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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Semidefiniteness of the Hessian and optimization

This question is for sure a duplicate, but different users seem to give different answers. The question is: suppose you find that the Hessian matrix for a function $f(\textbf{x})$ is semidefinite ...
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25 views

Finding the critical points of $f(x,y) = x y^2 - x^2 y + x y$

Trying to find the critical points of $f(x,y) = y^2x - yx^2 + xy$. I took partial derivative with respect to x, so $F_x = y^2 - 2xy + y$ $F_x = y(y - 2x + 1)$ Then with respect to y, $F_y = 2xy - ...
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27 views

Variational optimization problem with several constraints

I am looking for solutions, approaches or hints to solve this variational optimization problem: Let $f:\mathbb{R}\rightarrow [0,\infty)$ be such that $\int f(x)\,dx=1$ and $\int x\,f(x)\,dx=0$ and ...
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Find the largest range of values of the step size α for which the algorithm is globally convergent

Consider a fixed-step-size gradient algorithm applied to the following function f. $$ f(x)=1+2x_1+3(x_1^2+x_2^2 )+4x_1 x_2$$ Find the range of values of the step size α for which the fixed step ...
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Question about separability of convex envelopes

Given a function $f(\boldsymbol{x})$ defined on the hypercube $\boldsymbol{x} \in [0,1]^n$. Suppose $f(\boldsymbol{x})$ can be expressed as $f(\boldsymbol{x})=c(\boldsymbol{x})+g(\boldsymbol{x})$, ...
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Recovering the optimal primal solution from dual solution

I'm having trouble finding the optimal primal solution of a particular problem from its dual solution. Primal: $\texttt{Maximize} \ \ 10 x_1 + 24 x_2 + 20 x_3 + 20 x_4 + 25 x_5$ Subject to $x_1 + ...
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43 views

The limit of an argmax function

Consider the function $$f(x,n)=x(A-cn)\frac{1-x^{n}}{1-x^{n+1}},$$ where $n\in\{1,2,...\}$, $x\in\mathbb{R}_{\geq0}$, and $A>c>0$. Let $n^{*}(x)=\mbox{argmax}_{n}f(x,n)$ denote the $n$ that ...
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Linear Integer Optimzation Problem (scheduling problem)

Does any of you know how to get this done?
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25 views

Where can I find an algorithm to compute $\min_{x \in \Delta_n} \langle g , x - y \rangle_1 + c\lvert x - y\rvert_1^2$?

I wish to compute the minimizer of $$ \min_{x \in \Delta_n} \langle g , x - y \rangle + \frac{c}{2}\lvert x - y\rvert_1^2$$ where the subindex $1$ indicates that the norm is the $1$-norm and ...
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optima; value of a function

Suppose we have the following function $$Err(f) = \frac{1}{2}E|Y-f(X)| = P(Y=1,f(X)=-1) + P(Y=-1,f(X)=1),$$ where $Y, f(X) \in \{-1, 1\}$. How can find the optimal value of the above function, Err? I ...
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42 views

Minimize a non-convex function subject to linear dynamics constraint

I want to solve the following problem: $$\min\limits_{\bf u} \frac{\bf c^T {\bf x} (T_f)}{\| \bf c\|\|{\bf x} (T_f)\|}$$ subject to $$\dot{\bf x} (t) = A {\bf x}(t) + B {\bf u}(t)$$ $$x(0) = x_0$$ ...
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8 views

Calculating minimum optimization of wall panel leftovers in excel

6,4,2,8,8,3,8,9 Group the following values based on the constraints listed below. Each value can only be used once. There is no limit to the amount of values in a group. The sum of each group must ...
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1answer
52 views

If g(x) is the maximum value of f(t)

Let f be continuous on [a,b] and define a function g(x) on [a,b] as follows g(a)=f(a) and for a $\lt\ $x $\le\ $b then g(x) be the maximum value of f(t) on [a,x]. Prove that g(x) is continuous of ...
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229 views

I am trying to maximize an exponential function [closed]

I am looking for the value of $x$ that will maximize $y$ in the following equation $$ y=e^{-(x-a)^2/b} $$ where $a$ and $b$ are constants. Any help is appreciated
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52 views

linear approximation with respect to L1 norm

I am trying to solve this problem: Find the best $L^1$ linear approximation of $e^x$ on [0,1] i.e. minimize $\int_0^1|e^x-\alpha-\beta x| dx$ any hints how to proceed
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Explanation of strategies in infinite horizon dynamic programming problem

My question is regarding the Bellman equation regarding strategy $\sigma^{(1)}$ on the last 2 lines (I have attached pictures of the book below). If we know that all future states will have value of ...
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1answer
42 views

The meaning of 'worst case'

When giving bound on convergence rate, complexity and so on, people sometimes will specify it by 'worst case'. What is the meaning of 'worst case'?
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22 views

Finding a solution to matrix equation occurring inside an optimization problem

As a part of an optimization problem (while equating the derivative of the cost function to 0), I'm getting the following expression. $$-2XX^TC + 2XX^TACC^T + \gamma GA = 0,$$ where, $X, C, G$ are ...
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8 views

How to optimize this types of problems?

