Efficient model for interval goal programming with arbitrary penalty function

Penalty function is a key factor in interval goal programming (IGP), especially for decision makers weighing resources vis-à-vis goals. Many approaches (Chang et al. J Oper Res Soc 57:469–473, 2006; Chang and Lin Eur J Oper Res 199, 9–20, 2009; Jones et al. Omega 23, 41–48, 1995; Romero Eur J Oper Res 153, 675–686, 2004; Vitoriano and Romero J Oper Res Soc 50, 1280–1283, 1999)have been proposed for treating several types of penalty functions in the past several decades. The recent approach of Chang and Lin (Eur J Oper Res 199, 9–20, 2009) considers the S-shaped penalty function. Although there are many approaches cited in literature, all are complicated and inefficient. The current paper proposes a novel and concise uniform model to treat any arbitrary penalty function in IGP. The efficiency and usefulness of the proposed model are demonstrated in several numeric examples.     

Counterexamples to a triality theorem for quadratic-exponential minimization problems

The main goal of this note is to give a counterexample to the Triality Theorem in Gao and Ruan (Math Methods Oper Res 67:479–491, 2008). This is done first by considering a more general optimization problem with the aim to encompass several examples from Gao and Ruan (Math Methods Oper Res 67:479–491, 2008) and other papers by Gao and his collaborators (see f.i. Gao Duality principles in nonconvex systems. Theory, methods and applications. Kluwer, Dordrecht, 2000; Gao and Sherali Advances in applied mathematics and global optimization. Springer, Berlin, 2009). We perform a thorough analysis of the general optimization problem in terms of local extrema while presenting several counterexamples.     

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.     

On relations between chance constrained and penalty function problems under discrete distributions

We extend the theory of penalty functions to stochastic programming problems with nonlinear inequality constraints dependent on a random vector with known distribution. We show that the problems with penalty objective, penalty constraints and chance constraints are asymptotically equivalent under discretely distributed random parts. This is a complementary result to Branda (Kybernetika 48(1):105–122, 2012a), Branda and Dupa?ová (Ann Oper Res 193(1):3–19, 2012), and Ermoliev et al. (Ann Oper Res 99:207–225, 2000) where the theorems were restricted to continuous distributions only. We propose bounds on optimal values and convergence of optimal solutions. Moreover, we apply exact penalization under modified calmness property to improve the results.     

The Shapley value for shortest path games: a non-graph-based approach

The shortest path games are considered in this paper. The transportation of a good in a network has costs and benefits. The problem is to divide the profit of the transportation among the players. Fragnelli et al. (Math Methods Oper Res 52: 251–264, 2000) introduce the class of shortest path games and show it coincides with the class of monotone games. They also give a characterization of the Shapley value on this class of games. In this paper we consider further five characterizations of the Shapley value (Hart and Mas-Colell’s in Econometrica 57:589–614, 1989; Shapley’s in Contributions to the theory of games II, annals of mathematics studies, vol 28. Princeton University Press, Princeton, pp 307–317, 1953; Young’s in Int J Game Theory 14:65–72, 1985, Chun’s in Games Econ Behav 45:119–130, 1989; van den Brink’s in Int J Game Theory 30:309–319, 2001 axiomatizations), and conclude that all the mentioned axiomatizations are valid for the shortest path games. Fragnelli et al. (Math Methods Oper Res 52:251–264, 2000)’s axioms are based on the graph behind the problem, in this paper we do not consider graph specific axioms, we take $TU$ axioms only, that is we consider all shortest path problems and we take the viewpoint of an abstract decision maker who focuses rather on the abstract problem than on the concrete situations.     

