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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Counterfeit coin problem.

Problem $N$ coins, $N-1$ equal coins and one heavier counterfiet coin. With a balance beam given we want to find the counterfeit coin. We have two different goals: minimise the expected number of ...
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Optimization Word Problem, revenue

A sorority plans a bus trip to the Great Mall of America during Thanksgiving break. The bus they charter seats 44 and charge a flat rate of 350 dollars plus 35 dollars per person. However, for every ...
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187 views

Find the minumum using Newton-Raphson

I have the following function: $f(x) = 100(x_2 - x_1^2)^2 + (1-x_1)^2$ I have to find the minimum of this function using the Newton Raphson method. The point where I have to start is $x = [1.2$, ...
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organizing rectangles on top of each other

We have some rectangles that should be organized in a number of columns. Each column height should be in the range of $[H, H+d]$ in which $d$ is a small number relative to the height of the ...
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Constrained optimisation question

Since $f$ has a local extremum at $x_1$, then surely the LHS of equation (3) always zero? If so, then isn't lambda always simply zero too? But this cannot be, otherwise the last sentence of the ...
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maximum value of $f(x)= 13x^3 + 5y^2 + 6yz + 5z^2$ [closed]

Find the maximum and minimum values of $ f(x, y, z) = 13x^3 + 5y^2 + 6yz + 5z^2 $on the solid ball $x^2 + y^2 + z^2 ≤ 1$.
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Maximization problem on a graph

Consider a graph $G(V,E)$. Let degree of each vertex be denoted to $\beta(v) < d$. Maximize the following, where $\beta(v)$ is the only variable for all vertex $v\in V$. $$ \max \sum_{(u,v)\in E} ...
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Are my linear program equations correct?

Here's the problem: "An electronics company has a contract to deliver 21,475 radios within the next four weeks. The client is willing to pay 20 dollars for each radio delivered by the end of the first ...
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Operational Research. (Ressource Management)

I am looking for a solution that i know exists already in the field of "Operational Research"... I Just can't put my finger on the name of the thing. An heuristic to solve a very common and simple ...
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Find Maximum of Function $L(p)=\prod_{i=1}^{20}\left[ (1-p)A_{i}+pB_{i}\right]$

The $A_{i}'s$ and $B_{i}'s$ are known. I seek the $p$ which maximizes $L(p)$. I thought it might be easier to maximize $\log L(p)$ instead $L(p)$, but I think it is a dead end $\log ...
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What numerical methods could I use for this argmin problem?

I wish to solve the following using Numerical Methods: $$ \bar{m} = \underset{m \geq 0}{\text{argmin}} \left( \int_a^b \left( \frac{1}{\left(\sum_{i=1}^M \left(c_i^\alpha \cdot n^2 y^{-m-1} \cdot ...
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Orthogonal Procrustes Problem

The classical orthogonal Procrustes problem concerns finding the matrix $\Omega$ which minimizes $||A\Omega-B||_{F}$ subject to $\Omega'\Omega=I$, with A and B known matrices. Let A be the identity. I ...
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189 views

Real approximation to the maximum using Laplace's method integral

The Laplace's Method states that under some conditions, it holds that: $ \sqrt{\frac{2\pi}{M(-g''(x_0))}} h(x_0) e^{M g(x_0)} \approx \int_a^b\! h(x) e^{M g(x)}\, dx \text { as } M\to\infty$ Where ...
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400 views

Definition of tangent cone in continuous optimization .

Looking at the definition of tangent cone in continuous optimization : If $M$ is a open subset of $\mathbb R^n$ $x \in M$, The tangent cone of $M$ at $x$ is defined by $$\mathbb T (M, x) = \big\{d ...
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16 views

Global consistency of constraints in a MIP program

How does a Mixed Integer Programming (MIP) solver ensure global consistency of constraints while adding an additional constraint (during branch and bound). A naive method would be to add the ...
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Internals of a MIP Solver

I would like to learn about the internals of a Mixed Integer Programming (MIP) solver. Which concepts shall I read about? Are there a couple of standard books which can be a good start?
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gradient descent with respect to a matrix

I am trying to solve a maximize a scalar function f(X), where X is a matrix. I want to solve this using gradient descent, I have taken the derivative of f(X) w.r.t X. This seems a naive question, but ...
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Selecting a subset from a set such that a given quantity is minimized

Let $A$ be is a set of some $p$-dimensional points $x \in \mathbb{R}^p$. Let $d_x^A$ denote the mean Euclidean distance from the point $x$ to its $k$ nearest points in $A$ (others than $x$). Let $C ...
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Finding the smallest sub-family of subsets needed to form a new subset

TL/DR I have a universe $U$ of items $u_i$ and a family $F$ of subsets of $U$ (call them $P_j$ ⊆ $U$). Given this family of subsets, I would like to find the sub-family $C$ ⊆ $F$ of subsets that can ...
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171 views

Simple optimization problem.

