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We present a sufficient condition for monotonicity of the nonlinear nonsmooth system generated by Fischer–Burmeister function associated with nonlinear complementarity problem. Based on the presented condition, the nonlinear complementarity problem considered in this paper is equivalently formulated to a nonsmooth monotone system. We then propose a modified multivariate spectral gradient projection method for the resulting system, and establish the global convergence without smoothness and Lipschitz condition. Preliminary numerical experiments show that, compared to some existing methods, the proposed method is effective. [ABSTRACT FROM AUTHOR]

By equivalently transforming a class of weakly nonlinear complementarity problems into a modulus equation, and introducing a smoothing approximation of the absolute value function, a smoothing Newton method is established for solving the weakly nonlinear complementarity problem. Under some mild assumptions, the proposed method is shown to possess global convergence and locally quadratical convergence. Especially, the global convergence results do not need a priori existence of an accumulation point with some suitable conditions. Numerical results are given to show the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

We enrich the theory of variational inequalities in the case of an aggregative structure by implementing recent results obtained by using the Selten–Szidarovszky technique. We derive existence, semi-uniqueness and uniqueness results for solutions and provide a computational method. As an application we derive very powerful practical equilibrium results for Nash equilibria of sum-aggregative games and illustrate with Cournot oligopolies. [ABSTRACT FROM AUTHOR]

We consider a geometrical approach to the optimisation problems motivated by transportation system management. Here, we provide a comprehensive account of geometric programming based on the elementary Finsler geometry in R n . Then, we present a Finslerian dynamical model for the nonlinear complementarity problem of traffic equilibrium that can be applied to a variety of equilibrium problems. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the modulus-based matrix splitting iteration method for solving a class of nonlinear complementarity problems under a weakened condition, and present the general convergence conditions for the method in terms of spectral radius and matrix norm, respectively. Moreover, for some special cases of the method, we propose the concrete convergence conditions and optimal parameters. These convergence theories improve the existing results to some extent. The numerical experiments verify the validity and practicality of the presented results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a preconditioned general two-step modulus-based accelerated overrelaxation (MAOR) iteration method for solving a class of nonlinear complementarity problems. The convergence analysis and the condition of the iterative parameters are given when the system matrix is either positive definite or an H + -matrix. Numerical examples further illustrate that the proposed method is efficient and has better performance than some existing modulus-based iteration methods in aspects of the number of iteration steps and CPU time. [ABSTRACT FROM AUTHOR]

For constrained equations with nonisolated solutions and a certain family of Newton-type methods, it was previously shown that if the equation mapping is 2-regular at a given solution with respect to a direction which is interior feasible and which is in the null space of the Jacobian, then there is an associated large (not asymptotically thin) domain of starting points from which the iterates are well defined and converge to the specific solution in question. Under these assumptions, the constrained local Lipschitzian error bound does not hold, unlike the common settings of convergence and rate of convergence analyses. In this work, we complement those previous results by considering the case when the equation mapping is 2-regular with respect to a direction in the null space of the Jacobian which is in the tangent cone to the set, but need not be interior feasible. Under some further conditions, we still show linear convergence of order 1/2 from a large domain around the solution (despite degeneracy, and despite that there may exist other solutions nearby). Our results apply to constrained variants of the Gauss–Newton and Levenberg–Marquardt methods, and to the LP-Newton method. An illustration for a smooth constrained reformulation of the nonlinear complementarity problem is also provided. [ABSTRACT FROM AUTHOR]

For a class of second-order cone nonlinear complementarity problems, abbreviated as SOCNCPs, we establish the modulus-based matrix splitting relaxation iteration methods, which are obtained by reformulating equivalently SOCNCP as an implicit fixed-point equation based on Jordan algebra associated with the second-order cone. The global convergence theorems are given under suitable choices of the involved splitting matrix and parameter matrices. When the splitting matrix is symmetric positive definite, the strategy choice of the parameters is discussed. Numerical experiments have shown that the modulus-based iteration methods are effective for solving SOCNCPs. [ABSTRACT FROM AUTHOR]

In this paper we consider the nonlinear complementarity problem over circular cones (CCNCP) which contains a lot of circular cone optimization problems. We study a one-parametric class of smoothing functions which can be used to reformulate the CCNCP as a system of smooth nonlinear equations. Based on the equivalent reformulation, we propose a smoothing inexact Newton method to solve the CCNCP. In each iteration, the proposed method solves the nonlinear equations only approximately. Since the inexact direction is not necessarily descent, a new derivative-free nonmonotone line search is developed to ensure that the proposed method has global and local superlinear and quadratical convergence. Some numerical results are also reported. [ABSTRACT FROM AUTHOR]

