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

primal dual interior point methods for semidefinite programming convergence rates stability and numerical results primal dual interior point path following methods for semidefinite programming are considered several variants are discussed based on newton s method applied to three equations primal feasibility dual feasibility and some form of centering condition the focus is on three such algorithms called the xz xz zx and q methods for the xz zx and q algorithms the newton system is well defined and its jacobian is nonsingular at the solution under nondegeneracy assumptions the associated schur complement matrix has an unbounded condition number on the central path under the nondegeneracy assumptions and an additional rank assumption practical aspects are discussed including mehrotra predictor corrector variants and issues of numerical stability compared to the other methods considered the xz zx method is more robust with respect to its ability to step close to the boundary converges more rapidly and achieves higher accuracy

polynomial convergence of primal dual algorithms for semidefinite programming based on the monteiro and zhang family of directions this paper establishes the polynomial convergence of the class of primal dual feasible interior point algorithms for semidefinite programming sdp based on the monteiro and zhang family of search directions in contrast to monteiro and zhang s work math programming 81 1998 pp 281 299 here no condition is imposed on the scaling matrix that determines the search direction we show that the polynomial iteration complexity bounds of two well known algorithms for linear programming namely the short step path following algorithm of kojima mizuno and yoshise math programming 44 1989 pp 1 26 and monteiro and adler math programming 44 1989 pp 27 41 and pp 43 66 and the predictor corrector algorithm of mizuno todd and ye math oper res 18 1993 pp 945 981 carry over into the context of sdp since the monteiro and zhang family of directions includes the alizadeh haeberly and overton direction we establish for the first time the polynomial convergence of algorithms based on this search direction

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

an inverse problem for a parabolic variational inequality with an integro differential operator we consider the calibration of a levy process with american vanilla options the price of an american vanilla option as a function of the maturity and the strike satisfies a forward in time linear complementarity problem involving a partial integro differential operator it leads to a variational inequality in a suitable weighted sobolev space calibrating the levy process amounts to solving an inverse problem where the state variable satisfies the previously mentioned variational inequality we propose a regularized least square method after studying the variational inequality carefully we find necessary optimality conditions for the least square problem in this work we focus on the case when the volatility is bounded away from zero

quantitative stability analysis of stochastic quasi variational inequality problems and applications we consider a parametric stochastic quasi variational inequality problem sqvip for short where the underlying normal cone is defined over the solution set of a parametric stochastic cone system we investigate the impact of variation of the probability measure and the parameter on the solution of the sqvip by reformulating the sqvip as a natural equation and treating the orthogonal projection over the solution set of the parametric stochastic cone system as an optimization problem we effectively convert stability of the sqvip into that of a one stage stochastic program with stochastic cone constraints under some moderate conditions we derive holder outer semicontinuity and continuity of the solution set against the variation of the probability measure and the parameter the stability results are applied to a mathematical program with stochastic semidefinite constraints and a mathematical program with sqvip constraints

associative and jordan algebras and polynomial time interior point algorithms for symmetric cones we present a general framework whereby analysis of interior point algorithms for semidefinite programming can be extended verbatim to optimization problems over all classes of symmetric cones derivable from associative algebras in particular such analyses are extendible to the cone of positive semidefinite hermitian matrices with complex and quaternion entries and to the lorentz cone we prove the case of the lorentz cone by using the embedding of its associated jordan algebra in the clifford algebra as an example of such extensions we take monterio s polynomial time complexity analysis of the family of similarly scaled directions introduced by monteiro and zhang 1998 and generalize it to cone lp over all representable symmetric cones

cartesian p property and its applications to the semidefinite linear complementarity problem we introduce a cartesian p property for linear transformations between the space of symmetric matrices and present its applications to the semidefinite linear complementarity problem sdlcp with this cartesian p property we show that the sdlcp has gus property i e globally unique solvability and the solution map of the sdlcp is locally lipschitzian with respect to input data our cartesian p property strengthens the corresponding p properties of gowda and song 15 and allows us to extend several numerical approaches for monotone sdlcps to solve more general sdlcps namely sdlcps with the cartesian p property in particular we address important theoretical issues encountered in those numerical approaches such as issues related to the stationary points in the merit function approach and the existence of newton directions and boundedness of iterates in the non interior continuation method of chen and tseng 6

analysis of infeasible interior point paths arising with semidefinite linear complementarity problems we consider semidefinite monotone linear complementarity problems sdlcp in the space equation here n of real symmetric n n matrices equipped with the cone equation here n of all symmetric positive semidefinite matrices one may define weighted using any m equation here n as weight infeasible interior point paths by replacing the standard condition xy ri r 0 that defines the usual central path by xy yx 2 rm under some mild assumptions the most stringent is the existence of some strictly complementary solution of sdlcp these paths have a limit as r 0 and they depend analytically on all path parameters such as r and m even at the limit point r 0

a potential reduction newton method for constrained equations extending our previous work t wang r d c monteiro and j s pang math programming 74 1996 pp 159 195 this paper presents a general potential reduction newton method for solving a constrained system of nonlinear equations a major convergence result for the method is established specializations of the method to a convex semidefinite program and a monotone complementarity problem in symmetric matrices are discussed strengthened convergence results are established in the context of these specializations

