a new primal dual path following interior point algorithm for semidefinite optimization abstract in this paper we present a new primal dual path following interior point algorithm for semidefinite optimization the algorithm is based on a new technique for finding the search direction and the strategy of the central path at each iteration we use only full nesterov todd step moreover we obtain the currently best known iteration bound for the algorithm with small update method namely o n log n which is as good as the linear analogue

superlinear convergence of an infeasible predictor corrector path following interior point algorithm for a semidefinite linear complementarity problem using the helmberg kojima monteiro direction an interior point method ipm defines a search direction at each interior point of a region these search directions form a direction field which in turn gives rise to a system of ordinary differential equations odes the solutions of the system of odes can be viewed as underlying paths in the interior of the region in c k sim and g zhao math program ser a 110 2007 pp 475 499 these off central paths are shown to be well defined analytic curves and any of their accumulation points is a solution to a given monotone semidefinite linear complementarity problem sdlcp the study of these paths provides a way to understand how iterates generated by an interior point algorithm behave in this paper we give a sufficient condition using these off central paths that guarantees superlinear convergence of a predictor corrector path following interior point algorithm for sdlcp using the helmberg kojima monteiro hkm direction this sufficient condition is implied by a currently known sufficient condition for superlinear convergence using this sufficient condition we show that for any linear semidefinite feasibility problem superlinear convergence using the interior point algorithm with the hkm direction can be achieved for a suitable starting point we work under the assumption of strict complementarity

logarithmic barrier decomposition based interior point methods for stochastic symmetric programming abstract we introduce and study two stage stochastic symmetric programs with recourse to handle uncertainty in data defining deterministic symmetric programs in which a linear function is minimized over the intersection of an affine set and a symmetric cone we present a benders decomposition based interior point algorithm for solving these problems and prove its polynomial complexity our convergence analysis proved by showing that the log barrier associated with the recourse function of stochastic symmetric programs behaves a strongly self concordant barrier and forms a self concordant family on the first stage solutions since our analysis applies to all symmetric cones this algorithm extends zhao s results g zhao a log barrier method with benders decomposition for solving two stage stochastic linear programs math program ser a 90 2001 507 536 for two stage stochastic linear programs and mehrotra and ozevin s results s mehrotra m g ozevin decomposition based interior point methods for two stage stochastic semidefinite programming siam j optim 18 1 2007 206 222 for two stage stochastic semidefinite programs

a log barrier method with benders decomposition for solving two stage stochastic linear programs an algorithm incorporating the logarithmic barrier into the benders decomposition technique is proposed for solving two stage stochastic programs basic properties concerning the existence and uniqueness of the solution and the underlying path are studied when applied to problems with a finite number of scenarios the algorithm is shown to converge globally and to run in polynomial time

min max and min max regret versions of combinatorial optimization problems a survey abstract min max and min max regret criteria are commonly used to define robust solutions after motivating the use of these criteria we present general results then we survey complexity results for the min max and min max regret versions of some combinatorial optimization problems shortest path spanning tree assignment min cut min s t cut knapsack since most of these problems are np hard we also investigate the approximability of these problems furthermore we present algorithms to solve these problems to optimality

product form cholesky factorization in interior point methods for second order cone programming second order cone programming socp problems are typically solved by interior point methods as in linear programming lp interior point methods can in theory solve socps in polynomial time and can in practice exploit sparsity in the problem data specifically when cones of large dimension are present the density that results in the normal equations that are solved at each iteration can be remedied in a manner similar to the treatment of dense columns in an lp here we propose a product form cholesky factorization pfcf approach and show that it is more numerically stable than the alternative sherman morrison woodbury approach we derive several pfcf variants and compare their theoretical perfomance finally we prove that the elements of l in the cholesky factorizations ldlt that arise in interior point methods for socp are uniformly bounded as the duality gap tends to zero as long as the iterates remain is some conic neighborhood of the cental path

