dc programming and dca thirty years of developments the year 2015 marks the 30th birthday of dc difference of convex functions programming and dca dc algorithms which constitute the backbone of nonconvex programming and global optimization in this article we offer a short survey on thirty years of developments of these theoretical and algorithmic tools the survey is comprised of three parts in the first part we present a brief history of the field while in the second we summarize the state of the art results and recent advances we focus on main theoretical results and dca solvers for important classes of difficult nonconvex optimization problems and then give an overview of real world applications whose solution methods are based on dca the third part is devoted to new trends and important open issues as well as suggestions for future developments

nonparametric fuzzy regression k nn and kernel smoothing techniques abstract fuzzy regression without predefined functional form or nonparametric fuzzy regression is investigated the two most basic nonparametric regression techniques in statistics namely k nearest neighbor smoothing and kernel smoothing are fuzzified and analyzed algorithms are proposed to obtain the best smoothing parameters based on the minimization of cross validation criteria

schur subalgebras and an application to the symmetric group abstract let k be an infinite field of prime characteristic p and let d r be positive integers of the same parity satisfying a certain congruence condition we prove that the schur algebra s 2 d is isomorphic to a subalgebra of the form es 2 r e where e is a certain idempotent of s 2 r translating this result via ringel duality to the symmetric groups d and r we obtain lattice isomorphisms between specht modules between young modules and between permutation modules here modules labelled by the partitions r k k correspond to modules labelled by d k k this provides a representation theoretical interpretation for part of the fractal structures observed for the decomposition numbers of the symmetric groups corresponding to two part partitions

the velocity tracking problem for navier stokes flows with bounded distributed controls we present some systematic approaches to the mathematical analysis and numerical approximation of the time dependent optimal control problem of tracking the velocity for navier stokes flows in bounded two dimensional domains with bounded distributed controls we study the existence of optimal solutions and derive an optimality system from which optimal solutions may be determined we also define and analyze semidiscrete in time and fully space time discrete approximations of the optimality system and a gradient method for the solution of the fully discrete system the results of some computational experiments are provided

fuzzy nonparametric regression based on local linear smoothing technique in a great deal of literature on fuzzy regression analysis most of research has focused on some predefined parametric forms of fuzzy regression relationships especially on the fuzzy linear regression models in many practical situations it may be unrealistic to predetermine a fuzzy parametric regression relationship in this paper a fuzzy nonparametric model with crisp input and lr fuzzy output is considered and based on the distance measure for fuzzy numbers suggested by diamond p diamond fuzzy least squares information sciences 46 1988 141 157 the local linear smoothing technique in statistics with the cross validation procedure for selecting the optimal value of the smoothing parameter is fuzzified to fit this model some simulation experiments are conducted to examine the performance of the proposed method and three real world datasets are analyzed to illustrate the application of the proposed method the results demonstrate that the proposed method works quite well not only in producing satisfactory estimate of the fuzzy regression function but also in reducing the boundary effect significantly

a simultaneous projections method for linear inequalities abstract an iterative method for solving general systems of linear inequalities is considered the method a relaxed generalization of cimmino s scheme for solving linear systems was first suggested by censor and elfving each iterate is obtained as a convex combination of the orthogonal projections of the previous iterate on the half spaces defined by the linear inequalities the algorithm is particularly suitable for implementation on computers with parallel processors we prove convergence from any starting point for both consistent and nonconsistent systems to a feasible point in the first case and to a weighted least squares type solutions in the second

parallel projection methods and the resolution of ill posed problems abstract in this paper we consider a modification of the parallel projection method for solving overdetermined nonlinear systems of equations introduced recently by diniz ehrhardt and martinez 1 this method is based on the classical cimmino s algorithm for solving linear systems the components of the function are divided into small blocks as an attempt to correct the intrinsic ill conditioning of the system and the new iteration is a convex combination of the projections onto the linear manifolds defined by different blocks the modification suggested here was motivated by the application of the method to the resolution of a nonlinear fredholm first kind integral equation we prove convergence results and we report numerical experiments

an efficient algorithm for globally minimizing a quadratic function under convex quadratic constraints in this paper we investigate two approaches to minimizing a quadratic form subject to the intersection of finitely many ellipsoids the first approach is the d c difference of convex functions optimization algorithm abbr dca whose main tools are the proximal point algorithm and or the projection subgradient method in convex minimization the second is a branch and bound scheme using lagrangian duality for bounding and ellipsoidal bisection in branching the dca was first introduced by pham dinh in 1986 for a general d c program and later developed by our various work is a local method but from a good starting point it provides often a global solution this motivates us to combine the dca and our branch and bound algorithm in order to obtain a good initial point for the dca and to prove the globality of the dca in both approaches we attempt to use the ellipsoidal constrained quadratic programs as the main subproblems the idea is based upon the fact that these programs can be efficiently solved by some available polynomial and nonpolynomial time algorithms among them the dca with restarting procedure recently proposed by pham dinh and le thi has been shown to be the most robust and fast for large scale problems several numerical experiments with dimension up to 200 are given which show the effectiveness and the robustness of the dca and the combined dca branch and bound algorithm

