Maximal Monotone Operators and the Proximal Point Algorithm in the Presence of Computational Errors

In a finite-dimensional Euclidean space, we study the convergence of a proximal point method to a solution of the inclusion induced by a maximal monotone operator, under the presence of computational errors. Most results known in the literature establish the convergence of proximal point methods, when computational errors are summable. In the present paper, the convergence of the method is established for nonsummable computational errors. We show that the proximal point method generates a good approximate solution, if the sequence of computational errors is bounded from above by a constant.     

A proximal point algorithm for finding a common zero of a finite family of maximal monotone operators in the presence of computational errors

In a Hilbert space, we study the convergence of a proximal point method to a common zero of a finite family of maximal monotone operators under the presence of computational errors. Most results known in the literature establish the convergence of proximal point methods, when computational errors are summable. In the present paper, the convergence of the method is established for nonsummable computational errors. We show that the proximal point method generates a good approximate solution, if the sequence of computational errors is bounded from above by a constant.     

The prox-Tikhonov regularization method for the proximal point algorithm in Banach spaces

It is known, by Rockafellar (SIAM J Control Optim 14:877–898, 1976), that the proximal point algorithm (PPA) converges weakly to a zero of a maximal monotone operator in a Hilbert space, but it fails to converge strongly. Lehdili and Moudafi (Optimization 37:239–252, 1996) introduced the new prox-Tikhonov regularization method for PPA to generate a strongly convergent sequence and established a convergence property for it by using the technique of variational distance in the same space setting. In this paper, the prox-Tikhonov regularization method for the proximal point algorithm of finding a zero for an accretive operator in the framework of Banach space is proposed. Conditions which guarantee the strong convergence of this algorithm to a particular element of the solution set is provided. An inexact variant of this method with error sequence is also discussed.     

A proximal point algorithm for the monotone second-order cone complementarity problem

This paper is devoted to the study of the proximal point algorithm for solving monotone second-order cone complementarity problems. The proximal point algorithm is to generate a sequence by solving subproblems that are regularizations of the original problem. After given an appropriate criterion for approximate solutions of subproblems by adopting a merit function, the proximal point algorithm is verified to have global and superlinear convergence properties. For the purpose of solving the subproblems efficiently, we introduce a generalized Newton method and show that only one Newton step is eventually needed to obtain a desired approximate solution that approximately satisfies the appropriate criterion under mild conditions. Numerical comparisons are also made with the derivative-free descent method used by Pan and Chen (Optimization 59:1173–1197, 2010), which confirm the theoretical results and the effectiveness of the algorithm.     

Feasible Method for Generalized Semi-Infinite Programming

In this paper, we analyze the outer approximation property of the algorithm for generalized semi-infinite programming from Stein and Still (SIAM J. Control Optim. 42:769–788, 2003). A simple bound on the regularization error is found and used to formulate a feasible numerical method for generalized semi-infinite programming with convex lower-level problems. That is, all iterates of the numerical method are feasible points of the original optimization problem. The new method has the same computational cost as the original algorithm from Stein and Still (SIAM J. Control Optim. 42:769–788, 2003). We also discuss the merits of this approach for the adaptive convexification algorithm, a feasible point method for standard semi-infinite programming from Floudas and Stein (SIAM J. Optim. 18:1187–1208, 2007).     

Strong convergence of a proximal point algorithm with general errors

In this paper we construct a proximal point algorithm for maximal monotone operators with appropriate regularization parameters. We obtain the strong convergence of the proposed algorithm, which affirmatively answer the open question put forth by Boikanyo and Morosanu (Optim Lett 4:635–641, 2010).     

Returns to scale in multiplicative models in data envelopment analysis

One class of models introduced in DEA is called multiplicative models, in which, as shown by Banker and Maindiratta (Manag. Sci. 32:126–135, 1986), the piecewise linear frontiers usually employed in DEA are replaced by a frontier that is piecewise Cobb-Douglas(=log  linear). Banker and Maindiratta (Manag. Sci. 32:126–135, 1986) introduced a model to identify the most productive scale size pattern, and Banker et al. (Eur. J. Oper. Res. 154:345–362, 2004) presented a two-stage method for the identification of returns to scale (RTS) in multiplicative models. In this paper it is shown that both the RTS situation and the MPSS pattern could be determined by a single model in one step. The new method is important in the computational point of view.     