the price of robustness a robust approach to solving linear optimization problems with uncertain data was proposed in the early 1970s and has recently been extensively studied and extended under this approach we are willing to accept a suboptimal solution for the nominal values of the data in order to ensure that the solution remains feasible and near optimal when the data changes a concern with such an approach is that it might be too conservative in this paper we propose an approach that attempts to make this trade off more attractive that is we investigate ways to decrease what we call the price of robustness in particular we flexibly adjust the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations an attractive aspect of our method is that the new robust formulation is also a linear optimization problem thus we naturally extend our methods to discrete optimization problems in a tractable way we report numerical results for a portfolio optimization problem a knapsack problem and a problem from the net lib library

extension of primal dual interior point algorithms to symmetric cones in this paper we show that the so called commutative class of primal dual interior point algorithms which were designed by monteiro and zhang for semidefinite programming extends word for word to optimization problems over all symmetric cones the machinery of euclidean jordan algebras is used to carry out this extension unlike some non commutative algorithms such as the xs sx method this class of extensions does not use concepts outside of the euclidean jordan algebras in particular no assumption is made about representability of the underlying jordan algebra as a special case we prove polynomial iteration complexities for variants of the short semi long and long step path following algorithms using the nesterov todd xs or sx directions

on the implementation of interior point decomposition algorithms for two stage stochastic conic programs in this paper we develop a practical primal interior decomposition algorithm for two stage stochastic programming problems the framework of this algorithm is similar to the framework in s mehrotra and m g ozevin decomposition based interior point methods for two stage stochastic convex quadratic programs with recourse to appear in oper res s mehrotra and m g ozevin siam j optim 18 2007 pp 206 222 and g zhao math program 90 2001 pp 507 536 however their algorithm is altered in a simple yet fundamental way to achieve practical performance in particular this new algorithm weighs the log barrier terms in the second stage problems differently from the theoretical algorithms analyzed in s mehrotra and m g ozevin oper res to appear s mehrotra and m g ozevin siam j optim 18 2007 pp 206 222 and g zhao math program 90 2001 pp 507 536 we give a method for generating a suitable starting point a method for selecting a good starting barrier parameter a heuristic for first stage step length calculation without performing line searches and a method for adaptive addition of new scenarios over the course of the algorithm the decomposition algorithm is implemented to solve two stage stochastic conic programs with recourse whose underlying cones are cartesian products of linear second order and semidefinite cones the performance of primal decomposition method is studied on a set of randomly generated test problems as well as a two stage stochastic programming extension of the markowitz portfolio selection model the computational results show that an efficient and stable implementation of the primal decomposition method is possible these results also show that in problems with a large number of scenarios the adaptive addition of scenarios can yield computational savings of up to 80

primal dual path following algorithms for semidefinite programming this paper deals with a class of primal dual interior point algorithms for semidefinite programming sdp which was recently introduced by kojima shindoh and hara siam j optim 7 1997 pp 86 125 these authors proposed a family of primal dual search directions that generalizes the one used in algorithms for linear programming based on the scaling matrix x1 2s 1 2 they study three primal dual algorithms based on this family of search directions a short step path following method a feasible potential reduction method and an infeasible potential reduction method however they were not able to provide an algorithm which generalizes the long step path following algorithm introduced by kojima mizuno and yoshise progress in mathematical programming interior point and related methods n megiddor ed springer verlag berlin new york 1989 pp 29 47 in this paper we characterize two search directions within their family as being unique solutions of systems of linear equations in symmetric variables based on this characterization we present a simplified polynomial convergence proof for one of their short step path following algorithms and for the first time a polynomially convergent long step path following algorithm for sdp which requires an extra sqrt n factor in its iteration complexity order as compared to its linear programming counterpart where n is the number of rows or columns of the matrices involved

a topological characterization of the ω limit sets for analytic flows on the plane the sphere and the projective plane in this paper the limit sets for analytic and polynomial differential equations on the plane are characterized up to homeomorphisms the analogous problem is solved in full detail for analytic flows on the sphere and the projective plane we also explain how to carry on the same program for analytic flows defined on open subsets of these surfaces

