the nonlinear programming method of wilson han and powell with an augmented lagrangian type line search function part 2 an efficient implementation with linear least squares subproblems

applications of second order cone programming in a second order cone program socp a linear function is minimized over the intersection of an affine set and the product of second order quadratic cones socps are nonlinear convex problems that include linear and convex quadratic programs as special cases but are less general than semidefinite programs sdps several efficient primaldual interior point methods for socp have been developed in the last few years after reviewing the basic theory of socps we describe general families of problems that tan be recast as socps these include robust linear programming and robust leastsquares problems problems involving sums or maxima of norms or with convex hyperbolic constraints we discuss a variety of engineering applications such as filter design antenna array weight design truss design and grasping forte optimization in robotics we describe an efficient primaldual interior point method for solving socps which shares many of the features of primaldual interior point methods for linear program

a superlinearly convergent sequential quadratically constrained quadratic programming algorithm for degenerate nonlinear programming we present an algorithm that achieves superlinear convergence for nonlinear programs satisfying the mangasarian fromovitz constraint qualification and the quadratic growth condition this convergence result is obtained despite the potential lack of a locally convex augmented lagrangian the algorithm solves a succession of subproblems that have quadratic objectives and quadratic constraints both possibly nonconvex by the use of a trust region constraint we guarantee that any stationary point of the subproblem induces superlinear convergence which avoids the problem of computing a global minimum we compare this algorithm with sequential quadratic programming algorithms on several degenerate nonlinear programs

a branch and cut algorithm for nonconvex quadratically constrained quadratic programming we present a branch and cut algorithm that yields in finite time a globally e optimal solution with respect to feasibility and optimality of the nonconvex quadratically constrained quadratic programming problem the idea is to estimate all quadratic terms by successive linearizations within a branching tree using reformulation linearization techniques rlt to do so four classes of linearizations cuts depending on one to three parameters are detailed for each class we show how to select the best member with respect to a precise criterion the cuts introduced at any node of the tree are valid in the whole tree and not only within the subtree rooted at that node in order to enhance the computational speed the structure created at any node of the tree is flexible enough to be used at other nodes computational results are reported that include standard test problems taken from the literature some of these problems are solved for the first time with a proof of global optimality

a reduced order observer for descriptor systems abstract in this paper a reducer order observer is analyzed for so called generalized state space or descriptor systems based on the staircase form for generalized state space models a recursive algorithm is presented to construct a reduced order observer for a given observable descriptor system in the algorithm only the eigenfrequencies of the observer have to be specified

confluent hypergeometric functions on tube domains our problem specialized to the case of the siegel upper half space h of degree m concerns the fourier expansion of a series s z fl det z a det a a l here z is a variable on h l is a lattice in the space v of all real symmetric matrices of size m and fl ec it can be shown that this is convergent if re f l m and also that if re m 2 and r e f l m 2 it has a fourier expansion of the form

on the sequential quadratically constrained quadratic programming methods an iteration of the sequential quadratically constrained quadratic programming method sqcqp consists of minimizing a quadratic approximation of the objective function subject to quadratic approximation of the constraints followed by a line search in the obtained direction methods of this class are receiving attention due to the development of efficient interior point techniques for solving subproblems with this structure via formulating them as second order cone programs recently fukushima et al 2003 proposed a sqcqp method for convex minimization with twice continuously differentiable data their method possesses global and locally quadratic convergence and it is free of the maratos effect the feasibility of subproblems in their method is enforced by switching between the linear and quadratic approximations of the constraints this strategy requires computing a strictly feasible point as well as choosing some further parameters we propose a sqcqp method where feasibility of subproblems is ensured by introducing a slack variable and hence is automatic in addition we do not assume convexity of the objective function or twice differentiability of the problem data while our method has all the desirable convergence properties it is easier to implement among other things it does not require computing a strictly feasible point which is a nontrivial task in addition its global convergence requires weaker assumptions

the nonlinear programming method of wilson han and powell with an augmented lagrangian type line search function the paper represents an outcome of an extensive comparative study of nonlinear optimization algorithms this study indicates that quadratic approximation methods which are characterized by solving a sequence of quadratic subproblems recursively belong to the most efficient and reliable nonlinear programming algorithms available at present the purpose of this paper is to analyse the theoretical convergence properties and to investigate the numerical performance in more detail in part 1 the exactl 1 penalty function of han and powell is replaced by a differentiable augmented lagrange function for the line search computation to the able to prove the global convergence and to show that the steplength one is chosen in the neighbourhood of a solution in part 2 the quadratic subproblem is exchanged by a linear least squares problem to improve the efficiency and to test the dependence of the performance from different solution methods for the quadratic or least squares subproblems

sequential quadratically constrained quadratic programming norm relaxed algorithm of strongly sub feasible directions abstract in this paper we present a sequential quadratically constrained quadratic programming sqcqp norm relaxed algorithm of strongly sub feasible directions for the solution of inequality constrained optimization problems by introducing a new unified line search and making use of the idea of strongly sub feasible direction method the proposed algorithm can well combine the phase of finding a feasible point by finite iterations and the phase of a feasible descent norm relaxed sqcqp algorithm moreover the former phase can preserve the sub feasibility of the current iteration and control the increase of the objective function at each iteration only a consistent convex quadratically constrained quadratic programming problem needs to be solved to obtain a search direction without any other correctional directions the global superlinear and a certain quadratic convergence which is between 1 step and 2 step quadratic convergence properties are proved under reasonable assumptions finally some preliminary numerical results show that the proposed algorithm is also encouraging

