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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When change making problem has an optimal greedy solution?

A well-known Change-making problem, which asks how can a given amount of money be made with the least number of coins of given denominations for some sets of coins (50c, 25c, 10c, 5c, 1c) will ...
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Optimization of Frobenius Norm and Nuclear Norm

How to solve the following optimization problem, \begin{equation} \boldsymbol{\hat{x}} = argmin_{\boldsymbol{x}} \frac{1}{2} \| \boldsymbol{x - y} \|_F^2 + \lambda \| \boldsymbol{x} \|_{*} ...
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Maximizing inner product

Suppose we have two row vectors $a$ and $b$ of nonnegative real numbers such that, for $j<k$ $a_j\leq a_k$ and $b_j\leq b_k$. Let P be a permutation matrix. Can we prove (or disprove) that $$ ...
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1answer
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Convexity of problem with inverse matrix

I am trying to solve the next problem \begin{aligned} & \underset{P}{\text{maximize}} & & \log \det P \\ & \text{subject to} & & A^T P^{-1} + P^{-1} A \preceq 0 \\ ...
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1answer
12 views

An Optimal Value of a Diagonal Matrix $\Xi$ in $ H = U \Xi$

We have access to very accurate estimates of matrices $H$ and $U$ (both are $n \times k$, $n > k$) such that the following relationship holds $$ H = U \Xi$$ where $\Xi$ is a $k \times k$ diagonal ...
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Conversion into linear program

I have an optimisation problem with decision variables that are multiplied with another (a weighted average is calculated). I'd like to convert it into a linear program. I found this link that ...
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1answer
23 views

How to determine a function whose minima falls on a specified curve?

I have a family of curves given by $g(x,y)=C_0 yx^{-n}$. How can I determine the function $f(x,y)$ for the family of curves that satisfies the condition that the local minima $\frac{\partial ...
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1answer
46 views

Precalculus - connect 2 towns

A state highway department plans to construct a new road between towns $A$ and $B$. Town $A$ lies on an abandoned road that runs east-west. Town $B$ is $20$ miles north of the point on that road that ...
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Difficulty in understanding a solution: Constraint minimization of sum of Non-symmetric matrices

I am trying to understand why there is significance difference in the performance of two proposed solutions. Original question (Constraint minimization of sum of Non-symmetric matrices) ...
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24 views

Constraint optimization with Calculus of Variations. How to handle positive function constraint?

the I am attempting to maximize the functional $F[f]$ with a constrain that $f$ has to be non-negative and some other integral constraints. More, specifically, \begin{align*} &\max F[f]\\ ...
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14 views

convexity of inverse of a matrix

I know that the function $f(X)$ which maps matrix $X$ to $Tr((X)^{-1})$ is convex for symmetric positive definite $X$. This has also been answered in Is the trace of inverse matrix convex? for ...
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1answer
33 views

Finding extrema of a continuous, univariate function.

Problem: Let $f:[0,1]\to\mathbb{R}$ be given by $f(x)=a(x-b)^2+c$, where $a,b,c$ are parameters. Find the minimum and maximum of $f$ depending on the values of $a,b,c$. I understand how to do this, ...
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Latest evolutionary algorithms

I am required to build an evolutionary algorithm to approach modelling optimization problems. More precisely, let $f:\mathbb{R}^{n} \longrightarrow \mathbb{R}^{m}$ be a model that is dependent from ...
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15 views

Optimization of a Quadratic on a Linear Variety

We have a linear subspace $L_j := L[s_0,s_1,...,s_j]$ and a linear variety: $x_0 + L_j := [x_0 + y : y \in L_j]$ and a standard quadratic cost function $V(x) = a + b^Tx + 0.5x^TCx, \ \ \ C^T = C ...
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1answer
55 views

does the volume of a ball remain constant under deformation?

I'm a psychology student and was reading Piaget, he says that the volume of a sphere (ball of clay) remains constant if we deform the sphere into a roll for example, If you take the limit case of the ...
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23 views

Show that if $a_{i -1} + a_{i}$ is a maximum for all $2 \le i \le n$, then $\sum_{i=1}^{n} a_{i}$ is a maximum? [on hold]

In particular, I would like to show that if N_{i} is such that for given values of $N_{i+1}$ and $N_{i-1}$, $\ln\left(N_{i-1}\right) + \ln\left(N_{i}\right)$ is a maximum for each $2 \le i \le n$, ...
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1answer
40 views

What is the derivative of this?

I have a function of the following form: $J = \|W^TW-I\|_F^2$ Where, $W$ is a matrix and $F$ is the Frobenius Norm. How can I find the derivative of $\frac{\partial J}{\partial W}$ ?
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45 views

What is the derivative of this? [duplicate]

I have a function of the following form: $J = \|W^TW-I\|_F^2$ Where, $W$ is a matrix and $F$ is the Frobenius Norm. How can I find the derivative of $\frac{\partial J}{\partial W}$ ?
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7 views

How can I efficiently optimize stochastic multi-modal functions?