attractors and symmetries of vector fields the inverse problem abstract the inverse problem of constructing dynamical systems x with a prescribed compact attracting set k r n and non trivial dynamics on it is solved the boundaries of the attracting basins corresponding to different connected components of k and the relationship between the symmetries of k x and are also studied

adaptive selection of face coarse degrees of freedom in the bddc and the feti dp iterative substructuring methods we propose a class of method for the adaptive selection of the coarse space of the bddc and feti dp iterative substructuring methods the methods work by adding coarse degrees of freedom constructed from eigenvectors associated with intersections of selected pairs of adjacent substructures it is assumed that the starting coarse degrees of freedom are already sufficientto prevent relative rigid body motions in any selected pair of adjacent substructures a heuristic indicator of the the condition number is developed and a minimal number of coarse degrees of freedom is added to decrease the indicator under a given threshold it is shown numerically on 2d elasticity problems that the indicator based on pairs of substructures with common edges predicts the actual condition number reasonably well and that the method can select adaptively the hard part of the problem and concentrate computational work there to achieve good convergence of the iterations at a modest cost

interior point trajectories in semidefinite programming in this paper we study interior point trajectories in semidefinite programming sdp including the central path of an sdp this work was inspired by the seminal work of megiddo on linear programming trajectories progress in math programming interior point algorithms and related methods n megiddo ed springer verlag berlin 1989 pp 131 158 under an assumption of primal and dual strict feasibility we show that the primal and dual central paths exist and converge to the analytic centers of the optimal faces of respectively the primal and the dual problems we consider a class of trajectories that are similar to the central path but can be constructed to pass through any given interior feasible or infeasible point and study their convergence finally we study the derivatives of these trajectories and their convergence

a dwindling filter inexact projected hessian algorithm for large scale nonlinear constrained optimization in this paper we propose a dwindling filter inexact projected hessian algorithm for solving large scale nonlinear constrained optimization for large scale applications inexact projected hessian algorithm is needed to get search direction by solving one or more linear systems approximately using iterative linear algebra techniques the envelope of the dwindling filter becomes thinner and thinner as the step size approaches zero so that the new filter has more flexibility for the acceptance of the trial step compared with traditional filter under mild conditions global convergence and local superlinear convergence rate are obtained the numerical experiments are reported to show the effectiveness of the proposed algorithm for large scale problems

on the set of wild points of attracting surfaces in r3 abstract suppose that a closed surface s r 3 is an attractor not necessarily global for a discrete dynamical system assuming that its set of wild points w is totally disconnected we prove that up to an ambient homeomorphism it has to be contained in a straight line as a corollary we show that there exist uncountably many different 2 spheres in r 3 none of which can be realized as an attractor for a homeomorphism our techniques hinge on a quantity r k that can be defined for any compact set k r 3 and is related to how wildly it sits in r 3 we establish the topological results that i r w r s and ii any totally disconnected set having a finite r must be contained in a straight line up to an ambient homeomorphism the main result follows from these and the fact that attractors have a finite r

robust submodular observation selection in many applications one has to actively select among a set of expensive observations before making an informed decision for example in environmental monitoring we want to select locations to measure in order to most effectively predict spatial phenomena often we want to select observations which are robust against a number of possible objective functions examples include minimizing the maximum posterior variance in gaussian process regression robust experimental design and sensor placement for outbreak detection in this paper we present the submodular saturation algorithm a simple and efficient algorithm with strong theoretical approximation guarantees for cases where the possible objective functions exhibit submodularity an intuitive diminishing returns property moreover we prove that better approximation algorithms do not exist unless np complete problems admit efficient algorithms we show how our algorithm can be extended to handle complex cost functions incorporating non unit observation cost or communication and path costs we also show how the algorithm can be used to near optimally trade off expected case e g the mean square prediction error in gaussian process regression and worst case e g maximum predictive variance performance we show that many important machine learning problems fit our robust submodular observation selection formalism and provide extensive empirical evaluation on several real world problems for gaussian process regression our algorithm compares favorably with state of the art heuristics described in the geostatistics literature while being simpler faster and providing theoretical guarantees for robust experimental design our algorithm performs favorably compared to sdp based algorithms c 2008 andreas krause h brendan mcmahan carlos guestrin and anupam gupta krause mcmahan guestrin and gupta