In my Calculus I assigment, I'm stuck on the following : Find $M_1=(x_1,y_1)$ on $y=5x+6$ and $M_2=(x_2,y_2)$ on $y=-(x-3)^2+4$ such that the square of the distance between $M_1$ and $M_2$ is ...
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Minimizing a linear function on a strictly convex set.

All the theorems that I know considering the uniqueness of a solution to a minimization/maximization problem requires the strict convexity/strict concavity of the objective function. But consider the ...
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35 views

How do I find a minimum of a function satisfying conditions

Given the following function $y=(x_1-2)^2+(x_2-3)^2+(x_3-5)^2$ and the following conditions $x_3-x_2\geq2$ $x_2-x_1\geq2$ How do I find $x_1$, $x_2$ and $x_3$ such that the $y$ is minimum. I ...
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Optimization of quadratic fractions

Is there an efficient way (for example to convexify, lower bound (except special cases), or something like that) to optimize quadratic fractions? For example: $$ min_x \frac{x^\top A x + x^\top B ...
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Differentiability of the Value (Support) Function

Consider the following problem, \begin{align} c(y,\mathbf{w})=\inf_{\substack{\mathbf{x} \in \mathbb{R}^n_{+} \\ \text{s.t. }f(\mathbf{x}) \geq y }} \mathbf{w} \cdot \mathbf{x} \end{align} where ...
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Prove range of f',$\{f'(x),x\in X\}$ dense in $X^*$

Let $X$ be a Banach Space and let $f: X\rightarrow \Bbb R$ be a Fre'chet differentiable function. Suppose that $f$ is bounded from below on any bounded set and satisfies $lim_{||x||\rightarrow ...
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Proof that this matrix is TUM

Suppose that A is a mxn TU Matrix. (Totally unimodular). Proof that [A I]^T (so I mean the column vector with A the first element and I the second, where I is the identity matrix) is also TUM. Can ...
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57 views

Finding Range Of Formula With Constraint

$a$, $b$, $c$, $d$ are positive such that $a^{4}+b^{4}+c^{4}+d^{4}=4$ find the range of $a^{4}+64abcd$ in the case of maximum, it's not easy for me to adjust coefficients in AM-GM.
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Calculus 1 - Optimization Cylinder

A (right circular) cylindrical can has a volume of 60π cubic inches. Suppose that the metal used for the top and bottom of the can costs 4 cents per square inch, while material for the side of the ...
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How do I find a function to minimize another function?

I am given to constants $b, n \in \mathbb{N}$. The task is to find a function $r(b,n)$ such that $\text{range}(r)=[1,b]$ and the value of $\frac{b}{r(b,n)}(n+2^{r(b,n)})$ is minimal. Do I have to ...
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Lagrange multipliers - perturbation of constraints

I have been spending some time learning about Lagrange multipliers lately. Something is puzzling me though. Reading around (also on Wikipedia) I saw multiple time the interpretation that lagrange ...
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Third and higher order optimality conditions?

In the derivation of first and second order optimality criteria for a vector $X^*$ to be a local optimum to an unconstrained problem, we ignore the higher order terms of Taylor's expansion as we ...
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Maximizing a sum of inner products

Someone asked this question on a French maths forum here and it caught my attention. The question is the following: let $(E, \langle \cdot, \cdot \rangle)$ be a Euclidean vector space. Find the ...
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How do I find out if a critical point of a function is a maximum or a minimum?