ramified partition algebras for each natural number n poset t and t tuple of scalars q we introduce the ramified partition algebra q which is a physically motivated and natural generalization of the partition algebra 24 25 the partition algebra coincides with case t 1 for fixed n and t these algebras like the partition algebra have a basis independent of q we investigate their representation theory in case t 1 2 leq we show that q is quasi hereditary over field k when q 1 q 2 is invertible in k and k is such that certain finite group algebras over k are semisimple e g when k is algebraically closed characteristic zero under these conditions we determine an index set for simple modules of q and construct standard modules with this index set we show that there are unboundedly many choices of q such that q is not semisimple for sufficiently large n but that it is generically semisimple for all n we construct tensor space representations of certain non semisimple specializations of q and show how to use these to build clock model transfer matrices 24 in arbitrary physical dimensions

approximation algorithms for indefinite quadratic programming we consider approximation schemes for indefinite quadratic programming we argue that such an approximation can be found in polynomial time for fixed andt wheret denotes the number of negative eigenvalues of the quadratic term our algorithm is polynomial in 1 for fixedt and exponential int for fixed we next look at the special case of knapsack problems showing that a more efficient polynomial int approximation algorithm exists

on a subproblem of trust region algorithms for constrained optimization we study a subproblem that arises in some trust region algorithms for equality constrained optimization it is the minimization of a general quadratic function with two special quadratic constraints properties of such subproblems are given it is proved that the hessian of the lagrangian has at most one negative eigenvalue and an example is presented to show that the hessian may have a negative eigenvalue when one constraint is inactive at the solution

a trust region strategy for minimization on arbitrary domains we present a trust region method for minimizing a general differentiable function restricted to an arbitrary closed set we prove a global convergence theorem the trust region method defines difficult subproblems that are solvable in some particular cases we analyze in detail the case where the domain is a euclidean ball for this case we present numerical experiments where we consider different hessian approximations

on the solution of a two ball trust region subproblem in this paper we investigate the structure of a two ball trust region subproblem arising frequently in nonlinear parameter identification problems and propose a method for its solution the method decomposes the subproblem and allows the application of efficient well studied methods for the solution of trust region subproblems arising in unconstrained optimization in the discussion of the structure we focus on the case where both constraints are active and on the treatment of the unconstrained problem

computing a celis dennis tapia trust region step for equality constrained optimization we study an approach for minimizing a convex quadratic function subject to two quadratic constraints this problem stems from computing a trust region step for an sqp algorithm proposed by celis dennis and tapia 1985 for equality constrained optimization our approach is to reformulate the problem into a univariate nonlinear equation 0 where the function is continuous at least piecewise differentiable and monotone well established methods then can be readily applied we also consider an extension of our approach to a class of non convex quadratic functions and show that our approach is applicable to reduced hessian sqp algorithms numerical results are presented indicating that our algorithm is reliable robust and has the potential to be used as a building block to construct trust region algorithms for small sized problems in constrained optimization

newton s method with a model trust region modification a modified newton method for unconstrained minimization is presented and analyzed the modification is based upon the model trust region approach this report contains a thorough analysis of the locally constrained quadratic minimizations that arise as subproblems in the modified newton iteration several promising alternatives are presented for solving these subproblems in ways that overcome certain theoretical difficulties exposed by this analysis very strong convergence results are presented concerning the minimization algorithm in particular the explicit use of second order information is justified by demonstrating that the iterates converge to a point which satisfies the second order necessary conditions for minimization with the exception of very pathological cases this occurs whenever the algorithm is applied to problems with continuous second partial derivatives

on local solutions of the celis dennis tapia subproblem we discuss the distribution of the local solutions of the celis dennis tapia cdt subproblem which appears in some trust region algorithms for nonlinear optimization we also give some examples to show the differences between the cdt subproblem and the single ball constraint subproblem these results show that the complexity of the cdt subproblem does not depend on the complexity of the structure of the dual plane thus they provide the possibility to search for the global minimizer in the dual plane

an ellipsoidal branch and bound algorithm for global optimization a branch and bound algorithm is developed for global optimization branching in the algorithm is accomplished by subdividing the feasible set using ellipses lower bounds are obtained by replacing the concave part of the objective function by an affine underestimate a ball approximation algorithm obtained by generalizing of a scheme of lin and han is used to solve the convex relaxation of the original problem the ball approximation algorithm is compared to sedumi as well as to gradient projection algorithms using randomly generated test problems with a quadratic objective and ellipsoidal constraints