bandgap optimization of two dimensional photonic crystals using semidefinite programming and subspace methods in this paper we consider the optimal design of photonic crystal structures for two dimensional square lattices the mathematical formulation of the bandgap optimization problem leads to an infinite dimensional hermitian eigenvalue optimization problem parametrized by the dielectric material and the wave vector to make the problem tractable the original eigenvalue problem is discretized using the finite element method into a series of finite dimensional eigenvalue problems for multiple values of the wave vector parameter the resulting optimization problem is large scale and non convex with low regularity and non differentiable objective by restricting to appropriate eigenspaces we reduce the large scale non convex optimization problem via reparametrization to a sequence of small scale convex semidefinite programs sdps for which modern sdp solvers can be efficiently applied numerical results are presented for both transverse magnetic tm and transverse electric te polarizations at several frequency bands the optimized structures exhibit patterns which go far beyond typical physical intuition on periodic media design

on the p 2 and p 2 properties in the semidefinite linear complementarity problem abstract motivated by the so called p 2 property in the semidefinite linear complementarity problems in this article we introduce the concept of p 2 property for a linear transformation on the space of real n n symmetric matrices while these two properties turn out to be different we show that they are equivalent for the lyapunov transformation l a double sided multiplicative transformation m a and a particular class of stein transformations we also show that p 2 implies the ssm and q properties

a note on the existence of the alizadeh haeberly overton direction for semidefinite programming this note establishes a new sufficient condition for the existence and uniqueness of the alizadeh haeberly overton direction for semidefinite programming

on complexity analysis of the primal dual interior point method for semidefinite optimization problem based on a new proximity function abstract the purpose of this paper is to obtain new complexity results for a semidefinite optimization sdo problem we define a new proximity function for the sdo by a new kernel function furthermore we formulate an algorithm for a large update primal dual interior point method ipm for the sdo by using the proximity function and give its complexity analysis and then we show that the worst case iteration bound for our ipm is o n log n q 1 q log n where q 1

on the superlinear and quadratic convergence of primal dual interior point linear programming algorithms this paper presents a convergence rate analysis for interior point primal dual linear programming algorithms conditions that guarantee q superlinear convergence are identified in two distinct theories both state that under appropriate assumptions q superlinear convergence is achieved by asymptotically taking the step to the boundary of the positive orthant and letting the barrier parameter approach zero at a rate that is superlinearly faster than the convergence of the duality gap to zero the first theory makes no nondegeneracy assumption and explains why in recent numerical experimentation q superlinear convergence was always observed the second theory requires the restrictive assumption of primal nondegeneracy however it gives the surprising result that q superlinear convergence can still be attained even if centering is not phased out provided the iterates asymptotically approach the central path the latter theory is extended to produce a satisfactory q quadratic convergence theory it requir

strengthened existence and uniqueness conditions for search directions in semidefinite programming abstract primal dual interior point p d i p methods for semidefinite programming sdp are generally based on solving a system of matrix equations for a newton type search direction for a symmetrization of the optimality conditions these search directions followed the derivation of similar p d i p methods for linear programming lp among these a computationally interesting search direction is the aho direction however in contrast to the lp case existence and uniqueness of the aho search direction is not guaranteed under the standard nondegeneracy assumptions two different sufficient conditions are known that guarantee the existence and uniqueness independent of the specific linear constraints the first is given by shida shindoh kojima and is based on the semidefiniteness of the symmetrization of the product sx at the current iterate the second is a centrality condition given first by monteiro zanjacomo and then improved by monteiro todd in this paper we revisit and strengthen both of the above mentioned sufficient conditions we include characterizations for existence and uniqueness in the matrix equations arising from the linearization of the optimality conditions as well we present new results on the relationship between the kronecker product and the symmetric kronecker product that arise from these matrix equations we conclude with a proof that the existence and uniqueness of the aho direction is a generic property for every sdp problem and extend the results to the general monteiro zhang family of search directions

first and second order methods for semidefinite programming in this paper we survey the most recent methods that have been developed for the solution of semidefinite programs we first concentrate on the methods that have been primarily motivated by the interior point ip algorithms for linear programming putting special emphasis in the class of primal dual path following algorithms we also survey methods that have been developed for solving large scale sdp problems these include first order nonlinear programming nlp methods and more specialized path following ip methods which use the preconditioned conjugate gradient or residual scheme to compute the newton direction and the notion of matrix completion to exploit data sparsity

avoiding numerical cancellation in the interior point method for solving semidefinite programs the matrix variables in a primal dual pair of semidefinite programs are getting increasingly ill conditioned as they approach a complementary solution multiplying the primal matrix variable with a vector from the eigenspace of the non basic part will therefore result in heavy numerical cancellation this effect is amplified by the scaling operation in interior point methods a complete example illustrates these numerical issues in order to avoid numerical problems in interior point methods we propose to maintain the matrix variables in a cholesky form we discuss how the factors of the v space cholesky form can be updated after a main iteration of the interior point method with nesterov todd scaling an analogue for second order cone programming is also developed numerical results demonstrate the success of this approach

optimality conditions for nonconvex semidefinite programming this paper concerns nonlinear semidefinite programming problems for which no convexity assumptions can be made we derive first and second order optimality conditions analogous to those for nonlinear programming using techniques similar to those used in nonlinear programming we extend existing theory to cover situations where the constraint matrix is structurally sparse the discussion covers the case when strict complementarity does not hold the regularity conditions used are consistent with those of nonlinear programming in the sense that the conventional optimality conditions for nonlinear programming are obtained when the constraint matrix is diagonal

