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Questions tagged [optimization]

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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coordinate descent in very basic

I try to figure out how coordinate descent works from wiki https://en.wikipedia.org/wiki/Coordinate_descent From wiki example : the equation is $5x^2-6xy+5y^2$. Let $x = -0.5$ and $y =-1$ For the ...
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zdt4 test function [on hold]

please help me to run this code in Matlab: message error :Caused by: Failure in initial fitness function evaluation. GAMULTIOBJ cannot continue. ...
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Constrained Optimization using Lagrange Multipliers: Hard Problem

My problem is to maximize a the function below: $$ \frac{\left\langle\overrightarrow{\omega}_{m}^{^{^\ddagger}}\textbf{T}_{12} \overrightarrow{\omega}_{s}\right\rangle }{\sqrt{\left\langle \...
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Optimization of the limit of a sequence of nondifferentiable functions

Consider $N \in \mathbb{N}$ real numbers $a_1 \leq a_2 \leq \ldots \leq a_N.$ Moreover, consider the following sequence of functions: $$f_{k}(x) = \sum_{i=1}^N |x-a_i|^\frac{1}{k},$$ where ...
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Confusion on proving $\operatorname{conv}(E) \subseteq P$ via induction

Terms are translated from a different language, so I am not sure whether they coincide. Let $E$ represent the number of corners of $P(A,b)$ and $P(A,b)$ be a polytope. Prove that $\operatorname{conv}...
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Optimization problem: Find the point on the line $−x + 2y − 1 = 0$ that is closest to the point $(1, 2)$.

Find the point on the line $−x + 2y − 1 = 0$ that is closest to the point $(1, 2)$. I solved the optimization and got $x=14/10$ and $y = 1.7$ but my $y$ coordinate is not correct. can anyone ...
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Norm of $\|(x-x^*,y-y^*,z-z^*) \|\leq \|\nabla J(x,y,z)\|$

I want to prove that $$\|(x-x^*,y-y^*,z-z^*)\|\leq \|\nabla J(x,y,z)\|,$$ where $J(x,y,z)=\exp(\frac{x+y+z}{2016})+\frac{x^2+2y^2+3z^2}{2}$ and $(x^*,y^*,z^*)$ is the minimum of $J$. Also, I have $...
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How to show the optimization/ODE fixed point iteration steps converge?

I have $\vec{C} = G(\vec{\beta})$ by solving a system of ODE numerically. Thanks for the help of Robert the ODE can be found in this link: Solving a system of ODE Also $\vec{\beta}$ should satisfy $$...
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Optimization of the least used wire

question:There are two trees that are spaced 30 meters apart. The height of one of them is 12 meters and the other is 28 meters. The two trees should be kept by two wires so that both wires are ...
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How to optimize a series of equations whose outputs are a variable of the subsequent equatinos

The basic question is, given $f(x) = y$ and $f(y) = z$, how can you find $x$ such that $z$ is at its maximum? I can optimize each equation independently, but I do not know how to optimize when ...
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Procrustes Problem with Maximization (Instead of Minimization)

The classical (orthogonal) Procrustes problem is to solve the optimization problem $$ \begin{array}{rl} \min&\|\Omega{A}-B\|_F\\ \text{s.t.}&\Omega^\mathrm{T}\Omega=I \end{array} $$ The ...
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Find the minimum and maximum possible values of the conditional probability

Given two events $A$ and $B$, such that $P(A) = 0.3$ and $P(A ∩ B) = 0.1$. Find the minimum and maximum possible values of the conditional probability $P(A | B)$.
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Permutative Constraint on Image Approximation

Motivation I am trying to explore the idea of constraining the approximation of an image represented by an $m$-by-$n$ matrix $A$ by the values on a linearly-spaced interval of $mn$ elements $L$ ...
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Coming up with a cost function for optimization for a complex control system

I am relatively new to this topic. I understand the basics of optimization of control systems using a cost function and constraints and solve it as a minimization or maximization problem. I also ...
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mixed integer programming - turning conditional statements into inequalities

I have if statements in my constraints and I'm having trouble turning it into an inequality problem. The statement is as following: IF a>=x1, THEN f(x1,x2) = a+x2, Else f(x1,x2) = a. x1 and x2 are ...
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Steiner tree to minimise travelling distance: Building roads to connect a network of points

Suppose we have four points in a unit square, as described in the question here. We are tasked with building a network of roads that connect all the cities. The travelling distance (T) of this network ...
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Finding a constraint on one variable of a multivariable function to constrain the entire function

