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early reference for minimization of quadratic form
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vanamali@netzero.net
science forum beginner


Joined: 04 Jun 2006
Posts: 3

PostPosted: Sun Jun 04, 2006 12:51 pm    Post subject: early reference for minimization of quadratic form Reply with quote

I have a question on an early definitive reference to minimizing
quadratic forms subject to linear equality constraint. That is, the
solution to minimizing the quadratic form b^T R b subject to the
equality constraint C^T b = d is well known, and usually solved by the
method of Lagrange multipliers. I would like to know since when the
solution has been known to mathematicians. Even if the precise history
cannot be traced, is it possible to give a well-known text-book
reference prior to 1970 that gives the result ? Thanks.

vv
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Lee Rudolph
science forum Guru


Joined: 28 Apr 2005
Posts: 566

PostPosted: Sun Jun 04, 2006 1:05 pm    Post subject: Re: early reference for minimization of quadratic form Reply with quote

vanamali@netzero.net writes:

Quote:
I have a question on an early definitive reference to minimizing
quadratic forms subject to linear equality constraint. That is, the
solution to minimizing the quadratic form b^T R b subject to the
equality constraint C^T b = d is well known, and usually solved by the
method of Lagrange multipliers. I would like to know since when the
solution has been known to mathematicians. Even if the precise history
cannot be traced, is it possible to give a well-known text-book
reference prior to 1970 that gives the result ? Thanks.

If you have or can obtain access to MathSciNet, I suggest you search
there for starters. A quick-and-dirty search found a review of
Markowitz, Harry, "The optimization of a quadratic function subject
to linear constraints", Naval Res. Logist. Quart. 3 (1956), 111--133,
which *clearly* is a "definitive reference" though not a textbook.
I suspect, but cannot prove from the text of the Mathematical Reviews
review, that Richard Bellman's textbook _Introduction to matrix analysis_,
McGraw-Hill Book Co., Inc., New York-Toronto-London 1960, gives the
result.

Lee Rudolph
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vanamali@netzero.net
science forum beginner


Joined: 04 Jun 2006
Posts: 3

PostPosted: Wed Jun 07, 2006 9:27 am    Post subject: Re: early reference for minimization of quadratic form Reply with quote

Quote:
A quick-and-dirty search found a review of Markowitz, Harry, "The
optimization of a quadratic function subject to linear constraints",
Naval Res. Logist. Quart. 3 (1956), 111--133, which *clearly* is a
"definitive reference" though not a textbook.

I greatly appreciate your pointing out the paper by Markowitz. Someone
was kind enough to send me a scanned copy of the paper and I find that
his paper deals with linear inequality constraints rather than
equality.

Quote:
I suspect, but cannot prove from the text of the Mathematical Reviews
review, that Richard Bellman's textbook _Introduction to matrix analysis_,
McGraw-Hill Book Co., Inc., New York-Toronto-London 1960, gives the
result.

Alas, the only copy in our library is not in the shelf where it should
be. I have requested for a search and awaiting news about it.

Apparently the solution with equality constraint must have been known
for a very long time, as it is one of the simplest application of the
method of Lagrange multipilers.

vv
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