Solution Methods for Pseudomonotone Variational Inequalities

We extend some results due to Thanh-Hao (Acta Math. Vietnam. 31: 283–289, [2006]) and Noor (J. Optim. Theory Appl. 115:447–452, [2002]). The first paper established a convergence theorem for the Tikhonov regularization method (TRM) applied to finite-dimensional pseudomonotone variational inequalities (VIs), answering in the affirmative an open question stated by Facchinei and Pang (Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer, New York, [2003]). The second paper discussed the application of the proximal point algorithm (PPA) to pseudomonotone VIs. In this paper, new facts on the convergence of TRM and PPA (both the exact and inexact versions of PPA) for pseudomonotone VIs in Hilbert spaces are obtained and a partial answer to a question stated in (Acta Math. Vietnam. 31:283–289, [2006]) is given. As a byproduct, we show that the convergence theorem for inexact PPA applied to infinite-dimensional monotone variational inequalities can be proved without using the theory of maximal monotone operators. This research was supported in part by a grant from the National Sun Yat-Sen University, Kaohsiung, Taiwan. It has been carried out under the agreement between the National Sun Yat-Sen University, Kaohsiung, Taiwan and the University of Pisa, Pisa, Italy. The authors thank the anonymous referee for useful comments and suggestions.     

An augmented Lagrangian proximal alternating method for sparse discrete optimization problems

In this paper, we propose an augmented Lagrangian proximal alternating (ALPA) method for solving two classes of large-scale sparse discrete constrained optimization problems. Specifically, the ALPA method generates a sequence of augmented Lagrangian (AL) subproblems in the out iterations and utilizes a proximal alternating linearized minimization method and sparse projection techniques to solve these AL subproblems. And we study the first-order optimality conditions for these two classes of problems. Under some suitable assumptions, we show that any accumulation point of the sequence generated by the ALPA method satisfies the necessary first-order optimality conditions of these problems or is a local minimizer of these problems. The computational results with practical problems demonstrate that our method can find the suboptimal solutions of the problems efficiently and is competitive with some other local solution methods.

     

An alternative procedure for selecting a good value for the parameter <Emphasis Type="Italic">c</Emphasis> in RBF-interpolation

The impact of the scaling parameter c on the accuracy of interpolation schemes using radial basis functions (RBFs) has been pointed out by several authors. Rippa (Adv Comput Math 11:193–210, 1999) proposes an algorithm based on the idea of cross validation for selecting a good such parameter value. In this paper we present an alternative procedure, that can be interpreted as a refinement of Rippa’s algorithm for a cost function based on the euclidean norm. We point out how this method is related to the procedure of maximum likelihood estimation, which is used for identifying covariance parameters of stochastic processes in spatial statistics. Using the same test functions as Rippa we show that our algorithm compares favorably with cross validation in many cases and discuss its limitations. Finally we present some computational aspects of our algorithm.     

A Fréchet derivative-free cubically convergent method for set-valued maps

We introduce a new iterative method in order to approximate a locally unique solution of variational inclusions in Banach spaces. The method uses only divided differences operators of order one. An existence–convergence theorem and a radius of convergence are given under some conditions on divided difference operator and Lipschitz-like continuity property of set-valued mappings. Our method extends the recent work related to the resolution of nonlinear equation in Argyros (J Math Anal Appl 332:97–108, 2007) and has the following advantages: faster convergence to the solution than all the previous known ones in Argyros and Hilout (Appl Math Comput, 2008 in press), Hilout (J Math Anal Appl 339:53–761, 2008, Positivity 10:673–700, 2006), and we do not need to evaluate any Fréchet derivative. We provide also an improvement of the ratio of our algorithm under some center-conditions and less computational cost. Numerical examples are also provided.      