a superlinearly convergent primal dual infeasible interior point algorithm for semidefinite programming a primal dual infeasible interior point path following algorithm is proposed for solving semidefinite programming sdp problems if the problem has a solution then the algorithm is globally convergent if the starting point is feasible or close to being feasible the algorithm finds an optimal solution in at most o sqrt n l iterations where n is the size of the problem and l is the logarithm of the ratio of the initial error and the tolerance if the starting point is large enough then the algorithm terminates in at most o nl steps either by finding a solution or by determining that the primal dual problem has no solution of norm less than a given number moreover we propose a sufficient condition for the superlinear convergence of the algorithm in addition we give two special cases of sdp for which the algorithm is quadratically convergent

semidefinite inverse eigenvalue problems with prescribed entries and partial eigendata in this paper we study the semidefinite inverse eigenvalue problem of reconstructing a real n by n matrix c such that it is nearest to the original pre estimated real n by n matrix c o in the frobenius norm and satisfies the measured partial eigendata where the required matrix c should preserve the symmetry positive semidefiniteness and the prescribed entries of the pre estimated matrix c o we propose the alternating direction method of multipliers for solving the semidefinite inverse eigenvalue problem where three related iterative algorithms are presented we also extend our method to the case of lower bounds numerical experiments are reported to illustrate the efficiency of the proposed method for solving semidefinite inverse eigenvalue problems

existence and uniqueness of search directions in interior point algorithms for the sdp and the monotone sdlcp various search directions used in interior point algorithms for the semidefinite program sdp and the monotone semidefinite linear complementarity problem sdlcp are characterized by the intersection of a maximal monotone affine subspace and a maximal and strictly antitone affine subspace this observation provides a unified geometric view over the existence of those search directions

on two interior point mappings for nonlinear semidefinite complementarity problems extending our previous work monteiro and pang 1996 this paper studies properties of two fundamental mappings associated with the family of interior point methods for solving monotone nonlinear complementarity problems over the cone of symmetric positive semidefinite matrices the first of these maps lead to a family of new continuous trajectories which include the central trajectory as a special case these trajectories completely fill up the set of interior feasible points of the problem in the same way as the weighted central paths do the interior of the feasible region of a linear program unlike the approach based on the theory of maximal monotone maps taken by shida and shindoh 1996 and shida shindoh and kojima 1995 our approach is based on the theory of local homeomorphic maps in nonlinear analysis

polynomial convergence of a new family of primal dual algorithms for semidefinite programming this paper establishes the polynomial convergence of a new class of primal dual interior point path following feasible algorithms for semidefinite programming sdp whose search directions are obtained by applying newton s method to the symmetric central path equation p x p t 1 2 p t s p 1 p x p t 1 2 mu i 0 where p is a nonsingular matrix specifically we show that the short step path following algorithm based on the frobenius norm neighborhood and the semilong step path following algorithm based on the operator 2 norm neighborhood have o sqrt n l and o nl iteration complexity bounds respectively when p i this yields the first polynomially convergent semilong step algorithm based on a pure newton direction restricting the scaling matrix p at each iteration to a certain subset of nonsingular matrices we are able to establish an o n3 2l iteration complexity for the long step path following method the resulting subclass of search directions contains both the nesterov todd direction and the helmberg rendl vanderbei wolkowicz kojima shindoh hara monteiro direction

first and second order analysis of nonlinear semidefinite programs in this paper we study nonlinear semidefinite programming problems convexity duality and first order optimality conditions for such problems are presented a second order analysis is also given second order necessary and sufficient optimality conditions are derived finally sensitivity analysis of such programs is discussed

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

an inexact smoothing method for soccps based on a one parametric class of smoothing function in this paper we introduce a one parametric class of smoothing functions which contains the fischer burmeister smoothing function as special case based on this class of smoothing functions an inexact smoothing method for solving second order cone complementarity problems soccps is proposed in each iteration the corresponding linear system is solved only approximately under mild assumptions it is proved that the proposed method has global convergence and local superlinear convergence properties preliminary numerical results indicate that the method is effective for large scale problems

a scaled gauss newton primal dual search direction for semidefinite optimization interior point methods for semidefinite optimization sdo have recently been studied intensively due to their polynomial complexity and practical efficiency most of these methods are extensions of linear optimization lo algorithms as opposed to the lo case there are several different ways of constructing primal dual search directions in sdo the usual scheme is to apply linearization in conjunction with symmetrization to the perturbed optimality conditions of the sdo problem symmetrization is necessary since the linearized system is overdetermined a way of avoiding symmetrization is to find a least squares solution of the overdetermined system such a gauss newton direction was investigated by kruk et al the gauss newton direction in semidefinite programming research report corr 98 16 university of waterloo waterloo canada 1998 without giving any complexity analysis in this paper we present a similar direction where a local norm is used in the least squares formulation and we give a polynomial complexity analysis and computational evaluation of the resulting primal dual algorithm