Given that $min [ t_{f} - t_{0} ]$ such that $x(x'(t_{0}),y'(t_{0}),z'(t_{0}),t_{0}) = 0$ $y(x'(t_{0}),y'(t_{0}),z'(t_{0}),t_{0}) = 0$ $z(x'(t_{0}),y'(t_{0}),z'(t_{0}),t_{0}) = 0$ $x(t_{f}) = ...
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Can Three Equilateral Triangles with Sidelength $s$ Cover A Unit Square?

A previous question on the site asked for a short proof of the fact that three equilateral triangles with unit side length cannot be arranged to cover a square with unit side lengths. Given the truth ...
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38 views

Rate of Convergence of complicated sequence with interactions

I have been working on a problem where the sequence turns out to be so complex that i am unable to find its convergence rate with necessary and sufficient conditions on the parameters.After working ...
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33 views

The Euler-Poisson equation

$$\int_{0}^\pi (x''^2+4x^2) dt$$ $$ x(0)=x'(0)=0; x(\pi)=0;x'(\pi)=sinh(\pi)$$ This is The Euler-Poisson equation, i found: $$\frac {\partial f}{\partial x}-\frac {d}{dt} \frac{\partial f}{\partial ...
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How to maximize shipping box volume

Earlier last week I realized I needed to ship a large volume of things domestically. Of course, I decided that I wanted to do so as cheaply as possible. I first looked at USPS standard post rates. I ...
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27 views

lagrange method, linear constraints and unique global maximum

My book in linear programming states two things that I do not understand. We are working with the lagrange method with linear constraints.: From multivariate calculus we have that at a critical ...
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16 views

Help Required in eigenvectors for sparse matrix?

I have a large sparse matrix A(~400000,~400000) . If I randomly remove few rows from the matrix will there be considerable change in the eigenvalues and the eigenvector's compared to eigenvector's of ...
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1answer
24 views

Maximum Volume of a rectangular box in ellipsoid

This is the problem I am working on: Find the maximum volume of a rectangular box that can be inscribed in the ellipsoid: $x^2/25 + y^2/4 + z^2/49 = 1$ with sides parallel to the coordinate axis I ...
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28 views

Radio factory linear program

I need a help with this exercise. I’m supposed to write a liner program for the problem below and then solve it using simplex method, but I’ don’t know how to include all the factors into variables. ...
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22 views

Hammersley–Chapman–Robbins bound for Rice distribution

I am trying to evaluate the Hammersley–Chapman–Robbins bound for the variance of an unbiased estimate $\hat{\alpha}$ of $\alpha$ (for a given $\sigma$) for the Rice distribution: $$p(x|\alpha,\sigma) ...
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How to Solve this maximization Problem?

You are given two s: N and K. Lun the dog is interested in strings that satisfy the following conditions: The string has exactly N characters, each of which is either 'A' or 'B'. The string s ...
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Finding Maximum and Minimum for f(x,y)

The problem I am working on is: Find the maximum and minimum values of the function: $f(x,y) = -3x^2 - 14xy - 3y^2 -8$ on the disk: $x^2 + y^2 \leq 4$ The $-14xy$ term is severely throwing me for a ...
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How to minimize $x^2+4xy+5y^2-4x-6y+7$ without using calculus

I would like to find the smallest possible value of the function $$f(x,y)=x^2+4xy+5y^2-4x-6y+7$$ without taking any derivatives. My thoughts were to complete the square on both $x$ and $y$ and ...
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How to Solve this Linear Programming Problem?

$$\max[Z(x,y)=x+y]$$ $$-x+y\le 1$$ $$x\ge 0$$ $$y\ge 0$$ What i have done so far ? I tried simplex method , but i can't stop iterating . It really seems like a live lock . So how can i solve ...
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Maximizing the area of a triangle with its vertices on a parabola.

So, here's the question: I have the parabola $y=x^2$. Take the points $A=(-1.5, 2.25)$ and $B=(3, 9)$, and connect them with a straight line. Now, I am trying find out how to take a third point on ...
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Quasiconcavity of $g(x)=xf(K-x)$

The function $f(x)$ is strictly increasing, finite, positive and twice continuously differentiable on the compact interval $[0,K]$, and $f(0)=0$. I'm trying to either find a counterexample to, or a ...
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Question of proof of maxima related to quadratic form

Suppose $\bf{A}$ is a symmetric positive-definite matrix and now we want to maximize function $f(\bf{x})=\bf{x}^\rm{T}\bf{A}\bf{x}$ with restriction $\bf{x}^\rm{T}\bf{x}=\rm{1}$. Using Lagrange ...
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Showing CP-rule is not optimal for $P \mid p_j = 1, \text{ intree} \mid \sum C_j$.