On Ulm’s method using divided differences of order one

We provide new sufficient convergence conditions for the semilocal convergence of Ulm’s method (Tzv 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 in Burmeister (Z Angew Math Mech 52:101–110, 1972), Kornstaedt (Aequ Math 13:21–45, 1975), Moser (1973), and Potra and Pták (Cas Pest Mat 108:333–341, 1983) do not. We also show that under the same hypotheses and computational cost, finer error bounds can be obtained. Some error bounds are also shown to be sharp. Numerical examples are also provided further validating the results.     

Asymptotic expansions and convergence rates for Euler’s approximations of semigroups

We provide optimal bounds for errors in Euler’s approximations of semigroups in Banach algebras and of semigroups of operators in Banach spaces. Furthermore, we construct asymptotic expansions for such approximations with optimal bounds for remainder terms. The sizes of errors are controlled by smoothness properties of semigroups. In this paper we use Fourier–Laplace transforms and a reduction of the problem to the convergence rates and asymptotic expansions in the Law of Large Numbers. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-70/09. This paper was written in 2004. In the interim, several related articles were published; let us mention [14, 13, 15].     

An Explicit Formula for the Coefficients of the Saddle Point Method

Most standard textbooks about asymptotic approximations of integrals do not give explicit formulas for the coefficients of the asymptotic methods of Laplace and saddle point. In these techniques, those coefficients arise as the Taylor coefficients of a function defined in an implicit form, and the coefficients are not given by a closed algebraic formula. Despite this fact, we can extract from the literature some formulas of varying degrees of explicitness for those coefficients: Perron’s method (in Sitzungsber. Bayr. Akad. Wissensch. (Münch. Ber.), 191–219, 1917) offers an explicit computation in terms of the derivatives of an explicit function; in (de Bruijn, Asymptotic Methods in Analysis. Dover, New York, 1950) we can find a similar formula for the Laplace method which uses derivatives of an explicit function. Dingle (in Asymptotic Expansions: Their Derivation and Interpretation, Academic Press, New York, 1973) gives the coefficients of the saddle point method in terms of a contour integral. Perron’s method is rediscovered in (Campbell et al., Stud. Appl. Math. 77:151–172, 1987), but they also go farther and compute the above mentioned derivatives by means of a recurrence. The most recent contribution is (Wojdylo, SIAM Rev. 48(1):76–96, 2006), which rediscovers the Campbell, Fr?man and Walles’ formula and rewrites it in terms of Bell polynomials (in the Laplace method) using new ideas of combinatorial analysis which efficiently simplify and systematize the computations. In this paper we continue the research line of these authors. We combine the more systematic version of the saddle point method introduced in (López et al., J. Math. Anal. Appl. 354(1):347–359, 2009) with Wojdylo’s idea to derive a new and more explicit formula for the coefficients of the saddle point method, similar to Wojdylo’s formula for the coefficients of the Laplace method. As an example, we show the application of this formula to the Bessel function.     

Proximal-Like Algorithm Using the Quasi D-Function for Convex Second-Order Cone Programming

In this paper, we present a measure of distance in a second-order cone based on a class of continuously differentiable strictly convex functions on ℝ₊₊. Since the distance function has some favorable properties similar to those of the D-function (Censor and Zenios in J. Optim. Theory Appl. 73:451–464 [1992]), we refer to it as a quasi D-function. Then, a proximal-like algorithm using the quasi D-function is proposed and applied to the second-cone programming problem, which is to minimize a closed proper convex function with general second-order cone constraints. Like the proximal point algorithm using the D-function (Censor and Zenios in J. Optim. Theory Appl. 73:451–464 [1992]; Chen and Teboulle in SIAM J. Optim. 3:538–543 [1993]), under some mild assumptions we establish the global convergence of the algorithm expressed in terms of function values; we show that the sequence generated by the proposed algorithm is bounded and that every accumulation point is a solution to the considered problem. Research of Shaohua Pan was partially supported by the Doctoral Starting-up Foundation (B13B6050640) of GuangDong Province. Jein-Shan Chen is a member of the Mathematics Division, National Center for Theoretical Sciences, Taipei Office. The author’s work was partially supported by National Science Council of Taiwan.     

