Updating the regularization parameter in the adaptive cubic regularization algorithm

The adaptive cubic regularization method (Cartis et al. in Math. Program. Ser. A 127(2):245?C295, 2011; Math. Program. Ser. A. 130(2):295?C319, 2011) has been recently proposed for solving unconstrained minimization problems. At each iteration of this method, the objective function is replaced by a cubic approximation which comprises an adaptive regularization parameter whose role is related to the local Lipschitz constant of the objective??s Hessian. We present new updating strategies for this parameter based on interpolation techniques, which improve the overall numerical performance of the algorithm. Numerical experiments on large nonlinear least-squares problems are provided.     

General two-function topological minimax theorems

Using the level set method of Joó (Acta Math Hung 54(1–2):163–172, 1989), a general two-function topological minimax theorem are proved. The theorem improves and generalizes the known results shown by Cheng and Lin (Acta Math Hung 73(1–2):65–69, 1996), Lin and Cheng (Acta Math Hung 100(3):177–186, 2003), and Frenk and Kassay (Math Program Ser A 105(1):145–155, 2006).     

An inverse-free Newton-Jarratt-type iterative method for solving equations under the gamma condition

A local as well as a semilocal convergence analysis for Newton–Jarratt–type iterative method for solving equations in a Banach space setting is studied here using information only at a point via a gamma-type condition (Argyros in Approximate Solution of Operator Equations with Applications, [2005]; Wang in Chin. Sci. Bull. 42(7):552–555, [1997]). This method has already been examined by us in (Argyros et al. in J. Comput. Appl. Math. 51:103–106, [1994]; Argyros in Comment. Mat. XXIII:97–108, [1994]), where the order of convergence four was established using however information on the domain of the operator. In this study we also establish the same order of convergence under weaker conditions. Moreover we show that all though we use weaker conditions the results obtained here can be used to solve equations in cases where the results in (Argyros et al. in J. Comput. Appl. Math. 51:103–106, [1994]; Argyros in Comment. Mat. XXIII:97–108, [1994]) cannot apply. Our method is inverse free, and therefore cheaper at the second step in contrast with the corresponding two–step Newton methods. Numerical Examples are also provided.     

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).     

Non-Decaying Solutions to the Navier Stokes Equations in Exterior Domains

We establish a new theorem of existence (and uniqueness) of solutions to the Navier-Stokes initial boundary value problem in exterior domains. No requirement is made on the convergence at infinity of the kinetic field and of the pressure field. These solutions are called non-decaying solutions. The first results on this topic dates back about 40 years ago see the references (Galdi and Rionero in Ann. Mat. Pures Appl. 108:361–366, 1976, Arch. Ration. Mech. Anal. 62:295–301, 1976, Arch. Ration. Mech. Anal. 69:37–52, 1979, Pac. J. Math. 104:77–83, 1980; Knightly in SIAM J. Math. Anal. 3:506–511, 1972). In the articles Galdi and Rionero (Ann. Mat. Pures Appl. 108:361–366, 1976, Arch. Ration. Mech. Anal. 62:295–301, 1976, Arch. Ration. Mech. Anal. 69:37–52, 1979, Pac. J. Math. 104:77–83, 1980) it was introduced the so called weight function method to study the uniqueness of solutions. More recently, the problem has been considered again by several authors (see Galdi et al. in J. Math. Fluid Mech. 14:633–652, 2012, Quad. Mat. 4:27–68, 1999, Nonlinear Anal. 47:4151–4156, 2001; Kato in Arch. Ration. Mech. Anal. 169:159–175, 2003; Kukavica and Vicol in J. Dyn. Differ. Equ. 20:719–732, 2008; Maremonti in Mat. Ves. 61:81–91, 2009, Appl. Anal. 90:125–139, 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).     

Newton Polyhedra of Discriminants of Projections

For a system of polynomial equations, whose coefficients depend on parameters, the Newton polyhedron of its discriminant is computed in terms of the Newton polyhedra of the coefficients. This leads to an explicit formula (involving Euler obstructions of toric varieties) in the unmixed case, suggests certain open questions in general, and generalizes a number of similar known results (Gelfand et al. in Discriminants, resultants, and multidimensional determinants. Birkhäuser, Boston, 1994; Sturmfels in J. Algebraic Comb. 32(2):207–236, 1994; McDonald in Discrete Comput. Geom. 27:501–529, 2002; Gonzalez-Perez in Can. J. Math. 52(2):348-368, 2000; Esterov and Khovanskii in Funct. Anal. Math. 2(1), 2008).     

