linear programming

Discussion in 'C++' started by eric, Jun 20, 2011.

  1. eric

    eric Guest

    Dear c++ or advanced computer scienctist or mathmatician:
    on the book(Introduction to Algorithms 3rd ed) page 851, chapter 29
    Linear Programming
    I ask one of that book's author:
    -----------------------------------------------
    dear thomas:

    the the end of that page

    if a linear program has some feasible solutions but does not have a
    finite optimal objective value, we say that the linear program is
    unbounded

    but
    you want us to prove in Exercise 29.1-9
    to show
    that a linear program can have a finite optimal objective value even
    if
    the feasible region is not bounded

    which is very unlogical for me

    looking to hear from you soon
    Eric

    ------------------------------------------------------------------

    Eric, you'll have to think outside the box just a little bit. It may
    seem illogical, but there's a simple solution.

    Tom Cormen
    Professor and Chair
    Dartmouth College Department of Computer Science
    http://www.cs.dartmouth.edu/~thc/
    -----------------------------------------------------
    but I still not figure out
    plz give hint/suggestion/advice, thank your help/effort/time a lot in
    advance, Eric
     
    eric, Jun 20, 2011
    #1
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  2. * eric, on 20.06.2011 05:53:
    > Dear c++ or advanced computer scienctist or mathmatician:
    > on the book(Introduction to Algorithms 3rd ed) page 851, chapter 29
    > Linear Programming
    > I ask one of that book's author:
    > -----------------------------------------------
    > dear thomas:
    >
    > the the end of that page
    >
    > if a linear program has some feasible solutions but does not have a
    > finite optimal objective value, we say that the linear program is
    > unbounded
    >
    > but
    > you want us to prove in Exercise 29.1-9
    > to show
    > that a linear program can have a finite optimal objective value even
    > if
    > the feasible region is not bounded
    >
    > which is very unlogical for me
    >
    > looking to hear from you soon
    > Eric
    >
    > ------------------------------------------------------------------
    >
    > Eric, you'll have to think outside the box just a little bit. It may
    > seem illogical, but there's a simple solution.
    >
    > Tom Cormen
    > Professor and Chair
    > Dartmouth College Department of Computer Science
    > http://www.cs.dartmouth.edu/~thc/
    > -----------------------------------------------------
    > but I still not figure out
    > plz give hint/suggestion/advice, thank your help/effort/time a lot in
    > advance, Eric


    Let

    A = does not have finite objective value
    B = is unbounded

    One statement you're quoting is that A implies B.
    This means that in every case where A is true, B will necessarily be true.
    Another is that B does not imply A.
    This means that in some cases where B is true, A is not true.

    For example, if you're out in the rain without any protection, then you get wet.
    And in some cases when you get wet, you are not out in the rain without protection.


    Cheers & hth., and please post to a relevant group next time!,

    - Alf

    --
    blog at <url: http://alfps.wordpress.com>
     
    Alf P. Steinbach /Usenet, Jun 20, 2011
    #2
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  3. eric

    eric Guest

    On Jun 19, 11:36 pm, "Alf P. Steinbach /Usenet" <alf.p.steinbach
    > wrote:
    > * eric, on 20.06.2011 05:53:
    >
    >
    >
    > > Dear c++ or advanced computer scienctist or mathmatician:
    > >    on the book(Introduction to Algorithms 3rd ed) page 851, chapter29
    > > Linear Programming
    > > I ask one of that book's author:
    > > -----------------------------------------------
    > >     dear thomas:

    >
    > >    the the end of that page

    >
    > >   if a linear program has some feasible solutions but does not have a
    > >   finite optimal objective value, we say that the linear program is
    > >   unbounded

    >
    > >   but
    > >   you want us to prove in Exercise 29.1-9
    > >   to show
    > >   that a linear program can have a finite optimal objective value even
    > > if
    > >   the feasible region is not bounded

    >
    > >   which is very unlogical for me

    >
    > >   looking to hear from you soon
    > >   Eric

    >
    > > ------------------------------------------------------------------

    >
    > > Eric, you'll have to think outside the box just a little bit.  It may
    > > seem illogical, but there's a simple solution.

