on generalized variational inequalities involving relaxed lipschitz and relaxed monotone operators abstract we consider the solvability based on iterative algorithms of the generalized variational inequality gvi problems involving the relaxed lipschitz and relaxed monotone operators

discrete painleve equations coalescences limits and degeneracies abstract starting from the standard form of the five discrete painleve equations we show how one can obtain through appropriate limits a host of new equations which are also the discrete analogues of the continuous painleve equations a particularly interesting technique is the one based on the assumption that some simplification takes place in the autonomous form of the mapping following which the deautonomization leads to a new n dependence and introduces more new discrete painleve equations

quadratic relations in continuous and discrete painleve equations the quadratic relations between the solutions of a painleve equation and that of a different one or the same one with a different set of parameters are investigated in the continuous and discrete cases we show that the quadratic relations existing for the continuous pii piii pv and pvi have analogues as well as consequences in the discrete case moreover the discrete painleve equations have quadratic relations of their own without any reference to the continuous case

auxiliary principle and iterative algorithms for generalized set valued strongly nonlinear mixed variational like inequalities in this paper we extend the auxiliary principle technique to study a class of generalized set valued strongly nonlinear mixed variational like inequalities we prove the existence of a solution of the auxiliary problem for the generalized set valued strongly nonlinear mixed variational like inequalities construct the iterative algorithm for the generalized set valued strongly nonlinear mixed variational like inequalities and show the existence of a solution of the generalized set valued strongly nonlinear mixed variational like inequality by using the auxiliary principle technique we also prove the convergence of iterative sequences generated by the algorithm

degeneration through coalescence of the q painleve vi equation starting from the q discrete form of the painleve vi equation we obtain its degenerate forms by applying the procedure of coalescence of singularities the whole cascade of degenerate forms is thus obtained leading to new forms for the discrete painleve iv and v equations the lax pairs of these discrete painleve equations are explicitly constructed thus confirming their integrability

a new system of generalized mixed quasi variational inclusions with h η monotone operators abstract in this paper we introduce and study a new system of generalized mixed quasi variational inclusions with h monotone operators we prove the convergence of a new iterative algorithm for this system of generalized mixed quasi variational inclusions the results in this paper extend and improve some known results in the literature

existence of solutions and convergence of iterative algorithms for a system of generalized nonlinear mixed quasi variational inclusions in this paper we introduce and study a new system of generalized nonlinear mixed quasi variational inclusions in q uniformly smooth banach spaces we prove the existence and uniqueness of solutions for this system of generalized nonlinear mixed quasi variational inclusions we also prove the convergence of several new two step iterative algorithms with or without errors for this system of generalized nonlinear mixed quasi variational inclusions the results in this paper extend and improve some known results in the literature

a feasible descent sqp algorithm for general constrained optimization without strict complementarity in this paper a class of optimization problems with equality and inequality constraints is discussed firstly the original problem is transformed to an associated simpler problem with only inequality constraints and a parameter the later problem is shown to be equivalent to the original problem if the parameter is large enough but finite then a feasible descent sqp algorithm for the simplified problem is presented at each iteration of the proposed algorithm a master direction is obtained by solving a quadratic program which always has a feasible solution with two corrections on the master direction by two simple explicit formulas the algorithm generates a feasible descent direction for the simplified problem and a height order correction direction which can avoid the maratos effect without the strict complementarity then performs a curve search to obtain the next iteration point thanks to the new height order correction technique under mild conditions without the strict complementarity the globally and superlinearly convergent properties are obtained finally an efficient implementation of the numerical experiments is reported

a new class of completely generalized quasi variational inclusions in banach spaces in this paper a new notion of j proximal mapping for a nonconvex lower semicontinuous subdifferentiable proper functional on banach space is introduced the existence and lipschitz continuity of j proximal mapping of a lower semicontinuous subdifferentiable proper functional are proved by applying the concept we introduce and study a new class of completely generalized quasi variational inclusions in reflexive banach spaces a novel and innovative iterative algorithm for finding the approximate solutions is suggested and analyzed the convergence criteria of the iterative sequences generated by the new iterative algorithm is also given these algorithm and existence result generalize many known results under hilbert space setting in recent literature to reflexive banach spaces