Linear and nonlinear error bounds for lower semicontinuous functions

In this paper, we give some characterizations of linear and nonlinear error bounds for lower semicontinuous functions by a new notion, called subslope. And, extend some results of Azé and Corvellec (SIAM J Optim 12:913–927, 2002) and Corvellec and Motreanu (Math Program Ser A 114:291–319, 2008) slightly. Furthermore, we get a sufficient and necessary condition for global linear error bounds.     

A Characterization of the Locally Finite Networks Admitting Non-Constant Harmonic Functions of Finite Energy

We characterize the locally finite networks admitting non-constant harmonic functions of finite energy. Our characterization unifies the necessary existence criteria of Thomassen (J Comb Theory, Ser B 49:87?C102, 1990) and of Lyons and Peres (2011) with the sufficient criterion of Soardi (1991). We also extend a necessary existence criterion for non-elusive non-constant harmonic functions of finite energy due to Georgakopoulos (J Lond Math Soc, 2010).     

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.     

Algebraic Hecke characters and strictly compatible systems of abelian mod p Galois representations over global fields

In [10] (C R Acad Sci Paris Ser I Math 323(2) 117–120, 1996), [11] (Math Res Lett 10(1):71–83 2003), [12] (Can J Math 57(6):1215–1223 2005), Khare showed that any strictly compatible systems of semisimple abelian mod p Galois representations of a number field arises from a unique finite set of algebraic Hecke characters. In this article, we consider a similar problem for arbitrary global fields. We give a definition of Hecke character which in the function field setting is more general than previous definitions by Goss and Gross and define a corresponding notion of compatible system of mod p Galois representations. In this context we present a unified proof of the analog of Khare’s result for arbitrary global fields. In a sequel we shall apply this result to strictly compatible systems arising from Drinfeld modular forms, and thereby attach Hecke characters to cuspidal Drinfeld Hecke eigenforms.     

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.     

Optimal Transport, Convection, Magnetic Relaxation and Generalized Boussinesq Equations

We establish a connection between optimal transport theory (see Villani in Topics in optimal transportation. Graduate studies in mathematics, vol. 58, AMS, Providence, 2003, for instance) and classical convection theory for geophysical flows (Pedlosky, in Geophysical fluid dynamics, Springer, New York, 1979). Our starting point is the model designed few years ago by Angenent, Haker, and Tannenbaum (SIAM J. Math. Anal. 35:61–97, 2003) to solve some optimal transport problems. This model can be seen as a generalization of the Darcy–Boussinesq equations, which is a degenerate version of the Navier–Stokes–Boussinesq (NSB) equations. In a unified framework, we relate different variants of the NSB equations (in particular what we call the generalized hydrostatic-Boussinesq equations) to various models involving optimal transport (and the related Monge–Ampère equation, Brenier in Commun. Pure Appl. Math. 64:375–417, 1991; Caffarelli in Commun. Pure Appl. Math. 45:1141–1151, 1992). This includes the 2D semi-geostrophic equations (Hoskins in Annual review of fluid mechanics, vol. 14, pp. 131–151, Palo Alto, 1982; Cullen et al. in SIAM J. Appl. Math. 51:20–31, 1991, Arch. Ration. Mech. Anal. 185:341–363, 2007; Benamou and Brenier in SIAM J. Appl. Math. 58:1450–1461, 1998; Loeper in SIAM J. Math. Anal. 38:795–823, 2006) and some fully nonlinear versions of the so-called high-field limit of the Vlasov–Poisson system (Nieto et al. in Arch. Ration. Mech. Anal. 158:29–59, 2001) and of the Keller–Segel for Chemotaxis (Keller and Segel in J. Theor. Biol. 30:225–234, 1971; Jäger and Luckhaus in Trans. Am. Math. Soc. 329:819–824, 1992; Chalub et al. in Mon. Math. 142:123–141, 2004). Mathematically speaking, we establish some existence theorems for local smooth, global smooth or global weak solutions of the different models. We also justify that the inertia terms can be rigorously neglected under appropriate scaling assumptions in the generalized Navier–Stokes–Boussinesq equations. Finally, we show how a “stringy” generalization of the AHT model can be related to the magnetic relaxation model studied by Arnold and Moffatt to obtain stationary solutions of the Euler equations with prescribed topology (see Arnold and Khesin in Topological methods in hydrodynamics. Applied mathematical sciences, vol. 125, Springer, Berlin, 1998; Moffatt in J. Fluid Mech. 159:359–378, 1985, Topological aspects of the dynamics of fluids and plasmas. NATO adv. sci. inst. ser. E, appl. sci., vol. 218, Kluwer, Dordrecht, 1992; Schonbek in Theory of the Navier–Stokes equations, Ser. adv. math. appl. sci., vol. 47, pp. 179–184, World Sci., Singapore, 1998; Vladimirov et al. in J. Fluid Mech. 390:127–150, 1999; Nishiyama in Bull. Inst. Math. Acad. Sin. (N.S.) 2:139–154, 2007).     