If I've found the critical point of a function defined in some constraint (perhaps using Lagrange multipliers and the like); how do I find out if it's a relative/global maximum/minimum of a function ...
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Prison break: a minimisation problem

Consider a prison with $n$ prisoners. Each cell contains a phone which can be used to call any other cell. Each prisoner has a different piece of information which, when put together, will ...
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help me find the minimum cost

an insurance company employs full and part-time staff, who work 40 and 20 hours per week respectively. Full-time staff are paid $\$800$ per-week and part-time-staff $\$320.$ In addition, it is ...
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Constraints of a linear programming problem

QUESTION Sandy Arledge is the program scheduling manager for WCBN‐TV. Sandy would like to plan the schedule of television shows for next Wednesday evening. Of the nine possible one‐half hour ...
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Minimum of two convex functions

I'm having trouble showing the below statement is true. $\hat{\alpha}=\arg\min_\alpha \frac{1}{n} \sum_{i=1}^{n} f(u_i - h(v_i, \alpha))$ where $h(v_i, \alpha)$ is linear in $\alpha$. ...
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How can Hotelling reduce the Euler-Lagrange equation in his calculus of variations mine problem?

In a 1931 paper Hotelling gives the discounted profit of a mining operation as: $$P=\int_{0}^{\infty} \dot{x} p(x,\dot{x},t) e^{-rt} \:\:dt$$ Note that this is, for the most part, a typical calculus ...
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Expressing a vector as the best linear combination of “random” vectors

Suppose I have something like: $\vec{v} = \langle 1, 2, 3, 4, 5 \rangle$ and I have a set of vectors (these are all just made up numbers): $\vec{w_1} = \langle 3, 7, -2, -4, 8 \rangle$ $\vec{w_2} ...
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Proof of a simple optimisation trick

I found this nice result in this question: Simple optimization trick, where the author claims it is easy to prove. How would the proof look like? Let $f,g:X\to\Bbb R$ be two functions where $X$ is ...
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Maximum Theorem with concavity condition

I read on Wikipedia's entry for the Maximum Theorem that: If $f$ is concave and $C$ has a convex graph, then $f^*$ is concave and $C^*$ is convex-valued. I'm working on the first part right now ...
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Jacobian in Levenberg-Marquardt for 4-Parameter equation

I am trying to fully understand how I can use Levenberg-Marquardt to minimise a 4 parameter equation. There are lots of fancy programs to do this but the documentation about the mathematics is ...
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Maximum of a sequence $\left({n\choose k} \lambda^k\right)_k$

Is there an expression for the maximum of a sequence $\left({n\choose k} \lambda^k\right)_k$ (i.e. $\max_{k\in\{0,\ldots,n\}}{n\choose k}\lambda^k)$ in terms of elementary functions of $n$ and ...
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Profit Maximization Question of a Leontief (Perfect Complements) Production Function?

This is a question from my intermediate micro economics text book. Any help is very appreciated! Given Info: Company ST (a company which offers custom travel-planning services) is a ...
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Optimal consumption policy

I start with an initial capital C and at the beginning of day $n=1,...,N$ I observe the random variable $X_n$, where $\mathbb E X_n=\mu_n$. The $X_n$ are independent. I also choose $c_n$ on day $n$, ...
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How to express coprimality as a constraint in an optimization problem over integers?

I am currently working with an optimization problem that is defined over a a set of $D$-dimensional integer vectors where each component is bounded by $M$. Let us refer to this optimization problem ...
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Extremal constant width curves on sphere

Definition Given some length $w\in\mathbb R$, I'm interested in closed convex sets $S$ of points with the following properties: For all pairs of points from $S$, the distance between them will be ...
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Prove that $\{(x, y): x\in ri (dom f), y >f(x)\}\subset ri (epi f)$

$f: R^n \to (-\infty, \infty]$ is a convex function. Prove that $ri (epi f)=\{(x, y): x\in ri (dom f), y >f(x)\}$ I have used definition of convex functions to prove that the right-side is ...
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Orthogonal Procrustes Variant

(author note: this question was also asked on mathoverflow). The orthogonal Procrustes problem seeks a matrix $M$ that minimizes $||AM-B||_F$ subject to $M^TM=I$, where $M$ is $d\times d$ and both ...
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What is the shortest distance from the origin to the intersection of $xyz=a$ and $y=bx$?

Constraints: $a,b>0$ Here is what I have so far: In order to get the shortest distance from the origin, we set $f(x,y,z)=x^2+y^2+z^2$ subject to the constrains $xyz=a$ and $y=bx$. By Lagrange ...