I'm looking for methods for optimizing stochastic functions. I'm probably abusing the notation here, since this is a new field for me. By stochastic functions I mean functions whose output is a ...
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6 views

Proof of Optimality for Approximation of Probability Spaces by PCA

I have come across a theorem that states, that the $d$-dimensional subspace found by PCA is the optimal approximation of a probability space with such a plane, in the sense that it minimises the ...
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Determine $\max V(s,l)$

For \begin{align*}V(s,l)=(2s-210)(2l-297)\min\{s,l\}\end{align*}and \begin{align*}&0<s<\frac{210}{2}\\&0<l<\frac{297}{2}\end{align*} I tried to find the stationary points by ...
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17 views

Maximize a function on $H=\left\{x \right\}$

Let $f: \mathbb{R} \to \mathbb{R}$ be a function. How to find $\sup\limits_{x\in H}f(x)$, where $H=\left\{x \right\}$?
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How to minimize $ \int_{\Omega} \frac1{2 \theta} (u(x) - v(x))^2 + \lambda|\rho(v(x))| dx $

I'm new in optimizations and i am trying to understand how to obtain $ v $ that minimizes $ \int_{\Omega} \frac1{2 \theta} (u(x) - v(x))^2 + \lambda|\rho(v(x))| dx $ where $\rho(x)$ - continuous ...
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$F(x,y)=2x^4-3x^2y+y^2$. Show that $(0,0)$ is local minimum of the Reduction of F for every linear line that passes through $(0,0)$.

first of all I checked if (0,0) is critical point $Df(0,0)=(8x^3-6xy,-3x^3+2y)| = (0,0) $ now my idea was to replace $y$ with $xk$ because of the reduction of $F$ ,and find the hessian matrix to ...
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Find the equations that determine minimizing x for the following [on hold]

I don't quite understand what the question is asking and how to approach it. I am given the following two equations: i. P=1/2 $x^T A^T AX- x^TA^Tb$ ii. E=$||Ax-b||^2$ I could use some pointers ...
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1answer
13 views

Finding the minimum value of $1/2 (x_{1}^2+x_{2}^2)-x_{2}b_{2}$

I am trying to find the minimum value of the following: $1/2 (x_{1}^2+x_{2}^2)-x_{2}b_{2}$ I know this is equal to: $1/2 ([x_1 x_2] Id [x_1 x_2]^T )-[x_1 x_2] [0 b_2]^T$ To find minimum we have to ...
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1answer
34 views

Recover the inverse after interative solution of a linear system

I have solved the linear system $\mathbf{A} \mathbf{x} = \mathbf{b}$ with an iterative solver. The problem is well-posed ($\mathbf{A}$ is invertible, $\mathbf{b} \ne \mathbf{0}$, blah blah blah). ...
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1answer
35 views

Find $\min x^TAy+b^Tx+c^Ty$ subject to $1^Tx=1^Ty=1,x\ge 0,y\ge 0$

The problem seems to be easy but I can't find a solution :( Problem: Given $A\in\mathbb{R}^{m\times n}, A\ge 0, b\in\mathbb{R}^{m}, c\in\mathbb{R}^{n}$. Minimize $f(x,y) = x^TAy+b^Tx+c^Ty$ subject to ...
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1answer
39 views

minimizing linear combination of inner products

$\mathbf{x},\mathbf{y_1},\mathbf{y_2}\in \mathbb{R}^{m}$ and $\alpha_1,\alpha_2 \in \mathbb{R}$. Also $\|\mathbf{y_1}\|_2 = \|\mathbf{y_2}\|_2 = 1$ and $\alpha_1\geq\alpha_2\geq0$. How should we ...
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A good enough solution for the 0-1 multiple knapsack problem

Can you please explain, or point to an easy explanation of a good enough solution to the 0-1 multiple knapsack problem? This is a single constraint problem so only one-dimensional weights are to be ...
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Multivariable Optimization

I'm struggling to figure out how to go about this question: You are to produce a concrete box with an open top with a volume of 1$m^3$, having a wall and base thickness of $2$cm, by pouring concrete ...
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Arc length contest! Minimize the arc length of $f(x)$ when given 3 conditions.

Contest: Give an example(s) of a continuous function $f$ that satisfies three conditions: $f(x) \geq 0$ on the interval $0\leq x\leq 1$; $f(0)=0$ and $f(1)=0$; the area bounded by the graph of $f$ ...
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1answer
28 views

Using Lagrangian multipliers to check the solution

suppose we have an objective function $f(x)$ that we want to maximize subject to constraint $g(x)=c$. What I do next is set up Lagrangian optimization as follows and take the derivative ...
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1answer
28 views

how to find the solution of this cost function?