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

asymptotical periodicity for analytic triangular maps of type less than 2 abstract we prove that if f is an analytic triangular map of type less than 2 in the sharkovsky ordering then all points are asymptotically periodic for f the same is true if instead of being analytic f is just continuous but has the property that each fibre contains finitely many periodic points improving earlier counterexamples in kolyada 1992 16 and balibrea et al 2002 3 we also show that this need not be the case when f is a c map finally we remark that type less than 2 and closedness of periodic points are equivalent properties in the c 1 setting for triangular maps

on extending some primal dual interior point algorithms from linear programming to semidefinite programming this work concerns primal dual interior point methods for semidefinite programming sdp that use a search direction originally proposed by helmberg et al siam j optim 6 1996 pp 342 361 and kojima shindoh and hara siam j optim 7 1997 pp 86 125 and recently rediscovered by monteiro siam j optim 7 1997 pp 663 678 in a more explicit form in analyzing these methods a number of basic equalities and inequalities were developed in kojima shindoh and hara and also in monteiro through different means and in different forms in this paper we give a concise derivation of the key equalities and inequalities for complexity analysis along the exact line used in linear programming lp producing basic relationships that have compact forms almost identical to their counterparts in lp we also introduce a new formulation of the central path and variable metric measures of centrality these results provide convenient tools for deriving polynomiality results for primal dual algorithms extended from lp to sdp using the aforementioned and related search directions we present examples of such extensions including the long step infeasible interior point algorithm of zhang siam j optim 4 1994 pp 208 227

decomposition based interior point methods for stochastic quadratic second order cone programming zhao 2001 introduced a logarithmic barrier interior point method for solving two stage stochastic linear programs with recourse using benders decomposition this decomposition algorithm and its analysis have been extended by alzalg and ariyawansa 2014 for solving two stage stochastic programs with recourse over all symmetric cones in this paper we utilize their work to derive a logarithmic barrier decomposition based interior point algorithm for two stage stochastic convex quadratic second order cone programs with recourse

interior point methods for the monotone semidefinite linear complementarity problem in symmetric matrices the sdlcp semidefinite linear complementarity problem in symmetric matrices introduced in this paper provides a unified mathematical model for various problems arising from systems and control theory and combinatorial optimization it is defined as the problem of finding a pair x y of n times n symmetric positive semidefinite matrices which lies in a given n n 1 2 dimensional affine subspace fc of sc 2 and satisfies the complementarity condition x bullet y 0 where sc denotes the n n 1 2 dimensional linear space of symmetric matrices and x bullet y the inner product of x and y the problem enjoys a close analogy with the lcp in the euclidean space in particular the central trajectory leading to a solution of the problem exists under the nonemptiness of the interior of the feasible region and a monotonicity assumption on the affine subspace fc the aim of this paper is to establish a theoretical basis of interior point methods with the use of newton directions toward the central trajectory for the monotone sdlcp

note on a paper of j llibre and g rodriguez concerning algebraic limit cycles abstract in a recent paper of llibre and rodriguez j differential equations 198 2004 374 380 it is proved that every configuration of cycles in the plane is realizable up to homeomorphism by a polynomial vector field of degree at most 2 n r 1 where n is the number of cycles and r the number of primary cycles a cycle c is primary if there are no other cycles contained in the bounded region limited by c in this letter we prove the same theorem by using an easier construction but with a greater polynomial bound the vector field we construct has degree at most 4 n 1 by using the same technique we also construct r 3 polynomial vector fields realizing up to homeomorphism any configuration of limit cycles which can be linked and knotted in r 3 this answers a question of r sverdlove

