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Mathematics Stack Exchange

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.

In mathematics, computer science, economics, or management science, mathematical optimization (alternatively, optimization or mathematical programming) is the selection of a best element (with regard to some criteria) from some set of available alternatives.

An optimization problem can be represented in the following way: given a function $f:A\to\mathbb{R}$ from some set $A$ to the real numbers, we want to find an element $x_0\in A$ such that $f(x_0)\le f(x)$ for all $x \in A$ ("minimization") or such that $f(x_0)\ge f(x)$ for all $x \in A$ ("maximization").

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Home work on linear optimization problem

I'm sorry I could not upload my homework because I'm too new to post images. The image can be obtained here: http://i40.tinypic.com/2u5rl80.png. I'm also sorry to ask such a big question. The instructor didn't provide enough materials in class for…
user76300
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Is $g(x) : = \inf_{i \in I} f_i(x)$ convex or not?

Let $\{f_i : i \in I\}$ be a set (finite or infinite) of functions from a convex set $\mathcal{D}\subseteq \mathbb{R}^n$ to $\mathbb{R}$ which are convex and bounded on $\mathcal{D}$. Let $g(x) : = \inf_{i \in I} f_i(x)$. Is $g$ convex? Why or Why…
shk910
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Computing if a given grid can be escaped from

I am trying to see if there is a known solution to my problem, possibly lacking the right search terms or similar, The problem is given a circular or square grid of points size $X$, can you escape all the wires Size $Y$ and clearance between the…
Reroute
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Given $y = x^3 − 2x$ for $x \geq 0$, find the equation of the tangent line to $y$ where the absolute value of the slope is minimized.

I tried finding the derivative of this, and promptly got $y=$ about $0.816$, but I have no idea how to put that into equation form or if I'm even correct.
Ayanoria
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need help with raw distance function

I need help with the following function: $$ \rho(x,\theta)=\min_{\lambda\in [0,\infty]} d(x ,x+\lambda[\cos \theta, \sin \theta]^T),$$ such that $$ x+\lambda[\cos \theta,\sin \theta]^T\in \bigcup_i \mathcal{W}\mathcal{O}_i$$ how to write the 'min'…
CaTx
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Inquiry about the strongly convex and conjugate function

The easy statement tells me that the conjugate of a strongly convex function has a lipschitz continuous gradient. But I am thinking about the how could a conjugate of a function is strongly convex. I cannot figure this out. For example. For the…
Sustcer_Shuai
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Revised Simplex: reduced cost and related constraint

In the revised simplex method, you can get the reduced costs straightforward from the tableau. I know which they are, but I don't know which reduced cost I should "relate" to which constraint. Will the leftmost value relate to the 1st constraint,…
Izabela
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Understanding this definition of Polyhedron edge

Let $P=\{x:\;a_i^Tx\leq b_i,\;\forall i\}$ be a Polyhedron in $\mathbb{R}^n$. Let $x$ and $y$ be two distinct basic feasible solutions. Recall $I_x =\{i:\;a_i^Tx\leq b_i \}$ Suppose that rank $\{ a_i: i \in I_x ∩ I_y \} = n − 1$. Then the line…
user619755
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Sufficient conditions on $f$ to obtain that $\arg\max_{x \in K}f(x) \cap K' \neq \emptyset$ ($K$, $K'$ compact)

Let $f : \mathbb{R}^n \rightarrow \mathbb{R}$ (for some $n \in \mathbb{N}^\ast$) be a continuous function and $K, K'$ be two compact subspaces of $\mathbb{R}^n$ such that $K' \subset K$. I was wondering if there exists simple sufficient conditions…
deeppinkwater
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Necessary condition for constrained minimization

The first-order necessary condition for the constrained problem $\underset{x \in \Omega}{\min} f(x)$ is $$\nabla f(x^*)^Tp \geq 0$$ for all feasible directions $p$ at the solution $x^*$. Intuitively it makes sense that there are no descent…
harisf
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Polyhedron $P=P(A,b)$ is unbounded iff $P(A,0)\neq \{0\} $

I want to show that a non-empty polyhedron $P=P(A,b)$ is unbounded iff $P(A,0)\neq \{0\} $ Definition: A Polyhedron is the set of points, satisfying the inequalities $Ax≤b$, where $A$ is a matrix and $b$ a vector. For example, looking at the…
user619755
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Is there a way to rewrite this minimax problem more efficiently?

I have reduced a certain engineering problem to a few components, A fixed vector $\mathbf{k}^{3\times 1}$, Three fixed matrices $\mathbf{M}_1^{6\times 3}$, $\mathbf{M}_2^{6\times 3}$, $\mathbf{N}^{6\times 3}$ A freely chosen 'input' vector…
Sanchises
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Characterize infinite critical points and why this argument works

I'm asked to characterize all the critical points of this function $f(x,y)=x^2+4y^2-4xy+2x-4y+3$ So as always I find the partial derivatives and solve the system $f_x(x,y)=2x-4y+2=0$ $f_y(x,y)=-4x+8y-4=0$ This system has infinite solutions(i.e…
mmendina
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hessian matrix not positive definite at a minimum?

I have a function for which I want to find the global minimum. The function is: V = 0.8173894901710712` x[1]^2 - 0.6243937065879472` x[1]^3 - 0.054869713607121895` x[1]^4 - 0.5870835253855531 x[1]^2 x[2] + 0.6314675008831476` x[1]^3 x[2] +…
dbm
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Textbooks on modern optimization (on machine learning) with exercises

I know Boyd's famous Convex Optimization, but for me it's a little bit old because it was written in 2003 and some progresses have been made during this decade. The book Optimization for Machine Learning fill the gap nicely but it's not a textbook:…
Ziyuan
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