avoiding the maratos effect by means of a nonmonotone line search ii inequality constrained problems feasible iterates when solving inequality constrained optimization problems via sequential quadratic programming sqp it is potentially advantageous to generate iterates that all satisfy the constraints all quadratic programs encountered are then feasible and there is no need for a surrogate merit function 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 defined outside the feasible set it has recently been shown that this is indeed possible by means of a suitable perturbation of the original sqp iteration without losing superlinear convergence in this context the well known maratos effect is compounded by the possible infeasibility of the full step of one even close to a solution these difficulties have been accommodated by making use of a suitable modification of a bending technique proposed by mayne and polak requiring evaluation of the constraints function at an auxiliary point at each iteration in part i

a feasible trust region sequential quadratic programming algorithm an algorithm for smooth nonlinear constrained optimization problems is described in which a sequence of feasible iterates is generated by solving a trust region sequential quadratic programming sqp subproblem at each iteration and by perturbing the resulting step to retain feasibility of each iterate by retaining feasibility the algorithm avoids several complications of other trust region sqp approaches the objective function can be used as a merit function and the sqp subproblems are feasible for all choices of the trust region radius global convergence properties are analyzed under various assumptions on the approximate hessian under additional assumptions superlinear convergence to points satisfying second order sufficient conditions is proved

a working set sqcqp algorithm with simple nonmonotone penalty parameters in this paper we present a new sequential quadratically constrained quadratic programming sqcqp algorithm in which a simple updating strategy of the penalty parameter is adopted this strategy generates nonmonotone penalty parameters at early iterations and only uses the multiplier corresponding to the bound constraint of the quadratically constrained quadratic programming qcqp subproblem instead of the multipliers of the quadratic constraints which will bring some numerical advantages furthermore by using the working set technique we remove the constraints of the qcqp subproblem that are locally irrelevant and thus the computational cost could be reduced without assuming the convexity of the objective function or the constraints the algorithm is proved to be globally superlinearly and quadratically convergent preliminary numerical results show that the proposed algorithm is very promising when compared with the tested sqp algorithms

zeta functions and eisenstein series on metaplectic groups the main objectives of this paper are 1 to construct an euler product from a hecke eigenform of half integral weight 2 to prove its meromorphic continuation to the whole plane 3 to examine its relationship with the fourier coefficients of the form and 4 to apply these results to the investigations of eisenstein series of half integral weight the automorphic forms are holomorphic ones considered on the metaplectic cover m of g where g n sp n f with a totally real algebraic number field f the first three are generalizations of the corresponding results in the elliptic and hilbert modular cases obtained in 2 and 6 at the same time all four closely parallel similar results in the case of integral weight in 9 and s10 to describe our results let us take for simplicity f to be q in which case the weight k is half an odd positive integer we consider a holomorphic form f of weight k with respect to a principal congruence subgroup of sp n z of level n which is an eigenform in the sense that f it m 2 m f with a complex number 2 m where n is a multiple of 4 and t m is a hecke operator defined for each positive integer m in a natural way as our first main result we shall prove theorem 4 4 that

on the fundamental periods of automorphic forms of arithmetic type the automorphic forms to be investigated are defined with respect to a congruence subgroup f of an algebraic group g which consists of the invertible elements of a quaternion algebra b over a totally real algebraic number field e we have g glz r i where 6 resp 6 is the set of archimedean primes of e unramified resp ramified in b and n denotes the hamilton quaternions such a b is said to be of signature 5 6 we consider a holomorphic map h of h a into a complex vector space v such that

a sequential quadratically constrained quadratic programming method with an augmented lagrangian line search function based on an augmented lagrangian line search function a sequential quadratically constrained quadratic programming method is proposed for solving nonlinearly constrained optimization problems compared to quadratic programming solved in the traditional sqp methods a convex quadratically constrained quadratic programming is solved here to obtain a search direction and the maratos effect does not occur without any other corrections the active set strategy used in this subproblem can avoid recalculating the unnecessary gradients and approximate hessian matrices of the constraints under certain assumptions the proposed method is proved to be globally superlinearly and quadratically convergent as an extension general problems with inequality and equality constraints as well as nonmonotone line search are also considered

hyperdisk based large margin classifier we introduce a large margin linear binary classification framework that approximates each class with a hyperdisk the intersection of the affine support and the bounding hypersphere of its training samples in feature space and then finds the linear classifier that maximizes the margin separating the two hyperdisks we contrast this with support vector machines svms which find the maximum margin separator of the pointwise convex hulls of the training samples arguing that replacing convex hulls with looser convex class models such as hyperdisks provides safer margin estimates that improve the accuracy on some problems both the hyperdisks and their separators are found by solving simple quadratic programs the method is extended to nonlinear feature spaces using the kernel trick and multi class problems are dealt with by combining binary classifiers in the same ways as for svms experiments on a range of data sets show that the method compares favourably with other popular large margin classifiers

a superlinearly convergent sqp method without boundedness assumptions on any of the iterative sequences this paper is aimed to present a new sequential quadratic programming sqp algorithm for finding a solution to nonlinear constrained programming problems with weak conditions where the improved direction can be yielded by solving one quadratic programming qp and the correction direction can be obtained by solving another qp the main characters of the proposed algorithm are as follows first by limiting infeasibility of sqp iterates the boundedness of the iteration sequence can be obtained in the case of the feasible set being nonempty and bounded as well as the constraint functions being convex second global convergence can be proved under slater constraint qualification cq furthermore superlinear convergence can be ensured under suitable conditions third the proposed algorithm is further improved with a bidirectional line search technique finally some numerical experiments are operated to test the proposed algorithms and the results demonstrate that they are promising