a new exchange method for convex semi infinite programming in this paper we propose a new exchange method for solving convex semi infinite programming csip problems we introduce a new dropping rule in the proposed exchange algorithm which only keeps those active constraints with positive lagrange multipliers moreover we exploit the idea of looking for eta infeasible indices of the lower level problem as the adding rule in our algorithm hence the algorithm does not require to solve a maximization problem over the index set at each iteration it only needs to find some points such that a certain computationally easy criterion is satisfied under some reasonable conditions the new adding dropping rule guarantees that our algorithm provides an approximate optimal solution for the csip problem in a finite number of iterations in the numerical experiments we apply the proposed algorithm to solve some test problems from the literature including some medium sized problems from complex approximation theory and fir filter design we compare our algorithm with an existing central cutting plane algorithm and with the semi infinite solver fseminf in matlab toolbox and we find that our algorithm solves the csip problem much faster for the fir filter design problem we show that our algorithm solves the problem better than some algorithms that were technically established for the problem

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

a general stochastic outer approximations method optimization problems involving an infinite number of constraints are considered this paper presents a general stochastic outer approximations method which incorporates mechanisms for active search of relevant constraints and for dropping of irrelevant constraints the method extracts the characteristic features of several stochastic outer approximations algorithms suggested by wardi j optim theory appl 56 1988 pp 285 311 j optim theory appl 64 1990 pp 615 640 and furthermore develops the approach to get advantages of the eaves zangwill scheme similarly to gonzaga and polak siam j control optim 17 1979 pp 477 493 the method is based on the use of quasi optimality functions satisfying some general unrestricted assumptions these functions are usually employed in the stopping criteria of numerical techniques for solving simpler problems it is shown that the method s trajectories almost surely converge to the quasi optimal set following the proposed approach a stochastic algorithm for solving the approximation problem is constructed and studied the proposed general method can be considered as a developed eaves zangwill method applying the multistart technique at each iteration for the search of relevant constraints parameters

a gauss newton approach to solving generalized inequalities generalized inequalities are systems of the form gx k 0 where g maps between normed linear spaces and k denotes the partial order induced by the closed convex cone k e g k r m1 0 rm2 in this paper a gauss newton type algorithm is presented for minimizing the distance function rhox mbox dist gx k inf vert gx k vert k in k the technique globalizes the well known newton methods for solving generalized inequalities and overcomes the difficulties associated with subgradient methods for the global minimization of

a nonsmooth levenberg marquardt method for solving semi infinite programming problems in this paper we first transform the semi infinite programming problem into the kkt system by the techniques in d h li l qi j tam s y wu a smoothing newton method for semi infinite programming j global optim 30 2004 169 194 l qi s y wu g l zhou semismooth newton methods for solving semi infinite programming problems j global optim 27 2003 215 232 then a nonsmooth and inexact levenberg marquardt method is proposed for solving this kkt system based on h dan n yamashita m fukushima convergence properties of the inexact levenberg marquardt method under local error bound conditions optimim methods softw 11 2002 605 626 this method is globally and superlinearly even quadratically convergent finally some numerical results are given

some outer approximation methods for semi infinite optimization problems abstract the paper starts with a simple model and convergence theorem for outer approximation methods this general framework is used to unifyingly derive and modify certain exchange methods cutting methods and discretization methods for semi infinite programming problems by that in particular a cutting plane method for convex semi infinite programs is developed for a practically reasonable specification the method is more generally stated the subproblems in the given algorithm are moderately sized quadratic problems and each step of the algorithm can be performed by means of finitely many operations

approximation in normal linear spaces abstract a historical account is given of the development of methods for solving approximation problems set in normed linear spaces approximation of both real functions and real data is considered with particular reference to lp or lp and chebyshev norms as well as coverage of methods for the usual linear problems an account is given of the development of methods for approximation by functions which are nonlinear in the free parameters and special attention is paid to some particular nonlinear approximating families