robust discrete optimization and network flows we propose an approach to address data uncertainty for discrete optimization and network flow problems that allows controlling the degree of conservatism of the solution and is computationally tractable both practically and theoretically in particular when both the cost coefficients and the data in the constraints of an integer programming problem are subject to uncertainty we propose a robust integer programming problem of moderately larger size that allows controlling the degree of conservatism of the solution in terms of probabilistic bounds on constraint violation when only the cost coefficients are subject to uncertainty and the problem is a 0 1 discrete optimization problem on n variables then we solve the robust counterpart by solving at most n 1 instances of the original problem thus the robust counterpart of a polynomially solvable 0 1 discrete optimization problem remains polynomially solvable in particular robust matching spanning tree shortest path matroid intersection etc are polynomially solvable we also show that the robust counterpart of an np hard approximable 0 1 discrete optimization problem remains approximable finally we propose an algorithm for robust network flows that solves the robust counterpart by solving a polynomial number of nominal minimum cost flow problems in a modified network

second order cone programming second order cone programming socp problems are convex optimization problems in which a linear function is minimized over the intersection of an affine linear manifold with the cartesian product of second order lorentz cones linear programs convex quadratic programs and quadratically constrained convex quadratic programs can all be formulated as socp problems as can many other problems that do not fall into these three categories these latter problems model applications from a broad range of fields from engineering control and finance to robust optimization and combinatorial optimization on the other hand semidefinite programming sdp that is the optimization problem over the intersection of an affine set and the cone of positive semidefinite matrices includes socp as a special case therefore socp falls between linear lp and quadratic qp programming and sdp like lp qp and sdp problems socp problems can be solved in polynomial time by interior point methods the computational effort per iteration required by these methods to solve socp problems is greater than that required to solve lp and qp problems but less than that required to solve sdp s of similar size and structure because the set of feasible solutions for an socp problem is not polyhedral as it is for lp and qp problems it is not readily apparent how to develop a simplex or simplex like method for socp while socp problems can be solved as sdp problems doing so is not advisable both on numerical grounds and computational complexity concerns for instance many of the problems presented in the survey paper of vandenberghe and boyd vb96 as examples of sdps can in fact be formulated as socps and should be solved as such in 2 3 below we give socp formulations for four of these examples the convex quadratically constrained quadratic programming qcqp problem problems involving fractional quadratic functions rutcor rutgers university e mail alizadeh rutcor rutgers edu research supported in part by the u s national science foundation grant ccr 9901991 ieor columbia university e mail gold ieor columbia edu research supported in part by the department of energy grant de fg02 92er25126 national science foundation grants dms 94 14438 cda 97 26385 and dms 01 04282

a class of volumetric barrier decomposition algorithms for stochastic quadratic programming abstract ariyawansa and zhu have introduced a class of volumetric barrier decomposition algorithms k a ariyawansa y zhu a class of polynomial volumetric barrier decomposition algorithms for stochastic semidefinite programming available as technical report 2006 7 department of mathematics washington state university pullman wa submitted for publication available from for solving two stage stochastic semidefinite programs with recourse ssdps k a ariyawansa y zhu stochastic semidefinite programming a new paradigm for stochastic optimization 4or the quarterly journal of the belgian french and italian or societies in press available as technical report 2004 10 department of mathematics washington state university pullman wa october 2004 available from in this paper we utilize their work for ssdps to derive a class of volumetric barrier decomposition algorithms for solving two stage stochastic quadratic programs with recourse and to establish polynomial complexity of certain members of the class of algorithms

an interior point method for semidefinite programming we propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices

solving some large scale semidefinite programs via the conjugate residual method most current implementations of interior point methods for semidefinite programming use a direct method to solve the schur complement equation sce m delta y h in computing the search direction when the number of constraints is large the problem of having insufficient memory to store m can be avoided if an iterative method is used instead numerical experiments have shown that the conjugate residual cr method typically takes a huge number of steps to generate a high accuracy solution on the other hand it is difficult to incorporate traditional preconditioners into the sce except for block diagonal preconditioners we decompose the sce into a 2 2 block system by decomposing delta y similarly for h into two orthogonal components with one lying in a certain subspace that is determined from the structure of m numerical experiments on semidefinite programming problems arising from the lovasz theta function of graphs and maxcut problems show that high accuracy solutions can be obtained with a moderate number of cr steps using the proposed equation

