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Convex Analysisの二,三の進展について

Note: Professor Pach's number: [045]; Also in: Facsimile edition: World Classics in Mathematics Series, vol. 28, China Science Press, Beijing, 2006. Mimeographed from 100 Research Problems in Discrete Geometry (with W. O. J. Moser). Reference DCG-BOOK-2008-001 URL: http://www.math.nyu.edu/~pach/research_problems.html Record created on 2008-11-18, modified on 2017-05-12

Research Problems in Discrete Geometry

Reverse search for enumeration

Power networks are undergoing a fundamental transition, with traditionally passive consumers becoming ‘prosumers’ — proactive consumers with distributed energy resources, actively managing their consumption, production and storage of energy. A key question that remains unresolved is: how can we incentivize coordination between vast numbers of distributed energy resources, each with different owners and characteristics? Virtual power plants and peer-to-peer (P2P) energy trading offer different sources of value to prosumers and the power network, and have been proposed as different potential structures for future prosumer electricity markets. In this Perspective, we argue they can be combined to capture the benefits of both. We thus propose the concept of the federated power plant, a virtual power plant formed through P2P transactions between self-organizing prosumers. This addresses social, institutional and economic issues faced by top-down strategies for coordinating virtual power plants, while unlocking additional value for P2P energy trading. The rise of prosumers has led to creation of virtual power plants and peer-to-peer trading to help manage a diverse and distributed array of energy sources. This Perspective proposes the federated power plant, which combines these concepts to meet some of their individual challenges and offer new value.

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Using peer-to-peer energy-trading platforms to incentivize prosumers to form federated power plants

Introduction. 1. Mathematical Preliminaries. Submodular Systems and Base Polyhedra. 2. From Matroids to Submodular Systems. Matroids. Polymatroids. Submodular Systems. 3. Submodular Systems and Base Polyhedra. Fundamental Operations on Submodular Systems. Greedy Algorithm. Structures of Base Polyhedra. Intersecting- and Crossing-Submodular Functions. Related Polyhedra. Submodular Systems of Network Type. Neoflows. 4. The Intersection Problem. The Intersection Theorem. The Discrete Separation Theorem. The Common Base Problem. 5. Neoflows. The Equivalence of the Neoflow Problems. Feasibility for Submodular Flows. Optimality for Submodular Flows. Algorithms for Neoflows. Matroid Optimization. Submodular Analysis. 6. Submodular Functions and Convexity. Conjugate Functions and a Fenchel-Type Min-Max Theorem for Submodular and Supermodular Functions. Subgradients of Submodular Functions. The Lovasz Extensions of Submodular Functions. 7. Submodular Programs. Submodular Programs - Unconstrained Optimization. Submodular Programs - Constrained Optimization. Nonlinear Optimization with Submodular Constraints. 8. Separable Convex Optimization. Optimality Conditions. A Decomposition Algorithm. Discrete Optimization. 9. The Lexicographically Optimal Base Problem. Nonlinear Weight Functions. Linear Weight Functions. 10. The Weighted Max-Min and Min-Max Problems. Continuous Variables. Discrete Variables. 11. The Fair Resource Allocation Problem. Continuous Variables. Discrete Variables. 12. The Neoflow Problem with a Separable Convex Cost Function. References. Index.

Submodular functions and optimization

The maximum weight/cardinality stable set problem is to nd a maximum weight/cardinality stable set of a given graph. It is well known that these problems for general graphs belong to the class of NP-hard. However, for several classes of graphs, e.g., for perfect graphs and claw-free graphs and so on, these problem can be solved in polynomial time. For instance, Minty (1980), Sbihi (1980) and Lov asz and Plummer (1986) have proposed polynomial time algorithms nding a maximum cardinality stable set of a claw-free graph. Moreover, it has been believed that Minty's algorithm is the unique algorithm nding a maximum weight stable set of a claw-free graph up to date. Here we show that Minty's algorithm for the weighted version fails for some special cases, and give modi cations to overcome it.

A revision of Minty's algorithm for finding a maximum weight stable set of a claw-free graph

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

Discrete fixed point theorem reconsidered

Designing matching mechanisms under constraints: An approach from discrete convex analysis

In this paper we show that any rooted tree ofn vertices can be straight-line embedded into any setS ofn points in the plane in general position so that the image of the root is arbitrarily specified.

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Akihisa Tamura

Papers

A revision of Minty's algorithm for finding a maximum weight stable set of a claw-free graph

An Optimal Algorithm for Scanning All Spanning Trees of Undirected Graphs

Discrete fixed point theorem reconsidered

Designing matching mechanisms under constraints: An approach from discrete convex analysis

The rooted tree embedding problem into points in the plane