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

a norm relaxed method of feasible directions for finely discretized problems from semi infinite programming abstract in this paper a class of finely discretized semi infinite programming sip problems is discussed combining the idea of the norm relaxed method of feasible directions mfd and the technique of updating discretization index set we present a new algorithm for solving the discretized semi infinite dsi problems from sip at each iteration the iteration point is feasible for the discretized problem and an improved search direction is computed by solving only one direction finding subproblem i e a quadratic program and some appropriate constraints are chosen to reduce the computational cost a high order correction direction can be obtained by solving another quadratic programming subproblem with only equality constraints under weak conditions such as mangasarian fromovitz constraint qualification mfcq the proposed algorithm possesses weak global convergence moreover the superlinear convergence is obtained under linearly independent constraint qualification licq and other assumptions in the end some elementary numerical experiments are reported

benchmarking optimization software with performance profiles we propose performance profiles distribution functions for a performance metric as a tool for benchmarking and comparing optimization software we show that performance profiles combine the best features of other tools for performance evaluation

special functions arising from discrete painleve equations a survey this article is a survey on recent studies on special solutions of the discrete painleve equations especially on hypergeometric solutions of the q painleve equations the main part of this survey is based on the joint work k kajiwara t masuda m noumi y ohta y yamada hypergeometric solutions to the q painleve equations imrn 2004 47 2004 2497 2521 k kajiwara t masuda m noumi y ohta y yamada construction of hypergeometric solutions to the q painleve equations imrn 2005 24 2005 1439 1463 with kajiwara masuda ohta and yamada after recalling some basic facts concerning painleve equations for comparison we give an overview of the present status of studies on difference discrete painleve equations as a source of special functions

on sequential approximate simultaneous analysis and design in classical topology optimization summary we study the simultaneous analysis and design sand formulation of the classical topology optimization problem subject to linear constraints on material density variables based on a dual method in theory and a primal dual method in practice we propose a separable and strictly convex quadratic lagrange newton subproblem for use in sequential approximate optimization of the sand formulated classical topology design problem the sand problem is characterized by a large number of nonlinear equality constraints the equations of equilibrium that are linearized in the approximate convex subproblems the availability of cheap second order information is exploited in a lagrange newton sequential quadratic programming like framework in the spirit of efficient structural optimization methods the quadratic terms are restricted to the diagonal of the hessian matrix the subproblems have minimal storage requirements are easy to solve and positive definiteness of the diagonal hessian matrix is trivially enforced theoretical considerations reveal that the dual statement of the proposed subproblem for sand minimum compliance design agrees with the ever popular optimality criterion method which is a nested analysis and design formulation this relates in turn to the known equivalence between rudimentary dual sequential approximate optimization algorithms based on reciprocal and exponential intervening variables and the optimality criterion method copyright 2016 john wiley sons ltd

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

on the constant positive linear dependence condition and its application to sqp methods in this paper we introduce a constant positive linear dependence condition cpld which is weaker than the mangasarian fromovitz constraint qualification mfcq and the constant rank constraint qualification crcq we show that a limit point of a sequence of approximating karush kuhn tucker kkt points is a kkt point if the cpld holds there we show that a kkt point satisfying the cpld and the strong second order sufficiency conditions ssosc is an isolated kkt point we then establish convergence of a general sequential quadratical programming sqp method under the cpld and the ssosc finally we apply these results to analyze the feasible sqp method proposed by panier and tits in 1993 for inequality constrained optimization problems we establish its global convergence under the ssosc and a condition slightly weaker than the mangasarian fromovitz constraint qualification and we prove superlinear convergence of a modified version of this algorithm under the ssosc and a condition slightly weaker than the linear independence constraint qualification

line search filter methods for nonlinear programming local convergence a line search method is proposed for nonlinear programming using fletcher and leyffer s filter method which replaces the traditional merit function a simple modification of the method proposed in a companion paper siam j optim 16 2005 pp 1 31 introducing second order correction steps is presented it is shown that the proposed method does not suffer from the maratos effect so that fast local convergence to second order sufficient local solutions is achieved