an inverse problem for a parabolic variational inequality arising in volatility calibration with american options in finance the price of an american option is obtained from the price of the underlying asset by solving a parabolic variational inequality the free boundary associated with this variational inequality can be interpreted as the price for which the option should be exercised the calibration of volatility from the observations of the prices of an american option yields an inverse problem for the previously mentioned parabolic variational inequality after studying the variational inequality and the exercise price we give results concerning the sensitivity of the option price and of the exercise price with respect to the variations of the volatility the inverse problem is addressed by a least square method with suitable regularization terms we give necessary optimality conditions involving an adjoint state for a simplified inverse problem and we study the differentiability of the cost function optimality conditions are also given for the genuine calibration problem

efficient semidefinite branch and cut for map mrf inference we propose a branch and cut b c method for solving general map mrf inference problems the core of our method is a very efficient bounding procedure which combines scalable semidefinite programming sdp and a cutting plane method for seeking violated constraints in order to further speed up the computation several strategies have been exploited including model reduction warm start and removal of inactive constraints we analyze the performance of the proposed method under different settings and demonstrate that our method either outperforms or performs on par with state of the art approaches especially when the connectivities are dense or when the relative magnitudes of the unary costs are low we achieve the best reported results experiments show that the proposed algorithm achieves better approximation than the state of the art methods within a variety of time budgets on challenging non submodular map mrf inference problems

polynomiality of primal dual algorithms for semidefinite linear complementarity problems based on the kojima shindoh hara family of directions kojima shindoh and hara proposed a family of search directions for the semidefinite linear complementarity problem sdlcp and established polynomial convergence of a feasible short step path following algorithm based on a particular direction of their family the question of whether polynomiality could be established for any direction of their family thus remained an open problem this paper answers this question in the affirmative by establishing the polynomiality of primal dual interior point algorithms for sdlcp based on any direction of the kojima shindoh and hara family of search directions we show that the polynomial iteration complexity bounds of two well known algorithms for linear programming namely the short step path following algorithm of kojima et al and monteiro and adler and the predictor corrector algorithm of mizuno et al carry over to the context of sdlcp

exploiting sparsity in primal dual interior point methods for semidefinite programming the helmberg rendl vanderbei wolkowicz kojima shindoh hara monteiro and nesterov todd search directions have been used in many primal dual interior point methods for semidefinite programs this paper proposes an efficient method for computing the two directions when the semidefinite program to be solved is large scale and sparse

primal dual affine scaling algorithms fail for semidefinite programming in this paper we give an example of a semidefinite programming problem in which primal dual affine scaling algorithms using the hrvw ksh m mt and aho directions fail we prove that each of these algorithms can generate a sequence converging to a non optimal solution and that for the aho direction even its associated continuous trajectory can converge to a non optimal point in contrast with these directions we show that the primal dual affine scaling algorithm using the nt direction for the same semidefinite programming problem always generates a sequence converging to the optimal solution both primal and dual problems have interior feasible solutions and unique optimal solutions which satisfy strict complementarity and are nondegenerate everywhere

complementarity and nondegeneracy in semidefinite programming primal and dual nondegeneracy conditions are defined for semidefinite programming given the existence of primal and dual solutions it is shown that primal nondegeneracy implies a unique dual solution and that dual nondegeneracy implies a unique primal solution the converses hold if strict complementarity is assumed primal and dual nondegeneracy assumptions do not imply strict complementarity as they do in lp the primal and dual nondegeneracy assumptions imply a range of possible ranks for primal and dual solutions x and z this is in contrast with lp where nondegeneracy assumptions exactly determine the number of variables which are zero it is shown that primal and dual nondegeneracy and strict complementarity all hold generically numerical experiments suggest probability distributions for the ranks of x and z which are consistent with the nondegeneracy conditions

on the nesterov todd direction in semidefinite programming we study different choices of search direction for primal dual interior point methods for semidefinite programming problems one particular choice we consider comes from a specialization of a class of algorithms developed by nesterov and todd for certain convex programming problems we discuss how the search directions for the nesterov todd nt method can be computed efficiently and demonstrate how they can be viewed as newton directions this last observation also leads to convenient computation of accelerated steps using the mehrotra predictor corrector approach in the nt framework we also provide an analytical and numerical comparison of several methods using different search directions and suggest that the method using the nt direction is more robust than alternative methods

a spectral bundle method for semidefinite programming a central drawback of primal dual interior point methods for semidefinite programs is their lack of ability to exploit problem structure in cost and coefficient matrices this restricts applicability to problems of small dimension typically semidefinite relaxations arising in combinatorial applications have sparse and well structured cost and coefficient matrices of huge order we present a method that allows us to compute acceptable approximations to the optimal solution of large problems within reasonable time semidefinite programming problems with constant trace on the primal feasible set are equivalent to eigenvalue optimization problems these are convex nonsmooth programming problems and can be solved by bundle methods we propose replacing the traditional polyhedral cutting plane model constructed from subgradient information by a semidefinite model that is tailored to eigenvalue problems convergence follows from the traditional approach but a proof is included for completeness we present numerical examples demonstrating the efficiency of the approach on combinatorial examples

fast numerical solution of parabolic integrodifferential equations with applications in finance we numerically solve parabolic problems u t ca u 0 in 0 t times omega t infty where omega subset mathbb r is a bounded interval and ca is a strongly elliptic integrodifferential operator of order rho in 0 2 a discontinuous galerkin dg discretization in time and a wavelet discretization in space are used the densely populated matrices in the corresponding linear systems of equations are replaced by sparse ones using appropriate wavelet compression techniques the linear systems in each time step are solved by an incomplete gmres iteration under these conditions we show that the complexity of our algorithm is linear up to some logarithmic terms in the number of spatial degrees of freedom and present error estimates applications to purely discontinuous levy processes arising in finance are given

