round(f(x))? [on hold]

Let's say I have a function $f(x)$, and I round it to $\operatorname{round}(f(x))$. The difference is: $$g(x) = f(x) - \operatorname{round}(f(x))$$ Now I linearly scale $f(x)$ by $h(x) = mx + b$ ...
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Hyper-plane that separating hyper-cube.

Suppose $\Omega \in \mathbb{R^4}$ is closed unit ball in $ ||.||_{inf}$ i.e. Hyper-cube. 1) Am I right that there are L=16 extreme points of $\Omega$, all are vertices of the hyper-cube. 2)Is it ...
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Max/Min Notation Question

In a paper I'm currently reading it gives alpha to be the following value. $\alpha = \max_t \min_{t_j \in T_N} ||t-t_j||_2$ I am wondering what exactly this means? I have the following code: ...
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Determine the maximum cross‐sectional area.

The client wants to maximise the volume of a materials store to be constructed next to a 3 metre high stone wall (shown as OA in the cross section in the diagram). The roof (AB) and front (BC) are ...
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22 views

Above what order of magnitude a pure cutting-plane algorithm must be forgotten in favour of branch-and-cut?

Crawling the web on the subject of the cutting-plane algorithm, I have seen everywhere that a pure cutting-plane method cannot be used for numerical instability reasons after some iterations. But do ...
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24 views

How to find the smallest value by using Lagrange multiplicators?

Let $a$, $b$ and $c$ be positive constants. How one can find the smallest value of the sum of three numbers $x_1$, $x_2$ and $x_3$ at the surface $\dfrac{a}{x_1}+\frac{b}{x_2}+\frac{c}{x_3}=1$ by ...
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Homography between known and unknown rectangle corners

I would like to know if there is a solution for the problem of homography estimation in the special case in which one of the views is unknown but has some constraints, particularly if we know the ...
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Closest Positive-Definite Matrix Subject to a Contraint

Given a positive, semidefinite, real 2n by 2n matrix $A$, is there a formula or an algorithm that finds the closest (in some sense, preferably Frobenius distance) positive, semidefinite, real 2n by 2n ...
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Gradient descent in inequality constrained optimization problems

I want to solve an optimization problem using a gradient descent algorithm maximize $$ max \log( \frac{Ax + b}{ Cx + b} ) $$ $$s. t. \quad 0 \le x \le 1 $$ where x is a vector and the ...
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Global optimization

Assume that I want to find the global minimum of a non-linear, non-convex, multidimensional function subject to several restrictions. Could you recommend me any deterministic strategy which can ...
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formulate optimization problem in MATLAB [closed]

I have set of weights: lets say my long only portfolio looks like this id weight bweight GROUP A 0.25 0.3 T B 0.1 0.25 T C 0.05 0.25 E D 0.6 0.2 E I have a 4x4 cov as well I would like to add ...
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Find maximum of the function

I have the following target function $$ f(m,q)=\sum^{N}_{i=1}|m_i-q_i| $$, where $$m,q\in R^N$$ and $$\sum^{N}_{i=1}m_i=1, \forall m_i>0$$ $$\sum^{N}_{i=1}q_i=1, \forall q_i>0$$ I would like ...
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dual value of a linear constraint

Assume a minimization problem. The dual of an inequality '<' constraint is the marginal improvement in the objective function (ie marginal reduction) by marginally increasing the right-hand-side ...
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39 views

Question about the ellipsoid method

I have some technical question concerning the ellipsoid method Referring to the paper : http://paswkshop.comm.utoronto.ca/~weiyu/01658226.pdf It is mentioned in p.1317 at the last line in the left ...
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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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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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25 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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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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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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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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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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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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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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21 views

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 ...