I have the following cost function. $J = \sum_{i=1}^N a\, Trace(W^TX_iW) - b\, Trace(W^TY_iW)$ Where $X_i$ and $Y_i$ are symmetric matrices, $a$ and $b$ are scalars. How can I find W?
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Linear optimization with constraint on the sum on “consecutive” indexes

I am struggling with the following optimization problem. Given $\{\lambda_i\}_{i=1\ldots k} > 0$ consider $\max_{x_i \in A} \sum_{i=1}^k \lambda_i x_i$ where $A = \{x_i \in \mathbb{R} : ...
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1answer
29 views

Eigen value system? solution

I have the following system. $AW = \lambda B W$ Where $A,B,W$ are matrices and $\lambda$ is a scalar. The values of $A,B$ and $\lambda$ are known. $B$ is invertible. This is a solution to an ...
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Unique minimizer of $\|x\|_{\mathcal A}$ subject to $\Phi x=\Phi x_0$

I'm trying to understand the proof of Lemma 2.3 of the paper Simple bounds for recovering low-complexity models. The authors want to find bounds on the numbers of rows $m$ of $\Phi$ to ensure that ...
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How to know the rate of convergence of a majorization - minimization algorithm?

The basic idea of majorization-minimization (MM) principlein optimization is to convert a hard problem (for example, non-smooth) into a sequence of simpler ones (for example smooth). To minimize ...
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1answer
34 views

Founding maxima or minima to a function

$g(x)=e^{x-1}+x^{2}-3+2x$ How can I find when this function has maxima and minima? I found the derivative but I can't understand how find the solution when $g'(x)=0$. It's high school material.
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Network/graph theory -acyclic problem [closed]

Consider an acyclic directed network of n vertices, labeled $i=1...n$, and suppose that the labels are assigned such that all edges run from vertices with higher labels to vertices with lower. Show ...
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Remove minimal number of elements

Given the numbers $ 1,2,..,2n + 1 $ , $ n > 0$ , remove as few numbers as possible so that among the remaining numbers no number is equal to the sum of two other numbers. After removal of first ...
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Maximizing sum of sign functions

I want to solve the following optimization problem: $$ \max_{{\bf u}} \sum_{i=1}^N \left(a_i \textrm{ sign}({\bf u \ . x_i}) \right), $$ where ${\bf u}$ and ${\bf x_i}$ are $p\times1$ vectors, $a_i ...
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Analyze the variation of this function $f(x)=x \exp[a+b(x-1)] \, E_1[a+b (x-1)]$ w.r.t. $x$

Please, I need to analyse the variation of the following function w.r.t. $x$ : $f(x)=x \exp[a+b(x-1)] \, E_1[a+b (x-1)]$, where $E_1[a+b (x-1)]$ is the exponential integral, $b>a$, $a>0$, ...
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2answers
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Constraint minimization of sum of Non-symmetric matrices

I am trying to find closed form solution to following problem \begin{equation} \begin{array}{c} \text{min} \hspace{4mm} \big(\lambda_1\left( \mathbf{y}^T V^{(1)}\mathbf{x} \right)^2 + ...
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Assessing the “Quality ” of a solution sobtained by using lagrangian multipliers

I have an ill-defined question. I work in machine learning and am trying to learn the parameters of a model, such that my problem amounts to constrained optimization. That is, I have some training ...
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Local optimization with multiple starting values \approx global optimization?

I need to find the minimum/maximum of a nonlinear function but the constraints in the optimization problem make it tougher to solve (not a convex problem). I don't have a good global optimization ...
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1answer
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How many additional crews should be brought in to minimize the cost of an oil spill?

An oil spill has fouled $200$ miles of Pacific shoreline. The oil company responsible has been given $14$ days to clean up, after which a fine will be $10000$ \$/day. The local cleanup crew cleans $5$ ...
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1answer
20 views

How do I set a lower bound to the solution's norm in a QP problem

I know that LASSO-regularization can be used to scale into an $L_1$ upper bound for a solution. But what if I want the norm to be within a specific range $[a,b]$? ie. I also want to set a lower bound? ...
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Wrong ILP solution with LPSolve (simple example)

I added the following example into LPSolve and found a strange issue. I don't want S1 and S2 to overlap within certain margins. ...
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1answer
23 views
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Reducing an I-optimal problem to a Pareto-optimal problem

Given a set $\textbf y\subset\mathbb R^2$, let $y = (y_1,y_2), y'=(y'_1,y'_2)\in\textbf y$ be elements of that set, let $\alpha_{min}\in\mathbb R$, $\alpha_{min}<1$, $\alpha_{max}\in\mathbb R$, ...