iterative algorithm for finding approximate solutions of mixed quasi variational like inclusions in the present paper we introduce the concept of h relaxed strong convexity of a differentiable functional and extend ding and yao s auxiliary variational inequality technique x p ding j c yao existence and algorithm of solutions for mixed quasi variational like inclusions in banach spaces computers and mathematics with applications 49 2005 857 869 to develop iterative algorithms for finding the approximate solutions to the mixed quasi variational like inclusion problem in short mqvlip in a banach space on the one hand we establish a result on the existence of a solution to the equilibrium problem by virtue of well known brouwer s fixed point theorem moreover by using this result we derive the existence and uniqueness of a solution to the mqvlip and the existence of the approximate solutions generated by the algorithm for the mqvlip on the other hand we use the concepts of h relaxed strong convexity of a differentiable functional and h cocoercivity of a composite map to prove the strong convergence of the approximate solutions to the unique solution of the mqvlip

existence and algorithm of solutions for mixed quasi variational like inclusions in banach spaces in this paper we study a class of mixed quasi variational like inclusion with a bifunction in reflexive banach spaces by applying a minimax inequality due to the first author some existence and uniqueness theorems of solutions for the mixed quasi variational like inclusions are proved next by applying the auxiliary variational inequality technique we suggest an iterative algorithm to compute the approximate solutions of the mixed quasi variational like inclusions the convergence criteria of the algorithm is studied under quite mild conditions our results represent improvement and generalization of the previously known results in literature

a recursive quadratic programming algorithm that uses differentiable exact penalty functions in this paper a recursive quadratic programming algorithm for solving equality constrained optimization problems is proposed and studied the line search functions used are approximations to fletcher s differentiable exact penalty function global convergence and local superlinear convergence results are proved and some numerical results are given

the auxiliary principle and an algorithm for a system of generalized set valued mixed variational like inequality problems in banach spaces in this paper we introduce and study a new system of generalized set valued mixed variational like inequality problems sgsmvlip and its related auxiliary problems in reflexive banach spaces the auxiliary principle technique is applied to study the existence and an iterative algorithm of solutions for the system of generalized set valued mixed variational like inequality problems at first the existence and uniqueness of solutions of the auxiliary problems for sgsmvlip is shown next an iterative algorithm for solving sgsmvlip is constructed by using the existence and uniqueness result finally we prove the existence of solutions of sgsmvlip and discuss the convergence analysis of the algorithm these results improve unify and generalize many corresponding known results given in the literature

existence and iterative approximation of solutions of generalized mixed quasi variational like inequality problem in banach spaces in this paper some existence theorems for the mixed quasi variational like inequalities problem in a reflexive banach space are established the auxiliary principle technique is used to suggest a novel and innovative iterative algorithm for computing the approximate solution for the mixed quasi variational like inequalities problem consequently not only the existence of theorems of the mixed quasi variational like inequalities is shown but also the convergence of iterative sequences generated by the algorithm is also proven the results proved in this paper represent an improvement of previously known results

from continuous painleve iv to the asymmetric discrete painleve i we examine the discrete equation that is obtained from the schlesinger transformations of the continuous painleve iv equation the use of the schlesinger transformations naturally establishes the lax pair of the discrete equation which is an asymmetric form of d p and has as continuous limit we analyse the transformations of the discrete equation and show that they lead to a self dual description where the discrete equation and the schlesinger s play the same role finally using the relation between and asymmetric d we examine in detail the special solutions of both equations

a simple feasible sqp algorithm for inequality constrained optimization in this paper an sqp method generating feasible iterates is presented to solve the nonlinear inequality constrained optimization problems a new modified method which ensures the global and superlinear convergence is proposed by solving systems of linear equation instead of qp subproblems and linear squares problems here the search direction is a suitable combination of a descent direction a feasible direction and a second order revised direction the theoretical analysis shows that global and superlinear convergence can be induced under some suitable conditions