existence verification for higher degree singular zeros of nonlinear systems finding approximate solutions to systems of n nonlinear equations in n real variables is a much studied problem in numerical analysis somewhat more recently researchers have developed numerical methods to provide mathematically rigorous error bounds on such solutions we say that we verify existence of the solution within those bounds on the variables however when the jacobi matrix is singular at the solution no computational techniques to verify existence can handle the general case nonetheless computational verification that one or more solutions exists within a region in complex space containing the real bounds is possible by computing the topological degree in a previous paper we presented theory and algorithms for the simplest case when the rank defect of the jacobian matrix at the solution is 1 and the topological index is 2 here we generalize that result to arbitrary topological index d ge2 we present theory algorithms and experimental results we also present a heuristic for determining the degree obtaining a value that we can subsequently verify with our algorithms although execution times are slow compared to corresponding bound verification processes for nonsingular systems the order with respect to system size is still cubic

10e9 solution to the elliptic painleve equation a function formalism for sakai s elliptic painleve equation is presented this establishes the equivalence between the two formulations by sakai and by ohta ramani grammaticos we also give a simple geometric description of the elliptic painleve equation as a non autonomous deformation of the addition formula on elliptic curves by using these formulations we construct a particular solution of the elliptic painleve equation expressed in terms of the elliptic hypergeometric function 10e9

global convergence of a robust filter sqp algorithm abstract we present a robust filter sqp algorithm for solving constrained optimization problems this algorithm is based on the modified quadratic programming proposed by burke to avoid the infeasibility of the quadratic programming subproblem at each iteration compared with other filter sqp algorithms our algorithm does not require any restoration phase procedure which may spend a large amount of computation the main advantage of our algorithm is that it is globally convergent without requiring strong constraint qualifications such as mangasarian fromovitz constraint qualification mfcq and the constant rank constraint qualification crcq furthermore the feasible limit points of the sequence generated by our algorithm are proven to be the kkt points if some weaker conditions are satisfied numerical results are also presented to show the efficiency of the algorithm

a robust sequential quadratic programming method the sequential quadratic programming method developed by wilson han and powell may fail if the quadratic programming subproblems become infeasible or if the associated sequence of search directions is unbounded this paper considers techniques which circumvent these difficulties by modifying the structure of the constraint region in the quadratic programming subproblems furthermore questions concerning the occurrence of an unbounded sequence of multipliers and problem feasibility are also addressed

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 geometrical description of the discrete painleve vi and v equations we present a geometrical approach for the discrete painleve equations based on weyl groups the method relies on the bilinear formalism and assumes that the multidimensional function lives on the weight lattice of the appropriate affine weyl group the equations for the function a system of nonautonomous hirota miwa equations govern the evolution along the independent variable and the parameters of the equation the latter evolution induced by the schlesinger transformations in the present paper we analyse the case of the e 1 7 group using the geometrical description we derive the nonlinear discrete equations we find that in the case of the e 1 7 group these are the asymmetric q pvi and d pv that were recently proposed

riccati solutions of discrete painleve equations with weyl group symmetry of type e8 1 we present special solutions of the discrete painleve equations associated with a0 1 a0 1 and a0 1 surfaces these solutions can be expressed by solutions of linear difference equations here the a0 1 surface discrete painleve equation is the most generic difference equation as all discrete painleve equations can be obtained by its degeneration limit these special solutions exist when the parameters of the discrete painleve equation satisfy a particular constraint we consider that these special functions belong to the hypergeometric family although they seem to go beyond the known discrete and q discrete hypergeometric functions we also discuss the degeneration scheme of these solutions

augmented lagrangian methods under the constant positive linear dependence constraint qualification two augmented lagrangian algorithms for solving kkt systems are introduced the algorithms differ in the way in which penalty parameters are updated possibly infeasible accumulation points are characterized it is proved that feasible limit points that satisfy the constant positive linear dependence constraint qualification are kkt solutions boundedness of the penalty parameters is proved under suitable assumptions numerical experiments are presented

