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The starting point of this volume was a conference entitled
"Progress in Mathematical Programming," held at the Asilomar
Conference Center in Pacific Grove, California, March 1-4, 1987.
The main topic of the conference was developments in the theory and
practice of linear programming since Karmarkar's algorithm. There
were thirty presentations and approximately fifty people attended.
Presentations included new algorithms, new analyses of algorithms,
reports on computational experience, and some other topics related
to the practice of mathematical programming. Interestingly, most of
the progress reported at the conference was on the theoretical
side. Several new polynomial algorithms for linear program- ming
were presented (Barnes-Chopra-Jensen, Goldfarb-Mehrotra, Gonzaga,
Kojima-Mizuno-Yoshise, Renegar, Todd, Vaidya, and Ye). Other
algorithms presented were by Betke-Gritzmann, Blum,
Gill-Murray-Saunders-Wright, Nazareth, Vial, and Zikan-Cottle.
Efforts in the theoretical analysis of algo- rithms were also
reported (Anstreicher, Bayer-Lagarias, Imai, Lagarias,
Megiddo-Shub, Lagarias, Smale, and Vanderbei). Computational
experiences were reported by Lustig, Tomlin, Todd, Tone, Ye, and
Zikan-Cottle. Of special interest, although not in the main
direction discussed at the conference, was the report by Rinaldi on
the practical solution of some large traveling salesman problems.
At the time of the conference, it was still not clear whether the
new algorithms developed since Karmarkar's algorithm would replace
the simplex method in practice. Alan Hoffman presented results on
conditions under which linear programming problems can be solved by
greedy algorithms.
This volume presents a collection of papers on game theory
dedicated to Michael Maschler. Through his dedication and
contributions to game theory, Maschler has become an important
figure particularly in the area of cooperative games. Game theory
has since become an important subject in operations research,
economics and management science. As befits such a volume, the main
themes covered are cooperative games, coalitions, repeated games,
and a cost allocation games. All the contributions are
authoritative surveys of a particular topic, so together they will
present an invaluable overview of the field to all those working on
game theory problems.
Following Karmarkar's 1984 linear programming algorithm, numerous
interior-point algorithms have been proposed for various
mathematical programming problems such as linear programming,
convex quadratic programming and convex programming in general.
This monograph presents a study of interior-point algorithms for
the linear complementarity problem (LCP) which is known as a
mathematical model for primal-dual pairs of linear programs and
convex quadratic programs. A large family of potential reduction
algorithms is presented in a unified way for the class of LCPs
where the underlying matrix has nonnegative principal minors
(P0-matrix). This class includes various important subclasses such
as positive semi-definite matrices, P-matrices, P*-matrices
introduced in this monograph, and column sufficient matrices. The
family contains not only the usual potential reduction algorithms
but also path following algorithms and a damped Newton method for
the LCP. The main topics are global convergence, global linear
convergence, and the polynomial-time convergence of potential
reduction algorithms included in the family.
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