on the choice of the regularization parameter in nonlinear inverse problems this paper focuses on regularization techniques for nonlinear ill posed inverse problems tikhonov regularization and regularization due to the use of norm constraints are analyzed a model function technique is proposed to iteratively determine an optimal regularization parameter or the parameter characterizing the norm constraint and to estimate the error in the data if it is not known a priori

a trust region method for norm constrained problems a trust region method for the solution of nonlinear optimization problems with norm constraints is analyzed such optimization problems arise in parameter identification and nonlinear eigenvalue problems the characteristics of these applications motivate the formulation and analysis of the trust region method the algorithms studied here allow for inexact gradient information and the use of subspace methods for the approximate solution of subproblems characterizations and the descent properties of trust region steps are given criteria for the existence of successful iterations under inexact gradient information and under the use of subspace methods are established and global convergence of the method is proven

a parallel subgradient projections method for the convex feasibility problem abstract an iterative method to solve the convex feasibility problem for a finite family of convex sets is presented the algorithm consists in the application of a generalization of an acceleration procedure introduced by de pierro in a parallel version of the subgradient projections method proposed by censor and lent the generated sequence is shown to converge for any starting point some numerical results are presented

on schur algebras and related algebras ii this paper is a continuation of 7 the purpose of 7 was to set up in the context of reductive algebraic groups a theory of generalized schur algebras a special case of our construction gives the schur algebras s n r studied extensively in lo we concentrate here on the schur algebras themselves and show that our construction yields new results in their representation theory by means of the schur functor see 10 chap 6 we obtain new results in the representation theory of the symmetric groups the main results are as follows the schur algebras s n r defined over a principal ideal domain r have finite global dimension see also 1 i over a field or complete valuation ring r every finitely generated projective s n r module has a weyl module filtration we prove the irreducibility of the socle of certain weyl modules see also 15 we describe a connection between weights of projective indecomposable modules for the schur algebras in terms of the involution on irreducible representations of symmetric groups given by tensoring with the sign representation we show that over a complete valuation ring or a field every young module for a symmetric group has a specht series in particular the projective indecomposables have specht series finally we show that for r n and a complete valuation ring or field r the blocks of the schur algebra s n r are given by the rule embodied in nakayama s conjecture the notations and conventions are as in the first part of this work 7

on some properties of quadratic programs with a convex quadratic constraint in this paper we consider the problem of minimizing a possibly nonconvex quadratic function with a quadratic constraint we point out some new properties of the problem in particular in the first part of the paper we show that i given a kkt point that is not a global minimizer it is easy to find a better feasible point ii strict complementarity holds at the local nonglobal minimizer in the second part of this paper we show that the original constrained problem is equivalent to the unconstrained minimization of a piecewise quartic merit function using the unconstrained formulation we give in the nonconvex case a new second order necessary condition for global minimizers in the third part of this paper algorithmic applications of the preceding results are briefly outlined and some preliminary numerical experiments are reported

minimal representation of quadratically constrained convex feasible regions in this paper we are concerned with characterizing minimal representations of feasible regions defined by both linear and convex quadratic constraints we say that representation is minimal if every other representation has either more quadratic constraints or has the same number of quadratic constraints and at least as many linear constraints we will prove that a representation is minimal if and only if it contains no redundant constraints no pseudo quadratic constraints and no implicit equality constraints we define a pseudo quadratic constraint as a quadratic constraint that can be replaced by a finite number of linear constraints in order to prove the minimal representation theorem we also prove that if the surfaces of two quadratic constraints match on a ball then they match everywhere in this paper we also provide algorithms that can be used to detect implicit equalities and pseudoquadratic constraints the redundant constraints can be identified using the hypersphere directions hd method

local minimizers of quadratic functions on euclidean balls and spheres in this paper a characterization of the local nonglobal minimizes of a quadratic function defined on a euclidean ball or a sphere is given it is proven that there exists at most one local nonglobal minimizes and that the lagrange multiplier that corresponds to this minimizes is the largest solution of a nonlinear scalar equation an algorithm is proposed for computing the local nonglobal minimizes

computing the local nonglobal minimizer of a large scale trust region subproblem we propose two algorithms for computing the local nonglobal minimizer of a quadratic function subject to a constraint set that is a euclidean sphere we also discuss the case where the constraint set is a euclidean ball at each iteration of the algorithms we compute the two smallest eigenvalues of a parametric matrix using an arpack subroutine only matrix vector multiplications are required hence we are able to exploit the possible sparsity of the hessian matrix of the quadratic objective making the algorithms suitable for large problems this improves previous approaches based on matrix factorizations we also give a geometric relationship based on extremal ellipsoids between the global and the local nonglobal minimizers of the quadratic function under the given sphere constraint

