Yao's Millionaires' Problem: Difference between revisions

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In [[applied mathematics]], the '''fundamental theorem of linear programming''', in a weak formulation, states that the [[maxima and minima]] of a [[linear function]] over a [[convex polygon]]al region occur at the region's corners. Further, if an extreme value occurs at two corners, then it must also occur everywhere on the [[line segment]] between them.


==Statement==
Consider the optimization problem


:<math>\min c^T x \text{ subject to } x \in P</math>
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Where <math>P = \{x \in \mathbb{R}^n : Ax \leq b\}</math>. If <math>P</math> is a bounded polyhedron (and thus a polytope) and <math>x^\ast</math> is an optimal solution to the problem, then <math>x^\ast</math> is either an extreme point (vertex) of <math>P</math>, or lies on a face <math>F \subset P</math> of optimal solutions.
 
==Proof==
Suppose, for the sake of contradiction, that <math>x^\ast \in \mathrm{int}(P)</math>. Then there exists some <math>\epsilon > 0</math> such that the ball of radius <math>\epsilon</math> centered at <math>x^\ast</math> is contained in <math>P</math>, that is <math>B_{\epsilon}(x^\ast) \subset P</math>. Therefore,
 
:<math>x^\ast - \frac{\epsilon}{2} \frac{c}{||c||} \in P</math> and  
 
:<math>c^T\left( x^\ast - \frac{\epsilon}{2} \frac{c}{||c||}\right) = c^T x^\ast - \frac{\epsilon}{2} \frac{c^T c}{||c||} = c^T x^\ast - \frac{\epsilon}{2} ||c|| < c^T x^\ast.</math>
 
Hence <math>x^\ast</math> is not an optimal solution, a contradiction. Therefore, <math>x^\ast</math> must live on the boundary of <math>P</math>. If <math>x^\ast</math> is not a vertex itself, it must be the convex combination of vertices of <math>P</math>, say <math>x_1, ..., x_t</math>. Then <math>x^\ast = \sum_{i=1}^t \lambda_i x_i</math> with <math>\lambda_i \geq 0</math> and <math>\sum_{i=1}^t \lambda_i = 1</math>.  Observe that  
 
:<math>0=c^{T}\left(\left(\sum_{i=1}^{t}\lambda_{i}x_{i}\right)-x^{\ast}\right)=c^{T}\left(\sum_{i=1}^{t}\lambda_{i}(x_{i}-x^{\ast})\right)=\sum_{i=1}^{t}\lambda_{i}(c^{T}x_{i}-c^{T}x^{\ast}).</math>
 
Since <math>x^{\ast}</math> is an optimal solution, all terms in the sum are nonnegative. Since the sum is equal to zero, we must have that each individual term is equal to zero.  Hence, <math>c^{T}x^{\ast}=c^{T}x_{i}</math> for each <math>x_i</math>, so every <math>x_i</math> is also optimal, and therefore all points on the face whose vertices are <math>x_1, ..., x_t</math>, are optimal solutions.
 
==References==
* http://demonstrations.wolfram.com/TheFundamentalTheoremOfLinearProgramming/
 
{{Fundamental theorems}}
[[Category:Fundamental theorems]]
 
{{mathapplied-stub}}
{{comp-sci-stub}}

Latest revision as of 14:01, 9 March 2014


The feeling of freedom and self-control that you feel as you drive through the mountain trails can help you forget all the stress and worries of daily life. If you keep winning competitions, you will be invited to bigger and bigger races. If you go on a regular mountain biking, you will develop a healthier body and disposition. If you don’t need the overly aggressive styling of the Scapel 5 then you will find the Rush 6 a welcome alternative. However, due to its high price tag and light weight, it is seldom used for jumping and downhill races.

Adjust the tyre pressure too so it is at its optimum level. If you're planning on doing mostly road biking with the occasional cross country trip, and only once in a while going down an actual mountain. A wider tire will give you more traction when riding over rocks or in turns, you can run at lower pressure and the more cushioned the ride will feel, but they are heavier. You can travel on the equivalent of 10 cents a gallon of gas. It is a choice you make that determines the level of commitment you have.

If you beloved this short article and you would like to receive a lot more facts relating to Transfering to mountain bike sizing. kindly pay a visit to our own site. They have great suspension that fits the need of both amateur and professional bikers. People now prefer to have the simplest design in their bikes as long as it can get the job well done. The tube sits next to your mouth so you can sip as you ride. In no time you'll be cruising through the woods and getting into great shape. Re-lube the chain carefully, making sure each link is covered with oil, spin the pedals backwards a few times, then wipe off any excess oil from the chain with an old rag.

These people are discovering the benefits of having a strong fit body. He just stood there looking at the view from the top-his view and perspective changed by a few seconds and a climb up a little mountain. This article is to help you when buying a new bike. There is no shortage of videos of mountain bike crashes from around the world, with many riders with appropriate safety equipment escaping serious injury, while others were not that fortunate. Within 2 weeks of breathing exercises, the warm up period disappeared.

From there, there are levers designed to work with the different variations of brake calipers and dual control levers that control braking and shifting. There is anything from shocks to gears, special wheels, exclusive take care of bars that can be immediately switched to match the terrain you're riding on and so substantially much more. One of the main concerns for any cyclist is safety, having the right kit is a must. Whether it's a testament to South Dakota's commitment to recycling, or just whimsy, Bone Collector embodies the features of the best bike trails in the world: Sedona, Moab, Flagstaff, Whistler and Tahoe, without the long drive or expensive plane ticket. Part of Park City's distinction as the International Mountain Bike Association's first and only gold-level Ride Center is its abundance of beginner trails.

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