on the solution of large scale sdp problems by the modified barrier method using iterative solvers the limiting factors of second order methods for large scale semidefinite optimization are the storage and factorization of the newton matrix for a particular algorithm based on the modified barrier method we propose to use iterative solvers instead of the routinely used direct factorization techniques the preconditioned conjugate gradient method proves to be a viable alternative for problems with a large number of variables and modest size of the constrained matrix we further propose to avoid explicit calculation of the newton matrix either by an implicit scheme in the matrix vector product or using a finite difference formula this leads to huge savings in memory requirements and for certain problems to further speed up of the algorithm

maximum entropy estimation of density and regression functions abstract this paper presents an information efficient technique to determine the functional forms of density and regression functions by maximizing entropy subject to given side conditions a flexible functional form approach is established as special cases of this approach exponential polynomial cobb douglas translog generalized cobb douglas generalized leontief fourier flexible form and minflex laurent series etc can be derived in a monte carlo experiment the maximum entropy regression method shows good performance in terms of goodness of fit boxplots and isoquants produced by the maximum entropy method are compared with those produced by other well known functions

retrieving levy processes from option prices regularization of an ill posed inverse problem we propose a stable nonparametric method for constructing an option pricing model of exponential levy type consistent with a given data set of option prices after demonstrating the ill posedness of the usual and least squares version of this inverse problem we suggest to regularize the calibration problem by reformulating it as the problem of finding an exponential levy model that minimizes the sum of the pricing error and the relative entropy with respect to a prior exponential levy model we prove the existence of solutions for the regularized problem and show that it yields solutions which are continuous with respect to the data stable with respect to the choice of prior and which converge to the minimum entropy least squares solution of the initial problem when the noise level in the data vanishes

a maximum entropy approach to recovering information from multinomial response data abstract the classical maximum entropy me approach to estimating the unknown parameters of a multinomial discrete choice problem which is equivalent to the maximum likelihood multinomial logit ml estimator is generalized the generalized maximum entropy gme model includes noise terms in the multinomial information constraints each noise term is modeled as the mean of a finite set of a priori known points in the interval 1 1 with unknown probabilities where no parametric assumptions about the error distribution are made a gme model for the multinomial probabilities and for the distributions associated with the noise terms is derived by maximizing the joint entropy of multinomial and noise distributions under the assumption of independence the gme formulation reduces to the me in the limit as the sample grows large or when no noise is included in the entropy maximization further even though the gme and the logit estimators are conceptually different the dual gme model is related to a gener

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

general interior point maps and existence of weighted paths for nonlinear semidefinite complementarity problems extending the previous work of monteiro and pang 1998 this paper studies properties of fundamental maps that can be used to describe the central path of the monotone nonlinear com plementarity problems over the cone of symmetric positive semidefinite matrices instead of focusing our attention on a specific map as was done in the approach of monteiro and pang 1998 this paper considers a general form of a fundamental map and introduces conditions on the map that allow us to extend the main results of monteiro and pang 1998 to this general map each fundamental map leads to a family of weighted continuous trajectories which include the central trajectory as a special case hence for complementarity problems over the cone of symmetric positive semidefinite matrices the notion of weighted central path depends on the fundamental map used to represent the central path

solving semidefinite programming problems via alternating direction methods in this paper we consider an alternating direction algorithm for the solution of semidefinite programming problems sdp the main idea of our algorithm is that we reformulate the complementary conditions in the primal dual optimality conditions as a projection equation by using this reformulation we only need to make one projection and solve a linear system of equation with reduced dimension in each iterate we prove that the generated sequence converges to the solution of the sdp under weak conditions

a second order bundle method to minimize the maximum eigenvalue function in this paper we present a nonsmooth algorithm to minimize the maximum eigenvalue of matrices belonging to an affine subspace of n n symmetric matrices we show how a simple bundle method the approximate eigenvalue method can be used to globalize the second order method developed by m l overton in the eighties and recently revisited in the framework of the lagrangian theory with no additional assumption the resulting algorithm generates a minimizing sequence a geometrical and constructive proof is given to prove that quadratic convergence is achieved asymptotically some strict complementarity and non degeneracy assumptions are needed we also introduce new variants of bundle methods for semidefinite programming

a feasible interior point algorithm for nonconvex nonlinear programming in this paper a new method is presented to deal with nonconvex nonlinear inequality constrained problem by using the idea of e effective set it is only required to solve two systems of linear equations per single iteration the theoretical analysis shows that the algorithm is global convergence under some suitable conditions

centers of monotone generalized complementarity problems let c be a full dimensional closed pointed and convex cone in a finite dimensional real vector space e with an inner product of x y e and m a maximal monotone subset of e e this paper studies the existence and continuity of centers of the monotone generalized complementarity problem associated with c and m find x y m c c such that 0 here c y e 0 for all x c denotes the dual cone of c the main result of the paper unifies and extends some results established for monotone complementarity problems in euclidean space and monotone semidefinite linear complementarity problems in symmetric matrices

