Subdifferential and optimality conditions for the difference of set-valued mappings

In this paper, an existence theorem of the subgradients for set-valued mappings, which introduced by Borwein (Math Scand 48:189?C204, 1981), and relations between this subdifferential and the subdifferential introduced by Baier and Jahn (J Optim Theory Appl 100:233?C240, 1999), are obtained. By using the concept of this subdifferential, the sufficient optimality conditions for generalized D.C. multiobjective optimization problems are established. And the necessary optimality conditions, which are the generalizations of that in Gadhi (Positivity 9:687?C703, 2005), are also established. Moreover, by using a special scalarization function, a real set-valued optimization problem is introduced and the equivalent relations between the solutions are proved for the real set-valued optimization problem and a generalized D.C. multiobjective optimization problem.     

When a vector quasimonotone mapping is a vector monotone mapping

In this paper we provide a sufficient conditions that under them a vector quasimonotone set-valued mapping transfer to a vector monotone set-valued mapping. In fact this note is a vector version of the papers (Farajzadeh, J Ineq Appl 2012:192, 2012) and (Hadjisavvas, Appl Math Lett 19:913–915, 2006).     

Weakly relaxed α-pseudomonotonicity and equilibrium problem in Banach spaces

In this paper, we introduce the weakly relaxed ??-pseudomonotone mapping which generalizes the class of pseudomonotone mappings. Some examples are given which show that the weakly relaxed ??-pseudomonotonicity is a proper generalization of pseudomonotonicity for both scalar and vector valued bi-functions. Using the KKM technique, we establish the existence of solutions of equilibrium problems and vector equilibrium problems with weakly relaxed ??-pseudomonotonicity in the reflexive Banach spaces. The present work extends some corresponding results of the variational-like inequalities (Fang and Huang in J. Optim. Theory Appl. 118:327?C338, 2003; Lee and Lee in J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 11: 231?C242, 2004) to equilibrium problems.     

Adjustment Coefficient for Risk Processes in Some Dependent Contexts

Following Müller and Pflug (Insur Math Econ 28:381?C392, 2001) and Nyrhinen (Adv Appl Probab 30:1008?C1026, 1998; J Appl Probab 36:733?C746, 1999), we study the adjustment coefficient of ruin theory in a context of temporal dependency. We provide a consistent estimator for this coefficient, and perform some simulations.     

Continuity of the solution mapping to parametric generalized vector equilibrium problems

In this paper, two kinds of parametric generalized vector equilibrium problems in normed spaces are studied. The sufficient conditions for the continuity of the solution mappings to the two kinds of parametric generalized vector equilibrium problems are established under suitable conditions. The results presented in this paper extend and improve some main results in Chen and Gong (Pac J Optim 3:511–520, 2010), Chen and Li (Pac J Optim 6:141–152, 2010), Chen et al. (J Glob Optim 45:309–318, 2009), Cheng and Zhu (J Glob Optim 32:543–550, 2005), Gong (J Optim Theory Appl 139:35–46, 2008), Li and Fang (J Optim Theory Appl 147:507–515, 2010), Li et al. (Bull Aust Math Soc 81:85–95, 2010) and Peng et al. (J Optim Theory Appl 152(1):256–264, 2011).     

A note on the global convergence theorem of the scaled conjugate gradient algorithms proposed by Andrei

In (Andrei, Comput. Optim. Appl. 38:402?C416, 2007), the efficient scaled conjugate gradient algorithm SCALCG is proposed for solving unconstrained optimization problems. However, due to a wrong inequality used in (Andrei, Comput. Optim. Appl. 38:402?C416, 2007) to show the sufficient descent property for the search directions of SCALCG, the proof of Theorem?2, the global convergence theorem of SCALCG, is incorrect. Here, in order to complete the proof of Theorem?2 in (Andrei, Comput. Optim. Appl. 38:402?C416, 2007), we show that the search directions of SCALCG satisfy the sufficient descent condition. It is remarkable that the convergence analyses in (Andrei, Optim. Methods Softw. 22:561?C571, 2007; Eur. J. Oper. Res. 204:410?C420, 2010) should be revised similarly.     