integrable mappings and soliton equations ii abstract some simple solutions of discrete soliton equations are shown to satisfy 2d mappings we show that these belong to a recently introduced 18 parameter family of integrable reversible mappings of the plane thus lending weight to a previous conjecture we also give an example of an integrable mapping occuring in an exactly solvable model in statistical mechanics finally we discuss the notion of generalized reversibility

on combining feasibility descent and superlinear convergence in inequality constrained optimization extension of quasi newton techniques from unconstrained to constrained optimization via sequential quadratic programming sqp presents several difficulties among these are the possible inconsistency away from the solution of first order approximations to the constraints resulting in infeasibility of the quadratic programs and the task of selecting a suitable merit function to induce global convergence in ths case of inequality constrained optimization both of these difficulties disappear if the algorithm is forced to generate iterates that all satisfy the constraints and that yield monotonically decreasing objective function values feasibility of the successive iterates is in fact required in many contexts such as in real time applications or when the objective function is not well defined outside the feasible set it has been recently shown that this can be achieved while preserving local two step superlinear convergence in this note the essential ingredients for an sqp based method exhibiting the desired properties are highlighted correspondingly a class of such algorithms is described and analyzed tests performed with an efficient implementation are discussed

further applications of a splitting algorithm to decomposition in variational inequalities and convex programming a classical method for solving the variational inequality problem is the projection algorithm we show that existing convergence results for this algorithm follow from one given by gabay for a splitting algorithm for finding a zero of the sum of two maximal monotone operators moreover we extend the projection algorithm to solveany monotone affine variational inequality problem when applied to linear complementarity problems we obtain a matrix splitting algorithm that is simple and for linear quadratic programs massively parallelizable unlike existing matrix splitting algorithms this algorithm converges under no additional assumption on the problem when applied to generalized linear quadratic programs we obtain a decomposition method that unlike existing decomposition methods can simultaneously dualize the linear constraints and diagonalize the cost function this method gives rise to highly parallelizable algorithms for solving a problem of deterministic control in discrete time and for computing the orthogonal projection onto the intersection of convex sets

a superlinearly convergent feasible method for the solution of inequality constrained optimization problems when iteratively solving optimization problems arising from engineering design applications it is sometimes crucial that all iterates satisfy a given set of hard inequality constraints and generally desirable that the objective function value improve at each iteration in this paper we propose an algorithm of the successive quadratic programming sqp type which unlike other algorithms of this type does enjoy such properties under mild assumptions the new algorithm is shown to converge from any initial point locally superlinearly numerically tested it has proven to be competitive with the most successful currently available nonlinear programming algorithms while the latter do not exhibit the desired properties

existence and algorithms for the bilevel new generalized mixed equilibrium problems in banach spaces abstract in this paper we introduce and analyze the new generalized mixed equilibrium problems ngmep and the bilevel new generalized mixed equilibrium problems bngmep in banach spaces first by using a minimax inequality some new existence theorems of the solution and the behavior of solution set for the ngmep and the bngmep are obtained in both arbitrary banach spaces with the coercive conditions and a reflexive banach space without the coercive conditions next by using auxiliary principle technique some new iterative algorithms for solving the ngmep and the bngmep are suggested and analyzed the strong convergence of the iterative sequences generated by the algorithms are also proved in both arbitrary banach spaces with the coercive conditions and a reflexive banach space without the coercive conditions these results are new and generalize some recent results in this field

a superlinearly convergent norm relaxed sqp method of strongly sub feasible directions for constrained optimization without strict complementarity in this paper a kind of optimization problems with nonlinear inequality constraints is discussed combined the ideas of norm relaxed sqp method and strongly sub feasible direction method as well as a pivoting operation a new fast algorithm with arbitrary initial point for the discussed problem is presented at each iteration of the algorithm an improved direction is obtained by solving only one direction finding subproblem which possesses small scale and always has an optimal solution and to avoid the maratos effect another correction direction is yielded by a simple explicit formula since the line search technique can automatically combine the initialization and optimization processes after finite iterations the iteration points always get into the feasible set the proposed algorithm is proved to be globally convergent and superlinearly convergent under mild conditions without the strict complementarity finally some numerical tests are reported