a new proof of the strong duality theorem for semidefinite programming semidefinite programs are convex optimization problems arising in a wide variety of applications and are the extension of linear programming most methods for linear programming have been generalized to semidefinite programs just as in linear programming duality theorem plays a basic and an important role in theory as well as in algorithmics based on the discretization method and convergence property this paper proposes a new proof of the strong duality theorem for semidefinite programming which is different from other common proofs and is more simple

hyperbolic manifolds with convex boundary let m m be a 3 manifold which carries a hyperbolic metric with convex boundary we consider the hyperbolic metrics on m such that the boundary is smooth and strictly convex we show that the induced metrics on the boundary are exactly the metrics with curvature k 1 and that the third fundamental forms of m are exactly the metrics with curvature k 2 each is obtained exactly once

a sequential quadratic programming algorithm for nonconvex nonsmooth constrained optimization we consider optimization problems with objective and constraint functions that may be nonconvex and nonsmooth problems of this type arise in important applications many having solutions at points of nondifferentiability of the problem functions we present a line search algorithm for situations when the objective and constraint functions are locally lipschitz and continuously differentiable on open dense subsets of mathbb r n our method is based on a sequential quadratic programming sqp algorithm that uses an ell 1 penalty to regularize the constraints a process of gradient sampling gs is employed to make the search direction computation effective in nonsmooth regions we prove that our sqp gs method is globally convergent to stationary points with probability one and illustrate its performance with a matlab implementation

modifications of the first remez algorithm this paper proves the convergence of several modifications of the first remez algorithm for the solution of linear and nonlinear chebyshev approximation problems on compact b subset mathbb r s while the first remez algorithm in its original form requires the determination of the global maximum of the error function on all of b in each iteration the algorithms given here are based on its being sufficient to compute the maximum of the kth error function on a grid b k 1 where b k k geqq 0 is a prescribed sequence of finite point sets in b with density tending to zero interpreted differently some results on the discretization of chebyshev approximation problems which do not use full grids b k in b but only small subsets of them are provided the paper concludes with some numerical examples for the solution of linear multivariate problems

on a generic property of geodesic flows in this paper we are concerned with a sufficient condition for a riemannian metric on a compact simply connected manifold to have infinitely many geometrically distinct closed geodesics 1969 gromoll meyer proved in gm that for every riemannian metric on a compact simply connected manifold m there are infinitely many geometrically distinct closed geodesics if the sequence bi m f i of betti numbers of the free loop space m is unbounded for some field f using the theory of minimal models vigue poirrier sullivan proved that the rational cohomology algebra h m i q of m is generated by a single element if and only if the sequence bi m i q i of betti numbers of the free loop space m of m is bounded it is a conjecture that the same statement holds for all fields of prime characteristic partial results are due to mccleary ziller mz and halperin vigue poirrier hv now we turn to the manifolds for which the hypothesis of gromoll meyer s theorem does not hold for example spheres and projective spaces then stability properties of the closed geodesics become important a closed geodesic is hyperbolic if the linearized poincare map has no eigenvalue of norm 1 from the bumpy metrics theorem due to abraham ab and anosov an2 and from a pertubation result due to klingenberg takens kt one can conclude a c4 generic metric on a compact manifold has either a non hyperbolic closed geodesic of twist type or all closed geodesics are hyperbolic in the first case there are infinitely many geometrically distinct closed geodesics in every tubular neighborhood of the closed geodesic of twist type due to a theorem by moser mo in the second case there are infinitely many closed geodesics if m is simply connected due to results by hingston hi and the author ra1 hence it follows from these results that a c4 generic metric on a compact simply connected manifold has infinitely many geometrically distinct closed geodesics we remark that it is an open question whether there is a riemannian metric on a simply connected compact manifold all of whose closed geodesics are hyperbolic in ra2 we show that the examples of metrics on the 2

on the numerical treatment of linearly constrained semi infinite optimization problems abstract we consider the application of two primal algorithms to solve linear semi infinite programming problems depending on a real parameter combining a simplex type strategy with a feasible direction scheme we obtain a descent algorithm which enables us to manage the degeneracy of the extreme points efficiently the second algorithm runs a feasible direction method first and then switches to the purification procedure the linear programming subproblems that yield the search direction involve only a small subset of the constraints these subsets are updated at each iteration using a multi local optimization algorithm numerical test examples taken from the literature in order to compare the numerical effort with other methods show the efficiency of the proposed algorithms