A filter-line-search method for unconstrained optimization

A new filter-line-search algorithm for unconstrained nonlinear optimization problems is proposed. Based on the filter technique introduced by Fletcher and Leyffer (Math. Program. 91:239–269, 2002) it extends an existing technique of Wächter and Biegler (SIAM J. Comput. 16:1–31, 2005) for nonlinear equality constrained problem to the fully general unconstrained optimization problem. The proposed method, which differs from their approach, does not depend on any external restoration procedure. Global and local quadratic convergence is established under some reasonable conditions. The results of numerical experiments indicate that it is very competitive with the classical line search algorithm.     

On rigidity of hypersurfaces with constant curvature functions in warped product manifolds

In this paper, we first investigate several rigidity problems for hypersurfaces in the warped product manifolds with constant linear combinations of higher order mean curvatures as well as “weighted” mean curvatures, which extend the work (Brendle in Publ Math Inst Hautes Études Sci 117:247–269, 2013; Brendle and Eichmair in J Differ Geom 94(94):387–407, 2013; Montiel in Indiana Univ Math J 48:711–748, 1999) considering constant mean curvature functions. Secondly, we obtain the rigidity results for hypersurfaces in the space forms with constant linear combinations of intrinsic Gauss–Bonnet curvatures $L_k$ . To achieve this, we develop some new kind of Newton–Maclaurin type inequalities on $L_k$ which may have independent interest.     

An improved local convergence analysis for Newton–Steffensen-type method

We provide a local convergence analysis for Newton–Steffensen-type algorithm for solving nonsmooth perturbed variational inclusions in Banach spaces. Under new center–conditions and the Aubin continuity property, we obtain the linear local convergence of Newton–Steffensen method. Our results compare favorably with related obtained in (Argyros and Hilout, 2007 submitted; Hilout in J. Math. Anal. Appl. 339:753–761, 2008).     

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).     

On Ulm’s method for Fréchet differentiable operators

We provide new sufficient convergence conditions for the semilocal convergence of Ulm’s method (Izv. Akad. Nauk Est. SSR 16:403–411, 1967) in order to approximate a locally unique solution of an equation in a Banach space setting. We show that in some cases, our hypotheses hold true but the corresponding ones (Burmeister in Z. Angew. Math. Mech. 52:101–110, 1972; Kornstaedt in Aequ. Math. 13:21–45, 1975; Petzeltova in Comment. Math. Univ. Carol. 21:719–725, 1980; Potra and Ptǎk in Cas. Pest. Mat. 108:333–341, 1983; Ulm in Izv. Akad. Nauk Est. SSR 16:403–411, 1967) do not. We also show that under the same hypotheses and computational cost as (Burmeister in Z. Angew. Math. Mech. 52:101–110, 1972; Kornstaedt in Aequ. Math. 13:21–45, 1975; Petzeltova in Comment. Math. Univ. Carol. 21:719–725, 1980; Potra and Ptǎk in Cas. Pest. Mat. 108:333–341, 1983; Ulm in Izv. Akad. Nauk Est. SSR 16:403–411, 1967) finer error sequences can be obtained. Numerical examples are also provided further validating the results.     

Algorithms for lattice games

This paper provides effective methods for the polyhedral formulation of impartial finite combinatorial games as lattice games (Guo et al. Oberwolfach Rep 22: 23–26, 2009; Guo and Miller, Adv Appl Math 46:363–378, 2010). Given a rational strategy for a lattice game, a polynomial time algorithm is presented to decide (i) whether a given position is a winning position, and to find a move to a winning position, if not; and (ii) to decide whether two given positions are congruent, in the sense of misère quotient theory (Plambeck, Integers, 5:36, 2005; Plambeck and Siegel, J Combin Theory Ser A, 115: 593–622, 2008). The methods are based on the theory of short rational generating functions (Barvinok and Woods, J Am Math Soc, 16: 957–979, 2003).     

Error-tolerance pooling designs based on Johnson graphs

As an application of the new model for pooling designs proposed by the last two authors in Guo and Wang (J Combin Theory Ser A 118:2056–2058, 2011), we construct a family of pooling designs based on the $t$ -cliques of various sizes of the Johnson graph $J(n,t)$ . Its performance as a pooling design is better than that given in Bai et al. (Discrete Appl Math 157:3038–3045, 2009).