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In [[geometry]], the '''Chebyshev center''' of a bounded set <math>Q</math> having non-empty [[Interior (topology)|interior]] is the center of the minimal-radius ball enclosing the entire set <math>Q</math>, or, alternatively, the center of largest inscribed ball of <math>Q</math>.<ref>{{cite book|title=Convex optimization|year=2004|publisher=Cambridge|location=New York|isbn=978-0-521-83378-3|author=Boyd, Stephen P.; Vandenberghe, Lieven}}</ref>
 
In the field of [[parameter estimation]], the Chebyshev center approach tries to find an estimator <math> \hat x </math> for <math> x </math> given the feasibility set <math> Q </math>, such that <math>\hat x</math> minimizes the worst possible estimation error for x (e.g. best worst case).
 
== Mathematical representation ==
There exist several alternative representations for the Chebyshev center.
Consider the set <math>Q</math> and denote its Chebyshev center by <math>\hat{x}</math>. <math>\hat{x}</math> can be computed by solving:
 
: <math> \min_{{\hat x},r} \left\{ r:\left\| {\hat x} - x \right\|^2 \leq r, \forall x \in Q \right\} </math>
 
or alternatively by solving:
 
:<math> \operatorname*{\arg\min}_{\hat{x}} \max_{x \in Q} \left\| x - \hat x \right\|^2. </math><ref name="BV">{{cite book|title=Convex Optimization|first1=Stephen P.|last1=Boyd|first2=Lieven|last2=Vandenberghe|year=2004|publisher=Cambridge University Press|isbn=978-0-521-83378-3|url=http://www.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf|format=pdf|accessdate=October 15, 2011}}</ref>
 
Despite these properties, finding the Chebyshev center may be a hard numerical [[mathematical optimization|optimization problem]]. For example, in the second representation above, the inner maximization is [[nonconvex optimization|non-convex]] if the set ''Q'' is not [[convex set|convex]].
 
== Relaxed Chebyshev center ==
Let us consider the case in which the set <math>Q</math> can be represented as the intersection of <math>k</math> ellipsoids.  
 
: <math> \min_{\hat x} \max_x \left\{ \left\| {\hat x} - x \right\|^2 :f_i (x) \le 0,0 \le i \le k \right\} </math>
with
: <math> f_i (x) = x^T Q_i x + 2g_i^T x + d_i  \le 0,0 \le i \le k. \, </math>
 
By introducing an additional matrix variable <math>\Delta = x x^T </math>, we can write the inner maximization problem of the Chebyshev center as:
 
: <math> \min_{\hat x} \max_{(\Delta ,x) \in G} \left\{ \left\| {\hat x} \right\|^2  - 2{\hat x}^T x + \operatorname{Tr}(\Delta ) \right\} </math>
where <math>\operatorname{Tr}(\cdot)</math> is the [[trace (linear algebra)|trace operator]] and
: <math> G = \left\{(\Delta ,x):{\rm{f}}_i (\Delta ,x) \le 0,0 \le i \le k,\Delta  = xx^T \right\} </math>
: <math> f_i (\Delta ,x) = \operatorname{Tr}(Q_i \Delta ) + 2g_i^T x + d_i. </math>
 
Relaxing our demand on <math>\Delta</math> by demanding <math> \Delta \leq xx^T </math>, i.e. <math>xx^T - \Delta \in S_+</math> where <math>S_+</math> is the set of [[positive semi-definite matrix|positive semi-definite matrices]], and changing the order of the min max to max min (see the references for more details), the optimization problem can be formulated as:
 
: <math> RCC = \max_{(\Delta ,x) \in {T}} \left\{ - \left\| x \right\|^2  + \operatorname{Tr}(\Delta ) \right\} </math>
with
: <math> {T} = \left\{ (\Delta ,x):\rm{f}_i (\Delta ,x) \le 0,0 \le i \le k,\Delta \le xx^T  \right\}. </math>
 
This last '''convex''' optimization problem is known as the '''relaxed Chebyshev center''' (RCC).
The RCC has the following important properties:
* The RCC is an upper bound for the exact Chebyshev center.
* The RCC is unique.
* The RCC is feasible.
 
== Constrained least squares ==
With a few simple mathematical tricks, it can be shown that the well-known constrained [[least squares]] (CLS) problem is a relaxed version of the Chebyshev center.
 
The original CLS problem can be formulated as:
: <math> {\hat x}_{CLS}  = \operatorname*{\arg\min}_{x \in C} \left\| y - Ax \right\|^2 </math>
with
: <math> { C} = \left\{ x:f_i (x) = x^T Q_i x + 2g_i^T x + d_i  \le 0,1 \le i \le k \right\}
</math>
: <math> Q_i  \ge 0,g_i  \in R^m ,d_i  \in R. </math>
 
It can be shown that this problem is equivalent to the following optimization problem:
: <math> \max_{(\Delta ,{{x}}) \in {V}} \left\{ { - \left\| {{x}} \right\|^2  + \operatorname{Tr}(\Delta )} \right\} </math>
with
: <math> V = \left\{ \begin{array}{c}
(\Delta ,x):x \in C{\rm{ }} \\
\operatorname{Tr}(A^T A\Delta ) - 2y^T A^T x + \left\| y \right\|^2  - \rho  \le 0,\rm{  }\Delta  \ge xx^T  \\
\end{array} \right\}.</math>
 
One can see that this problem is a relaxation of the Chebyshev center (though different than the RCC described above).
 
