192 research outputs found
On Equivalence of M-concavity of a Set Function and Submodularity of Its Conjugate
A fundamental theorem in discrete convex analysis states that a set function
is M-concave if and only if its conjugate function is submodular.
This paper gives a new proof to this fact
Fundamentals in Discrete Convex Analysis
This talk describes fundamental properties of M-convex and L-convex functions that play the central roles in discrete convex analysis.
These concepts were originally introduced in combinatorial optimization, but turned out to be relevant in economics.
Emphasis is put on discrete duality and conjugacy respect to the Legendre-Fenchel transformation.
Monograph information:
http://www.misojiro.t.u-tokyo.ac.jp/~murota/mybooks.html#DCAsiam200
Discrete Midpoint Convexity
For a function defined on a convex set in a Euclidean space, midpoint
convexity is the property requiring that the value of the function at the
midpoint of any line segment is not greater than the average of its values at
the endpoints of the line segment. Midpoint convexity is a well-known
characterization of ordinary convexity under very mild assumptions. For a
function defined on the integer lattice, we consider the analogous notion of
discrete midpoint convexity, a discrete version of midpoint convexity where the
value of the function at the (possibly noninteger) midpoint is replaced by the
average of the function values at the integer round-up and round-down of the
midpoint. It is known that discrete midpoint convexity on all line segments
with integer endpoints characterizes L-convexity, and that it
characterizes submodularity if we restrict the endpoints of the line segments
to be at -distance one. By considering discrete midpoint convexity
for all pairs at -distance equal to two or not smaller than two,
we identify new classes of discrete convex functions, called local and global
discrete midpoint convex functions, which are strictly between the classes of
L-convex and integrally convex functions, and are shown to be
stable under scaling and addition. Furthermore, a proximity theorem, with the
same small proximity bound as that for L-convex functions, is
established for discrete midpoint convex functions. Relevant examples of
classes of local and global discrete midpoint convex functions are provided.Comment: 39 pages, 6 figures, to appear in Mathematics of Operations Researc
Note on Minimization of Quasi M-convex Functions
For a class of discrete quasi convex functions called semi-strictly quasi
M-convex functions, we investigate fundamental issues relating to
minimization, such as optimality condition by local optimality, minimizer cut
property, geodesic property, and proximity property. Emphasis is put on
comparisons with (usual) M-convex functions. The same optimality
condition and a weaker form of the minimizer cut property hold for
semi-strictly quasi M-convex functions, while geodesic property and
proximity property fail
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