existence and algorithm of solutions for nonlinear mixed variational like inequalities in banach spaces in this paper we study a new class of nonlinear mixed variational like inequalities in reflexive banach spaces by applying a minimax inequality due to author we first prove an existence uniqueness theorem of solutions for the nonlinear mixed variational like inequalities next we extend the auxiliary problem technique under reflexive banach space setting to suggest and analyze an innovative iterative algorithm to compute the approximate solutions of the nonlinear mixed variational like inequalities the convergence criteria of the iterative algorithm is also given these results improve and generalize many known results in recent literature

parametric completely generalized mixed implicit quasi variational inclusions involving h maximal monotone mappings a new class of parametric completely generalized mixed implicit quasi variational inclusions involving h maximal monotone mappings is introduced by applying resolvent operator technique of h maximal monotone mapping and the property of fixed point set of set valued contractive mappings the behavior and sensitivity analysis of the solution set of the parametric completely generalized mixed implicit quasi variational inclusions involving h maximal monotone mappings are studied the continuity and lipschitz continuity of the solution set with respect to the parameter are proved under suitable assumptions our approach and results are new and improve unify and extend previous many known results in this field

from continuous to discrete painleve equations abstract discrete analogues of the painleve equations have recently been derived using either a method based on orthogonal polynomials or the singularity confinement method here we analyze additional methods for deriving integrable non autonomous discrete equations these methods are based on the discrete akns approach on the use of backlund and schlesinger transforms of the continuous painleve equations and on discrete analogues of miura transformations

general iterative algorithms for solving mixed quasi variational like inclusions in this paper we extend the auxiliary variational inequality technique due to ding and yao x p ding j c yao existence and algorithm of solutions for mixed quasi variational like inclusions in banach spaces comput math appl 49 2005 857 869 to develop iterative algorithms for finding the approximate solutions of a mixed quasi variational like inclusion problem in short mqvlip in the setting of banach spaces we first establish a result on the existence of a solution of the equilibrium problem by virtue of the fan kkm lemma then by using this result and a result by ding and tan x p ding k k tan a minimax inequality with applications to existence of equilibrium point and fixed point theorems colloq math 63 2 1992 233 247 we derive the existence of a unique solution of mqvlip and the existence of approximate solutions generated by the proposed algorithms moreover we also provide the new criteria for convergence of approximate solutions to the exact solution of mqvlip

a simple feasible sqp method for inequality constrained optimization with global and superlinear convergence in this paper a simple feasible sqp method for nonlinear inequality constrained optimization is presented at each iteration we need to solve one qp subproblem only after solving a system of linear equations a new feasible descent direction is designed the maratos effect is avoided by using a high order corrected direction under some suitable conditions the global and superlinear convergence can be induced in the end numerical experiments show that the method in this paper is effective

auxiliary principle and iterative algorithm for a new system of generalized mixed equilibrium problems in banach spaces abstract in this paper we study a new system of generalized mixed equilibrium problems involving skew symmetric bifunctions sgmep in reflexive banach spaces a system of auxiliary mixed equilibrium problems samep for solving the sgmep is introduced and the existence and uniqueness of the solutions of the samep is first proved next by using the auxiliary principle technique a new iterative algorithm to compute the approximate solutions of the sgmep is suggested and analyzed finally the strong convergence of the iterative sequences generated by the algorithm is also proved under quite mild conditions these results improve unify and generalize some known results in recent literature

on the local convergence of quasi newton methods for constrained optimization we consider the application of a general class of quasi newton methods to the solution of the classical equality constrained nonlinear optimization problem specifically we develop necessary and sufficient conditions for the q superlinear convergence of such methods and present a companion linear convergence theorem the essential conditions relate to the manner in which the hessian of the lagrangian function is approximated

general algorithm of solutions for nonlinear variational inequalities in banach space abstract in this paper we shall study a class of general nonlinear variational inequalities in reflexive banach spaces by applying a minimax inequality obtained by the author some existence uniqueness theorems of solutions for the general nonlinear variational inequalities are proved next by applying the auxiliary problem technique we suggest an innovative iterative algorithm to compute the approximate solutions of the general nonlinear variational inequality finally the convergence criteria is also discussed

