/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

1 Defining Network Flow A flow network is a directed graph G = (V,E) in which each edge (u, v) ∈ E has non-negative capacity c(u, v) ≥ 0. We require that if (u, v) ∈ E, then (v, u) / ∈ E. That is, if an edge exists, then the edge between the same vertices going the reverse direction does not exist. Every flow network has a source s and a sink t, and we assume that for every v ∈ V , there is some path s→ · · · → v → · · · → t. Note that this implies that flow networks are connected. Informally, the intuition behind network flow is to think of the edges as pipes and the weights on the edges as the capacity its corresponding pipe per unit of time. The question we are trying to ask is: how many units of water can we send from the source to the sink per unit of time? Formally, a flow in G is a function f : V × V → R that satisfies the following: • Capacity constraint. For all u, v ∈ V , we require 0 ≤ f(u, v) ≤ c(u, v). Our pipe cannot hold more than is allowed as dictated by its capacity. • Flow conservation. For u ∈ V − {s, t}, we require ∑

Network Flows

Introduction to linear and nonlinear programming

A theory of “discrete convex analysis” is developed for integer-valued functions defined on integer lattice points. The theory parallels the ordinary convex analysis, covering discrete analogues of the fundamental concepts such as conjugacy, subgradients, the Fenchel min-max duality, separation theorems and the Lagrange duality framework for convex/nonconvex optimization. The technical development is based on matroid-theoretic concepts, in particular, submodular functions and exchange axioms. Sections 1–4 extend the conjugacy relationship between submodularity and exchange ability, deepening our understanding of the relationship between convexity and submodularity investigated in the eighties by A. Frank, S. Fujishige, L. Lovasz and others. Sections 5 and 6 establish duality theorems for M- and L-convex functions, namely, the Fenchel min-max duality and separation theorems. These are the generalizations of the discrete separation theorem for submodular functions due to A. Frank and the optimality criteria for the submodular flow problem due to M. Iri-N. Tomizawa, S. Fujishige, and A. Frank. A novel Lagrange duality framework is also developed in integer programming. We follow Rockafellar’s conjugate duality approach to convex/nonconvex programs in nonlinear optimization, while technically relying on the fundamental theorems of matroid-theoretic nature.

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Discrete Convex Analysis

Reverse search for enumeration

The concept of M-convex function, introduced by Murota 1996, is a quantitative generalization of the set of integral points in an integral base polyhedron as well as an extension of valuated matroid of Dress and Wenzel 1990. In this paper, we extend this concept to functions on generalized polymatroids with a view to providing a unified framework for efficiently solvable nonlinear discrete optimization problems.

/pdf/m-convex-function-on-generalized-polymatroid-3lfjngkeuy.pdf

M-Convex Function on Generalized Polymatroid

Let G be an undirected graph with V vertices and E edges. Many algorithms have been developed for enumerating all spanning trees in G. Most of the early algorithms use a technique called "backtracking." Recently, several algorithms using a different technique have been proposed by Kapoor and Ramesh (1992), Matsui (1993), and Shioura and Tamura (1993). They find a new spanning tree by exchanging one edge of a current one. This technique has the merit of enabling us to compress the whole output of all spanning trees by outputting only relative changes of edges. Kapoor and Ramesh first proposed an O(N + V + E)-time algorithm by adopting such a "compact" output, where N is the number of spanning trees. Another algorithm with the same time complexity was constructed by Shioura and Tamura. These are optimal in the sense of time complexity but not in terms of space complexity because they take O(VE) space. We refine Shioura and Tamura's algorithm and decrease the space complexity from O(VE) to O(V + E) while preserving the time complexity. Therefore, our algorithm is optimal in the sense of both time and space complexities.

An Optimal Algorithm for Scanning All Spanning Trees of Undirected Graphs

Relationship of M-/L-convex functions with discrete convex functions by Miller and Favati-Tardella

Efficient allocation of indivisible goods is an important problem in mathematical economics and operations research, where the concept of Walrasian equilibrium plays a fundamental role. As a sufficient condition for the existence of a Walrasian equilibrium, the concept of gross substitutes condition for valuation functions is introduced by Kelso and Crawford (1982). Since then, several variants of gross substitutes condition as well as a discrete concavity concept, called M ♮ -concavity, have been introduced to show the existence of an equilibrium in various models. In this paper, we survey the relationship among Kelso and Crawford's gross substitutes condition and its variants, and discuss the connection with M ♮ -concavity. We also review various characterizations and properties of these concepts.

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Gross substitutes condition and discrete concavity for multi-unit valuations: a survey

http://www.dais.is.tohoku.ac.jp/~shioura/papers/do09.pdf

Akiyoshi Shioura

Papers

M-Convex Function on Generalized Polymatroid

An Optimal Algorithm for Scanning All Spanning Trees of Undirected Graphs

Relationship of M-/L-convex functions with discrete convex functions by Miller and Favati-Tardella

Gross substitutes condition and discrete concavity for multi-unit valuations: a survey

New algorithms for convex cost tension problem with application to computer vision