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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Stuck formulating constrained optimization problem with Simplex

I have an exercise to solve, and it is a constrained optimization problem. Here it is: "A company makes large championship trophies for youth athletic leagues. At the moment they are planning ...
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Matrix transformation for linear state-space systems

In http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-241j-dynamic-systems-and-control-spring-2011/lecture-notes/MIT6_241JS11_lec12.pdf on pages 11-12 it is said: For a stable ...
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2answers
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Precalc Optimization?

I need help with an optimization problem. I have a rectangle space being fenced. Three sides are fenced with a material costing 4 dollars and the last side costs 16 dollars. I was given that the area ...
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2answers
67 views

League of Legends optimal items

In the popular game League of Legends, your effective amount of hit points ($E$) against physical damage is a function of your actual hit points ($H$) and the amount of armor ($A$) you have. $$E = ...
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14 views

Fixed point of a value function

Suppose that we solved a dynamic optimization problem and figured out that the value function takes the form: $v(p)=h(p)+\alpha(p)v(g_1(p))+(1-\alpha(p))v(g_2(p))$ where we have explicit expressions ...
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41 views

Minimum variance, fixed mean , discrete random variable

Consider the ordered set $\mathcal{S}$ $=$ $\{0,a_i,a_2,\ldots,a_n\}$, where $a_i$ are all stricly positive real numbers and $a_i< a_{i+1}$ forall indices i. What is the random variable $X$ which ...
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1answer
80 views

Find the Lagrange multipliers with one constraint: $f(x,y,z) = xyz$ and $g(x,y,z) = x^2+2y^2+3z^2 = 6$

Where $f(x,y,z) = xyz$ and the constraint is $g(x,y,z) = x^2+2y^2+3z^2 = 6$ I have tried this problem like three or four times and not gotten the solution, I even asked this question once and got the ...
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23 views

Maximum and Sets of vertex-disjoint paths in a not-directed graph

Let's consider a weighted graph $G = (V,E)$ not directed. In this graph, there are several sinks $S$, which are vertices. Let's consider one vertex $V$ of this graph (which is a source). The problem ...
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14 views

Vertex-cover-like problem with reduction to maximum flow

I am trying to solve the following problem: Solve the following problem by reducing it to the computation of a maximum s-t-flow: Let G be an undirected graph, $c:V\rightarrow\mathbb{Z}$ and ...
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1answer
543 views

Using gradient descent and Newton's method combined

I have this function $f(\mathrm{X})$ where $\mathrm{X=A+B+C}$ where $\mathrm{A}$ is a diagonal element with variable $a$ on its diagonal. $\mathrm{B}$ is another diagonal matrix with variable $b$ on ...
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2answers
69 views

Finite Difference Approximation of Derivative [closed]

I want to build a finite-difference approximation of this derivative: $\frac{\partial^2T }{\partial x^2}$ There are given an error of approximation: $O(\Delta x^{4})$ and nodal values of function:$ ...
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1answer
71 views

Method to find the extremal values of $xyz$ subject to $x^2+2y^2+3z^2=a$

This question has been asked before but I want to lay out my method and get feedback on reasoning and process this took me a long to put together as I am new to the formatting: Let the function $f$ ...
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2answers
153 views

Global maximum and minimum of $f(x,y,z)=xyz$ with the constraint $x^2+2y^2+3z^2=6$ with Lagrange multipliers?

The global maximum and the global minimum of the function $f(x,y,z)=xyz$ with the constraint $x^2+2y^2+3z^2=6$ can be found using Lagrange multipliers. $\nabla f = \lambda \nabla g$ ...
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0answers
35 views

Finding the Maximum and Minimum values w/constraint [duplicate]

I apologize I have asked this question before but it died and I just got around to working it out based on the suggestions so here it is. Let the function $f$ be defined as $f$($x$,$y$,$z$) $=$ ...
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1answer
33 views

Proportion in sets

We have $3$ sets of positive integers. $$A = \{x_1,y_2,z_3\},\quad{} B = \{x_2,y_2,z_2\}, \quad{} C= \{x,y,z\}$$ Which proportion do we use for adding $A$ and $B$ ($x_1+x_2$ and so on), so the ...
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24 views

A question on Edgeworth Expansion

I'm working Edgeworth Expansion. I couldn't understand one thing . Can you help me about that please. $$Z= \frac{\sqrt {n} (\bar {x} -\mu)}{\sigma}$$ converges in distribution to N(0,1) I have ...
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1answer
39 views

Rate distortion function with infinite distortion

I am working through the problems in Elements of Information Theory by Cover and Thomas and have come across the following problem I couldn't answer. The problem is to find the rate distortion ...
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2answers
54 views

Without Extreme Value Theorem, how do we find absolute extrema?