    >
    > > Tom Cormen
    > > Professor and Chair
    > > Dartmouth College Department of Computer Science
    > >http://www.cs.dartmouth.edu/~thc/
    > > -----------------------------------------------------
    > > but I still not figure out
    > > plz give hint/suggestion/advice, thank your help/effort/time a lot in
    > > advance, Eric

    >
    > Let
    >
    >    A = does not have finite objective value
    >    B = is unbounded
    >
    > One statement you're quoting is that A implies B.
    > This means that in every case where A is true, B will necessarily be true..
    > Another is that B does not imply A.
    > This means that in some cases where B is true, A is not true.
    >
    > For example, if you're out in the rain without any protection, then you get wet.
    > And in some cases when you get wet, you are not out in the rain without protection.
    >
    > Cheers & hth., and please post to a relevant group next time!,
    >
    > - Alf
    >
    > --
    > blog at <url:http://alfps.wordpress.com>


    ----------------------------------------------
    I think you reply a little bit too fast.
    Could I ask you, are you computer science/math student/worker?
    in that book, that page, that sentence
    there is no word(s) (implies)

    which part do you think it can be treated as (implies)?
    "We say" ?

    why it can not be treated as (equal or defined it)

    even that sentence should be treated as your way (implies) not
    mine(equal)
    then
    Can you show out what is the definition of bounded or unbounded, /*
    since we need that definition but authors did not gave out */?

    looking to see any computer scientist/mathmatician 's advice/
    suggestion/hint and thanks a lot in advance, Eric
     
    eric, Jun 20, 2011
    #3
  4. * eric, on 20.06.2011 09:58:
    > in that book, that page, that sentence
    > there is no word(s) (implies)


    "if" - "then" expresses an implication

    use e.g. Wikipedia to learn about logical implications


    cheers & hth., & please next time post in an appropriate group,

    - alf

    --
    blog at <url: http://alfps.wordpress.com>
     
    Alf P. Steinbach /Usenet, Jun 20, 2011
    #4
  5. eric

    gwowen Guest

    On Jun 20, 7:36 am, "Alf P. Steinbach /Usenet" <alf.p.steinbach
    > wrote:
    > Let
    >
    >    A = (the program) does not have finite objective value
    >    B = (the program) is unbounded
    >
    > One statement you're quoting is that A implies B.


    That's not right. The statement he quotes gives B as the *definition*
    of A, so A <=> B
     
    gwowen, Jun 20, 2011
    #5
  6. eric

    gwowen Guest

    On Jun 20, 4:53 am, eric <> wrote:

    >  if a linear program has some feasible solutions but does not have a
    >  finite optimal objective value, we say that the linear program is
    >  unbounded


    Here, he's talking about the linear program being unbounded.

    > if the feasible region is not bounded


    Here, he's talking about the feasible region being unbounded.
    They're not the same thing.

    Guru Meditation: Maximise (7-x) subject to (x>0).
     
    gwowen, Jun 20, 2011
    #6
  7. eric

    Paul Guest

    "eric" <> wrote in message
    news:...
    > Dear c++ or advanced computer scienctist or mathmatician:
    > on the book(Introduction to Algorithms 3rd ed) page 851, chapter 29
    > Linear Programming
    > I ask one of that book's author:
    > -----------------------------------------------
    > dear thomas:
    >
    > the the end of that page
    >
    > if a linear program has some feasible solutions but does not have a
    > finite optimal objective value, we say that the linear program is
    > unbounded
    >
    > but
    > you want us to prove in Exercise 29.1-9
    > to show
    > that a linear program can have a finite optimal objective value even
    > if
    > the feasible region is not bounded
    >
    > which is very unlogical for me
    >

    Maybe a 'feasible region' is not bounded but the progam is bounded.
     
    Paul, Jun 20, 2011
    #7
  8. * gwowen, on 20.06.2011 12:07:
    > On Jun 20, 7:36 am, "Alf P. Steinbach /Usenet"<alf.p.steinbach
    > > wrote:
    >> Let
    >>
    >> A = (the program) does not have finite objective value
    >> B = (the program) is unbounded
    >>
    >> One statement you're quoting is that A implies B.

    >
    > That's not right. The statement he quotes gives B as the *definition*
    > of A, so A<=> B


    An "if-then" is a one way implication, not two-way.

    For example, IF the text is so vague/informal as to not make that distinction,
    THEN it's meaningless to ponder very fine details of meaning in the text.


    Cheers & hth.,

    - Alf

    --
    blog at <url: http://alfps.wordpress.com>
     
    Alf P. Steinbach /Usenet, Jun 20, 2011
    #8
  9. eric

    gwowen Guest

    On Jun 20, 1:09 pm, "Alf P. Steinbach /Usenet" <alf.p.steinbach
    > wrote:

    > > That's not right.  The statement he quotes gives B as the *definition*
    > > of A, so A<=>  B

    >
    > An "if-then" is a one way implication, not two-way.


