Pseudo-orthogonality for graph 1-Laplacian eigenvectors and applications to higher Cheeger constants and data clustering

The data clustering problem consists in dividing a data set into prescribed groups of homogeneous data. This is an NP-hard problem that can be relaxed in the spectral graph theory, where the optimal cuts of a graph are related to the eigenvalues of graph 1-Laplacian. In this paper, we first give new notations to describe the paths, among critical eigenvectors of the graph 1-Laplacian, realizing sets with prescribed genus. We introduce the pseudo-orthogonality to characterize m₃(G), a special eigenvalue for the graph 1-Laplacian. Furthermore, we use it to give an upper bound for the third graph Cheeger constant h₃(G), that is, h₃(G) 6 m₃(G). This is a first step for proving that the k-th Cheeger constant is the minimum of the 1-Laplacian Raylegh quotient among vectors that are pseudo-orthogonal to the vectors realizing the previous k - 1 Cheeger constants. Eventually, we apply these results to give a method and a numerical algorithm to compute m₃(G), based on a generalized inverse power method.     

Logarithmic Sobolev,isoperimetry and transport inequalities on graphs

In this paper,we study some functional inequalities(such as Poincaré inequality,logarithmic Sobolev inequality,generalized Cheeger isoperimetric inequality,transportation-information inequality and transportation-entropy inequality) for reversible nearest-neighbor Markov processes on connected finite graphs by means of(random) path method.We provide estimates of the involved constants.     

Multi-way dual Cheeger constants and spectral bounds of graphs

We introduce a set of multi-way dual Cheeger constants and prove universal higher-order dual Cheeger inequalities for eigenvalues of normalized Laplace operators on weighted finite graphs. Our proof proposes a new spectral clustering phenomenon deduced from metrics on real projective spaces. We further extend those results to a general reversible Markov operator and find applications in characterizing its essential spectrum.     

Random walks and local cuts in graphs

For a specified subset S of vertices in a graph G we consider local cuts that separate a subset of S. We consider the local Cheeger constant which is the minimum Cheeger ratio over all subsets of S, and we examine the relationship between the local Cheeger constant and the Dirichlet eigenvalue of the induced subgraph on S. These relationships are summarized in a local Cheeger inequality. The proofs are based on the methods of establishing isoperimetric inequalities using random walks and the spectral methods for eigenvalues with Dirichlet boundary conditions.     

On a weighted total variation minimization problem

The present article is devoted to the study of a constrained weighted total variation minimization problem, which may be viewed as a relaxation of a generalized Cheeger problem and is motivated by landslide modeling. Using the fact that the set of minimizers is invariant by a wide class of monotone transformations, we prove that level sets of minimizers are generalized Cheeger sets and obtain qualitative properties of the minimizers: they are all bounded and all achieve their essential supremum on a set of positive measure.     

Quasi-concave minimization subject to linear constraints

A finite algorithm is presented for solving the quasi-concave minimization problem subject to linear constraints. The concept of an extreme point is generalized to that of an extreme facet of a polyhedron. Then a search routine is developed for the detection of an extreme facet of the feasible region relative to the polyhedron defined by the current set of cuts. After identifying an extreme facet we cut it off by a cut developed for this purpose. We call this cut the facet cut. The method is both compatible with other cutting procedures and is finite..     

双圆弧扦值算法又一法——坐标法

本文介绍的方法是用坐标来进行计算 ,既方便 ,又可行 .用一组首尾相接彼此相切的园弧段去逼近给定的曲线使机床切割的机件或计算机绘制的图形进一步满足精度要求 .     

On a parallel genetic–tabu search based algorithm for solving the graph colouring problem

In this paper, we are interested in a particular combinatorial optimisation problem (COP), namely the graph colouring problem (GCP). To solve the GCP, we present a parallel approach adopting an efficient strategy. A brief survey on known methods for solving the GCP enables us to justify our approach which is based on a hybrid method, starting from a set of solutions initialized by the so-called RLF colouring method and combining both a genetic algorithm and the tabu search. A parallelising strategy is then applied. The performances of our method were evaluated through a series of experimentations achieved on an IBM SP2 multiprocessor. The processed graphs were chosen from two benchmark sets. The first, taken from the Internet, involves graphs whose chromatic numbers are known and the second involves random generated graphs. The analysis of the results proves the interest of our approach.     