locating leak detecting sensors in a water distribution network by solving prize collecting steiner arborescence problems we consider the problem of optimizing a novel acoustic leakage detection system for urban water distribution networks the system is composed of a number of detectors and transponders to be placed in a choice of hydrants such as to provide a desired coverage under given budget restrictions the problem is modeled as a particular prize collecting steiner arborescence problem we present a branch and cut and bound approach taking advantage of the special structure at hand which performs well when compared to other approaches furthermore using a suitable stopping criterion we obtain approximations of provably excellent quality in most cases actually optimal solutions the test bed includes the real water distribution network from the lausanne region as well as carefully randomly generated realistic instances

decomposition based interior point methods for two stage stochastic convex quadratic programs with recourse zhao showed that the log barrier associated with the recourse function of two stage stochastic linear programs behaves as a strongly self concordant barrier and forms a self concordant family on the first stage solutions in this paper we show that the recourse function is also strongly self concordant and forms a self concordant family for the two stage stochastic convex quadratic programs with recourse this allows us to develop bender s decomposition based linearly convergent interior point algorithms an analysis of such an algorithm is given in this paper

on the markus neumann theorem abstract a well known result by l markus 6 later extended by d a neumann 7 states that two continuous flows on a surface are equivalent if and only if there is a surface homeomorphism preserving orbits and time directions of their separatrix configurations in this paper we present several examples showing that as originally formulated the markus neumann theorem needs not work besides we point out the gap in its proof and show how to restate it in a correct and slightly more general way

a characterization of theω limit sets of planar continuous dynamical systems abstract if the family of curves g i i and the set of points s are given we find necessary and sufficient conditions for the existence of a c k map f from the plane into itself 0 k if k 0 we also assume that f is locally lipschitz such that the curves from g and the points from s are respectively nondegenerate trajectories and singular points of the system x f x and additionally s i i is the limit set of some trajectory of the system in such cases we also provide a detailed geometric description of s i i after contraction of the connected components of s

realization problems for limit cycles of planar polynomial vector fields abstract we show that for any finite configuration of closed curves r 2 one can construct an explicit planar polynomial vector field that realizes up to homeomorphism as the set of its limit cycles with prescribed periods multiplicities and stabilities the only obstruction given on this data is the obvious compatibility relation between the stabilities and the parity of the multiplicities the constructed vector fields are darboux integrable and admit a polynomial inverse integrating factor

global optimization of mixed integer nonlinear programs a theoretical and computational study this work addresses the development of an efficient solution strategy for obtaining global optima of continuous integer and mixed integer nonlinear programs towards this end we develop novel relaxation schemes range reduction tests and branching strategies which we incorporate into the prototypical branch and bound algorithm in the theoretical algorithmic part of the paper we begin by developing novel strategies for constructing linear relaxations of mixed integer nonlinear programs and prove that these relaxations enjoy quadratic convergence properties we then use lagrangian linear programming duality to develop a unifying theory of domain reduction strategies as a consequence of which we derive many range reduction strategies currently used in nonlinear programming and integer linear programming this theory leads to new range reduction schemes including a learning heuristic that improves initial branching decisions by relaying data across siblings in a branch and bound tree finally we incorporate these relaxation and reduction strategies in a branch and bound algorithm that incorporates branching strategies that guarantee finiteness for certain classes of continuous global optimization problems in the computational part of the paper we describe our implementation discussing wherever appropriate the use of suitable data structures and associated algorithms we present computational experience with benchmark separable concave quadratic programs fractional 0 1 programs and mixed integer nonlinear programs from applications in synthesis of chemical processes engineering design just in time manufacturing and molecular design