matrix lifting semi definite programming for detection in multiple antenna systems this paper presents a computationally efficient decoder for multiple antenna systems the proposed algorithm can be used for any constellation qam or psk and any labeling method the decoder is based on matrix lifting semi definite programming sdp the strength of the proposed method lies in a new relaxation approach applied to the previous work by mobasher this results in a reduction of the number of variables from nk 1 nk 2 2 in the previous work by mobasher to 2n k 2 in the new method where n is twice the number of transmit antennas and k is the number of constellation points in each real dimension it is shown that this reduction in the number of variables results in a significant computational complexity reduction compared to the previous work by mobasher moreover the proposed method offers a better symbol error rate performance as compared to some known and recent sdp based quasi maximum likelihood detection methods reported in the literature

on the algebraic riccati inequality arising in cone preserving time delay systems abstract this paper studies the problem of riccati stability of a pair of matrices for a matrix pair a b it was recently shown that if its corresponding time delay system is internally positive meaning that a is metzler and b is nonnegative then the pair a b is diagonally riccati stable if and only if a b is hurwitz we extend this to the case when the pair a b corresponds to a time delay system with a more general cone preserving property we show that if the time delay system relating to the pair a b is invariant on a symmetric cone the corresponding algebraic riccati inequality admits positive definite solutions which can be constructed via the scaling transformation on the euclidean jordan algebra associated with the symmetric cone for the special case when the symmetric cone is the positive semi definite cone an application to a class of stochastic systems is discussed

local and superlinear convergence of a primal dual interior point method for nonlinear semidefinite programming in this paper we consider a primal dual interior point method for solving nonlinear semidefinite programming problems we propose primal dual interior point methods based on the unscaled and scaled newton methods which correspond to the aho hrvw ksh m and nt search directions in linear sdp problems we analyze local behavior of our proposed methods and show their local and superlinear convergence properties

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

sensitivity analysis in linear programming and semidefinite programming using interior point methods we analyze perturbations of the right hand side and the cost parameters in linear programming lp and semidefinite programming sdp we obtain tight bounds on the perturbations that allow interior point methods to recover feasible and near optimal solutions in a single interior point iteration for the unique nondegenerate solution case in lp we show that the bounds obtained using interior point methods compare nicely with the bounds arising from using the optimal basis we also present explicit bounds for sdp using the monteiro zhang family of search directions and specialize them to the aho h k m and nt directions

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

exploiting sparsity in semidefinite programming via matrix completion i general framework a critical disadvantage of primal dual interior point methods compared to dual interior point methods for large scale semidefinite programs sdps has been that the primal positive semidefinite matrix variable becomes fully dense in general even when all data matrices are sparse based on some fundamental results about positive semidefinite matrix completion this article proposes a general method of exploiting the aggregate sparsity pattern over all data matrices to overcome this disadvantage our method is used in two ways one is a conversion of a sparse sdp having a large scale positive semidefinite matrix variable into an sdp having multiple but smaller positive semidefinite matrix variables to which we can effectively apply any interior point method for sdps employing a standard block diagonal matrix data structure the other way is an incorporation of our method into primal dual interior point methods which we can apply directly to a given sdp in part ii of this article we will investigate an implementation of such a primal dual interior point method based on positive definite matrix completion and report some numerical results

a primal dual potential reduction method for problems involving matrix inequalities we describe a potential reduction method for convex optimization problems involving matrix inequalities the method is based on the theory developed by nesterov and nemirovsky and generalizes gonzaga and todd s method for linear programming a worst case analysis shows that the number of iterations grows as the square root of the problem size but in practice it appears to grow more slowly as in other interior point methods the overall computational effort is therefore dominated by the least squares system that taust be solved in each iteration a type of conjugate gradient algorithm can be used for this purpose which results in important savings for two reasons first it allows us to take advantage of the special structure the problems often have e g lyapunov or algebraic riccati inequalities second we show that the polynomial bound on the number ofiterations remains valid even if the conjugate gradient algorithm is not run until completion which in practice can greatly reduce the computational effort per iteration we describe in detail how the algorithm works for optimization problems with l lyapunov inequaliues each of size m we prove an overall worst case operation count of o mssll5 the average case complexity appears to be closer to o m4ll5 this estimate is justified by extensive numerical experimentation and is consistent with other researchers experience with the practical performance of interior point algorithms for linear programming this result means that the computational cost of extending current control theory based on the solution of lyapunov or riccati equations to a theory that is based on the solution of multiple coupled lyapunov or riccati inequalities is modest

semi parametric estimation of american option prices abstract we introduce a novel semi parametric estimator of american option prices in discrete time the specification is based on a parameterized stochastic discount factor and is nonparametric w r t the historical dynamics of the markovian state variables the historical transition density estimator minimizes a distance built on the kullback leibler divergence from a kernel transition density subject to the no arbitrage restrictions for a non defaultable bond the underlying asset and some american option prices we use dynamic programming to make explicit the nonlinear restrictions on the euclidean and functional parameters coming from option data we study asymptotic and finite sample properties of the estimators

the structure of us food demand abstract an exactly aggregable system of gorman engel curves for us food consumption is developed and implemented box cox transformations on prices and income nest functional form the model nests rank up to rank three the model is estimated by nonlinear three stage least squares with annual time series data on 21 foods 17 nutrients age and race demographics and the distribution of income for 1919 1941 and 1947 2000 results are consistent with full rank three point estimates for the box cox parameters on income and prices are 0 86 and 1 09 respectively strongly rejecting zero and one in both cases no statistical evidence of serial correlation specification errors or parameter instability is found

