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

Convex Analysisの二,三の進展について

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

For a given pair of finite point setsP andQ in some Euclidean space we consider two problems: Problem 1 of finding the minimum Euclidean norm point in the convex hull ofP and Problem 2 of finding a minimum Euclidean distance pair of points in the convex hulls ofP andQ. We propose a finite recursive algorithm for these problems. The algorithm is not based on the simplicial decomposition of convex sets and does not require to solve systems of linear equations.

A recursive algorithm for finding the minimum norm point in a polytope and a pair of closest points in two polytopes

The minimum-norm point problem which arises in portfolio selections is discussed and an interior point algorithm to solve the problem is proposed in this paper. Three kinds of problems, the mean-variance, the index matching and the multiple factor models are viewed as variants of the minimum-norm point problem. Results of the computational experiments are attached to show the proposed algorithm as a very powerful tool for large scale portfolio optimization.

An interior point algorithm for large scale portfolio optimization

We present an algorithm for finding a nearest pair of points in two convex sets of R n , and therefore, their distance. The algorithm is based on the fixed-point theory of nonexpansive operators on a Hilbert space. Its practical implementation requires a fast projection algorithm. We introduce such a procedure for convex polyhedra. This algorithm effects a local search in the faces using visibility as a guide for finding the global minimum. After studying the convergence of both algorithms, we detail computer experiments on polyhedra (projection and distance). In the case of distances, these experiments show a sublinear time complexity relative to the total number of vertices.

An iterative algorithm for finding a nearest pair of points in two convex subsets of Rn

A batrnet We give a dual algorithm for the problem oC finding the minimum-norm point in the convex hull of a given finite set oC points in a Euclidean space. Our algorithm repeatedly rotates a separating supportinghyperplane and in finitely many steps finds the Carthest separating supporting-hyperplane, whose minimumnorm point is the desired minimum-norm point in the polytope. During the execution oC the algorithm the distance of the separating supporting-hyperplane monotonically increases. The algorithm is closely related to P. Wolfe's primal algorithm which find. a .equence of norm-decreMing points in the given polytope. Computational experiments are carried out to .how the behavior of our algorithm.

A dual algorithm for finding the minimum-norm point in a polytope

We propose a separating-hyperplane algorithm for finding a nearest pair of points in two poly­ topes, where each polytope is expressed as the convex hull of given points in a Euclidian space. The proposed algorithm is an extension of the authors dual algorithm for finding t.he minimum-norm point in a polytope.

/pdf/a-dual-algorithm-for-finding-a-nearest-pair-of-points-in-two-3i4q9zv8c9.pdf

A dual algorithm for finding a nearest pair of points in two polytopes

We propose an efficient algorithm for finding the minimum-norm point in the intersection of a polyhedron and a hyperplane, where the polyhedron is expressed as the sum of the convex hull of a finite point set and the conical hull of a finite direction vector set. Our algorithm solves the minimum-norm point problem by directly treating the given original data, the points, the direction vectors and the hyperplane, without computing points and direction vectors which generate the intersection of the polyhedron and the hyperplane. We also show that our algorithm is practically efficient by computational experiments.

An algorithm for finding the minimum-norm point in the intersection of a convex polyhedron and a hyperplane

Abstract We consider the problem of allocating divisible/indivisible goods to agents according to agents’ ordinal preferences. Hashimoto et al. [15] provided a nonalgorithmic and axiomatic characterization of well-studied probabilistic serial (PS) mechanism. Recently, Fujishige et al. [12] generalized the PS mechanism where goods are enlarged from a fixed set to a family of sets which is a polytope defined by a system of linear inequalities associated with submodular functions. The above extended PS (EPS) greatly improved the flexibility of allocations. Based on these two results, in this paper, we investigate the nonalgorithmic and axiomatic characterization of EPS. We show that the EPS rule is the only mechanism satisfying the ordinal fairness and a newly defined non-wastefulness . The submodularity plays a crucial role in our arguments. 

/pdf/a-simple-characterization-of-assignment-mechanisms-on-set-30zwyv92.pdf

A Simple Characterization of Assignment Mechanisms on Set Constraints

https://link.springer.com/content/pdf/10.1007/s10878-023-01063-0.pdf

Simultaneous eating algorithm and greedy algorithm in assignment problems

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