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% Creation Date: Tue Oct  6 10:47:46 MDT 1992

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%%%     author          = "Eberhard Kranich",
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%%%     filename        = "intbib.ltx",
%%%     address         = "FB WiWi and OR (Prof. Dr. Gal)
%%%                        FernUniversit{\"a}t Hagen
%%%                        Postfach 940
%%%                        D-5800 Hagen 1
%%%                        Germany",
%%%     checksum        = "37758 285 1398 15088",
%%%     email           = "WI805@DHAFEU11 (EARN/Bitnet)",
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%%%     keywords        = "bibliography, interior point, mathematical
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%%%                        a bibliography of interior point methods
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% Eberhard Kranich, Rosenstr. 58, D--5800 Hagen, West--Germany
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% last edit : August 23, 1991
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\begin{document}
   \begin{titlepage}
      \vspace*{\fill}
      {\centering {\large {\bf  INTERIOR POINT METHODS FOR \\
                                MATHEMATICAL PROGRAMMING~: \\
                                A BIBLIOGRAPHY  \vspace{4ex} \\} }
          Eberhard Kranich\footnote{FB
          Wirtschaftswissenschaften und Operations Research,
          FernUniversit{\"a}t Hagen, P.O. Box 940,
          D--5800 Hagen, West--Germany} \vspace{6ex}\\
          {\bf Diskussionsbeitrag Nr. 171\\ 1991 \vspace{\fill} \\}
      }
      {\noindent Diskussionsbeitr{\"a}ge des FB WiWi der FernUniversit{\"a}t
      Hagen \vspace{1ex}\\
      Herausgegeben vom Dekan des Fachbereichs \vspace{2ex}}\\
   \end{titlepage}

   \begin{abstract}
      Nothing is older than a yesterday's newspaper! In this sense, this
      bibliography is incomplete, of course, because of the enormous
      interest and ongoing research in interior--point methods for
      mathematical programming. These activities were initiated by
      Karmarkar's polynomial--time linear programming algorithm in 1984,
      resulting in a flood of research papers, articles, talks, developments
      of commercial software and so on. This working paper should be regarded
      as a first trial to obtain a survey over the research reports, journal
      articles, etc., published so far, resp.~known to the author.
      \vspace{2ex}\\

      {\bf Acknowledgements :}
      \vspace{2ex}\\

      {\bf Key Words :} Karmarkar's algorithm, interior--point methods,
      projective scaling algorithm, affine scaling algorithm,
      path--following methods, logarithmic barrier function algorithm,
      potential reduction algorithm, trajectory--following method,
      polynomial--time algorithm, method of centers, gravitational method,
      box method, mathematical programming.
      \vspace{2ex}\\

      {\bf Remark :} The reader is kindly requested to send those references
      to the authors which are not listed or incorrectly cited in this
      bibliography (if possible, a copy of the corresponding report or the
      article's citation with the (complete!) name of the journal, with a
      short abstract, cross--references and keywords), or to send a copy of
      his own reference list (can be handwritten!) in order to complete the
      current bibliography\footnote{e--mail : WI805@DHAFEU11 or
      WI811@DHAFEU11 (EARN), FAX : (0)2331~804313}. The actual list will be
      published elsewhere from time to time, possibly by e--mail.
      \vspace{2ex}\\

      {\bf Last edit~: \today}
   \end{abstract}

%   \newpage
%
%   \vspace*{\fill}
%   {\noindent {\em To Gerlinde : There is nothing in the world that can keep
%                   us apart... }}
%   \vspace{10ex}\\

   \newpage
   \vspace*{12ex}
   {\noindent {\Huge {\bf Introduction} \vspace{2ex}\\}}

   Interior--point methods for mathematical programming are known for a long
   time. With respect to solving nonlinear programming problems they have
   turned out to be a reliable tool, see e.g. Fiacco and McCormick%
   \cite{int:Fiacco3}, Frisch\cite{int:Frisch3,int:Frisch1},Huard%
   \cite{int:Huard1}, and Lootsma\cite{int:Lootsma1}. But in view of solving
   linear programming problems they could not compete with G.~B.~Dantzig's
   simplex method. They were only regarded as theoretical alternative
   solution methods to, never as practical substitutes for the simplex
   method, see e.g. Brown and Koopmans\cite{int:Brown1},
   Caroll\cite{int:Caroll1}, Dikin\cite{int:Dikin1,int:Dikin2},
   Frisch\cite{int:Frisch2}, and Parisot\cite{int:Parisot1}. That these
   methods have failed in linear programming practice at that time can be
   justified by the lack of modern numerical linear algebra\footnote{%
   A.~George and J.~W.~H.~Liu, {\em Computer Solutions of Large Sparse
   Positive Definite Systems}, Prentice Hall, 1981.} and sparse matrix
   technologies\footnote{S.~Pissanetzky, {\em Sparse Matrix Technology},
   Academic Press, 1984.}, see e.~g.~Marsten et al.\cite{int:Marsten6}.
   \vspace{2ex}\\