discretization method for semi definite programming semi definite programs are convex optimization problems arising in a wide variety of applications and the extension of linear programming most methods for linear programming have been generalized to semi definite programs this paper discusses the discretization method in semi definite programming the convergence and the convergent rate of error between the optimal value of the semi definite programming problems and the optimal value of the discretized problems are obtained an approximately optimal division is given under certain assumptions with the significance of the convergence property the duality result in semi definite programs is proved in a simple way which is different from the other common proofs

a new exact penalty method for semi infinite programming problems in this paper we consider a class of nonlinear semi infinite optimization problems these problems involve continuous inequality constraints that need to be satisfied at every point in an infinite index set as well as conventional equality and inequality constraints by introducing a novel penalty function to penalize constraint violations we form an approximate optimization problem in which the penalty function is minimized subject to only bound constraints we then show that this penalty function is exact that is when the penalty parameter is sufficiently large any local solution of the approximate problem can be used to generate a corresponding local solution of the original problem on this basis the original problem can be solved as a sequence of approximate nonlinear programming problems we conclude the paper with some numerical results demonstrating the applicability of our approach to pid control and filter design

a constraint linearization method for nondifferentiable convex minimization this paper presents a readily implementable algorithm for solving constrained minimization problems involving possibly nonsmooth convex functions the constraints are handled as in the successive quadratic approximations methods for smooth problems an exact penalty function is employed for stepsize selection a scheme for automatic limitation of penalty growth is given global convergence of the algorithm is established as well as finite termination for piecewise linear problems numerical experience is reported

superlinearly convergent approximate newton methods for lc 1 optimization problems in the literature the proof of superlinear convergence of approximate newton or sqp methods for solving nonlinear programming problems requires twice smoothness of the objective and constraint functions sometimes the second order derivatives of those functions are required to be lipschitzian in this paper we present approximate newton or sqp methods for solving nonlinear programming problems whose objective and constraint functions have locally lipschitzian derivatives and establishq superlinear convergence of these methods under the assumption that these derivatives are semismooth this assumption is weaker than the second order differentiability the extended linear quadratic programming problem in the fully quadratic case is an example of nonlinear programming problems whose objective functions have semismooth but not smooth derivatives

geodesics on s2 and periodic points of annulus homeomorphisms we show that an area preserving homeomorphism of the open or closed annulus which has at least one periodic point must in fact have infinitely many interior periodic points a consequence is the theorem that every smooth riemannian metric ons2 with positive gaussian curvature has infinitely many distinct closed geodesics

a note on an implementation of a method for quadratic semi infinite programming for convex quadratic semi infinite programming problems afortran package is described a first coarse grid is successively refined in such a way that the solution on the foregoing grids can be used on the one hand as starting points for the subsequent grids and on the other hand to considerably reduce the number of constraints which have to be considered in the subsequent problems this enables an efficient treatment of large problems with moderate storage requirements powell s 1983 numerically stable convex quadratic programming implementation is used to solve the qp subproblems

geodesic flows with positive topological entropy twist maps and hyperbolicity we prove a perturbation lemma for the derivative of geodesic flows in high dimension this implies that a c 2 generic riemannian metric has a nontrivial hyperbolic basic set in its geodesic flow

a globally convergent algorithm with adaptively refined discretization for semi infinite optimization problems arising in engineering design although most of the algorithms that have been proposed for the solution of semi infinite optimization problems make use at each iteration of a set of local maximizers over the range of the independent parameter the question of suitably approximating such maximizers is generally left aside it has been suggested that this issue can be addressed by means of an adaptively refined discretization of the interval of variation of the independent parameter the algorithm proposed in the paper makes use of such a technique and by means of a certain memory mechanism avoids the potential lack of convergence suffered by an existing algorithm while requiring a relatively small number of gradient evaluations