Cheeger’s cut,maxcut and the spectral theory of 1-Laplacian on graphs

This is primarily an expository paper surveying up-to-date known results on the spectral theory of 1-Laplacian on graphs and its applications to the Cheeger cut, maxcut and multi-cut problems. The structure of eigenspace, nodal domains, multiplicities of eigenvalues, and algorithms for graph cuts are collected.     

A logarithmic barrier cutting plane method for convex programming

The paper presents a logarithmic barrier cutting plane algorithm for convex (possibly non-smooth, semi-infinite) programming. Most cutting plane methods, like that of Kelley, and Cheney and Goldstein, solve a linear approximation (localization) of the problem and then generate an additional cut to remove the linear program's optimal point. Other methods, like the central cutting plane methods of Elzinga-Moore and Goffin-Vial, calculate a center of the linear approximation and then adjust the level of the objective, or separate the current center from the feasible set. In contrast to these existing techniques, we develop a method which does not solve the linear relaxations to optimality, but rather stays in the interior of the feasible set. The iterates follow the central path of a linear relaxation, until the current iterate either leaves the feasible set or is too close to the boundary. When this occurs, a new cut is generated and the algorithm iterates. We use the tools developed by den Hertog, Roos and Terlaky to analyze the effect of adding and deleting constraints in long-step logarithmic barrier methods for linear programming. Finally, implementation issues and computational results are presented. The test problems come from the class of numerically difficult convex geometric and semi-infinite programming problems.This work was completed under the support of a research grant of SHELL.On leave from the Eötvös University, Budapest, and partially supported by OTKA No. 2116.     

Projected viscosity subgradient methods for variational inequalities with equilibrium problem constraints in Hilbert spaces

In this paper, we introduce and study some low computational cost numerical methods for finding a solution of a variational inequality problem over the solution set of an equilibrium problem in a real Hilbert space. The strong convergence of the iterative sequences generated by the proposed algorithms is obtained by combining viscosity-type approximations with projected subgradient techniques. First a general scheme is proposed, and afterwards two practical realizations of it are studied depending on the characteristics of the feasible set. When this set is described by convex inequalities, the projections onto the feasible set are replaced by projections onto half-spaces with the consequence that most iterates are outside the feasible domain. On the other hand, when the projections onto the feasible set can be easily computed, the method generates feasible points and can be considered as a generalization of Maingé’s method to equilibrium problem constraints. In both cases, the strong convergence of the sequences generated by the proposed algorithms is proven.     

Dual Bounds and Optimality Cuts for All-Quadratic Programs with Convex Constraints

A central problem of branch-and-bound methods for global optimization is that often a lower bound do not match with the optimal value of the corresponding subproblem even if the diameter of the partition set shrinks to zero. This can lead to a large number of subdivisions preventing the method from terminating in reasonable time. For the all-quadratic optimization problem with convex constraints we present optimality cuts which cut off a given local minimizer from the feasible set. We propose a branch-and-bound algorithm using optimality cuts which is finite if all global minimizers fulfill a certain second order optimality condition. The optimality cuts are based on the formulation of a dual problem where additional redundant constraints are added. This technique is also used for constructing tight lower bounds. Moreover we present for the box-constrained and the standard quadratic programming problem dual bounds which have under certain conditions a zero duality gap.     

Total variation and Cheeger sets in Gauss space

The aim of this paper is to study the isoperimetric problem with fixed volume inside convex sets and other related geometric variational problems in the Gauss space, in both the finite and infinite dimensional case. We first study the finite dimensional case, proving the existence of a maximal Cheeger set which is convex inside any bounded convex set. We also prove the uniqueness and convexity of solutions of the isoperimetric problem with fixed volume inside any convex set. Then we extend these results in the context of the abstract Wiener space, and for that we study the total variation denoising problem in this context.     