In this paper, we present a one-layer recurrent neural network (NN) for solving convex optimization problems by using the Mangasarian and Solodov (MS) implicit Lagrangian function. In this paper by using Krush–Kuhn–Tucker conditions and MS function the NN model was derived from an unconstrained minimization problem. The proposed NN model is one layer and compared to the available NNs for solving convex optimization problems, which has a better performance in convergence time. The proposed NN model is stable in the sense of Lyapunov and globally convergent to optimal solution of the original problem. Finally, simulation results on several numerical examples are presented and the validity of the proposed NN model is demonstrated. [ABSTRACT FROM AUTHOR]

Based on the generalized Fischer–Burmeister smoothing function, we propose a regularization Newton method for solving the nonlinear complementarity problem. The proposed method views the regularization parameter as an independent variable. Moreover, it solves a perturbed Newton equation to obtain the search direction and adopts a simple nonmonotone line search scheme to improve the numerical results. Under suitable assumptions, we prove that our method has global and local quadratic convergence and the regularization parameter converges to zero globally Q-linearly. Numerical results shows that there does exist new smoothing function which is better than the Fischer–Burmeister smoothing function. [ABSTRACT FROM AUTHOR]

Having in mind singular solutions of smooth reformulations of complementarity problems, arising unavoidably when the solution in question violates strict complementarity, we study the behavior of Newton-type methods near singular solutions of nonlinear equations, assuming that the operator of the equation possesses a strongly semismooth derivative, but is not necessarily twice differentiable. These smoothness restrictions give rise to peculiarities of the analysis and results on local linear convergence and asymptotic acceptance of the full step, the issues addressed in this work. Moreover, we consider not only the basic Newton method, but also some stabilized versions of it intended for tackling singular (including nonisolated) solutions. Applications to nonlinear complementarity problems are also dealt with. [ABSTRACT FROM AUTHOR]

In this paper, using the sign patterns of the solution of the equivalent modulus equation, the resolution of the nonlinear complementarity problem shrinks to find the zero of a differentiable nonlinear function. Then, a sign-based Newton's method is established by applying the Newton's iteration. The theoretical analysis for the sign patterns of the solution of the equivalent modulus equation is given under the assumption of strictly complementarity. Moreover, by using the known modulus-based matrix splitting iteration method to detect the sign patterns of the solution of the equivalent modulus equation, a practical sign-detection Newton's method is proposed. Numerical examples show that the new methods are efficient and accelerate the convergence performance with higher precision and less CPU time than the existing modulus-based matrix splitting iteration method and the projection-based matrix splitting iteration method, especially for the large sparse problems. [ABSTRACT FROM AUTHOR]

As is well-known, Jacobian smoothing method is a popular one to solve nonlinear complementarity problems, in which the Jacobian consistency is stressed. By investigating an element of related functions’ B-differential, a smoothing Levenberg-Marquardt(LM) method is proposed based on a Chen-Harker-Kanzow-Smale(CHKS) smoothing function, which satisfies a property called strongly Jacobian consistency. Finally, the numerical experiments illustrate the effectiveness of the given method. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of nonlinear complementarity problems associated with circular cone (CCNCP for short), which is a type of non-symmetric cone complementarity problems. Useful properties of the circular cone are investigated, which help to reformulate equivalently CCNCP as an implicit fixed-point equation. Based on the implicit fixed-point equation and splittings of the system matrix, we establish a class of relaxation modulus-based matrix splitting iteration methods for solving such a complementarity problem. The convergence of the proposed modulus-based matrix splitting iteration methods has been analyzed and the strategy choice of the parameters are discussed when the splitting matrix is symmetric positive definite. Numerical experiments have shown that the modulus-based iteration methods are effective for solving CCNCP. [ABSTRACT FROM AUTHOR]

In this paper, we establish a two-step modulus-based matrix splitting iteration method for solving a class of nonlinear complementarity problems with the system matrix being an H+-matrix. The convergence analysis of the proposed method is given. Numerical examples show that the proposed method is efficient. [ABSTRACT FROM AUTHOR]