a projected lagrangian algorithm for semi infinite programming a globally convergent algorithm is presented for the solution of a wide class of semi infinite programming problems the method is based on the solution of a sequence of equality constrained quadratic programming problems and usually has a second order convergence rate numerical results illustrating the method are given

exact penalty functions and stability in locally lipschitz programming in this paper we extend the theory of exact penalty functions for nonlinear programs whose objective functions and equality and inequality constraints are locally lipschitz arbitrary simple constraints are also allowed assuming a weak stability condition we show that for all sufficiently large penalty parameter values an isolated local minimum of the nonlinear program is also an isolated local minimum of the exact penalty function a tight lower bound on the parameter value is provided when certain first order sufficiency conditions are satisfied we apply these results to unify and extend some results for convex programming since several effective algorithms for solving nonlinear programs with differentiable functions rely on exact penalty functions our results provide a framework for extending these algorithms to problems with locally lipschitz functions

simple closed geodesics on most alexandrov surfaces abstract we study the existence of simple closed geodesics on most in the sense of baire category alexandrov surfaces with curvature bounded below compact and without boundary we show that it depends on both the curvature bound and the topology of the surfaces

on the mathematical foundations of nondifferentiable optimization in engineering design it is shown by example that a large class of engineering design problems can be transcribed into the form of a canonical optimization problem with inequality constraints involving mar functions such problems are commonly referred to as semi infinite optimization problems the bulk of this paper is devoted to the development of a mathematical theory for the construction of first order nondifferentiable optimization algorithms related to phase i phase ii methods of feasible directions which solve these semi infinite optimization problems the applicability of the theory is illustrated with examples that are relevant to engineering design

conditional subgradient optimization theory and applications abstract we generalize the subgradient optimization method for nondifferentiable convex programming to utilize conditional subgradients firstly we derive the new method and establish its convergence by generalizing convergence results for traditional subgradient optimization secondly we consider a particular choice of conditional subgradients obtained by projections which leads to an easily implementable modification of traditional subgradient optimization schemes to evaluate the subgradient projection method we consider its use in three applications uncapacitated facility location two person zero sum matrix games and multicommodity network flows computational experiments show that the subgradient projection method performs better than traditional subgradient optimization in some cases the difference is considerable these results suggest that our simply modification may improve subgradient optimization schemes significantly this finding is important as such schemes are very popular especially in the context of lagrangean relaxation

an approximation technique for the numerical solution of a stefan problem a nonlinear approximation technique for the numerical solution of certain free boundary problems is proposed the method is shown for a degenerate one dimensional stefan problem for this problem an error estimate which is independent of the used algorithm is derived numerical examples are discussed

a sensitivity approach to optimal spline robot trajectories abstract a robot trajectory planning problem is considered using smooth interpolating cubic splines as joint space trajectories the path is parameterized in terms of time intervals between knots a minimum time optimization problem is formulated under maximum torque and velocity constraints and is solved by means of a first order derivative type algorithm for semi infinite nonlinear programming feasible directions in the parameter space are generated using sensitivity coefficients of the active constraints numerical simulations are reported for a two link scara robot the proposed approach can be used for optimizing more general objective functions under different types of constraints

curve shortening and the topology of closed geodesics on surfaces we study flat knot types of geodesics on compact surfaces m2 for every flat knot type and any riemannian metric g we introduce a conley index associated with the curve shortening flow on the space of immersed curves on m2 we conclude existence of closed geodesics with prescribed flat knot types provided the associated conley index is nontrivial

on the extension of constrained optimization algorithms from differentiable to nondifferentiable problems this paper presents three general schemes for extending differentiable optimization algorithms to nondifferentiable problems it is shown that the armijo gradient method phase i phase ii methods of feasible directions and exact penalty function methods have conceptual analogs for problems with locally lipschitz functions and implementable analogs for problems with semismooth functions the exact penalty method has required the development of a new optimality condition

the maximum entropy method on the mean applications to linear programming and superresolution in this paper we give two different results we propose new methods to solve classical optimization problems in linear programming we also obtain precise quantitative results for the superresolution phenomenon as observed earlier by practical searchers on specific algorithms the common background of our work is the generalized moment problem which is known to be connected with linear programming and superresolution we describe the maximum entropy method on the mean that provides solution to the problem and leads to computational criteria to decide the existence of solutions or not