Truncated regularized Newton method for convex minimizations

Recently, Li et al. (Comput. Optim. Appl. 26:131–147, 2004) proposed a regularized Newton method for convex minimization problems. The method retains local quadratic convergence property without requirement of the singularity of the Hessian. In this paper, we develop a truncated regularized Newton method and show its global convergence. We also establish a local quadratic convergence theorem for the truncated method under the same conditions as those in Li et al. (Comput. Optim. Appl. 26:131–147, 2004). At last, we test the proposed method through numerical experiments and compare its performance with the regularized Newton method. The results show that the truncated method outperforms the regularized Newton method. The work was supported by the 973 project granted 2004CB719402 and the NSF project of China granted 10471036.     

Efficient three-step iterative methods with sixth order convergence for nonlinear equations

In this paper, we present two new three-step iterative methods for solving nonlinear equations with sixth convergence order. The new methods are obtained by composing known methods of third order of convergence with Newton’s method and using an adequate approximation for the derivative, that provides high order of convergence and reduces the required number of functional evaluations per step. The first method is obtained from Potra-Pták’s method and the second one, from Homeier’s method, both reaching an efficiency index of 1.5651. Our methods are comparable with the method of Parhi and Gupta (Appl Math Comput 203:50–55, 2008). Methods proposed by Kou and Li (Appl Math Comput 189:1816–1821, 2007), Wang et al. (Appl Math Comput 204:14–19, 2008) and Chun (Appl Math Comput 190:1432–1437, 2007) reach the same efficiency index, although they start from a fourth order method while we use third order methods and simpler arithmetics. We prove the convergence results and check them with several numerical tests that allow us to compare the convergence order, the computational cost and the efficiency order of our methods with those of the original methods.     

Solving molecular distance geometry problems by global optimization algorithms

In this paper we consider global optimization algorithms based on multiple local searches for the Molecular Distance Geometry Problem (MDGP). Three distinct approaches (Multistart, Monotonic Basin Hopping, Population Basin Hopping) are presented and for each of them a computational analysis is performed. The results are also compared with those of two other approaches in the literature, the DGSOL approach (Moré, Wu in J. Glob. Optim. 15:219–234, 1999) and a SDP based approach (Biswas et al. in An SDP based approach for anchor-free 3D graph realization, Technical Report, Operations Research, Stanford University, 2005).     

A convergence analysis for directional two-step Newton methods

We present a Kantorovich-type semilocal convergence analysis of the Newton–Josephy method for solving a certain class of variational inequalities. By using a combination of Lipschitz and center-Lipschitz conditions, and our new idea of recurrent functions, we provide an analysis with the following advantages over the earlier works (Wang 2009, Wang and Shen, Appl Math Mech 25:1291–1297, 2004) (under the same or less computational cost): weaker sufficient convergence conditions, larger convergence domain, finer error bounds on the distances involved, and an at least as precise information on the location of the solution.     

A categorical approach to Weyl modules

Global and local Weyl modules were introduced via generators and relations in the context of affine Lie algebras in [CP2] and were motivated by representations of quantum affine algebras. In [FL] a more general case was considered by replacing the polynomial ring with the coordinate ring of an algebraic variety and partial results analogous to those in [CP2] were obtained. In this paper we show that there is a natural definition of the local and global Weyl modules via homological properties. This characterization allows us to define the Weyl functor from the category of left modules of a commutative algebra to the category of modules for a simple Lie algebra. As an application we are able to understand the relationships of these functors to tensor products, generalizing results in [CP2] and [FL]. We also analyze the fundamental Weyl modules and show that, unlike the case of the affine Lie algebras, the Weyl functors need not be left exact.