an improved prediction correction method for monotone variational inequalities with separable operators in this paper we study the variational inequality problems with a particular splitting structure in which the mapping f does not have an explicit form and only its function values can be employed in the numerical methods for solving such problems studies and applications of such problems can be found in fukushima 1992 3 glowinski 1984 4 glowinski 1989 5 and xu 2007 15 the paper he et al 2006 6 presents an efficient prediction correction method for such problems based on the predictor method in the latter this paper presents two classes of correction methods which are more convenient to be carried out than that in this reference numerical experiments show that the new method is effective

self concordance and decomposition based interior point methods for the two stage stochastic convex optimization problem we study the two stage stochastic convex optimization problem whose first and second stage feasible regions admit a self concordant barrier we show that the barrier recourse functions and the composite barrier functions for this problem form self concordant families these results are used to develop prototype primal interior point decomposition algorithms that are more suitable for a heterogeneous distributed computing environment we show that the worst case iteration complexity of the proposed algorithms is the same as that for the short and long step primal interior algorithms applied to the extensive formulation of this problem the generality of our results allows the possibility of using barriers other than the standard log barrier in an algorithmic framework

decomposition based interior point methods for two stage stochastic semidefinite programming we introduce two stage stochastic semidefinite programs with recourse and present an interior point algorithm for solving these problems using bender s decomposition this decomposition algorithm and its analysis extend zhao s results math program 90 2001 pp 507 536 for stochastic linear programs the convergence results are proved by showing that the logarithmic barrier associated with the recourse function of two stage stochastic semidefinite programs with recourse is a strongly self concordant barrier on the first stage solutions the short step variant of the algorithm requires o sqrt p kr ln mu 0 epsilon newton iterations to follow the first stage central path from a starting value of the barrier parameter mu 0 to a terminating value e the long step variant requires o p kr ln mu 0 epsilon damped newton iterations the calculation of the gradient and hessian of the recourse function and the first stage newton direction decomposes across the second stage scenarios

configurations of limit cycles and planar polynomial vector fields abstract we show that every finite configuration of disjoint simple closed curves of the plane is topologically realizable as the set of limit cycles of a polynomial vector field moreover the realization can be made by algebraic limit cycles and we provide an explicit polynomial vector field exhibiting any given finite configuration of limit cycles

a feasible primaldual interior point method for linear semidefinite programming in this paper we consider a feasible primaldual interior point method for linear semidefinite programming problem sdp based on alizadehhaeberlyoverton aho direction monteiro 1997 firstly and by a new and simple technique we establish the existence and uniqueness of optimal solution of the perturbed problem sdp and its convergence to optimal solution of sdp next we present new different alternatives to calculate the displacement step after we establish the convergence of the obtained algorithm and we show that its complexity is o nln 1 x0 s0 finally we present some numerical simulations which show the effectiveness of the algorithm developed in this work

robust optimization of contaminant sensor placement for community water systems we present a series of related robust optimization models for placing sensors in municipal water networks to detect contaminants that are maliciously or accidentally injected we formulate sensor placement problems as mixed integer programs for which the objective coefficients are not known with certainty we consider a restricted absolute robustness criteria that is motivated by natural restrictions on the uncertain data and we define three robust optimization models that differ in how the coefficients in the objective vary under one set of assumptions there exists a sensor placement that is optimal for all admissible realizations of the coefficients under other assumptions we can apply sorting to solve each worst case realization efficiently or we can apply duality to integrate the worst case outcome and have one integer program the most difficult case is where the objective parameters are bilinear and we prove its complexity is np hard even under simplifying assumptions we consider a relaxation that provides an approximation giving an overall guarantee of near optimality when used with branch and bound search we present preliminary computational experiments that illustrate the computational complexity of solving these robust formulations on sensor placement applications

volumetric barrier decomposition algorithms for stochastic quadratic second order cone programming ariyawansa and zhu 2011 have derived volumetric barrier decomposition algorithms for solving two stage stochastic semidefinite programs and proved polynomial complexity of certain members of the algorithms in this paper we utilize their work to derive volumetric barrier decomposition algorithms for solving two stage stochastic convex quadratic second order cone programming and establish polynomial complexity of certain members of the proposed algorithms