Harmonic mappings in Bergman spaces

In this paper, we investigate the properties of mappings in harmonic Bergman spaces. First, we discuss the coefficient estimate, the Schwarz-Pick Lemma and the Landau-Bloch theorem for mappings in harmonic Bergman spaces in the unit disk $\mathbb D $ of $\mathbb C $ . Our results are generalizations of the corresponding ones in Chen et al. (Proc Am Math Soc 128:3231–3240, 2000), Chen et al. (J Math Anal Appl 373:102–110, 2011), Chen et al. (Ann Acad Sci Fenn Math 36:567–576, 2011). Then, we study the Schwarz-Pick Lemma and the Landau-Bloch theorem for mappings in harmonic Bergman spaces in the unit ball $\mathbb B ^{n}$ of $\mathbb C ^{n}$ . The obtained results are generalizations of the corresponding ones in Chen and Gauthier (Proc Am Math Soc 139:583–595 2011). At last, we get a characterization for mappings in harmonic Bergman spaces on $\mathbb B ^{n}$ in terms of their complex gradients.     

Approximating common fixed points of non-self asymptotically nonexpansive mapping in Banach spaces

In this paper, we consider a composite iterative algorithm with errors for approximating a common fixed points of non-self asymptotically nonexpansive mappings in the framework of Hilbert spaces. Our results improve and extend Chidume et al. (J. Math. Anal. Appl. 280:364–374, [2003]), Shahzad (Nonlinear Anal. 61:1031–1039, [2005]), Su and Qin (J. Appl. Math. Comput. 24:437–448, [2007]) and many others.     

An augmented Lagrangian approach for sparse principal component analysis

Principal component analysis (PCA) is a widely used technique for data analysis and dimension reduction with numerous applications in science and engineering. However, the standard PCA suffers from the fact that the principal components (PCs) are usually linear combinations of all the original variables, and it is thus often difficult to interpret the PCs. To alleviate this drawback, various sparse PCA approaches were proposed in the literature (Cadima and Jolliffe in J Appl Stat 22:203–214, 1995; d’Aspremont et?al. in J Mach Learn Res 9:1269–1294, 2008; d’Aspremont et?al. SIAM Rev 49:434–448, 2007; Jolliffe in J Appl Stat 22:29–35, 1995; Journée et?al. in J Mach Learn Res 11:517–553, 2010; Jolliffe et?al. in J Comput Graph Stat 12:531–547, 2003; Moghaddam et?al. in Advances in neural information processing systems 18:915–922, MIT Press, Cambridge, 2006; Shen and Huang in J Multivar Anal 99(6):1015–1034, 2008; Zou et?al. in J Comput Graph Stat 15(2):265–286, 2006). Despite success in achieving sparsity, some important properties enjoyed by the standard PCA are lost in these methods such as uncorrelation of PCs and orthogonality of loading vectors. Also, the total explained variance that they attempt to maximize can be too optimistic. In this paper we propose a new formulation for sparse PCA, aiming at finding sparse and nearly uncorrelated PCs with orthogonal loading vectors while explaining as much of the total variance as possible. We also develop a novel augmented Lagrangian method for solving a class of nonsmooth constrained optimization problems, which is well suited for our formulation of sparse PCA. We show that it converges to a feasible point, and moreover under some regularity assumptions, it converges to a stationary point. Additionally, we propose two nonmonotone gradient methods for solving the augmented Lagrangian subproblems, and establish their global and local convergence. Finally, we compare our sparse PCA approach with several existing methods on synthetic (Zou et?al. in J Comput Graph Stat 15(2):265–286, 2006), Pitprops (Jeffers in Appl Stat 16:225–236, 1967), and gene expression data (Chin et?al in Cancer Cell 10:529C–541C, 2006), respectively. The computational results demonstrate that the sparse PCs produced by our approach substantially outperform those by other methods in terms of total explained variance, correlation of PCs, and orthogonality of loading vectors. Moreover, the experiments on random data show that our method is capable of solving large-scale problems within a reasonable amount of time.     