scaling corrections site percolation and ising model in three dimensions using finite size scaling techniques we obtain accurate results for critical quantities of the ising model and the site percolation in three dimensions we pay special attention to parametrizing the corrections to scaling which is necessary to bring the systematic errors below the statistical ones

on a unified approach to transformations and elementary solutions of painleve equations an algorithmic method is developed for investigating the transformation properties of second order equations of painleve type this method which utilizes the singularity structure of these equations yields explicit transformations which relate solutions of the painleve equations ii vi with different parameters these transformations easily generate rational and other elementary solutions of the equations the relationship between painleve equations and certain new equations quadratic in the second derivative of painleve type is also discussed

a method combining norm relaxed qp subproblems with systems of linear equations for constrained optimization based on the ideas of norm relaxed sequential quadratic programming sqp method and the strongly sub feasible direction method we propose a new sqp algorithm for the solution of nonlinear inequality constrained optimization unlike the previous work at each iteration the norm relaxed quadratic programming subproblem nrqps in our algorithm only consists of the constraints corresponding to an estimate of the active set and the high order correction direction used to avoid the maratos effect is obtained by solving a system of linear equations sle which also only consists of such a subset of constraints and gradients moreover the line search technique can effectively combine the initialization process with the optimization process and therefore if the starting point is not feasible the iteration points always get into the feasible set after a finite number of iterations the global convergence is proved under the mangasarian fromovitz constraint qualification mfcq and the superlinear convergence is obtained without assuming the strict complementarity finally the numerical experiments show that the proposed algorithm is effective and promising for the test problems

feasible generalized monotone line search sqp algorithm for nonlinear minimax problems with inequality constraints in this paper the nonlinear minimax problems with inequality constraints are discussed and a sequential quadratic programming sqp algorithm with a generalized monotone line search is presented at each iteration a feasible direction of descent is obtained by solving a quadratic programming qp to avoid the maratos effect a high order correction direction is achieved by solving another qp as a result the proposed algorithm has global and superlinear convergence especially the global convergence is obtained under a weak mangasarian fromovitz constraint qualification mfcq instead of the linearly independent constraint qualification licq at last its numerical effectiveness is demonstrated with test examples

a sequential equality constrained quadratic programming algorithm for inequality constrained optimization in this paper the feasible type sqp method is improved a new sqp algorithm is presented to solve the nonlinear inequality constrained optimization as compared with the existing sqp methods per single iteration in order to obtain the search direction it is only necessary to solve equality constrained quadratic programming subproblems and systems of linear equations under some suitable conditions the global and superlinear convergence can be induced

a qp free globally convergent locally superlinearly convergent algorithm for inequality constrained optimization an algorithm is proposed for the minimization of a smooth function subject to smooth inequality constraints unlike sequential quadratic programming type methods this algorithm does not involve the solution of quadratic programs but merely that of linear systems of equations locally the iteration can be viewed as a perturbation of a quasi newton iteration on both the primal and dual variables for the solution of the equalities in the kuhn tucker first order conditions of optimality it is observed that provided the current iterate is feasible and the current multiplier estimates are strictly positive the primal component of the quasi newton direction is a direction of descent for the objective function this fact is used to induce global convergence without the need of a surrogate merit function a careful bending of the search direction prevents any maratos like effect and local superlinear convergence is proven while the algorithm requires that an initial feasible point be available the succe

existence and algorithm of solutions for generalized strongly nonlinear mixed variational like inequalities in banach spaces in this paper we introduce and study a class of generalized strongly nonlinear mixed variational like inequalities in reflexive banach spaces the auxiliary principle technique is applied to study the existence and iterative algorithm of solutions for generalized strongly nonlinear mixed variational like inequalities first the existence of solutions of the auxiliary problems for the generalized strongly nonlinear mixed variational like inequalities is shown second an iterative algorithm for solving the generalized strongly nonlinear mixed variational like inequalities is constructed by using this existence result finally not only the existence of solutions of the generalized strongly nonlinear mixed variational like inequalities is shown but also the convergence of iterative sequences generated by the algorithm is also proven