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.     

One-Parameter Families of Feasible Sets in Semi-infinite Optimization

Feasible sets in semi-infinite optimization are basically defined by means of infinitely many inequality constraints. We consider one-parameter families of such sets. In particular, all defin-ing functions - including those defining the index set of the inequality constraints - will depend on a parameter. We note that a semi-infinite problem is a two-level problem in the sense that a point is feasible if and only if all global minimizers of a corresponding marginal function are nonnegative. For a quite natural class of mappings we characterize changes in the global topological structure of the corresponding feasible set as the parameter varies. As long as the index set (-mapping) of the inequality constraints is lower semicontinuous, all changes in topology are those which generically appear in one-parameter sets defined by finitely many constraints. In the case, however, that some component of the mentioned index set is born (or vanishes), the topological change is of global nature and is not controllable. In fact, the change might be as drastic as that when adding or deleting an (arbitrary) inequality constraint.     

Differentiability of p-Harmonic Functions on Metric Measure Spaces

We study p-harmonic functions on metric measure spaces, which are formulated as minimizers to certain energy functionals. For spaces supporting a p-Poincaré inequality, we show that such functions satisfy an infinitesmal Lipschitz condition almost everywhere. This result is essentially sharp, since there are examples of metric spaces and p-harmonic functions that fail to be locally Lipschitz continuous on them. As a consequence of our main theorem, we show that p-harmonic functions also satisfy a generalized differentiability property almost everywhere, in the sense of Cheeger’s measurable differentiable structures.     

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.     

Existence and stability of solitary-wave solutions of equations of Benjamin-Bona-Mahony type

We consider solitary-wave solutions of equations of Benjamin-Bona-Mahony type. We show that for a large class of equations of BBM type, there do exist stable sets consisting of solitary-wave profile functions. In the case of generalized BBM equations, we found that there are profile functions of stable solitary waves that are not the minimizers of the associated variational problem. Such a phenomenon is not known to exist for equations of Korteweg-de Vries type.     

On Equilibrium Shape of Charged Flat Drops

The equilibrium shapes of two‐dimensional charged, perfectly conducting liquid drops are governed by a geometric variational problem that involves a perimeter term modeling line tension and a capacitary term modeling Coulombic repulsion. Here we give a complete explicit solution to this variational problem. Namely, we show that at fixed total charge a ball of a particular radius is the unique global minimizer among all sufficiently regular sets in the plane. For sets whose area is also fixed, we show that balls are the only minimizers if the charge is less than or equal to a critical charge, while for larger charge minimizers do not exist. Analogous results hold for drops whose potential, rather than charge, is fixed. © 2018 Wiley Periodicals, Inc.     

Optimality and duality in nonsmooth multiobjective fractional programming problem with constraints

In this paper, we establish some quotient calculus rules in terms of contingent derivatives for the two extended-real-valued functions defined on a Banach space and study a nonsmooth multiobjective fractional programming problem with set, generalized inequality and equality constraints. We define a new parametric problem associated with these problem and introduce some concepts for the (local) weak minimizers to such problems. Some primal and dual necessary optimality conditions in terms of contingent derivatives for the local weak minimizers are provided. Under suitable assumptions, sufficient optimality conditions for the local weak minimizers which are very close to necessary optimality conditions are obtained. An application of the result for establishing three parametric, Mond–Weir and Wolfe dual problems and several various duality theorems for the same is presented. Some examples are also given for our findings.

     

Equivalence of the generalized complementarity problem to differentiable unconstrained minimization

We consider an unconstrained minimization reformulation of the generalized complementarity problem (GCP). The merit function introduced here is differentiable and has the property that its global minimizers coincide with the solutions of GCP. Conditions for its stationary points to be global minimizers are given. Moreover, it is shown that the level sets of the merit function are bounded under suitable assumptions. We also show that the merit function provides global error bounds for GCP. These results are based on a condition which reduces to the condition of the uniform P-function when GCP is specialized to the nonlinear complementarity problem. This condition also turns out to be useful in proving the existence and uniqueness of a solution for GCP itself. Finally, we obtain as a byproduct an error bound result with the natural residual for GCP.We thank Jong-Shi Pang for his valuable comments on error bound results with the natural residual for the nonlinear complementarity problem. We are also grateful to the anonymous referees for some helpful comments. The research of the second author was supported in part by the Science Research Grant-in-Aid from the Ministry of Education, Science, and Culture, Japan.     

Mixed Type Duality for Set-valued Optimization Problems via Higher-order Radial Epiderivatives

In this article, we introduce a notion of higher-order radial epiderivative for set-valued maps and study its properties. A generalized concept of higher-order strict minimizers in set-valued optimization is proposed as well. By virtue of the radial epiderivative, we establish a mixed dual problem, and then weak, strong, and converse duality theorems are obtained in dealing with generalized strict minimizers.     