Multiple positive periodic solutions for an n-species competition predator-prey system on time scales

In this paper, we consider an n-species competition predator-prey system on time scales with Holling-type II functional response and multiple exploited (or harvesting) terms, which contains n?1 competing preys and one predator. By using the continuation theorem based on Gaines and Mawhin’s coincidence degree theory, easily verifiable criteria are established for global existence of multiple positive periodic solutions to the above system. In addition, our results generalize the corresponding results of Zhang and Hou (Nonlinear Anal. (RWA) 11:1560–1571, 2010), Fan and Wang (J. Math. Anal. Appl. 262:179–190, 2001), Ding and Lu (Appl. Math. Model. 33:2748–2756, 2009).     

The strong convergence of a three-step algorithm for the split feasibility problem

Based on the very recent work by Dang and Gao (Invers Probl 27:1–9, 2011) and Wang and Xu (J Inequal Appl, doi:10.1155/2010/102085, 2010), and inspired by Yao (Appl Math Comput 186:1551–1558, 2007), Noor (J Math Anal Appl 251:217–229, 2000), and Xu (Invers Probl 22:2021–2034, 2006), we suggest a three-step KM-CQ-like method for solving the split common fixed-point problems in Hilbert spaces. Our results improve and develop previously discussed feasibility problem and related algorithms.     

Foulkes characters, Eulerian idempotents, and an amazing matrix

John Holte (Am. Math. Mon. 104:138?C149, 1997) introduced a family of ??amazing matrices?? which give the transition probabilities of ??carries?? when adding a list of numbers. It was subsequently shown that these same matrices arise in the combinatorics of the Veronese embedding of commutative algebra (Brenti and Welker, Adv. Appl. Math. 42:545?C556, 2009; Diaconis and Fulman, Am. Math. Mon. 116:788?C803, 2009; Adv. Appl. Math. 43:176?C196, 2009) and in the analysis of riffle shuffling (Diaconis and Fulman, Am. Math. Mon. 116:788?C803, 2009; Adv. Appl. Math. 43:176?C196, 2009). We find that the left eigenvectors of these matrices form the Foulkes character table of the symmetric group and the right eigenvectors are the Eulerian idempotents introduced by Loday (Cyclic Homology, 1992) in work on Hochschild homology. The connections give new closed formulae for Foulkes characters and allow explicit computation of natural correlation functions in the original carries problem.     

On the lower semicontinuity of the solution mappings to a parametric generalized strong vector equilibrium problem

In this paper, under new assumptions, which do not contain any information about the solution set, the lower semicontinuity of solution mappings to a parametric generalized strong vector equilibrium problem are established by using a scalarization method. These results extend and generalize the corresponding ones in Gong and Yao (J Optim Theory Appl 138:197–205, 2008), Chen and Li (Pac J Optim 6:141–151, 2010) and Li et al. (2012, submitted). Some examples are given to illustrate our results.     