== RCC vs. CLS ==
A solution set <math> (x,\Delta) </math> for the RCC is also a solution for the CLS, and thus <math> T \in V </math>.
This means that the CLS estimate is the solution of a looser relaxation than that of the RCC.
Hence the '''CLS is an upper bound for the RCC''', which is an upper bound for the real Chebyshev center.
 
== Modeling constraints ==
Since both the RCC and CLS are based upon relaxation of the real feasibility set <math>Q</math>, the form in which <math>Q</math> is defined affects its relaxed versions. This of course affects the quality of the RCC and CLS estimators.
As a simple example consider the linear box constraints:
: <math> l \leq a^T x \leq u </math>
which can alternatively be written as
: <math> (a^T x - l)(a^T x - u) \leq 0. </math>
It turns out that the first representation results with an upper bound estimator for the second one, hence using it may dramatically decrease the quality of the calculated estimator.
 
This simple example shows us that great care should be given to the formulation of constraints when relaxation of the feasibility region is used.
 
== See also ==
* [[Bounding sphere]]
* [[Smallest-circle problem]]
* [[Centre (geometry)]]
* [[Centroid]]
 
== References ==
{{Reflist}}
* Y. C. Eldar, A. Beck, and M. Teboulle, [http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4471880 "A Minimax Chebyshev Estimator for Bounded Error Estimation,"] IEEE Trans. Signal Processing, 56(4): 1388–1397 (2007).
* A. Beck and Y. C. Eldar, [http://dx.doi.org/10.1137/060656784 "Regularization in Regression with Bounded Noise: A Chebyshev Center Approach,"] SIAM J. Matrix Anal. Appl. 29 (2): 606–625 (2007).
 
[[Category:Estimation theory]]
[[Category:Geometric centers]]
[[Category:Mathematical optimization]]

Latest revision as of 16:10, 14 October 2014

Although we often keep attorneys in substantial esteem as careers go in our society, couple of us determine what they really do. It's not till we wind up a part of legitimate process that we realize how challenging the industry of law may be! If you need legal services, these report will bring you began in the correct path.

Tend not to www.youtube.com/watch?v=EFq8Vu4YIj0 feel compelled to use a legal representative because you achieved a couple of times and got some valuable advice. You must indication a legal contract only after you decide on costs and feel comfortable along with your legal professional. If you are hesitating since you have often heard poor aspects of this lawyer or consider the service fees are way too high, maintain hunting.

Be crystal clear about your spending budget. Even if you sense fairly positive that your circumstance will succeed, you still need to take into account the cost included. As you may interview prospective lawyers ask about their fees beforehand. Talk about your goals and your price range with any legal professional you are considering selecting. A precise knowledge of the fiscal facets of picking a attorney is important to the situation.

If you want a great legal professional to your enterprise, make use of your group. You could potentially question your banker, partners, insurance broker or perhaps your representatives when they know anything good legal professionals in the area. Do not wait to point this attorney to folks you understand when you have a good expertise.

Satisfy and job interview www.youtube.com/watch?v=EFq8Vu4YIj0 a few attorneys prior to deciding to employ 1. Speaking to each and every actually can provide the chance to see for those who have excellent a connection. It is additionally an excellent possibility to inquire further important questions that can help you make a well informed selection. Several lawyers are likely to use a 30 second conference with you free of charge.

Do not ignore the significance of your lawyer's individuality. An competitive legal representative is an advantages but an experienced and assured legal professional could help you win your scenario also. You need to talk with distinct lawyer and discover one who has the kind of persona and vision you require for the scenario.

When someone you care about passes away at the job, family typically is to be paid settlement. Worker's payment is designed for the personnel and for his immediate family members. Question your lawyer about the lawful treatment along with the documentation you need to take care of.

Legal professionals keep negligence insurance coverage for situations by which they neglected to follow the guidelines. Nevertheless, legal representatives dislike for such rates to increase for that reason, they will probably use legitimately liable techniques for your scenario. Do not forget that your lawyer would want to do right on your part.

Do not allow your legal representative impress you by using complicated authorized conditions. In case your lawyer utilizes terms you may not fully grasp, cease them and ask for a description. You have to know that some attorneys will make use of this procedure to make you sense powerless and offer them selves because the perfect answer to your issues.

Can be your legal representative a good listener? Your lawyer must know as far as possible concerning your scenario. In case your lawyer does not question questions or has a difficult time remembering the important points of your circumstance, you should look at employing someone that is actually a better listener which is prepared to take the time to acquaint on their own with the condition.

Before getting a attorney, you must take into account how much cash and time you are likely to put money into one. Also, consider the dollars you can expect to get rid of from taking off work into account. You will need to take all economic elements of getting a legal professional under consideration. There is no point in spending numerous 1000 money to hire a legal professional when your circumstance entails considerably less funds.

Using the assistance of a qualified attorney is the best practice to travel through authorized courtroom proceedings. When you understand how special the jargon is and just how complex the limitless rules are, you will be extremely pleased you've received a fantastic legal representative on your team. Give yourself a fighting chance with authorized issues and hire the proper legal professional.

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