A computational study of a solver system for processing two-stage stochastic LPs with enhanced Benders decomposition

We report a computational study of two-stage SP models on a large set of benchmark problems and consider the following methods: (i) Solution of the deterministic equivalent problem by the simplex method and an interior point method, (ii) Benders decomposition (L-shaped method with aggregated cuts), (iii) Regularised decomposition of Ruszczy??ski (Math Program 35:309?C333, 1986), (iv) Benders decomposition with regularization of the expected recourse by the level method (Lemaréchal et al. in Math Program 69:111?C147, 1995), (v) Trust region (regularisation) method of Linderoth and Wright (Comput Optim Appl 24:207?C250, 2003). In this study the three regularisation methods have been introduced within the computational structure of Benders decomposition. Thus second-stage infeasibility is controlled in the traditional manner, by imposing feasibility cuts. This approach allows extensions of the regularisation to feasibility issues, as in Fábián and Sz?ke (Comput Manag Sci 4:313?C353, 2007). We report computational results for a wide range of benchmark problems from the POSTS and SLPTESTSET collections and a collection of difficult test problems compiled by us. Finally the scale-up properties and the performance profiles of the methods are presented.     

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).     

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).     

Defect and Area in Beltrami–Klein Model of Hyperbolic Geometry

Ungar (Beyond the Einstein addition law and its gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrouector Spaces, 2001; Comput Math Appl 49:187–221, 2005; Comput Math Appl 53, 2007) introduced into hyperbolic geometry the concept of defect based on relativity addition of A. Einstein. Another approach is from Karzel (Resultate Math. 47:305–326, 2005) for the relation between the K-loop and the defect of an absolute plane in the sense (Karzel in Einführung in die Geometrie, 1973). Our main concern is to introduce a systematical exact definition for defect and area in the Beltrami–Klein model of hyperbolic geometry. Combining the ideas and methods of Karzel and Ungar give an elegant concept for defect and area in this model. In particular we give a rigorous and elementary proof for the defect formula stated (Ungar in Comput Math Appl 53, 2007). Furthermore, we give a formulary for area of circle in the Beltrami–Klein model of hyperbolic geometry.     

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.     

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).     

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.     

Minimax inequalities for set-valued mappings

In this paper, by using the finite intersection property, we first obtain two types of minimax inequalities for set-valued mappings, which improve and generalize the corresponding results in Ferro (J Optim Theory Appl 60:19?C31, 1989) and Li et?al. (J Math Anal Appl 281:707?C723, 2003). Then, by using the Ky Fan lemma and the Kakutani?CFan?CGlicksberg fixed point theorem, we also investigate some Ky Fan minimx inequalities for set-valued mappings.     

General convergence conditions of Newton’s method for m-Fréchet differentiable operators

We present a local as well as a semilocal convergence analysis for Newton’s method for approximating a locally unique solution of a nonlinear equation in a Banach space setting. Our hypotheses involve m-Fréchet-differentiable operators and general Lipschitz-type hypotheses, where m≥2 is a positive integer. The new convergence analysis unifies earlier results; it is more flexible and provides a finer convergence analysis than in earlier studies such as Argyros in J. Comput. Appl. Math. 131:149–159, 2001, Argyros and Hilout in J. Appl. Math. Comput. 29:391–400, 2009, Argyros and Hilout in J. Complex. 28:364–387, 2012, Argyros et al. Numerical Methods for Equations and Its Applications, CRC Press/Taylor & Francis, New York, 2012, Gutiérrez in J. Comput. Appl. Math. 79:131–145, 1997, Ren and Argyros in Appl. Math. Comput. 217:612–621, 2010, Traub and Wozniakowski in J. Assoc. Comput. Mech. 26:250–258, 1979. Numerical examples are presented further validating the theoretical results.     

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.     

Estimating upper bounds on the limit points of majorizing sequences for Newton’s method

We present new sufficient conditions for the semilocal convergence of Newton’s method to a locally unique solution of an equation in a Banach space setting. Upper bounds on the limit points of majorizing sequences are also given. Numerical examples are provided, where our new results compare favorably to earlier ones such as Argyros (J Math Anal Appl 298:374–397, 2004), Argyros and Hilout (J Comput Appl Math 234:2993-3006, 2010, 2011), Ortega and Rheinboldt (1970) and Potra and Pták (1984).     

A supplement to a regularization method for the proximal point algorithm

The purpose of this paper is to show that the iterative scheme recently studied by Xu (J Glob Optim 36(1):115–125, 2006) is the same as the one studied by Kamimura and Takahashi (J Approx Theory 106(2):226–240, 2000) and to give a supplement to these results. With the new technique proposed by Maingé (Comput Math Appl 59(1):74–79, 2010), we show that the convergence of the iterative scheme is established under another assumption. It is noted that if the computation error is zero or the approximate computation is exact, our new result is a genuine generalization of Xu’s result and Kamimura–Takahashi’s result.     

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.     

A Parallel Splitting Method for Separable Convex Programs

In this paper, we propose a new parallel splitting augmented Lagrangian method for solving the nonlinear programs where the objective function is separable with three operators and the constraint is linear. The method is an improvement of the method of He (Comput. Optim. Appl., 2(42):195–212, 2009), where we generate a predictor using the same parallel splitting augmented Lagrangian scheme as that in He (Comput. Optim. Appl., 2(42):195–212, 2009), while adopting a new strategy to get the next iterate. Under the mild assumptions of convexity of the underlying mappings and the non-emptiness of the solution set, we prove that the proposed algorithm is globally convergent. We apply the new method in the area of image processing and to solve some quadratic programming problems. The preliminary numerical results indicate that the new method is efficient.     

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.     

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.     

An explicit formula for the linearization coefficients of Bessel polynomials

We prove a single sum formula for the linearization coefficients of the Bessel polynomials. In two special cases we show that our formula reduces indeed to Berg and Vignat??s formulas in their proof of the positivity results about these coefficients (Constr. Approx. 27:15?C32, 2008). As a bonus we also obtain a generalization of an integral formula of Boros and Moll (J. Comput. Appl. Math. 106:361?C368, 1999).     

A Note on the Global Convergence of the Quadratic Hybridization of Polak–Ribière–Polyak and Fletcher–Reeves Conjugate Gradient Methods

A revision on condition (27) of Lemma 3.2 of Babaie-Kafaki (J. Optim. Theory Appl. 154(3):916–932, 2012) is made. Throughout, we use the same notation and equation numbers as in Babaie-Kafaki (J. Optim. Theory Appl. 154(3):916–932, 2012).     

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.