I have to find and classify the critical points of the following functions and then state which relative extrema are absolute extrema. $$f(x,y) = x^3 - y^3 - 2xy + 6$$ $$f(x,y) = xy + 2x - ...
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2answers
287 views

Simplex method - multiple optimal solutions?

I have to solve this optimization problem: $$\begin{array}{ll}\text{minimize} & z= x_1 - x_2 + 3x_3\\\\ \text{subject to} &x_1-x_2+x_3-x_4=2\\ & 2x_1-2x_2-x_3+x_5=0\\ & x_1, x_2, x_3, ...
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1answer
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Tree-width of a quadratic pseudo-Boolean function

A pseudo-Boolean function $f : \mathbb{B}^n \mapsto \mathbb{R}$ is of the following form. $$ f \left(x_1, \ldots, x_n\right) = \sum_{S\subseteq V} c_S \prod_{j \in S} x_j $$ Here $c_S \in ...
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3answers
100 views

max of $e$ with $a+b+c+d+e=8$ and $a^2+b^2+c^2+d^2+e^2=16$ [closed]

Given that a,b,c,d,e are real number such that: $\begin{cases} a+b+c+d+e=8\\ a^2+b^2+c^2+d^2+e^2=16 \end{cases}$ determine the maximun value of $e$. I started like that : ...
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65 views

Constrained LQR with a fixed terminal state. Can MPC be applied to this problem?

I am interested in solving the constrained LQR problem with discrete finite time when the target $x$ value is given, but the final $u$ could be anything s.t. constraints. $$\text{minimize }J = ...
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1answer
17 views

Min and max with two constraints (Lagrange Multipliers)

I should find the minimum and maximum values of f(x,y,z)=x+y+z given the constraints x^2+y^2+z^2=1 and x−y-z=1 I found here a same exercise, but I don't know how the define the value of x, y, z.
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49 views

Dynamic “Assignment Problem” (Hungarian Algorithm Extension?)

TL;DR: Trying to optimize assignments using Hungarian algorithm, but cannot determine costs until all assignments have been made due to dependencies. Using the terminology from Wikipedia's Assignment ...
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1answer
54 views

What is the difference between moving-horizon DP and MPC?

What is the difference between moving-horizon DP (dynamic programming) and MPC (modelbased predictive control)? In both cases, the system input at time $t$ is determined by solving a finite-time ...
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9 views

Arrow-Debreu Equilibrium pricing

I have this problem in asset pricing that I don't know how to solve. Here it is: Consider an economy with a complete set of Securities and $N$ states of the world Tomorrow. Assume that there are two ...
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Determine if fitted line is actually one line

I am trying to fit multiple lines through many data points in 3d space. My working method is sequential RANSAC, which now is fast enough and fits some lines, but produces some lines that don't fit one ...
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Indefinite Boolean Quadratic Programming: number of minima

The Boolean Quadratic Programming problem is defined as: $\min_{x} f(x) = x^TQx + c^Tx$ s.t. $ x \in \{0,1\}^n$ It is a well-studied NP-Hard problem with many approximation algorithms proposed. I ...
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How to distinguish as Linear or Non-linear constraints

In optimization problem $$\arg\max_{\substack{l_1 \in [0, 1],\cdots,l_{M} \in [0,1]}} T$$ I have $m=1\cdots,M$ constraints such that $$\ln \left[a_m\right] + \ln \left[ \left(\frac{l_m}{l_m + ...
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Value of Lagrangian Multiplier

I have a two dimensional optimization problem of the form $$ v = \max_{x,y} f(x,y)+g(x,y) $$ Both $g,f$ are concave and continuously differentiable. Assume the solution can be reached by first order ...
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How to calculate the hessian of a matrix function

I am estimating a model minimizing the following objective function, $ M(\theta) = (Z'G(\theta))'W(Z'G(\theta))$ $Z$ is an $N \times L$ matrix of data, and $W$ is an $L\times L$ weight matrix, ...
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1answer
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Transforming a $0$-$1$ knapsack problem into the standard form

I have the following $0$-$1$knapsack problem: $$\begin{align*} &\mathrm{Max} : \quad z= 3x_1 -4x_2+5x_3+7x_4-6x_5+x_6\\ & \mathrm{subject\ to}: -2x_1 +x_2 +10x_3 +3x_4 -5x_5+12x_6 \leq 4 ...
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4answers
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Compute the minimum value of $a^n + b^n + c^n$ subject to $a^2 + b^2 + c^2 = 1 $

Assume that $a,b,c$ are non-negative real numbers and $n$ is a natural number $n \ge 3$. What is $f(n)=$ the minimum value of $a^n + b^n + c^n$ ? I find ; $$f(3) = \frac{1}{\sqrt{3}}\qquad ...
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28 views

Is this Feasibility problem NP-hard?