Tower-of-sets analysis for the Kise–Ibaraki–Mine algorithm

Kise et al. (Oper. Res. 26:121–126, 1978) give an O(n ²) time algorithm to find an optimal schedule for the single-machine number of late jobs problem with agreeable job release dates and due dates. Li et al. (Oper. Res. 58:508–509, 2010a) point out that their proof of optimality for their algorithm is incorrect by giving a counter-example. In this paper, using the concept of “tower-of-sets” from Lawler (Math. Comput. Model. 20:91–106, 1994), we construct the tower-of-sets of the early job set generated by the algorithm. Then we give a correct proof of optimality for the algorithm and show a new result that the early job set by the algorithm obtains not only the maximum number of jobs but also the smallest total processing time among all optimal schedules. The result can be applied to study the problems of the hard real-time systems.     

A Kantorovich-type analysis of Broyden’s method using recurrent functions

We provide a semilocal convergence analysis for Broyden’s method for approximating locally unique solutions of nonlinear operator equations. Using the majorant principle we show that under the same or weaker hypotheses, in combination with our new idea of recurrent functions, we can find weaker sufficient conditions for the convergence of Broyden’s method as well as finer error bounds on the distances involved, and a more precise information on the location of the solution than before (Broyden, Math. Comput. 19:577–593, 1965; Chen, Ann. Inst. Stat. Math. 42:387–401, 1990; Dennis, Nonlinear Functional Analysis and Applications, pp. 425–472, Academic Press, San Diego, 1971; Li and Fukushima, Ann. Oper. Res. 103:71–97, 2001). Numerical examples are also provided involving polynomial, integral, and differential equations.     

Diffusion approximations for open Jackson networks with reneging

We consider generalized Jackson networks with reneging in which the customer patience times follow a general distribution that unifies the patience time without scaling adopted by Ward and Glynn (Queueing Syst 50:371–400, 2005) and the patience time with hazard rate scaling and unbounded support adopted by Reed and Ward (Math Oper Res 33:606–644, 2008). The diffusion approximations for both the queue length process and the abandonment-count process are established under the conventional heavy traffic limit regime. In light of the recent work by Dai and He (Math Oper Res 35:347–362, 2010), the diffusion approximations are obtained by the following four steps: first, establishing the stochastic boundedness for the queue length process and the virtual waiting time process; second, obtaining the $C$ -tightness and fluid limits for the queue length process and the abandonment-count process; then third, building an asymptotic relationship between the abandonment-count process and the queue length process in terms of the customer patience time. Finally, the fourth step is to get the diffusion approximations by invoking the continuous mapping theorem.     

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).     

On the Chvátal–Gomory closure of a compact convex set

In this paper, we show that the Chvátal–Gomory closure of any compact convex set is a rational polytope. This resolves an open question of Schrijver (Ann Discret Math 9:291–296, 1980) for irrational polytopes, and generalizes the same result for the case of rational polytopes (Schrijver in Ann Discret Math 9:291–296, 1980), rational ellipsoids (Dey and Vielma in IPCO XIV, Lecture Notes in Computer Science, vol 6080. Springer, Berlin, pp 327–340, 2010) and strictly convex bodies (Dadush et al. in Math Oper Res 36:227–239, 2011).     

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.     

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.     

A note on “A decision model for ranking suppliers in the presence of cardinal and ordinal data, weight restrictions, and nondiscriminatory factors”

Farzipoor Saen (Ann. Oper. Res. 172(1):177–192, 2009) proposed a method based on Data Envelopment Analysis (DEA) for selecting the best supplier in the presence of cardinal and ordinal data, weight restrictions, and non-discriminatory factors. In the present note, we show that the DEA method proposed by Farzipoor Saen (Ann. Oper. Res. 172(1):177–192, 2009) cannot be used for selecting the best supplier in the presence of imprecise data. It is also shown that there are computational errors in calculating the amounts of the preference intensity parameter and the ratio parameter in Farzipoor Saen’s paper.     

Mixed Type Duality in Multiobjective Fractional Programming Under Generalized ρ-univex Function

In this paper, three approaches given by Dinklebaeh (Manag Sci 13(7):492–498, 1967) and Jagannathan (Z Oper Res 17:618–630, 1968) for both primal and mixed type dual of a non differentiable multiobjective fractional programming problem in which the numerator of objective function contains square root of positive semi definite quadratic form are introduced. Also, the necessary and sufficient conditions of efficient solution for fractional programming are established and a parameterizations technique is used to established duality results under generalized ρ-univexity assumption.     