I have a function. L(x,y). L(x,y)= 8.5(xy) -3(x+y) + 1. Now I want to let my variables only take values between 0, and 1. The problem is as follows. For what values of Y, is L(x,y) < 0. That is, ...
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Maximizing Area of a quadrilateral inside of a square

The square ABCD has point M located on side AB and point N on side CD. Lines CM and BN intersect at point U. Lines DM and AN intersect at point V. Determine where points M and N should be placed to ...
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Lagrange dual of quadratic optimization problem with quadratic equality constraints

What is the Lagrange dual of the following optimization problem in $w \in \mathbb{R}^2$? $$\begin{array}{ll} \text{minimize} & w^T Q \, w\\ \text{subject to} & w_1^2 = 1\\ & w_2^2 = 1\end{...
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SDP formulation of dual norm

I know that the dual norm of a matrix can be formulated as a semidefinite program (SDP), i.e., $\|X\|_{2,*}$ is the solution to the following SDP in $Y$: $$\begin{array}{ll} \text{maximize} & Y^T ...
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Finding out maximum value of a multivarivable function with inequality constraints.

If x,y and z are 3 uniform random variables within [0,2pi) and |x-y| is bounded between 60 and 150 degrees,then what shall be the maximum value of F(x,y,z) =(c1/c2)^2, given c1 equals cos(x-z) and c2 ...
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Optimization without complex numbers

We need to find a minimum of functions: (1+$\sqrt x$)$^2$+$y^2$ Due to the fact that the function has a square root, the optimization algorithm goes into the area of complex numbers. How to make so ...
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integer programming???

I have a math problem and I would like to solve it, but I'm not sure what area to look under. Basically given $x \in \mathbb{R}^k$ (for my purposes, $x \in \mathbb{Q}^k$ since I am using a computer) ...
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Dual of conic program

Let $A$ be an $m \times n$ matrix (over $\mathbb{R}$), $b \in \mathbb{R}^m$, $c \in \mathbb{R}^n$ and $K \subseteq \mathbb{R}^n $ is a closed, convex, pointed cone with non-empty interior. We define a ...
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Strong Duality: If Dual is optimal then primal is optimal

Strong duality states that if the Primal has an optimal solution then the Dual has an optimal solution. Is the converse of this statement true? To me it would seem intuitive that it is, but I just ...
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LQR with derivative-dependent performance?

Given a standard LTI system with $$ \dot{x} = A x + B u $$ The standard LQR finds the control gain $K$ of the state feedback $u = -Kx$ such that $$ J_1 = \int_0^\infty \big( x^T Q x + u^T R u \big) ...
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Minimum value of $\cos x+\sin x$ for $0 \le x \le 1$

What will be the minimum value of $$\cos x+\sin x$$ for $0\le x \le 1$? The answer is $1$. I tried finding it's minima, but there is none for critical point. Which other approach shall I try?
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MINLP problem with linear constraints and piecewise term

I have an optimization problem where the objective function is non-linear but (I think) differentiable, there are linear constraints, and all parameters are integers. To make matters more difficult, ...
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A Maximize or Explore Problem over a Finite Time Series

I recently read "Reinforcement Learning" Bardo and Sutton which motivated me to come up with this problem (which I hope is well posed): The Problem Some sort of reward maximizing agent finds itself ...
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Optimization with strict inequality

I'm very familiar with using Lagrangeans to solve optimization problems with weak inequalities, but I just realized that I don't know how to solve simple optimization problems with strict inequalities....
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How to solve a one variable problem using artificial bee colony algorithm [on hold]

As I am doing my research based on ABC Algorithm first to understand that algorithm I wanted to solve any multi variable problem and a one variable problem,if someone can clearly state an optimization ...
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Using the least squares method for problems with two independent variables

This may be quite a specific question and I apologise however I have struggled to find any information regarding a method. I have 6 given $P_i$ values and 6 given $E_i$ and $F_i$ values and I want to ...
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Matrix optimization involve inverse

Can anyone help me to solve the following optimization:(where A is a strictly PSD with size n-by-n,x_i,y_i are both vectors of size n) enter image description here
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Least action and partial order

I am fascinated by minimizing principles; my favourite is the least action principle, $$ S[q]=\int_{t_1}^{t_2}L(q,\dot{q}) \ dt , $$ which states that the trajectory of a physical system will be such ...
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Equivalent maximization of $\log (f(x)) $ when $ f(x) $ is non-convex

I would like to know whether the following is equivalent. $$ \max_{x,y,z} \log (f(x,y,z)) = \max_{x,y,z} f(x,y,z) $$ In principle, I need to solve $ \max_{x,y,z} \log (f(x,y,z)) $ where $ f(x,y,z) $ ...
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Is consensus ADMM performance affected by how the objective function/ constraints are split?