I am trying to solve a combinatorial optimization problem (a feasibility problem) and I have very little idea of solving such problems. The problem is as follows: Solve for $\phi$; \begin{equation} ...
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1answer
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Asymmetric Least Squares Conversion from Equation to Matrix

In solving for asymmetric least squares baseline correction as defined in the article by Eilers and Boelens, the general equation is defined as: $$S = \displaystyle\sum_i w_i (y_i-z_i)^2 + \lambda ...
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Maximizing distance between points

I asked a similar question on SciComp, but it is a little out of the domain, so I thought I'd give it a try here as well. Give n points, I would like to place them in a periodic box (periodic such ...
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Shortest distance between two lines in 3-dimensional space [closed]

Can someone explain to me how to solve this question? Find the shortest distance between the lines $L_1 = \left\{t \begin{bmatrix} 1\\ 1\\ 1\end{bmatrix} : t \in \mathbb{R}\right\}$ and $L_2 = ...
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1answer
34 views

Finding extremas

Finding this quite tricky. $$e^{x}(\cos(x)-\sin(x))=0$$ Solve for $x$. This is an derivative of the original function: $$f(x)=e^x\cos(x)$$ And I am trying to find the extremas.
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KKT conditions with two inequality constrains

I need to minimize $f(x,y,z)=x^{2}+2y^{2}+3z^{2}$ subject to \begin{align*} &x-y-2z\leq 12\\ &x+2y-3z\leq 8. \end{align*} So I wrote the lagrangian of $f$. \begin{align*} ...
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455 views

Maximizing a function while minimizing one part of the same function

I have a function with two variables say $f(x,y)=f_1(x)-f_2(x,y)$ where $f_1(x)$ is the well known quadratic-form function in x while $f_2(x,y)$ is also a quadratic function in both x and y but not ...
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Optimal control with state-dependnet solution

I'm trying to solve the following control problem $$ \begin{eqnarray*} \max & & \int_{0}^{T}\sum_{i=1}^{2}-c_{i}(x_{i}-u_{i})^{+}\\ s.t. & & ...
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1answer
473 views

Find pressure in a sinusoidal function

Tiffany is a model rocket enthusiast. She has been working on a pressurized rocket filled with laughing gas. According to her design, if the atmospheric pressure exerted on the rocket is less than 10 ...
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What is the purpose of the 1/2 factor in SVM minimalisation equations?

The objective function for Support Vector Machines is in most sources formulated as: $\min\limits_{w,w_{0}} \frac{1}{2}||w||^2 + C\sum\limits_{i=1}^{N}\xi_{i}$ What is the signifance of the ...
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2answers
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utility function question from my textbook

Suppose there are two goods with prices $ p₁ = 2, p₂ = 5, $ the income is $ M = 40 $ and the utility function is $ U (x₁, x₂) = (x₁)^⅓ . (x₂)^ ½, $ Find the optimum consumption plan. Attempt: I do ...
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SQP with non linear constraints

I am asked to solve a $SQP$ problem using Matlab, in the instructions for writing the code I am supposed to create two functions one for the function $f$ to minimize and one for the constraints $c_i$, ...
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221 views

Extreme of $\cos(A)\cos(B)\cos(C)$ in a triangle without calculus.

If $A,B,C$ angles of a triangle, show extreme value of $$\cos(A)\cos(B)\cos(C)$$ I have tried using $A+B+C=\pi$, and applying all and any trig formulas, also AM-GM, but nothing helps. On this topic ...
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15 views

subdifferential of $\max_{i=1,\cdots,k} x_i+\frac{1}{2}\|x\|_2^2,\ \ \ x\in \mathbb{R}^n$

How to find the subdifferential of $$f(x) = \max_{i=1,\cdots,k} x_i+\frac{1}{2}\|x\|_2^2,\ \ \ x\in \mathbb{R}^n$$ My derivation is: $\nabla \frac{1}{2}\|x\|_2^2=\nabla \frac{1}{2}x^Tx=x$ ...
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19 views

Why do not we check second derivative test when solving constrained problems

when we are max/min an objective function z(x,y) subject to a constraint for example $$x^2+y^2\leq1$$ Then the solution will be in 2 steps : First : getting critical points inside this domain . ...
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16 views

Diagonal Newton's method for unconstrained optimisation.

Assume you are minimising a convex function $f$. The function is twice-differentiable. The well-known Newton's method consists in starting form some point $x_0$ and then using the iteration below. $$ ...
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1answer
32 views

Optimizing overlap between two reference frames

Let me share this little optimization problem with you: I have two orthonormal sets of vectors on $\mathbb{R}^3$, related by some Euler angles $(\alpha,\beta,\gamma)$ (corresponding to those of the ...