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 semidefinite quadratic linear programs using sdpt3 this paper discusses computational experiments with linear optimization problems involving semidefinite quadratic and linear cone constraints sqlps many test problems of this type are solved using a new release of sdpt3 a matlab implementation of infeasible primal dual path following algorithms the software developed by the authors uses mehrotra type predictor corrector variants of interior point methods and two types of search directions the hkm and nt directions a discussion of implementation details is provided and computational results on problems from the sdplib and dimacs challenge collections are reported

low complexity and high resolution line spectral estimation using cyclic minimization the line spectral estimation problem has applications in radar wireless communications spectroscopy and power electronics among others the signal is modeled as a sparse linear combination of complex sinusoids and the problem target is to estimate the number of sinusoids in the mixture and respective parameters of each individual sinusoid such as magnitude phase and frequency in this work we first introduce a novel formulation of the rank function which involves the solution of a multi convex optimization problem using the multi convex reformulation of the rank function an over parameterization of the line estimation problem is proposed together with a cyclic minimization procedure to obtain a solution the cyclic method iterates between the optimization of the signal subspace and null space and stops when the two are orthogonal every limit point of the sequence of iterates is shown to be a stationary point of the original problem numerical experiments show that only a small number of iterations is required for convergence the signal subspace optimization is a semidefinite program sdp and the null space optimization has a closed form solution the cost per iteration of a general purpose interior point method ipm to solve the sdp is a quartic function of the problem dimension it is shown that by exploiting the problem structure the cost per iteration of the ipm may be reduced to a cubic function of the problem dimension comparable to the cost of the alternating direction method of multipliers admm however contrarily to the latter the former achieves the precision necessary for fine frequency localization a large set of numerical experiments show the effectiveness of the proposed approach

approximating the little grothendieck problem over the orthogonal and unitary groups the little grothendieck problem consists of maximizing sum ij c ij x ix j ijcijxixj for a positive semidefinite matrix c over binary variables x i in pm 1 xi 1 in this paper we focus on a natural generalization of this problem the little grothendieck problem over the orthogonal group given c in mathbb r dn times dn c rdn dn a positive semidefinite matrix the objective is to maximize sum ij text tr left c t ij o io j t right ijtrcijtoiojt restricting o i oi to take values in the group of orthogonal matrices mathcal o d od where c ij cij denotes the ij th d times d d d block of c we propose an approximation algorithm which we refer to as orthogonal cut to solve the little grothendieck problem over the group of orthogonal matrices mathcal o d od and show a constant approximation ratio our method is based on semidefinite programming for a given d ge 1 d 1 we show a constant approximation ratio of alpha mathbb r d 2 r d 2 where alpha mathbb r d r d is the expected average singular value of a d times d d d matrix with random gaussian mathcal n left 0 frac 1 d right n0 1d i i d entries for d 1 d 1 we recover the known alpha mathbb r 1 2 2 pi r 1 2 2 approximation guarantee for the classical little grothendieck problem our algorithm and analysis naturally extends to the complex valued case also providing a constant approximation ratio for the analogous little grothendieck problem over the unitary group mathcal u d ud orthogonal cut also serves as an approximation algorithm for several applications including the procrustes problem where it improves over the best previously known approximation ratio of frac 1 2 sqrt 2 122 the little grothendieck problem falls under the larger class of problems approximated by a recent algorithm proposed in the context of the non commutative grothendieck inequality nonetheless our approach is simpler and provides better approximation with matching integrality gaps finally we also provide an improved approximation algorithm for the more general little grothendieck problem over the orthogonal or unitary group with rank constraints recovering when d 1 d 1 the sharp known ratios

maximum entropy in the problem of moments the maximum entropy approach to the solution of underdetermined inverse problems is studied in detail in the context of the classical moment problem in important special cases such as the hausdorff moment problem we establish necessary and sufficient conditions for the existence of a maximum entropy solution and examine the convergence of the resulting sequence of approximations a number of explicit illustrations are presented in addition to some elementary examples we analyze the maximum entropy reconstruction of the density of states in harmonic solids and of dynamic correlation functions in quantum spin systems we also briefly indicate possible applications to the lee yang theory of ising models to the summation of divergent series and so on the general conclusion is that maximum entropy provides a valuable approximation scheme a serious competitor of traditional pade like procedures

information theoretic measures of the income distribution in food demand a new method to nest estimate and test the rank and functional form of the income terms in an incomplete system of demand equations is developed information theory is employed to infer the u s income distribution from data on quintile and top five percentile income ranges and intra quintile and top five percentile mean incomes maximum entropy income distributions are combined with data on the u s demand for 21 food items to estimate u s food demand over the period 1919 1995 excluding 1942 1946

analyticity of the central path at the boundary point in semidefinite programming abstract in this paper we study the limiting behavior of the central path for semidefinite programming sdp we show that the central path is an analytic function of the barrier parameter even at the limit point provided that the semidefinite program has a strictly complementary solution a consequence of this property is that the derivatives of any order of the central path have finite limits as the barrier parameter goes to zero