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

coalescence cascades and special function solutions for the continuous and discrete painleve equations we present a systematization of the special function type solutions of continuous and discrete painleve equations our method is to start from and construct its special solutions from the solutions of the hypergeometric equation and by coalescence obtain the lower painleve equations together with their special solutions in the discrete case we study the symmetric one component three point mapping forms of discrete painleve equations starting from d

hypergeometric solutions to the q painleve equation of type a 1 4 we consider the q painleve equation of type a 1 4 a version of q painleve v equation and construct a family of solutions expressible in terms of certain basic hypergeometric series we also present the determinant formula for the solutions

self duality and schlesinger chains for the asymmetric d pii and q piii equations we analyse two asymmetric discrete painleve equations namely d pii and q piii which are known to be discrete forms of piii and pvi respectively we show that both equations are self dual this means that the same equation governs the evolution along the discrete independent variable and the transformations under the action of the schlesinger transforms along the parameters of the discrete painleve a bilinear formulation of the self dual equation is given as a system of nonautonomous hirota miwa equations

on the global convergence of a filter sqp algorithm a mechanism for proving global convergence in sqp filter methods for nonlinear programming nlp is described such methods are characterized by their use of the dominance concept of multiobjective optimization instead of a penalty parameter whose adjustment can be problematic the main point of interest is to demonstrate how convergence for nlp can be induced without forcing sufficient descent in a penalty type merit function the proof relates to a prototypical algorithm within which is allowed a range of specific algorithm choices associated with the hessian matrix representation updating the trust region radius and feasibility restoration

extracting clusters from aggregate panel data we propose a method that finds homogeneous trajectories in aggregate panel data the sqp algorithm is used to maximize the log likelihood function the p values are computed from the pseudo inverse hessian matrix a case study shows the application of the method in market segmentation this paper introduces a new application of the sequential quadratic programing sqp algorithm to the context of clustering aggregate panel data the optimization applies the sqp method in parameter estimation the method is illustrated on synthetic and empirical data sets distinct models are estimated and compared with varying numbers of clusters explanatory variables and data aggregation results show a good performance of the sqp algorithm for synthetic and empirical data sets synthetic data sets were simulated assuming two segments and two covariates and the correlation between the two covariates was controlled in three scenarios 0 00 no correlation 0 25 weak correlation and 0 50 moderate correlation the sqp algorithm identifies the correct number of segments for these three scenarios based on all information criteria aic aic3 and bic and retrieves the unobserved heterogeneity in preferences the empirical case study applies the sqp algorithm to consumer purchase data to find market segments results for the empirical data set can provide insights for retail category managers because they are able to compute the impact on the marginal shares caused by a change in the average price of one brand or product

infeasibility detection and sqp methods for nonlinear optimization this paper addresses the need for nonlinear programming algorithms that provide fast local convergence guarantees regardless of whether a problem is feasible or infeasible we present a sequential quadratic programming method derived from an exact penalty approach that adjusts the penalty parameter automatically when appropriate to emphasize feasibility over optimality the superlinear convergence of such an algorithm to an optimal solution is well known when a problem is feasible the main contribution of this paper however is a set of conditions under which the superlinear convergence of the same type of algorithm to an infeasible stationary point can be guaranteed when a problem is infeasible numerical experiments illustrate the practical behavior of the method on feasible and infeasible problems

special function solutions for asymmetric discrete painleve equations we study the solutions of a family of discrete painleve equations the equations that we examine are given as a system of two first order non autonomous mappings the solutions we are interested in are the ones obtained whenever the painleve equation can be reduced to a discrete riccati equation which can be linearized through a cole hopf transformation the special solutions thus obtained involve generalizations or reductions of the hypergeometric and q hypergeometric function

special function solutions of the discrete painleve equations in this paper we present a review of the special solutions of discrete difference and q discrete painleve equations in terms of discrete special functions these solutions exist whenever the parameters of the painleve equation satisfy a particular constraint the discrete special functions belong to the hypergeometric family although in some cases they seem to go beyond the known discrete and q discrete hypergeometric functions the equations studied in this paper are chosen on the basis of our recent classification of discrete painleve equation with the help of affine weyl groups

on the superlinear local convergence of a filter sqp method transition to superlinear local convergence is shown for a modified version of the trust region filter sqp method for nonlinear programming introduced by fletcher leyffer and toint 8 hereby the original trust region sqp steps can be used without an additional second order correction the main modification consists in using the lagrangian function value instead of the objective function value in the filter together with an appropriate infeasibility measure moreover it is shown that the modified trust region filter sqp method has the same global convergence properties as the original algorithm in 8