Nonzero-sum constrained discrete-time Markov games: the case of unbounded costs

In this paper, we consider discrete-time \(N\) -person constrained stochastic games with discounted cost criteria. The state space is denumerable and the action space is a Borel set, while the cost functions are admitted to be unbounded from below and above. Under suitable conditions weaker than those in (Alvarez-Mena and Hernández-Lerma, Math Methods Oper Res 63:261–285, 2006) for bounded cost functions, we also show the existence of a Nash equilibrium for the constrained games by introducing two approximations. The first one, which is as in (Alvarez-Mena and Hernández-Lerma, Math Methods Oper Res 63:261–285, 2006), is to construct a sequence of finite games to approximate a (constrained) auxiliary game with an initial distribution that is concentrated on a finite set. However, without hypotheses of bounded costs as in (Alvarez-Mena and Hernández-Lerma, Math Methods Oper Res 63:261–285, 2006), we also establish the existence of a Nash equilibrium for the auxiliary game with unbounded costs by developing more shaper error bounds of the approximation. The second one, which is new, is to construct a sequence of the auxiliary-type games above and prove that the limit of the sequence of Nash equilibria for the auxiliary-type games is a Nash equilibrium for the original constrained games. Our results are illustrated by a controlled queueing system.     

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.     

Gaussian skewness approximation for dynamic rate multi-server queues with abandonment

The multi-server queue with non-homogeneous Poisson arrivals and customer abandonment is a fundamental dynamic rate queueing model for large scale service systems such as call centers and hospitals. Scaling the arrival rates and number of servers arises naturally when a manager updates a staffing schedule in response to a forecast of increased customer demand. Mathematically, this type of scaling ultimately gives us the fluid and diffusion limits as found in Mandelbaum et al., Queueing Syst 30:149–201 (1998) for Markovian service networks. The asymptotics used here reduce to the Halfin and Whitt, Oper Res 29:567–588 (1981) scaling for multi-server queues. The diffusion limit suggests a Gaussian approximation to the stochastic behavior of this queueing process. The mean and variance are easily computed from a two-dimensional dynamical system for the fluid and diffusion limiting processes. Recent work by Ko and Gautam, INFORMS J Comput, to appear (2012) found that a modified version of these differential equations yield better Gaussian estimates of the original queueing system distribution. In this paper, we introduce a new three-dimensional dynamical system that is based on estimating the mean, variance, and third cumulant moment. This improves on the previous approaches by fitting the distribution from a quadratic function of a Gaussian random variable.     

Second order symmetric duality in multiobjective programming involving cones

An attempt is made to remove certain omissions and inconsistencies in the recent work of Mishra and Lai (European J. Oper. Res., 178:20–26, 2007).     

Modularity and monotonicity of games

The purpose of this paper is twofold. First, we generalize Kajii et al. (J Math Econ 43:218–230, 2007) and provide a condition under which for a game \(v\) , its Möbius inverse is equal to zero within the framework of the \(k\) -modularity of \(v\) for \(k \ge 2\) . This condition is more general than that in Kajii et al. (J Math Econ 43:218–230, 2007). Second, we provide a condition under which for a game \(v\) , its Möbius inverse takes non-negative values, and not just zero. This paper relates the study of totally monotone games to that of \(k\) -monotone games. Furthermore, this paper shows that the modularity of a game is related to \(k\) -additive capacities proposed by Grabisch (Fuzzy Sets Syst 92:167–189, 1997). To illustrate its application in the field of economics, we use these results to characterize a Gini index representation of Ben-Porath and Gilboa (J Econ Theory 64:443–467, 1994). Our results can also be applied to potential functions proposed by Hart and Mas-Colell (Econometrica 57:589–614, 1989) and further analyzed by Ui et al. (Math Methods Oper Res 74:427–443, 2011).     

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.