Strong CHIP, normality, and linear regularity of convex sets

We extend the property (N) introduced by Jameson for closed convex cones to the normal property for a finite collection of convex sets in a Hilbert space. Variations of the normal property, such as the weak normal property and the uniform normal property, are also introduced. A dual form of the normal property is derived. When applied to closed convex cones, the dual normal property is the property (G) introduced by Jameson. Normality of convex sets provides a new perspective on the relationship between the strong conical hull intersection property (strong CHIP) and various regularity properties. In particular, we prove that the weak normal property is a dual characterization of the strong CHIP, and the uniform normal property is a characterization of the linear regularity. Moreover, the linear regularity is equivalent to the fact that the normality constant for feasible direction cones of the convex sets at is bounded away from 0 uniformly over all points in the intersection of these convex sets.

     

Strong regularity and fuzzy strong regularity in semihypergroups

In this paper we introduce the concept of a fuzzy strongly regular relation on a semihypergroup, and prove a few results concerning this concept.     

Strong regularity of parametric interval matrices

Several verifiable sufficient and necessary conditions for strong regularity of parametric interval matrices are given. The conditions are formulated in terms of the spectral radius, positive definiteness and eigenvalues of certain real matrices and a scaled maximum norm.     

Strong convergence of the CQ method for fixed point iteration processes

Strong convergence theorems are obtained for the CQ method for an Ishikawa iteration process, a contractive-type iteration process for nonexpansive mappings, and the proximal point algorithm for maximal monotone operators in Hilbert spaces.     

Strong regularity and generalized inverses in jordan systems

A notion of generalized inverse extending that of Moore—Penrose inverse for continuous linear operators between Hilbert spaces and that of group inverse for elements of an associative algebra is defined in any Jordan triple system (J, P). An element a?J has a (unique) generalized inverse if and only if it is strongly regular, i.e., a?P(a)²J. A Jordan triple system J is strongly regular if and only if it is von Neumann regular and has no nonzero nilpotent elements. Generalized inverses have properties similar to those of the invertible elements in unital Jordan algebras. With a suitable notion of strong associativity, for a strongly regular element a?J with generalized inverse b the subtriple generated by {a, b} is strongly associative     

Strong calmness of perturbed KKT system for a class of conic programming with degenerate solutions

ABSTRACT

This paper is concerned with the strong calmness of the KKT solution mapping for a class of canonically perturbed conic programming, which plays a central role in achieving fast convergence under situations when the Lagrange multiplier associated to a solution of these conic optimization problems is not unique. We show that the strong calmness of the KKT solution mapping is equivalent to a local error bound for the solutions to the perturbed KKT system, and is also equivalent to the pseudo-isolated calmness of the stationary point mapping along with the calmness of the multiplier set mapping at the corresponding reference point. Sufficient conditions are also provided for the strong calmness by establishing the pseudo-isolated calmness of the stationary point mapping in terms of the noncriticality of the associated multiplier, and the calmness of the multiplier set mapping in terms of a relative interior condition for the multiplier set. These results cover and extend the existing ones in Hager and Gowda [Stability in the presence of degeneracy and error estimation. Math Program. 1999;85:181–192]; Izmailov and Solodov [Stabilized SQP revisited. Math Program. 2012;133:93–120] for nonlinear programming and in Cui et al. [On the asymptotic superlinear convergence of the augmented Lagrangian method for semidefinite programming with multiple solutions. 2016, arXiv: 1610.00875v1]; Zhang and Zhang [Critical multipliers in semidefinite programming. 2018, arXiv: 1801.02218v1] for semidefinite programming.     

OnBMO regularity for linear elliptic systems

Summary We prove a refinement of Campanato's result on local and global (under Dirichlet boundary conditions) BMO regularity for the gradient of solutions of linear elliptic systems of second order in divergence form: we just need that the coefficients are «small multipliers of BMO()», a class neither containing, nor contained in . We also prove local and global L^p regularity: this result neither implies, nor follows by the classical one by Agmon, Douglis and Nirenberg.Work partially supported by M.P.I.Project 40% «Equazioni di evoluzione e applicazioni fisico-matematiche».     

