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vanamali@netzero.net science forum beginner
Joined: 04 Jun 2006
Posts: 3

Posted: Sun Jun 04, 2006 12:51 pm Post subject:
early reference for minimization of quadratic form



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 wellknown textbook
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

Posted: Sun Jun 04, 2006 1:05 pm Post subject:
Re: early reference for minimization of quadratic form



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 wellknown textbook
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 quickanddirty search found a review of
Markowitz, Harry, "The optimization of a quadratic function subject
to linear constraints", Naval Res. Logist. Quart. 3 (1956), 111133,
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_,
McGrawHill Book Co., Inc., New YorkTorontoLondon 1960, gives the
result.
Lee Rudolph 

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vanamali@netzero.net science forum beginner
Joined: 04 Jun 2006
Posts: 3

Posted: Wed Jun 07, 2006 9:27 am Post subject:
Re: early reference for minimization of quadratic form



Quote:  A quickanddirty search found a review of Markowitz, Harry, "The
optimization of a quadratic function subject to linear constraints",
Naval Res. Logist. Quart. 3 (1956), 111133, 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_,
McGrawHill Book Co., Inc., New YorkTorontoLondon 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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