   A new polynomial--time linear programming interior point--method was
   presented by N.~K.~Karmarkar in 1984, which was claimed to be up to 100
   times faster than the simplex method, in practice and for any instance.
   Karmarkar's announcement stirred the interest not only of the popular
   press, see e.g.~Angier\cite{int:Angier1}, Emmett\cite{int:Emmett1},
   Fricker\cite{int:Fricker1}, Garfunkel et al.\cite{int:Garfunkel1},
   Gleick\cite{int:Gleick1}, Kolata\cite{int:Kolata1},
   Philpott\cite{int:Philpott1}, and Stipp\cite{int:Stipp1}, but also of the
   operations research community, see e.g.~the surveys of Bazaraa et
   al.\cite{int:Bazaraa1}, pp.~416--418, Freund\cite{int:Freund16}, Gill et
   al.\cite{int:Gill2}, Goldfarb and Todd\cite{int:Goldfarb8},
   Megiddo\cite{int:Megiddo4,int:Megiddo6}, Monma\cite{int:Monma1},
   Todd\cite{int:Todd2,int:Todd7}, Zimmermann\cite{int:Zimmermann1}, and, in
   some sense, den Hertog\cite{int:Hertog1}.
   \vspace{2ex}\\

   Since Karmarkar's publications\cite{int:Karmarkar1,int:Karmarkar2}, a
   great amount of contributions has been made towards the theoretical
   analysis and practical implementations of interior--point methods, see
   e.g.~Anstreicher\cite{int:Anstreicher10}, Bayer and Lagarias\cite%
   {int:Bayer2,int:Bayer3}, Lagarias\cite{int:Lagarias2}, and Monma\cite{%
   int:Monma2}, and the investigations are still going on, see the
   David~II--Report\cite{int:David1}, and Resende and Wright\cite{%
   int:Resende3}. The importance of interior--points methods is reflected by
   numerous talks, workshops and minisymposia held at various operations
   research or optimization meetings, see e.g.~the programs of the annual
   meetings of ORSA/TIMS or SIAM, Dolecki\cite{int:Dolecki1}, Freund\cite{%
   int:Freund16}, Megiddo\cite{int:Megiddo5,int:Megiddo10}, and Tovey\cite{%
   int:Tovey1,int:Tovey2}.
   \vspace{3ex}\\