Action minimizers under topological constraints in the planar equal-mass four-body problem

It is shown that in the planar equal-mass four-body problem, there exist two sets of action minimizers connecting two planar boundary configurations with fixed symmetry axes and specific order constraints: a double isosceles configuration and an isosceles trapezoid configuration, while order constraints are introduced on the boundary configurations. By applying the level estimate method, these minimizers are shown to be collision-free and they can be extended to two new sets of periodic or quasi-periodic orbits.     

Minimization of Gerstewitz functionals extending a scalarization by Pascoletti and Serafini

Scalarization in vector optimization is often closely connected to the minimization of Gerstewitz functionals. In this paper, the minimizer sets of Gerstewitz functionals are investigated. Conditions are given under which such a set is nonempty and compact. Interdependencies between solutions of problems with different parameters or with different feasible point sets are shown. Consequences for the parameter control in scalarization methods are proved. It is pointed out that the minimization of Gerstewitz functionals is equivalent to an optimization problem which generalizes the scalarization by Pascoletti and Serafini. The results contain statements about minimizers of certain Minkowski functionals and norms. Some existence results for solutions of vector optimization problems are derived.     

Higher-order optimality conditions for strict local minima

In this work, we study a nonsmooth optimization problem with generalized inequality constraints and an arbitrary set constraint. We present necessary conditions for a point to be a strict local minimizer of order k in terms of higher-order (upper and lower) Studniarski derivatives and the contingent cone to the constraint set. In the same line, when the initial space is finite dimensional, we develop sufficient optimality conditions. We also provide sufficient conditions for minimizers of order k using the lower Studniarski derivative of the Lagrangian function. Particular interest is put for minimizers of order two, using now a special second order derivative which leads to the Fréchet derivative in the differentiable case.     

Unconstrained Methods for Generalized Complementarity Problems

1.IntroductionThecomplementarityproblem,aspecialcaseofvariationalinequalityproblem,hasmanyapplicationsindifferentfieldssuchasmathematicalprogramming,gametheory,economics.Generally,thestandardcomplementarityproblemhasthefollowingform:y=F(x),x20,y20,(y,x)=0,(1.1)where(.,.)denotestheinnerproducts.WhenF(x)isanaffinefunctionofx,itreducestothelinearcomplementarityproblemwhichisdenotedbyLCP.Otherwisewecallitthenonlinearco7nplementaritypro6lemorsimplyNCP.Thecomplementarityproblemhajsattractedmanyr…     

On the best case performance of hit and run methods for detecting necessary constraints

The hit and run methods are probabilistic algorithms that can be used to detect necessary (nonredundant) constraints in systems of linear constraints. These methods construct random sequences of lines that pass through the feasible region. These lines intersect the boundary of the region at twohit-points, each identifying a necessary constraint. In order to study the statistical performance of such methods it is assumed that the probabilities of hitting particular constraints are the same for every iteration. An indication of the best case performance of these methods can be determined by minimizing, with respect to the hit probabilities, the expected value of the number of iterations required to detect all necessary constraints. We give a set of isolated strong local minimizers and prove that for two, three and four necessary constraints the set of local minimizers is the complete set of global minimizers. We conjecture that this is also the case for any number of necessary constraints. The results in this paper also apply to sampling problems (e.g., balls from an urn) and to the coupon collector's problem.     

Convergence of relaxed minimizers in set optimization

In this paper we investigate, in a unified way, the stability of several relaxed minimizers of set optimization problems. To this end, we introduce a topology on vector ordered spaces from which we derive a concept of convergence that allows us to study both the upper and the lower stability of the sets of relaxed minimizers we consider.     

Efficient solutions in generalized linear vector optimization

This paper establishes several new facts on generalized polyhedral convex sets and shows how they can be used in vector optimization. Among other things, a scalarization formula for the efficient solution sets of generalized linear vector optimization problems is obtained. We also prove that the efficient solution set of a generalized linear vector optimization problem in a locally convex Hausdorff topological vector space is the union of finitely many generalized polyhedral convex sets and it is connected by line segments.     

Existence and stability of standing waves for nonlinear fractional Schrödinger equations with Hartree type nonlinearity

In this paper, we consider the nonlinear fractional Schrödinger equations with Hartree type nonlinearity. We obtain the existence of standing waves by studying the related constrained minimization problems via applying the concentration-compactness principle. By symmetric decreasing rearrangements, we also show that the standing waves, up to translations and phases, are positive symmetric nonincreasing functions. Moreover, we prove that the set of minimizers is a stable set for the initial value problem of the equations, that is, a solution whose initial data is near the set will remain near it for all time.     

Certain properties of generalized α-formulas

The problem of realization of Boolean functions by generalized α-formulas is considered. The notion of a universal set of generalized α-formulas is introduced for a given set of Boolean functions. Universal sets of generalized α-formulas are constructed for the set of constant-preserving Boolean functions.     

覆盖广义粗糙集的模糊性

在研究覆盖广义粗糙集的基础上,利用两个距离函数Hamming和Euclidean距离函数,结合模糊集的最近寻常集,引入了覆盖广义粗糙集模糊度的概念,给出了一种模糊度计算方法,并证明了该模糊度的一些重要性质。这些结果在覆盖广义粗糙集的理论研究和应用都发挥着一定作用。