This paper establishes a theoretical framework of infeasible Mehrotra-type predictor–corrector algorithm for nonmonotone nonlinear complementarity problems over symmetric cones which can be regarded as an extension the Mehrotra’s algorithm proposed by Salahi et al. (On Mehrotra-type predictor–corrector algorithms. SIAM J Optim 18(4):1377–1397, 2005) from nonnegative orthant to symmetric cone. The iteration complexity of the algorithm is estimated, and some numerical results are provided. The numerical results show that the algorithm is efficient and reliable. [ABSTRACT FROM AUTHOR]

In this paper, the convergence conditions of the modulus-based matrix splitting iteration method for nonlinear complementarity problem of H-matrices are weakened. The convergence domain given by the proposed theorems is larger than the existing ones. Numerical examples show the advantages of the new theorems. [ABSTRACT FROM AUTHOR]

This study concerns about modeling the mixed equilibrium (ME) problem, including user equilibrium, system optimum, and Cournot-Nash players, with general common constraints (CCs) on transportation networks. The CCs capture the interactions of the decision variables of different players in ME, which could be internal interactions such as road link capacity constraints or external such as emission or congestion control policies. It is shown that ME with CCs can be modeled as a generalized Nash equilibrium problem (GNEP). The study proves that, under certain conditions, the GNEP-based ME is jointly convex, which can be reformulated as a variational inequality (VI). We then study the solution existence, uniqueness and solving method for the VI-based model, followed by a discussion on its potential applications. Numerical tests are conducted with common nonlinear link emission constraints as the CCs on a simple two-node, three-link network first, and then on the Nguyen Dupus network. The results show that modeling users' route choice behavior with CCs is more general in evaluating system performance, planning link capacities, and making congestion or emission control related policies. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give a generalized version of Caristi fixed point theorems in pseudo-metric spaces. Our results generalize and improve many of well-known theorems. As an application of our results, we give a new existence theorem to the generalized nonlinear complementarity problem and a solution of differential inclusion in the distributions setting. [ABSTRACT FROM AUTHOR]

Recently, the tensor complementarity problem has been investigated in the literature. An important question involving the property of global uniqueness and solvability for a class of tensor complementarity problems was proposed by Song and Qi (J Optim Theory Appl, 165:854-873, ). In the present paper, we give an answer to this question by constructing two counterexamples. We also show that the solution set of this class of tensor complementarity problems is nonempty and compact. In particular, we introduce a class of related structured tensors and show that the corresponding tensor complementarity problem has the property of global uniqueness and solvability. [ABSTRACT FROM AUTHOR]

Smoothing Newton methods have been successfully applied to solve the nonlinear complementarity problem (NCP). In this paper, we first study some properties of the generalized Fischer-Burmeister smoothing function. Based on this function, we then design a smoothing inexact Newton method for the NCP. At each iteration, a system of linear equations is solved only inexactly.Moreover, our method uses a nonmonotone line search technique which is much simpler than existing nonmonotone line searches used in smoothing Newton methods. Under suitable assumptions, we prove that the proposed method is globally and locally superlinearly convergent. Some numerical results are also reported. [ABSTRACT FROM AUTHOR]

In this paper, we propose a box-constrained differentiable penalty method for nonlinear complementarity problems, which not only inherits the same convergence rate as the existing $$\ell _\frac{1}{p}$$ -penalty method but also overcomes its disadvantage of non-Lipschitzianness. We introduce the concept of a uniform $$\xi $$ - $$P$$ -function with $$\xi \in (1,2]$$ , and apply it to prove that the solution of box-constrained penalized equations converges to that of the original problem at an exponential order. Instead of solving the box-constrained penalized equations directly, we solve a corresponding differentiable least squares problem by using a trust-region Gauss-Newton method. Furthermore, we establish the connection between the local solution of the least squares problem and that of the original problem under mild conditions. We carry out the numerical experiments on the test problems from MCPLIB, and show that the proposed method is efficient and robust. [ABSTRACT FROM AUTHOR]

This work deals with the analysis and numerical resolution of a broad class of complementarity problems on spaces of symmetric matrices. The complementarity conditions are expressed in terms of the Loewner ordering or, more generally, with respect to a dual pair of Loewnerian cones. [ABSTRACT FROM AUTHOR]

In this paper, we study some properties of quasiconvex nonlinear complementarity functions. However, we prove that a nonlinear complementarity function cannot be pseudoconvex. As a consequence of this, we show that every convex nonlinear complementarity function is nondifferentiable. Furthermore, some properties of homogeneous nonlinear complementarity functions are proved. [ABSTRACT FROM AUTHOR]