   In order to get an overview, interior--point methods can be broadly
   classified into the following categories, see e.g.~den Hertog\cite{%
   int:Hertog1}, whereby the selected citations are thought as examples~:
   \begin{enumerate}
      \item Projective scaling methods, introduced by
            Karmarkar\cite{int:Karmarkar1,int:Karmarkar2} and further
            investigated and extended e.g.~by Akguel\cite{int:Akguel1},
            Anstreicher\cite{int:Anstreicher3,int:Anstreicher4,%
            int:Anstreicher9,int:Anstreicher12,int:Anstreicher8},
            Asic et al.\cite{int:Asic2}, Bazaraa et al.\cite{int:Bazaraa1},
            Blair\cite{int:Blair1}, Dennis et al.\cite{int:Dennis1}, Dodani
            and Babu\cite{int:Dodani3}, Ferris and Philpott%
            \cite{int:Ferris1}, Franklin\cite{int:Franklin1}, Gay\cite{%
            int:Gay3}, de Ghellinck and Vial\cite{int:Ghellinck1}, Goldfarb
            and Mehrotra\cite{int:Goldfarb5,int:Goldfarb4}, Gonzaga\cite{%
            int:Gonzaga5,int:Gonzaga2}, Hettich and Margraff\cite{%
            int:Hettich1}, Hooker\cite{int:Hooker1}, Kojima\cite{%
            int:Kojima1}, Kojima and Tone\cite{int:Kojima2},
            McDiarmid\cite{int:McDiarmid1}, Mitchell\cite{int:Mitchell1},
            Mitchell and Todd\cite{int:Mitchell5,int:Mitchell6},
            Paris\cite{int:Paris1}, Powell\cite{int:Powell2},
            Roos\cite{int:Roos1},
            Shanno\cite{int:Shanno1}, Shanno and Marsten\cite{int:Shanno7},
            Shub\cite{int:Shub1}, Steger\cite{int:Steger1}, Todd\cite{%
            int:Todd8}, Todd and Burell\cite{int:Todd14},
            Tomlin\cite{int:Tomlin2}, Vial\cite{int:Vial3},
            Ye\cite{int:Ye17,int:Ye18}, Ye and Kojima\cite{int:Ye20},
            Zimmermann\cite{int:Zimmermann3}.
      \item 'Pure' affine scaling methods, originally proposed by Dikin%
            \cite{int:Dikin1,int:Dikin2,int:Dikin3} about twenty years ago,
            independently rediscovered by e.g.~Barnes\cite{int:Barnes3},
            Cavalier and Soyster\cite{int:Cavalier3}, Monma and
            Morton\cite{int:Monma3}, Vanderbei et al.\cite{int:Vanderbei7},
            and further investigated by e.g.~Adler et al.\cite{int:Adler10},
            Barnes et al.\cite{int:Barnes11} (belongs in principle to the
            class of path--following methods), Cavalier and
            Schall\cite{int:Cavalier2}, Christiansen and Kortanek\cite{%
            int:Christiansen1,int:Christiansen2}, Hettich and Ries\cite{%
            int:Hettich2}, Kortanek and Shi\cite{int:Kortanek3}, Marsten et
            al.\cite{int:Marsten5}, Mehrotra\cite{int:Mehrotra9}, Monteiro
            et al.\cite{int:Monteiro7} (the polynomiality is derived from
            the polynomial result of path--following methods~!),
            Ponnambalam and Vanelli\cite{int:Ponnambalam2}, Resende and
            Veiga\cite{int:Resende4}, Sherali\cite{int:Sherali1} (in
            principle a path--following method),
            Strang\cite{int:Strang1,int:Strang2,int:Strang3}, Todd\cite{%
            int:Todd10}, Wei\cite{int:Wei1,int:Wei2}.\\
            Megiddo and Shub\cite{int:Megiddo12} have shown that algorithms
            belonging to this class can be supposed to be non--polynomial.
      \item Path-- or trajectory--following methods, initiated by Gill et
            al.\cite{int:Gill7}, showing a certain equivalence to
            Karmarkar's algorithm, and further investigated by
            e.g.~Anstreicher et al.\cite{int:Anstreicher15}, Ben--Daya and
            Shetty\cite{int:BenDaya2,int:BenDaya1}, Goldfarb and Liu\cite{%
            int:Goldfarb3}, Gonzaga\cite{int:Gonzaga6,int:Gonzaga7}, Gonzaga
            and Todd\cite{int:Gonzaga12}, den Hertog et al.\cite{%
            int:Hertog4}, Kojima et al.\cite{int:Kojima4},
            Lustig\cite{int:Lustig3}, Lustig et al.\cite{int:Lustig7,%
            int:Lustig8}, Marsten et al.\cite{int:Marsten6},
            Marxen\cite{int:Marxen1}, McShane et al.\cite{int:McShane1},
            Megiddo\cite{int:Megiddo11}, Mehrotra\cite{int:Mehrotra10},
            Mehrotra and Sun\cite{int:Mehrotra14}, Monteiro and Adler\cite{%
            int:Monteiro3,int:Monteiro4}, Osborne\cite{int:Osborne1,%
            int:Osborne2}, Renegar\cite{int:Renegar1}, Roos\cite{int:Roos7,%
            int:Roos8},Todd\cite{int:Todd13}, Todd and Vial\cite{int:Todd20},
            Tseng\cite{int:Tseng1}, Vaidya\cite{int:Vaidya4}, Zhu and
            Kortanek\cite{int:Zhu1}.
      \item Affine scaling methods applied to a potential function,
            described in Anstreicher and Bosch\cite{int:Anstreicher13},
            Freund\cite{int:Freund2}, Gonzaga\cite{int:Gonzaga11,%
            int:Gonzaga4,int:Gonzaga8}, Ye\cite{int:Ye9}.
      \item Method of centers, described e.g. in Boggs et al.\cite{%
            int:Boggs2,int:Boggs1}, Jarre\cite{int:Jarre6,int:Jarre3,%
            int:Jarre1}, Jarre et al.\cite{int:Jarre5}, Mehrotra and
            Sun\cite{int:Mehrotra11}, Sonnevend\cite{int:Sonnevend2,%
            int:Sonnevend3,int:Sonnevend7}, Witzgall et
            al.\cite{int:Witzgall1}.
      \item Gravitational method, developed in Murty\cite{int:Murty3,%
            int:Murty5} and implemented by Chang and Murty\cite{int:Chang1}.
      \item Box method, as described in Zikan and Cottle\cite{int:Zikan1,%
            int:Zikan2,int:Zikan3}.
   \end{enumerate}
   Algorithms belonging to the class~4 are the most promising, because they
   do not loose their theoretical complexity in practice. Unfortunately, no
   computational experiences are published, at least as yet.
   \vspace{4ex}\\

   During the last few years practical implementations of linear programming
   interior--point methods were developed, which are superior to the simplex
   method, not necessarily for each special problem, but at least for a set of
   problems, such as Gay's {\em netlib} problem collection, see e.g.~%
   Lustig\cite{int:Lustig6}, Marsten et al.\cite{int:Marsten5,int:Marsten6},
   McShane et al.\cite{int:McShane1}. For solving everyday linear
   programming problems there exists a rule of thumb~: If the problem size is
   small, i.e.~the number of variables is lower than 500, the simplex method
   is superior in most cases and consequently 'still alive', otherwise solve
   the problem via an interior--point method.\\
   Nowadays, there exist at least two commercial implementations. The first
   one is OB1, developed by Shanno and Marsten\cite{int:Shanno8}, see also
   e.g.~Lustig et al.\cite{int:Lustig11}, and the second is the AT~\&~T KORBX
   Linear Programming System\cite{int:At1,int:Cheng4}, a software/hardware
   combination.
   \vspace{4ex}\\

   A concluding remark~: Talks, workshops and minisymposia are handled as
   technical reports, because I am rather sure that to each talk there
   exists an underlying report.

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