search directions in the sdp and the monotone sdlcp generalization and inexact computation we propose a family of directions that generalizes many directions proposed so far in feasible interior point methods for the sdp semidefinite programming and for the monotone sdlcp semidefinite linear complementarity problem we derive the family from the helmberg rendl vanderbei wolkowicz kojima shindoh hara monteiro direction by relaxing its centrality equation into a centrality inequality using this family we present a predictor corrector infeasible interior point method to provide a theoretical basis for inexact computation of directions in primal dual interior point methods for the sdp

generalized linear complementarity problems we introduce the concept of the generalized monotone linear complementarity problem glcp in order to unify lp convex qp monotone lcp and mixed monotone lcp we establish the basic properties of glcp and develop canonical forms for its representation we show that the glcp reduces to a monotone lcp in the same variables this implies that many results which hold true for monotone lcp extend to glcp in particular any interior point algorithm for monotone lcp extends to an interior point algorithm for glcp

semidefinite programming and matrix scaling over the semidefinite cone abstract let e be the hilbert space of real symmetric matrices with block diagonal form diag a m where a is n n and m is an l l diagonal matrix with the inner product x y trace xy we assume n l 1 i e allow n 0 or l 0 given x e we write x 0 x 0 if it is positive semidefinite positive definite let q e e be a symmetric positive semidefinite linear operator and min x 0 5 trace xqx x 1 x 0 the problem of testing if 0 is a significant problem called homogeneous programming on the one hand the feasibility problem in semidefinite programming sdp can be formulated as a homogeneous programming problem on the other hand it is related to the generalization of the classic problem of matrix scaling let 0 1 be a given accuracy u qe e e the identity matrix in e and n n l we describe a path following algorithm that in case 0 in o n ln n u newton iterations produces d 0 d 1 such that d if 0 in o n ln n u ln ln 1 newton iterations the algorithm produces d 0 such that dqde e where d is the operator that maps w e to d1 2wd1 2 moreover we use the algorithm to prove 0 if and only if there exists d 0 such that dqde e if and only if there exists d 0 such that qd 0 thus via this duality the matrix scaling problem is a natural dual to the feasibility problem in sdp this duality also implies that in blum et al bull ams 21 1989 1 real number model of computation the decision problem of testing the solvability of matrix scaling is both in np and co np although the above complexities can be deduced from our path following algorithm for general self concordant homogeneous programming and for matrix scaling obtained in scaling dualities and self concordant homogeneous programming in finite dimensional spaces technical report lcsr tr 359 department of computer science rutgers university new brunswick nj 1999 for the problems considered here the present analysis is quite elementary short and complete this simplicity is mainly due to a new inequality derived in this paper that relates the norm of scaled quantities at two successive newton iterations and implies a quadratic rate of convergence the present algorithm is not only a simple path following algorithm for testing the solvability of feasibility problem in sdp but is also capable of testing solvability of the matrix scaling problem when n 0 the algorithm reduces to the diagonal matrix scaling linear programming algorithm of khachiyan and kalantari siam j optim 4 1992 668 as in the case of lp the algorithm of this paper can be used to solve the general sdp problem

an interior point perspective on sensitivity analysis in semidefinite programming we study the asymptotic behavior of the interior point bounds arising from the work of yildirim and todd on sensitivity analysis in semidefinite programming in comparison with the optimal partition bounds we introduce a weaker notion of nondegeneracy and discuss its implications for perturbations of the right hand side vector or the cost matrix we show that the interior point bounds evaluated on the central path using the monteiro zhang family of search directions converge as the duality gap tends to zero to the symmetrized version of the optimal partition bounds under mild nondegeneracy assumptions furthermore our analysis does not assume strict complementarity as long as the central path converges to the analytic center in a relatively controlled manner we also show that the same convergence results carry over to iterates lying in an appropriate very narrow central path neighborhood if the nesterov todd direction is used to evaluate the interior point bounds we extend our results to the case of simultaneous perturbations of the right hand side vector and the cost matrix we also provide examples illustrating that our assumptions in general cannot be weakened

linear systems in jordan algebras and primal dual interior point algorithms we discuss a possibility of the extension of a primal dual interior point algorithm suggested recently by alizadeh et al 1994 we consider optimization problems defined on the intersection of a symmetric cone and an affine subspace the question of solvability of a linear system arising in the implementation of the primal dual algorithm is analyzed a nondegeneracy theory for the considered class of problems is developed the jordan algebra technique suggested by faybusovich 1995 plays major role in the present paper

a predictor corrector interior point algorithm for the semidefinite linear complementarity problem using the alizadeh haeberly overton search direction this paper proposes a globally convergent predictor corrector infeasible interior point algorithm for the monotone semidefinite linear complementarity problem using the alizadeh haeberly overton search direction and shows its quadratic local convergence under the strict complementarity condition

a new o nl iteration predictor corrector algorithm with wide neighborhood for semidefinite programming in this paper we extend the ai zhang predictor corrector method to the class of semidefinite programming first we define a new wide neighborhood n t b another key ingredient of our method is that we treat the classical newton direction as the sum of two other directions corresponding to respectively the negative part and the positive part of the right hand side we prove that besides the predictor steps each corrector step also reduces the duality gap by a rate of 1 1o n then the method enjoys the low iteration bound of o nl which is better than that of usual wide neighborhood algorithm o nl where n is the dimension of the problem and l x 0 t s 0 e with e the required precision and x 0 y 0 s 0 the initial interior solution