We are asked to find a counterexample that shows that the Critical Path rule is not optimal for $P \mid p_j = 1, \text{ intree} \mid \sum C_j$. However, after trying for two hours, I don't think I'll ...
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measurable selection for almost-minimizers of an irregular functional

I'm faced with the following problem: I have a functional $F$ defined on $H^1$ curves $[0,1] \rightarrow \Omega \subset \mathbb{R}^n$ where $\Omega$ is either a compact subset or the whole ...
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1answer
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Simply function F and find alpha for which F will be min

I have point coordinates like [x, y], where x and y are positive natural numbers. I need to ...
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2answers
56 views

Find all minima and maxima of $f(x) = (x+1)^{1/3}(x^2-2x+1)^{1/5}$

Find all minima and maxima of : $f(x) = (x+1)^{1/3}(x^2-2x+1)^{1/5}$ I feel kind of ashamed by posting such a question but I am not able to get the right answer. I've tried to do it the Fichtenholtz ...
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Find max/min of $f(x,y,z)$ on closed unit ball $B$ in $\mathbb{R^3}$

$f(x,y,z) = 3x - 2y + z$ Let $B$ be a closed unit ball in $\mathbb{R^3}$, find the max/min of f on $B$. We first need to observe $(a)$ the behavior of $f$ in $B^0$ $(b)$ the behavior of $f$ on ...
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Combinatorics : Minimization of the number of common objects between subsets

Let's consider the following setup. I have access to $N$ objects. Thanks to these objects, I can build up sub-packets containing $k$ such objects. I know that there exists a total of $\displaystyle ...
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Can we not determine the minimum and maximum value of a function by just obtaining the value of the function at the critical points?

Can we do the above or is it neccessary to find the sign of the second derivative. I find no problem with any of the above but the answer of the following question left me confused. "The real number ...
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find min max of function on unit ball

$g(a,b,c)=3a-2b+c$, B is a closed unit ball in $\mathbb R^3$. Find the max/min of g on B. What is the behavior of $g$ on the open unit ball, and the boundary of the unit ball? I think the unit ball ...
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Maximum of a 3-variable equation with constraints on the variables

So I need find the maximum of the equation $$ (10x + 10)[ (0.03y + 0.1) (0.1z + 0.5) + 1] $$ given that $0 \leq y \leq 30, \; x \geq 0,\; z \geq 0$, and $x + y + z = 100$. I'm not exactly sure how ...
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Reference material on Alternating Minimization Algorithm

I am looking for some good reference material (book/paper) for learning Alternating Minimization Algorithm. Any recommendation from optimization experts will be much appreciated. Thank you.
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There exists a descending chain of symmetry groups from a formal language string down to its smallest grammar.

Background. Let $\tau \in G_i$ be a permutation in the symmetry group of the smallest grammar $g_i$. Then $\tau$ permutes each set of positioned (within $g_i$) symbols $\{x_1^{(1)}, x_1^{(2)}, \dots, ...
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What is the derivative of a matrix w.r.t itself?

what is the derivative of \begin{equation}\partial \frac{x^TVx}{\partial V} \end{equation} where V is a matrix and x is a vector. In general what is the right way to calculate matrix derivatives w.r.t ...
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Solving non-linear optimization using generalized reduced gradient (GRG) method

Consider the following elementary maximization problem: \begin{align} f{=}\mathrm{argmax}_{y_{l,c}, p_{l,c}}~\sum_{l=1}^{L}\sum_{c=1}^{C} y_{l,c}\text{log}_2\left(1+\frac{p_{l,c}}{I_{l,c}}\right) ...
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Question about duality in nonlinear optimization

Let $f(x)$ and $h(x)$ be functions from $\mathbb{R}^n$ to $\mathbb{R}$ and consider the minimization problem $$ {\rm minimize} ~~~ f(x)$$ $$~~~~~~~~~{\rm subject ~to}~~h(x)=0.$$ Suppose the minimum is ...
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2answers
99 views

Optimization of parameter for recursive Cauchy sequence

I have the following recursive sequence I'm analyzing: $$V_0 = 50, V_1 = (1-10k)V_0,$$ $$V_{n+1} = (1-10k)V_n - 5kV_{n-1}$$ where $k > 0$ is a parameter that I'm investigating by running ...