Wikipedia:Reference desk/Archives/Mathematics/2013 September 9

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September 9

Operations research: issue with the dual problem in linear programming

We have:

${\begin{array}{rcrrrcr}\min &&z=&0.3x_{1}&+0.9x_{2}&&\\{\text{subject to:}}&&&x_{1}&+x_{2}&\geq &800\\&&&-7x_{1}&+10x_{2}&\geq &0\\&&&3x_{1}&-x_{2}&\geq &0\\&&&x_{1},&x_{2}&\geq &0\end{array}}$

To which the optimal solution is $x_{1}={\frac {8000}{17}},x_{2}={\frac {5600}{17}},z={\frac {7440}{17}}$. If I'm not mistaken, the dual problem is

${\begin{array}{rcrrrrcl}\max &&w=&800y_{1}&&&&\\{\text{subject to:}}&&&y_{1}&-7y_{2}&+3y_{3}&\leq &0.3\\&&&y_{1}&+10y_{2}&-y_{3}&\leq &0.9\\&&&y_{1},&y_{2},&y_{3}&\geq &0\end{array}}$

The issue is that the dual problem is unbound, in contradiction to the fact that if the dual is unbounded, then the primal must be infeasible.

Where am I wrong? Thanks, Oh, well (talk) 00:46, 9 September 2013 (UTC)

You can use the dual of a problem to generate a check on the solution to the primal problem and vice versa. In this case, reading from the solution to the primal problem, multiply the first constraint in the dual by 8000/17, the second by 5600/17 and add. The result is 800y₁+0y₂+1082.352941y₃≤437.6470588. So w≤437.6470588 in the dual problem, not unbounded. I used Excel to compute this instead of Alpha, so maybe the problem is the way you entered the parameters there. --RDBury (talk) 01:59, 9 September 2013 (UTC)

Looking for a video series to teach myself calculus.

Anyone know of a good one? Goodbye Galaxy (talk) 16:56, 9 September 2013 (UTC)

You might try Khan Academy. --RDBury (talk) 19:25, 9 September 2013 (UTC)

I haven't watched these specific lectures, but MIT OpenCourseWare has lectures here, [1] for single variable, and here, [2] for multivariable. They both also feature lecture notes. I watched through one of their courses on electromagnetism several years ago, it was good, I'm guessing these will be too:-)Phoenixia1177 (talk) 06:01, 12 September 2013 (UTC)

n-body problem

For special cases or the general case of the n-body problem: (1) What if anything do we know about whether there exists a positive-measure range of initial conditions that lead to convergence to an attractor? I.e., in the absence of collisions, is it known that from some range of initial conditions none of the objects ever gets flung away to infinity?

(2) What if anything do we know about whether there exists a positive-measure range of initial conditions that lead to divergence? I.e., in the absence of collisions, is it known that from some range of initial conditions some of the objects get flung away to infinity?

(I can't seem to find the answer in n-body problem or three-body problem.) Duoduoduo (talk) 17:44, 9 September 2013 (UTC)

It seems to me that (2) is rather trivial, it's quite easy to start with some initial condition which is guaranteed to lead to some of the bodies being ejected. This is how the early solar system cleared itself from the debris left after the formation of the planets, these debris got ejected. In case of the current Solar system, there exists a range of initial conditions were Mars gets ejected from the Solar System, see e.g. here. Count Iblis (talk) 18:35, 9 September 2013 (UTC)

In the three body problem, there has been some work on sufficient conditions for escape of one of the bodies. In general if one body has an initial velocity greater than the escape velocity of the system, it will escape (ignoring collisions). The more interesting case is for sub-escape velocity--see [3], [4], or [5] for examples of approaches. This paper has a nice discussion of some conditions in which the three body problem is stable. --Mark viking (talk) 19:05, 9 September 2013 (UTC)