In this paper, we consider the splitting method and the two-stage splitting method for solving a class of nonlinear complementarity problems with the coefficient matrix being an $$H$$ -matrix. Convergence result for the splitting method is presented when the splitting is $$H$$ -splitting. Moreover, for the two-stage splitting method, we estimate weighted max-norm bounds for iteration errors, and thereby, we show that the sequence generated by the two-stage iteration scheme converges to the unique solution of the nonlinear complementarity problem without any restriction on the initial vector. Numerical results show that both methods are efficient for solving the class of nonlinear complementarity problems. [ABSTRACT FROM AUTHOR]

In the context of complementarity problems, various concepts of solution regularity are known, each of them playing a certain role in the related theoretical and algorithmic developments. Despite the existence of rich literature on this subject, it appears that the exact relations between some of these regularity concepts remained unknown. In this note, we not only summarize the existing results on the subject but also establish the missing relations filling all the gaps in the current understanding of how different regularity concepts relate to each other. In particular, we demonstrate that strong regularity is in fact equivalent to nonsingularity of all matrices in the natural outer estimates of the generalized Jacobians of the most widely used residual mappings for complementarity problems. On the other hand, we show that CD-regularity of the natural residual mapping does not imply even BD-regularity of the Fischer-Burmeister residual mapping. As a result, we provide the complete picture of relations between the most important regularity conditions for mixed complementarity problems, with a special emphasis on those conditions used to justify the related numerical methods. A special attention is paid to the particular cases of a nonlinear complementarity problem and of a Karush-Kuhn-Tucker system. [ABSTRACT FROM AUTHOR]

Nagurney (1999) used variational inequalities to study economic equilibrium and financial networks and applied the modified projection method to solve the problem. In this paper, we formulate the problem as a nonlinear complementarity problem. The complementarity model is just the KKT condition for the model of Nagurney (1999). It is a simpler model than that of Nagurney (1999). We also establish sufficient conditions for existence and uniqueness of the equilibrium pattern, which are weaker than those in Nagurney (1999). Finally, we apply a smoothing Newton-type algorithm to solve the problem and report some numerical results. [ABSTRACT FROM AUTHOR]

Based on a smoothing symmetric disturbance FB-function, a smoothing inexact Newton method for solving the nonlinear complementarity problem with P-function was proposed. It was proved that under mild conditions, the given algorithm performed global and superlinear convergence without strict complementarity. For the same linear complementarity problem(LCP), the algorithm needs similar iteration times to the literature. However, its accuracy is improved by at least 4 orders with calculation time reduced by almost 50%, and the iterative number is insensitive to the size of the LCP. Moreover, fewer iterations and shorter time are required for solving the problem by using inexact Newton methods for different initial points. [ABSTRACT FROM AUTHOR]

We propose a smoothing Newton method for solving nonlinear complementarity problems based on a new smoothing function. We adopt a variant merit function. Under suitable conditions, we obtain the global convergence and local quadratic convergence of the proposed algorithm. Some numerical results indicate that our algorithm is effective for the given problems. [ABSTRACT FROM AUTHOR]

The nonlinear complementarity or NCP functions were introduced by Mangasarian and these functions are proved to be useful in constrained optimization and elsewhere. Interestingly enough there are only two general methods to derive such functions, while the known or used NCP functions are either individual constructions or modifications of the few individual NCP functions such as the Fischer-Burmeister function. In the paper we analyze the elementary properties of NCP functions and the various techniques used to obtain such functions from old ones. We also prove some new nonexistence results on the possible forms of NCP functions. Then we develop and analyze several new methods for the construction of nonlinear complementarity functions that are based on various geometric arguments or monotone transformations. The appendix of the paper contains the list and source of the known NCP functions. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new generalized penalized Fischer-Burmeister merit function, and show that the function possesses a system of favorite properties. Moreover, for the merit function, we establish the boundedness of level set under a weaker condition. We also propose a derivative-free algorithm for nonlinear complementarity problems with a nonmonotone line search. More specifically, we show that the proposed algorithm is globally convergent and has a locally linear convergence rate. Numerical comparisons are also made with the merit function used by Chen (J Comput Appl Math 232:455-471, ), which confirm the superior behaviour of the new merit function. [ABSTRACT FROM AUTHOR]