I just wanted to know whether Consensus ADMM convergence results are affected by the way we split the objective function. For example, will the results be affected if I split f=f1+f2+f3+f4 as (f1+f2) &...
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Given $a+b+c+d=4$, find the minimum value of $\Sigma_{cyc}\frac{a}{b^3+4}$

Given $a+b+c+d=4$, find the minimum value of $\Sigma_{cyc}\frac{a}{b^3+4}$ I'm pretty sure this has something to do with Holder's inequality, but I don't know how to solve this. By guessing I found $a=...
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Correctness of integer reformulation in the FICO MIP quick reference

I have stumbled upon an industry quick reference for MIP formulation by FICO: However, after checking their writing on section 2.3 Maximum value. It seem that there are problems with their ...
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Textbook Recommendations: Solving Systems of Matrix ODEs

This is in reference to the works of Trendafilov whose approaches to multivariate statistical problems boil down to solving a dynamical system involving matrices. Question: Can anyone suggest a book/...
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1answer
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Showing $\sum_{k \geq 0} kA^{k}$ converges while $\vert \vert A\vert \vert < 1$

Let $\vert \vert \cdot \vert \vert$ be a matrix norm on $A$ where $\vert \vert A\vert \vert < 1$. Show that $\sum_{k \geq 0} k A^{k}$ converges. My ideas: Let $m<l$ $1.$ Let $\vert\vert\sum_{k=...
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Operating with penalized regression models

Assume we are working with a penalized linnear regression model. We have the following optimization problem: \begin{equation} \min_{\beta}\left\{\left\lVert y-X\beta\right\rVert_2^2+\lambda\sum_{j=1}^...
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Maximizing $x^T A^T B \, x$ over the unit Euclidean sphere

Is there an algorithm to solve the QCQP $$\begin{array}{ll} \text{maximize} & x^T A^T B \, x\\ \text{subject to} & \|x\|_2 = 1\end{array}$$ when $A^T B$ is not necessarily symmetric? When $A^...
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The Proximity Operator of a Function with Multiple Affine Mapping

Let $f(\mathbf{x}) = g(\mathbf{A}\mathbf{x})$, where $\mathbf{A} \in \mathbb{R}^{M \times N}$ is a linear transformation satisfying $\mathbf{A}\mathbf{A}^T = \mathbf{I}$. Then for any $\mathbf{x} \in \...
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How can I design a (PID) Controller if I don't have a reference signal?

I have been trying to control lateral and longitudinal movement of a robot for an autonomous lane keeper project. I have no problem with the lateral movement, however I couldn2t figure out exactly how ...
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1answer
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Multivariate convex optimization problem involving logarithms 2

This is an extension to previous question in Multivariate convex optimization problem involving logarithms. Thanks a lot to David M. for the answer to there. Now, I like to extend the question a ...
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Multivariate convex optimization problem involving logarithms

Question-1: $$\min_{a, b} \sum_{i=1}^K b_i f(\frac{a_i}{b_i}) $$ s.t. $$ f(x) = (1+x) \log(1+x) -\log(x) - (1+x) \log(2)$$ $$ \sum_{i=1}^K a_i = 1.$$ $$ \sum_{i=1}^K b_i = 1.$$ $$ a,b > 0. $$ ...
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Maximizing the gain of a matrix

I am interested in following problem: $$\max_x\left(\frac{\|y\|}{\|x\|}\right) \qquad \text{subject to} \qquad y = Mx$$ Where $y$ and $x$ are vectors of same shape and $M$ is a square matrix and $|...
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Is this binary optimization problem stated in a correct way?

A city is divided in 6 districts and it is planning the location of firefighter stations such that every district is covered. However, not every district needs its own station as one station in a ...
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Confused about Nesterov momentum gradient descent algorithm

I've found a variety of variations of writing Nesterov but I cannot understand why they cannot simply be expanded into a one liner. Here is one I found that can just be re-arranged, can someone ...
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Optimal Value of a Cost Function as a Function of the Constraining variable

Consider the optimization problem : $ \textrm{min } f(\mathbf{x}) $ $ \textrm{subject to } \sum_i b_ix_i \leq a $ Using duality and numerical methods (with subgradient method) i.e. $d = \...