a one phase algorithm for semi infinite linear programming we present an algorithm for solving a large class of semi infinite linear programming problems this algorithm has several advantages it handles feasibility and optimality together it has very weak restrictions on the constraints it allows cuts that are not near the most violated cut and it solves the primal and the dual problems simultaneously we prove the convergence of this algorithm in two steps first we show that the algorithm can find ane optimal solution after finitely many iterations then we use this result to show that it can find an optimal solution in the limit we also estimate how good ane optimal solution is compared to an optimal solution and give an upper bound on the total number of iterations needed for finding ane optimal solution under some assumptions this algorithm is generalized to solve a class of nonlinear semi infinite programming problems applications to convex programming are discussed

descent methods for composite nondifferentiable optimization problems we present a framework for the development of globally defined descent algorithms for the minimization of non differentiable objective functionsf h o f withh convex within our structure the global convergence properties of the cauchy modified newton gauss newton and variable metric methods are easily established along with that of several new approaches examples illustrating the calculational techniques are provided

an implementation of a discretization method for semi infinite programming for linear semi infinite programming problems a discretization method is presented a first coarse grid is successively refined in such a way that the solution on the foregoing grids can be used on the one hand as starting points for the subsequent grids and on the other hand to considerably reduce the number of constraints which have to be considered in the subsequent problems this enables an efficient treatment of large problems with moderate storage requirements a numerically stable simplex algorithm is used to solve the lp subproblems numerical examples from bivariate chebyshev approximation are presented

exact penalty functions in proximal bundle methods for constrained convex nondifferentiable minimization this paper presents new versions of proximal bundle methods for solving convex constrained nondifferentiable minimization problems the methods employ 1 or exact penalty functions with new penalty updates that limit unnecessary penalty growth in contrast to other methods some of them are insensitive to problem function scaling global convergence of the methods is established as well as finite termination for polyhedral problems some encouraging numerical experience is reported the ideas presented may also be used in variable metric methods for smooth nonlinear programming

an aggregate subgradient method for nonsmooth and nonconvex minimization a class of implementable algorithms is described for minimizing any convex not necessarily differentiable functionf of several variables the methods require only the calculation off and one subgradient off at designated points they generalize lemarechal s bundle method more specifically instead of using all previously computed subgradients in search direction finding subproblems that are quadratic programming problems the methods use an aggregate subgradient which is recursively updated as the algorithms proceed each algorithm yields a minimizing sequence of points and iff has any minimizers then this sequence converges to a solution of the problem particular members of this algorithm class terminate whenf is piecewise linear the methods are easy to implement and have flexible storage requirements and computational effort per iteration that can be controlled by a user

relaxation based bounds for semi infinite programs finite formulations are presented for the calculation of lower and upper bounds on the optimal solution value of semi infinite programs sips involving smooth potentially nonconvex objective function and constraints the lower bounding problem is obtained by a formulation that combines the first and second order kkt necessary conditions of the lower level problem with a discretization of the parameter set the resulting mathematical program with equilibrium constraints mpec is a relaxation of the original sip and furnishes valid lower bounds if the parameter set is subdivided the optimal solution value of the lower bounding problem converges to the optimal solution value of the sip the upper bounding problem is based on convex and linear relaxations of the lower level problem resulting in a restriction of the sip if the parameter set is subdivided the constructed relaxations converge to the original lower level program the existence of sip slater points ensures convergence of the upper bounding problems to the optimal solution value of the sip several alternatives for the upper bounding problem are presented and discussed numerical results are presented for a number of test problems from the literature

a nonmonotonic trust region algorithm for a class of semi infinite minimax programming based on the discretization methods for solving semi infinite programming problems this paper presents a new nonmonotonic trust region algorithm for a class of semi infinite minimax programming problem under some mild assumptions the global convergence of the proposed algorithm is given numerical tests are reported that show the efficiency of the proposed method