In this paper we propose a class of new large-update primal-dual interior-point algorithms for P( κ) nonlinear complementarity problem (NCP), which are based on a class of kernel functions investigated by Bai et al. in their recent work for linear optimization (LO). The arguments for the algorithms are followed as Peng et al.'s for P( κ) complementarity problem based on the self-regular functions [Peng, J., Roos, C., Terlaky, T.: Self-Regularity: A New Paradigm for Primal-Dual Interior-Point Algorithms, Princeton University Press, Princeton, 2002]. It is worth mentioning that since this class of kernel functions includes a class of non-self-regular functions as special case, so our algorithms are different from Peng et al.'s and the corresponding analysis is simpler than theirs. The ultimate goal of the paper is to show that the algorithms based on these functions have favorable polynomial complexity. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the globalization of some kind of modified Levenberg-Marquardt methods for nonsmooth equations and their applications to nonlinear complementarity problems. In these modified Levenberg-Marquardt methods, only an approximate solution of a linear system at each iteration is required. Under some mild assumptions, the global convergence is shown. Finally, numerical results show that the present methods are promising. [ABSTRACT FROM AUTHOR]

Improving transportation system is essential for all people in each city since transport plays a very important role. Using mathematical programming approach transport problem is an effective way to improve transportation system. In this paper, the traffic equilibrium problem (TEP) with a general nonadditive route cost function is studied. We formulate the route cost function for each route as a disutility function, which can evaluate route cost function flexibly and analyze the route toll conveniently. Furthermore, we present the TEP with a nonlinear complementary problem (NCP) formulation. The monotonicity and the existence with the NCP formulation are also given under relative assumptions. [ABSTRACT FROM AUTHOR]

In this paper, based on a p-norm with p being any fixed real number in the interval (1,+∞), we introduce a family of new smoothing functions, which include the smoothing symmetric perturbed Fischer function as a special case. We also show that the functions have several favorable properties. Based on the new smoothing functions, we propose a nonmonotone smoothing Newton algorithm for solving nonlinear complementarity problems. The proposed algorithm only need to solve one linear system of equations. We show that the proposed algorithm is globally and locally superlinearly convergent under suitable assumptions. Numerical experiments indicate that the method associated with a smaller p, for example p=1.1, usually has better numerical performance than the smoothing symmetric perturbed Fischer function, which exactly corresponds to p=2. [ABSTRACT FROM AUTHOR]

This paper is concerned with the algorithm of the supply chain network equilibrium model and its equivalent supernetwork based traffic network equilibrium model with elastic demands. Both models are further written as nonlinear complementarity problems. Semismooth least squares reformulations of the complementarity problems are presented and their convergence properties are investigated. Considering the drawbacks of Quasi-Newton method (using the Fischer-Burmeister function), a semi-smooth Levenberg-Marquardt-type method is proposed to solve the problems. Numerical examples show that the Levenberg-Marquardt-type method can solve the supply chain network equilibrium model and its equivalent supernetwork based traffic network equilibrium model significantly, and is more efficient than the Quasi Newton method and the modified projection method. Furthermore, the Levenberg-Marquardt-type method with the equivalent supernetwork based complementarity formulation can be implemented faster than with the supply chain network equilibrium complementarity formulation. [ABSTRACT FROM AUTHOR]

family of merit functions are proposed, which are the generalization of several existing merit functions. A number of favorable properties of the proposed merit functions are established. By using these properties, a merit function method for solving nonlinear complementarity problem is investigated, and the global convergence of the proposed algorithm is proved under some standard assumptions. Some preliminary numerical results are given. [ABSTRACT FROM AUTHOR]

The nonlinear complementarity problem (denoted by NCP( F)) can be reformulated as the solution of a nonsmooth system of equations. In this paper, we propose a new smoothing and regularization Newton method for solving nonlinear complementarity problem with P-function ( P-NCP). Without requiring strict complementarity assumption at the P-NCP solution, the proposed algorithm is proved to be convergent globally and superlinearly under suitable assumptions. Furthermore, the algorithm has local quadratic convergence under mild conditions. Numerical experiments indicate that the proposed method is quite effective. In addition, in this paper, the regularization parameter ε in our algorithm is viewed as an independent variable, hence, our algorithm seems to be simpler and more easily implemented compared to many previous methods. [ABSTRACT FROM AUTHOR]

In this paper, we give a modified Broyden-like method for nonlinear complementarity problems, where we consider the smoothing parameter as an independent variable in the smoothing equation and adopt a modified nonmonotone step-length decision. We also analyze the global and local convergence of the modified method. Numerical experiments are also shown. [ABSTRACT FROM AUTHOR]