On the Gauss–Newton method

We provide a new semilocal convergence analysis of the Gauss–Newton method (GNM) for solving nonlinear equation in the Euclidean space. Using a combination of center-Lipschitz, Lipschitz conditions, and our new idea of recurrent functions, we provide under the same or weaker hypotheses than before (Ben-Israel, J. Math. Anal. Appl. 15:243–252, 1966; Chen and Nashed, Numer. Math. 66:235–257, 1993; Deuflhard and Heindl, SIAM J. Numer. Anal. 16:1–10, 1979; Guo, J. Comput. Math. 25:231–242, 2007; Häußler, Numer. Math. 48:119–125, 1986; Hu et al., J. Comput. Appl. Math. 219:110–122, 2008; Kantorovich and Akilov, Functional Analysis in Normed Spaces, Pergamon, Oxford, 1982), a finer convergence analysis. The results can be extended in case outer or generalized inverses are used. Numerical examples are also provided to show that our results apply, where others fail (Ben-Israel, J. Math. Anal. Appl. 15:243–252, 1966; Chen and Nashed, Numer. Math. 66:235–257, 1993; Deuflhard and Heindl, SIAM J. Numer. Anal. 16:1–10, 1979; Guo, J. Comput. Math. 25:231–242, 2007; Häußler, Numer. Math. 48:119–125, 1986; Hu et al., J. Comput. Appl. Math. 219:110–122, 2008; Kantorovich and Akilov, Functional Analysis in Normed Spaces, Pergamon, Oxford, 1982).     

Further variations on the theme of FC-groups

Groups that are FC, or more generally satisfy any of the weakenings of the FC-condition considered in de Giovanni (Serdica Math. J. 28:241?C254, 2002) and Robinson et?al. (J. Algebra 326:218?C226, 2011), have local systems consisting of normal finite-by-nilpotent subgroups. Apart from generalizing results from de Giovanni (Serdica Math. J. 28:241?C254, 2002) and Robinson et al. (J. Algebra 326:218?C226, 2011) to the more general context of locally (normal and finite-by-nilpotent) groups, we partially settle an open problem raised in Robinson et?al. (J. Algebra 326:218?C226, 2011) concerning the isomorphism of maximal p-subgroups, but in this more general setting of locally (normal and finite-by-nilpotent) groups.     

On some characterizations of preinvex fuzzy mappings

Jeyakumar (Methods Oper. Res. 55:109–125, 1985) and Weir and Mond (J. Math. Anal. Appl. 136:29–38, 1988) introduced the concept of preinvex function. The preinvex functions have some interesting properties. For example, every local minimum of a preinvex function is a global minimum and nonnegative linear combinations of preinvex functions are preinvex. Invex functions were introduced by Hanson (J. Math. Anal. Appl. 80:545–550, 1981) as a generalization of differentiable convex functions. These functions are more general than the convex and pseudo convex ones. The type of invex function is equivalent to the type of function whose stationary points are global minima. Under some conditions, an invex function is also a preinvex function. Syau (Fuzzy Sets Syst. 115:455–461, 2000) introduced the concepts of pseudoconvexity, invexity, and pseudoinvexity for fuzzy mappings of one variable by using the notion of differentiability and the results proposed by Goestschel and Voxman (Fuzzy Sets Syst. 18:31–43, 1986). Wu and Xu (Fuzzy Sets Syst 159:2090–2103, 2008) introduced the concepts of fuzzy pseudoconvex, fuzzy invex, fuzzy pseudoinvex, and fuzzy preinvex mapping from \(\mathbb{R}^{n}\) to the set of fuzzy numbers based on the concept of differentiability of fuzzy mapping defined by Wang and Wu (Fuzzy Sets Syst. 138:559–591, 2003). In this paper, we present some characterizations of preinvex fuzzy mappings. The necessary and sufficient conditions for differentiable and twice differentiable preinvex fuzzy mapping are provided. These characterizations correct and improve previous results given by other authors. This fact is shown with examples. Moreover, we introduce additional conditions under which these results are valid.     

An improved local convergence analysis for a two-step Steffensen-type method

We study a class of Steffensen-type algorithm for solving nonsmooth variational inclusions in Banach spaces. We provide a local convergence analysis under ω-conditioned divided difference, and the Aubin continuity property. This work on the one hand extends the results on local convergence of Steffensen’s method related to the resolution of nonlinear equations (see Amat and Busquier in Comput. Math. Appl. 49:13–22, 2005; J. Math. Anal. Appl. 324:1084–1092, 2006; Argyros in Southwest J. Pure Appl. Math. 1:23–29, 1997; Nonlinear Anal. 62:179–194, 2005; J. Math. Anal. Appl. 322:146–157, 2006; Rev. Colomb. Math. 40:65–73, 2006; Computational Theory of Iterative Methods, 2007). On the other hand our approach improves the ratio of convergence and enlarges the convergence ball under weaker hypotheses than one given in Hilout (Commun. Appl. Nonlinear Anal. 14:27–34, 2007).     