an approach to identification for robust control given measured data generated by a discrete time linear system we propose a model consisting of a linear time invariant system affected by norm bounded perturbation under mild assumptions the plants belonging to the resulting uncertain family form a convex set the approach depends on two key parameters an a priori given bound of the perturbation and the input used to generate the data it turns out that the size of the uncertain family can be reduced by intersecting the model families obtained by making use of different inputs the model validation problem in this identification scheme is analyzed for a given energy level the invalidation problem yields the family of those models which can never be invalidated for any possible input of fixed energy and any possible perturbation this leads to the intersection of all uncertain families a consequence of the invalidation problem is that for finite length measurements not all models can be invalidated using fixed energy inputs

an algorithm to determine boundedness of quadratically constrained convex quadratic programmes abstract in this paper we present an algorithm which can be used to determine whether or not a convex quadratic objective function is bounded from below over a feasible region defined by convex quadratic constraints if there are m constraints and n variables the algorithm terminates after at most min m 1 n 1 iterations each iteration requires the identification of implicit equality constraints in a system of homogeneous linear inequality and equality constraints and the identification can be completed by the solution of linear programmes in addition the algorithm has the advantage of providing a mechanism to reduce both the number of constraints and the dimension of the problem

a note on quadratic forms we extend an interesting theorem of yuan 12 for two quadratic forms to three matrices let c 1 c 2 c 3 be three symmetric matrices in n n if max x t c 1 x x t c 2 x x t c 3 x 0 for all x n it is proved that there exist t i 0 i 1 2 3 such that i 1 3 t i 1 and i 1 3 t i c i has at most one negative eigenvalue

a constrained least squares regularization method for nonlinear ill posed problems this paper deals with a method for solving ill posed nonlinear hilbert space operator equations f x y regularization is obtained by solving a constrained least squares regularization problem min f x y 2 quad text subject to j x leqq beta 2 beta serves as a regularization parameter and j x is a quadratic penalty functional to robustly and efficiently solve this regularization problem we apply a trust region method at each iteration the quadratic penalty constraint is retained a gauss newton approximation to the objective functional is taken and we add a quadratic trust region constraint the resulting quadratic subproblem is then reformulated as a nonlinear complementarily problem and solved using newton s method this paper applies methods to find approximate solutions to a severely ill posed nonlinear first kind integral equation arising in geophysics the method of generalized cross validation gcv is used to pick the regularization parameter when random er

mesh independence for nonlinear least squares problems with norm constraints if one solves an infinite dimensional optimization problem by introducing discretizations and applying a solution method to the resulting finite dimensional problem one often observes the very stable behavior of this method with respect to varying discretizations the most striking observation is the constancy of the number of iterations needed to satisfy a given stopping criterion in this paper an analysis of these phenomena is given and the so called mesh independence for non linear least squares problems with norm constraints ncnlls is proved a gauss newton method for the solution of ncnlls is discussed and its convergence properties are analyzed the mesh independence is proven in its sharpest formulation sufficient conditions for the mesh independence to hold are related to conditions guaranteeing convergence of the gauss newton method the results are demonstrated on a two point boundary value problem

row action methods for huge and sparse systems and their applications this paper brings together and discusses theory and applications of methods identified and labelled as row action methods for linear feasibility problems find x in bf r n such that ax leqq b linearly constrained optimization problems minimize f x subject to ax leqq b and some interval convex programming problems minimize f x subject to c leqq ax leqq b the main feature of row action methods is that they are iterative procedures which without making any changes to the original matrix a use the rows of a one row at a time such methods are important and have demonstrated effectiveness for problems with large or huge matrices which do not enjoy any detectable or usable structural pattern apart from a high degree of sparaseness fields of application where row action methods are used in various ways include image reconstruction from projection operations research and game theory learning theory pattern recognition and transportation theory a row action method for the

quadratically constrained convex quadratic programmes faulty feasible regions abstract we are concerned with the problem of minimizing a convex quadratic function subject to convex quadratic constraints in particular we are concerned with the application of the method of analytic centres to problems having a faulty feasible region i e a region that is either unbounded or not full dimensional each iteration of a method of centres must determine an analytic centre of a truncated feasible region if the feasible region is faulty then the truncated region may have no interior it may have no analytic centre it may have an infinity of centres or it may have a unique centre thus the difficulty caused by faulty feasible regions is that even when the problem has a solution the method of centres may fail typically faulty regions have been dealt with by the direct method of adding more constraints and variables often involving a big m constant our method is unique in that it results in a reduction in the number of constraints and effectively in the number of variables

fixed point quasi newton methods this article studies iterative methods defined by x k 1 phi x k e k where x k in mathbb r n and e k lie on a space of parameters sufficient conditions are established for local convergence and for convergence at an ideal linear or superlinear rate a theory of least change secant update methods for this class of processes is developed several examples are given showing a wide range of applications of the new theory key words quasi