    Usually yes. But this is phrased like'If "something" satifies
    "condition A", then we say it is "A-ified".' In English-as-she-is-
    used-by-mathematicians that's *a definition* of "A-ified". It's two
    way. As a native English speaking mathematician, you really should
    take me word on this.

    e.g. Definition of continuity under arbitrary topologies: "If the
    preimage under f() of any open set is itself, we say f() is
    continuous".
     
    gwowen, Jun 20, 2011
    #9
  10. eric

    gwowen Guest

    On Jun 20, 1:59 pm, gwowen <> wrote:

    > e.g. Definition of continuity under arbitrary topologies: "If the
    > preimage under f() of any open set is itself, we say f() is
    > continuous".


    Ooops missed a word ... e.g. Definition of continuity under arbitrary
    topologies: "If the preimage under f() of any open set is itself open,
    we say f() is continuous".
     
    gwowen, Jun 20, 2011
    #10
  11. * gwowen:
    > Usually yes. But this is phrased like'If "something" satifies
    > "condition A", then we say it is "A-ified".' In English-as-she-is-
    > used-by-mathematicians that's *a definition* of "A-ified".


    Uh, most mathematicians I know would have used "if and only if" or
    any of its alternatives in that case; in writing it would've been 'iff'.

    --
    Martijn van Buul -
     
    Martijn van Buul, Jun 20, 2011
    #11
  12. eric

    gwowen Guest

    On Jun 20, 4:13 pm, Martijn van Buul <> wrote:
    > * gwowen:
    >
    > > Usually yes.  But this is phrased like'If "something" satifies
    > > "condition A", then we say it is "A-ified".'  In English-as-she-is-
    > > used-by-mathematicians that's *a definition* of "A-ified".

    >
    > Uh, most mathematicians I know would have used "if and only if" or
    > any of its alternatives in that case; in writing it would've been 'iff'.


    Not in undergraduate lecture notes, which are much more informal.
    Anyway, go find one of the numerous mathematicians of your
    acquaintance and ask them what the definition an "unbounded linear
    program" is. Then ask them what the definition of "unbounded feasible
    region" is.

    Alternatively, go and read this incredibly basic introduction to
    linear programming -- http://www.caam.rice.edu/~yzhang/caam378/Notes/chap1.pdf
    (specifically S1.5 on pg 11)

    Then, come back, and admit that I know of what I speak.
     
    gwowen, Jun 20, 2011
    #12
  13. eric

    Kai-Uwe Bux Guest

    [OT] Re: linear programming

    eric wrote:

    > Dear c++ or advanced computer scienctist or mathmatician:
    > on the book(Introduction to Algorithms 3rd ed) page 851, chapter 29
    > Linear Programming
    > I ask one of that book's author:
    > -----------------------------------------------
    > dear thomas:
    >
    > the the end of that page
    >
    > if a linear program has some feasible solutions but does not have a
    > finite optimal objective value, we say that the linear program is
    > unbounded
    >
    > but
    > you want us to prove in Exercise 29.1-9
    > to show
    > that a linear program can have a finite optimal objective value even
    > if
    > the feasible region is not bounded
    >
    > which is very unlogical for me


    Notice the difference between

    a) a program is unbounded.
    b) a program has an unbounded feasible region.

    Any unbounded program has an unbounded feasible region. The exercise asks
    you to give an example showing that the converse does not hold. For this,
    think geometrically and draw some linear programs in dimension 2. You can
    find such examples there.


    Best,

    Kai-Uwe Bux
     
    Kai-Uwe Bux, Jun 20, 2011
    #13
  14. * gwowen:
    > On Jun 20, 4:13?pm, Martijn van Buul <> wrote:
    >> * gwowen:
    >>
    >> > Usually yes. ?But this is phrased like'If "something" satifies
    >> > "condition A", then we say it is "A-ified".' ?In English-as-she-is-
    >> > used-by-mathematicians that's *a definition* of "A-ified".

    >>
    >> Uh, most mathematicians I know would have used "if and only if" or
    >> any of its alternatives in that case; in writing it would've been 'iff'.

    >
    > Not in undergraduate lecture notes, which are much more informal.


    Your problem, not mine. Don't make any claims then. You're the one who takes
    an informal explanation out of context, not me.

    > Anyway, go find one of the numerous mathematicians of your
    > acquaintance and ask them what the definition an "unbounded linear
    > program" is. Then ask them what the definition of "unbounded feasible
    > region" is.


    Irrelevant. I didn't question the validity of that statement. I questioned
    the validity of your "argument".

    > Then, come back, and admit that I know of what I speak.


    Oh, I'll admit what I already know:

    You're awfully full of yourself, but have very little to show. But that
    applies to a lot of people here, so you're in good company.

    --
    Martijn van Buul -
     
    Martijn van Buul, Jun 20, 2011
    #14
  15. eric

    Kai-Uwe Bux Guest

    [OT] Re: linear programming

    Martijn van Buul wrote:

    > * gwowen:
    >> Usually yes. But this is phrased like'If "something" satifies
    >> "condition A", then we say it is "A-ified".' In English-as-she-is-
    >> used-by-mathematicians that's *a definition* of "A-ified".