In this paper we propose a non-interior-point smoothing algorithm for solving the monotone nonlinear complementarity problem (NCP). The proposed algorithm is simpler than many existing non-interior-point smoothing algorithms in the sense that it only needs to solve one system of linear equations and to perform one line search at each iteration. We show that the proposed algorithm is globally convergent under the assumption that the NCP concerned has a nonempty solution set. Such assumption is weaker than those required by most other non-interior-point smoothing algorithms. In particular, we prove that the solution obtained by the proposed algorithm is a maximally complementary solution of the NCP concerned. Preliminary numerical results are reported. [ABSTRACT FROM AUTHOR]

By using the Fischer-Burmeister function to reformulate the nonlinear complementarity problem (NCP) as a system of semismooth equations and using Kanzow’s smooth approximation function to construct the smooth operator, we propose a smoothing trust region algorithm for solving the NCP withP0 functions. We prove that every accumulation point of the sequence generated by the algorithm is a solution of the NCP. Under a nonsingularity condition, local Q-superlinear/Q-quadratic convergence of the algorithm is established without the strict complementarity condition. [ABSTRACT FROM AUTHOR]

In this paper we focus on the problem of identifying the index sets P(x):={i|x[sub i]>0}, N(x):={i|F[sub i](x)>0} and C(x):={i|x[sub i]=F[sub i](x)=0} for a solution x of the monotone nonlinear complementarity problem NCP(F). The correct identification of these sets is important from both theoretical and practical points of view. Such an identification enables us to remove complementarity conditions from the NCP and locally reduce the NCP to a system which can be dealt with more easily. We present a new technique that utilizes a sequence generated by the proximal point algorithm (PPA). Using the superlinear convergence property of PPA, we show that the proposed technique can identify the correct index sets without assuming the nondegeneracy and the local uniqueness of the solution. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new smooth function that possesses a property not satisfied by the existing smooth functions. Based on this smooth function, we discuss the existence and continuity of the smoothing path for solving the P 0 function nonlinear complementarity problem ( NCP). Using the characteristics of the new smooth function, we investigate the boundedness of the iteration sequence generated by the non-interior continuation methods for solving the P 0 function NCP under the assumption that the solution set of the NCP is nonempty and bounded. We show that the assumption that the solution set of the NCP is nonempty and bounded is weaker than those required by a few existing continuation methods for solving the NCP [ABSTRACT FROM AUTHOR]

Smoothing Newton methods for nonlinear complementarity problems NCP( F) often require F to be at least a P0-function in order to guarantee that the underlying Newton equation is solvable. Based on a special equation reformulation of NCP( F), we propose a new smoothing Newton method for general nonlinear complementarity problems. The introduction of Kanzow and Pieper's gradient step makes our algorithm to be globally convergent. Under certain conditions, our method achieves fast local convergence rate. Extensive numerical results are also reported for all complementarity problems in MCPLIB and GAMSLIB libraries with all available starting points. [ABSTRACT FROM AUTHOR]

In a recent paper, Gowda and Ravindran (Algebraic univalence theorems for nonsmooth functions, Research Report, Department of Mathematics and Statistics, University of Maryland, Baltimore, MD 21250, March 15, 1998) introduced the concepts of H-differentiability and H-differential for a function f : Rn → Rn and showed that the Fréchet derivative of a Fréchet differentiable function, the Clarke generalized Jacobian of a locally Lipschitzian function, the Bouligand subdifferential of a semismooth function, and the C-differential of a C-differentiable function are particular instances of H-differentials. In this paper, we consider two applications of H-differentiability. In the first application, we derive a necessary optimality condition for a local minimum of an H-differentiable function. In the second application, we consider a nonlinear complementarity problem corresponding to an H-differentiable function f and show how, under appropriate conditions on an H-differential of f, minimizing a merit function corresponding to f leads to a solution of the nonlinear complementarity problem. These two applications were motivated by numerous studies carried out for C1, convex, locally Lipschitzian, and semismooth function by various researchers. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear complementarity problem is transformed into the least squares problem with nonnegative constraints, and a SQP algorithm for this reformulation based on a damped Gauss-Newton type method is presented. It is shown that the algorithm is globally and locally superlinearly (quadratically) convergent without the assumption of monotonicity. [ABSTRACT FROM AUTHOR]

In this paper we introduce a general line search scheme which easily allows us to define and analyze known and new semismooth algorithms for the solution of nonlinear complementarity problems. We enucleate the basic assumptions that a search direction to be used in the general scheme has to enjoy in order to guarantee global convergence, local superlinear/quadratic convergence or finite convergence. We examine in detail several different semismooth algorithms and compare their theoretical features and their practical behavior on a set of large-scale problems. [ABSTRACT FROM AUTHOR]