Linear SPDEs Driven by Stationary Random Distributions

Using tools from the theory of stationary random distributions developed in It? (Mem. Coll. Sci., Univ. Kyoto, Ser. A: Math., 28:209?C223,?1954) and Yaglom (Theory Probab. Appl., 2:273?C320,?1957), we introduce a new class of processes which can be used as a model for the noise perturbing an SPDE. This type of noise is not necessarily Gaussian, but it includes the spatially homogeneous Gaussian noise introduced in Dalang (Electron. J. Probab. 4(6)?1999), and the fractional noise considered in Balan and Tudor (Stoch. Process. Appl., 120:2468?C2494,?2010). We derive some general conditions for the existence of a random field solution of a linear SPDE with this type of noise, under some mild conditions imposed on the Green function of the differential operator which appears in this equation. This methodology is applied to the study of the heat and wave equations (possibly replacing the Laplacian by one of its fractional powers), extending in this manner the results of Balan and Tudor (Stoch. Process. Appl., 120:2468?C2494,?2010) to the case H<1/2.     

Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities

In this paper, we analyze and discuss the well-posedness of two new variants of the so-called sweeping process, introduced by Moreau in the early 70s (Moreau in Sém Anal Convexe Montpellier, 1971) with motivation in plasticity theory. The first new variant is concerned with the perturbation of the normal cone to the moving convex subset \(C(t)\) , supposed to have a bounded variation, by a Lipschitz mapping. Under some assumptions on the data, we show that the perturbed differential measure inclusion has one and only one right continuous solution with bounded variation. The second variant, for which a large analysis is made, concerns a first order sweeping process with velocity in the moving set \(C(t)\) . This class of problems subsumes as a particular case, the evolution variational inequalities [widely used in applied mathematics and unilateral mechanics (Duvaut and Lions in Inequalities in mechanics and physics. Springer, Berlin, 1976]. Assuming that the moving subset \(C(t)\) has a continuous variation for every \(t\in [0,T]\) with \(C(0)\) bounded, we show that the problem has at least a Lipschitz continuous solution. The well-posedness of this class of sweeping process is obtained under the coercivity assumption of the involved operator. We also discuss some applications of the sweeping process to the study of vector hysteresis operators in the elastoplastic model (Krej?? in Eur J Appl Math 2:281–292, 1991), to the planning procedure in mathematical economy (Henry in J Math Anal Appl 41:179–186, 1973 and Cornet in J. Math. Anal. Appl. 96:130–147, 1983), and to nonregular electrical circuits containing nonsmooth electronic devices like diodes (Acary et al. Nonsmooth modeling and simulation for switched circuits. Lecture notes in electrical engineering. Springer, New York 2011). The theoretical results are supported by some numerical simulations to prove the efficiency of the algorithm used in the existence proof. Our methodology is based only on tools from convex analysis. Like other papers in this collection, we show in this presentation how elegant modern convex analysis was influenced by Moreau’s seminal work.     

On Bernstein-type properties of complete spacelike hypersurfaces immersed in a generalized Robertson–Walker spacetime

We apply the well know Omori?CYau generalized maximum principle (Omori in J Math Soc Jpn 19:205?C214, 1967; Yau in Commun Pure Appl Math 28:201?C228, 1975), as well as a suitable extension of it that was established in a joint work with Caminha (Caminha and de Lima in Gen Relat Grav 41:173?C189, 2009), in order to investigate Bernstein-type properties of complete spacelike hypersurfaces immersed in a generalized Robertson?CWalker spacetime, which is supposed to obey a standard convergence condition.