The Cheeger constant of a Jordan domain without necks

We show that the maximal Cheeger set of a Jordan domain \(\Omega \) without necks is the union of all balls of radius \(r = h(\Omega )^{-1}\) contained in \(\Omega \). Here, \(h(\Omega )\) denotes the Cheeger constant of \(\Omega \), that is, the infimum of the ratio of perimeter over area among subsets of \(\Omega \), and a Cheeger set is a set attaining the infimum. The radius r is shown to be the unique number such that the area of the inner parallel set \(\Omega ^r\) is equal to \(\pi r^2\). The proof of the main theorem requires the combination of several intermediate facts, some of which are of interest in their own right. Examples are given demonstrating the generality of the result as well as the sharpness of our assumptions. In particular, as an application of the main theorem, we illustrate how to effectively approximate the Cheeger constant of the Koch snowflake.     

An Outer Approximation Algorithm Guaranteeing Feasibility of Solutions and Approximate Accuracy of Optimality

We treat a concave programming problem with a compact convex feasible set. Assuming the differentiability of the convex functions which define the feasible set, we propose two solution methods. Those methods utilize the convexity of the feasible set and the property of the normal cone to the feasible set at each point over the boundary. Based on the proposed two methods, we propose a solution algorithm. This algorithm takes advantages over classical methods: (1) the obtained approximate solution is always feasible, (2) the error of such approximate value can be evaluated properly for the optimal value of such problem, (3) the algorithm does not have any redundant iterations.     

Coloring mixed hypertrees

A mixed hypergraph is a triple (V,C,D) where V is its vertex set and C and D are families of subsets of V, called C-edges and D-edges, respectively. For a proper coloring, we require that each C-edge contains two vertices with the same color and each D-edge contains two vertices with different colors. The feasible set of a mixed hypergraph is the set of all k's for which there exists a proper coloring using exactly k colors. A hypergraph is a hypertree if there exists a tree such that the edges of the hypergraph induce connected subgraphs of the tree.We prove that feasible sets of mixed hypertrees are gap-free, i.e., intervals of integers, and we show that this is not true for precolored mixed hypertrees. The problem to decide whether a mixed hypertree can be colored by k colors is NP-complete in general; we investigate complexity of various restrictions of this problem and we characterize their complexity in most of the cases.     

On the convergence of a class of outer approximation algorithms for convex programs

This paper presents a new class of outer approximation methods for solving general convex programs. The methods solve at each iteration a subproblem whose constraints contain the feasible set of the original problem. Moreover, the methods employ quadratic objective functions in the subproblems by adding a simple quadratic term to the objective function of the original problem, while other outer approximation methods usually use the original objective function itself throughout the iterations. By this modification, convergence of the methods can be proved under mild conditions. Furthermore, it is shown that generalized versions of the cut construction schemes in Kelley-Cheney-Goldstein's cutting plane method and Veinott's supporting hyperplane method can be incorporated with the present methods and a cut generated at each iteration need not be retained in the succeeding iterations.     

An analytic center quadratic cut method for the convex quadratic feasibility problem

 We consider a quadratic cut method based on analytic centers for two cases of convex quadratic feasibility problems. In the first case, the convex set is defined by a finite yet large number, N, of convex quadratic inequalities. We extend quadratic cut algorithm of Luo and Sun [3] for solving such problems by placing or translating the quadratic cuts directly through the current approximate center. We show that, in terms of total number of addition and translation of cuts, our algorithm has the same polynomial worst case complexity as theirs [3]. However, the total number of steps, where steps consist of (damped) Newton steps, function evaluations and arithmetic operations, required to update from one approximate center to another is , where ε is the radius of the largest ball contained in the feasible set. In the second case, the convex set is defined by an infinite number of certain strongly convex quadratic inequalities. We adapt the same quadratic cut method for the first case to the second one. We show that in this case the quadratic cut algorithm is a fully polynomial approximation scheme. Furthermore, we show that, at each iteration, k, the total number steps (as described above) required to update from one approximate center to another is at most , with ε as defined above. Received: April 2000 / Accepted: June 2002 Published online: September 5, 2002 Key words. convex quadratic feasibility problem – interior-point methods – analytic center – quadratic cuts – potential function