    >
    > Uh, most mathematicians I know would have used "if and only if" or
    > any of its alternatives in that case; in writing it would've been 'iff'.


    That does not match my experience. In _defintions_ you don't say "if and
    only if". Examples:

    A group G is called commutative or abelian if ab=ba for all a,b in G.

    Or:

    A group G is called commutative or abelian if its group law is
    commutative.

    I am not aware of a mathematician who would use "if and only if" within a
    definition, e.g.:

    A group G is called abelian if and only if ab=ba for all a,b in G.

    Of course, there is this mixture of theorem and definition, where you say
    something like: "For a ring R, the following three conditions are equivalent
    ....; if any of them holds for R, we call R noetherian." Note, however, the
    "if" in the definition part of that statement. This is, what I think feels
    most natural to a mathematician.


    To verify this point, you can check for instance in the following standard
    literature for the definition of a commutative group:

    Isaacs: Algebra, a graduate course (page 11)
    Bourbaki: Elements of Mathematic, Algebra 1 Chapters 1 - 3;
    1.4.2 (page 31)
    Lang: Algebra (3rd ed), page 4 bottom
    Rotman: Advanced Modern Algebra; Groups I Chapter 2 page 52

    In all cases, you will find "if" used instead of "if and only if". I cite
    those books not (just) as an apeal to authority. The graduate courses based
    upon those books shape the mathematical jargon of trained mathematicians.
    Hence, linguistic inclinations are picked up from those patterns.


    Of course when it comes to theorems, mathematicians do distinguish very
    precisely a mere implication from an equivalence.



    Best,

    Kai-Uwe Bux
     
    Kai-Uwe Bux, Jun 20, 2011
    #15
  16. eric

    eric Guest

    On Jun 20, 12:56 pm, Kai-Uwe Bux <> wrote:
    > Martijn van Buul wrote:
    > > * gwowen:
    > >> Usually yes.  But this is phrased like'If "something" satifies
    > >> "condition A", then we say it is "A-ified".'  In English-as-she-is-
    > >> used-by-mathematicians that's *a definition* of "A-ified".

    >
    > > Uh, most mathematicians I know would have used "if and only if" or
    > > any of its alternatives in that case; in writing it would've been 'iff'..

    >
    > That does not match my experience. In _defintions_ you don't say "if and
    > only if". Examples:
    >
    >   A group G is called commutative or abelian if ab=ba for all a,b in G.
    >
    > Or:
    >
    >   A group G is called commutative or abelian if its group law is
    >   commutative.
    >
    > I am not aware of a mathematician who would use "if and only if" within a
    > definition, e.g.:
    >
    >   A group G is called abelian if and only if ab=ba for all a,b in G.
    >
    > Of course, there is this mixture of theorem and definition, where you say
    > something like: "For a ring R, the following three conditions are equivalent
    > ...; if any of them holds for R, we call R noetherian." Note, however, the
    > "if" in the definition part of that statement. This is, what I think feels
    > most natural to a mathematician.
    >
    > To verify this point, you can check for instance in the following standard
    > literature for the definition of a commutative group:
    >
    >   Isaacs: Algebra, a graduate course (page 11)
    >   Bourbaki: Elements of Mathematic, Algebra 1 Chapters 1 - 3;
    >             1.4.2 (page 31)
    >   Lang: Algebra (3rd ed), page 4 bottom
    >   Rotman: Advanced Modern Algebra; Groups I Chapter 2 page 52
    >
    > In all cases, you will find "if" used instead of "if and only if". I cite
    > those books not (just) as an apeal to authority. The graduate courses based
    > upon those books shape the mathematical jargon of trained mathematicians.
    > Hence, linguistic inclinations are picked up from those patterns.
    >
    > Of course when it comes to theorems, mathematicians do distinguish very
    > precisely a mere implication from an equivalence.
    >
    > Best,
    >
    > Kai-Uwe Bux

    -----------------
    Dear Kai-Uwe Bux:

    I am waiting these 4 authores reply my email about this question. I
    hope their explanation/sulution match your suggestion/interpretation.
    And no matter what is these 4 authores going to reply me, I thanks a
    lot your help.

    sincerely, Eric
    -----------------
     
    eric, Jun 21, 2011
    #16
  17. eric

    gwowen Guest

    On Jun 21, 6:30 am, Gareth Owen <> wrote:

    > But please have the humility not to "correct" someone for whom it *is*
    > there first language,

    ^^^^^

    Ha! Skitt's Law strikes again.
     
    gwowen, Jun 21, 2011
    #17
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