The equivalence of “strong calmness” and “calmness” in optimal control theory

In an earlier paper optimality conditions expressed in terms of a Lipschitz continuous function which satisfies a condition resembling the Hamilton-Jacobi-Bellman equation were derived. It was shown that a certain hypothesis, strong calmness, is the weakest hypothesis under which such developments are possible. In the present paper it is shown that strong calmness is equivalent to calmness for a wide class of problems. Support is thereby given to strong calmness as being a reasonable hypothesis, since calmness is apparently the weakest known hypothesis assuring normality, in the sense that the Pontryagin maximum principle applies with the cost multiplier nonzero.     

Hilbert空间中求解分裂可行问题CQ算法的强收敛性

在Hilbert空间中,为了研究分裂可行问题迭代算法的强收敛性,提出了一种新的CQ算法.首先利用CQ算法构造了一个改进的Halpern迭代序列； 然后通过把分裂可行问题转化为算子不动点， 在较弱的条件下， 证明了该序列强收敛到分裂可行问题的一个解. 推广了Wang和Xu的有关结果.     

On coercivity and regularity for linear elliptic systems

We introduce a nonlinear method to study a ??universal?? strong coercivity problem for monotone linear elliptic systems by compositions of finitely many constant coefficient tensors satisfying the Legendre?CHadamard strong ellipticity condition. We give conditions and counterexamples for universal coercivity. In the case of non-coercive systems we give examples to show that the corresponding variational integral may have infinitely many nowhere C ¹ minimizers on their supports. For some universally coercive systems we also present examples with affine boundary values which have nowhere C ¹ solutions.     

Asymptotic linear bounds for the Castelnuovo-Mumford regularity

We prove asymptotic linear bounds for the Castelnuovo-Mumford regularity of certain filtrations of homogeneous ideals whose Rees algebras need not be Noetherian.

     

Strong labelings of linear forests

A （p, q）-graph G is called super edge-magic if there exists a bijective function f ： V（G） U E（G） →{1, 2 p＋q} such that f（u）＋ f（v）＋f（uv） is a constant for each uv C E（G） and f（Y（G）） = {1,2,...,p}. In this paper, we introduce the concept of strong super edge-magic labeling as a particular class of super edge-magic labelings and we use such labelings in order to show that the number of super edge-magic labelings of an odd union of path-like trees （mT）, all of them of the same order, grows at least exponentially with m.     

On the regularity of discrete linear systems

In this paper we have introduced a new regularity coefficient of time varying discrete linear system. On the base of this coefficient we have characterized the regularity of homogeneous discrete time varying linear systems by nonhomogeneous ones. Moreover we provided bounds for the regularity coefficient in terms of the existing in the literature regularity coefficients.     

Strong unicity and alternation for linear optimization

We consider linear semi-infinite optimization problems and prove characterizations for strong unicity. One of these is a weak alternation property. This result suggests the introduction of the property of regular strong unicity, which is equivalent to a stronger alternation property. The theorems are used in order to prove a Haar-type theorem for linear optimization problems. An application to best Chebyshev approximation is given.Communicated by A. V. Fiacco     

Strong duality for inexact linear programming

In this paper, we apply the Dubovitskii-Milyutin approach to derive strong duality theorems for inexact linear programming problems. Inexact linear programming deals with the standard linear problem in which the data is not well known and it is supposed to lie in certain given convex sets. The case of parametric dependence of the data is particularly analyzed and relations with semi-infinite and

semi-definite programming are also commented.     

Strong unicity in normed linear spaces

We consider best approximations in a real or complex normed linear space by elements of a finite dimensional subspace. It is the purpose of this paper to characterize, when a best approximation to a given element is strongly unique.