a relaxed cutting plane method for semi infinite semi definite programming in this paper we develop two discretization algorithms with a cutting plane scheme for solving combined semi infinite and semi definite programming problems i e a general algorithm when the parameter set is a compact set and a typical algorithm when the parameter set is a box set in the m dimensional space we prove that the accumulation point of the sequence points generated by the two algorithms is an optimal solution of the combined semi infinite and semi definite programming problem under suitable assumption conditions two examples are given to illustrate the effectiveness of the typical algorithm

on the bridge between combinatorial optimization and nonlinear optimization a family of semidefinite bounds for 0 1 quadratic problems leading to quasi newton methods this article presents a family of semidefinite programming bounds obtained by lagrangian duality for 0 1 quadratic optimization problems with linear or quadratic constraints these bounds have useful computational properties they have a good ratio of tightness to computing time they can be optimized by a quasi newton method and their final tightness level is controlled by a real parameter these properties are illustrated on three standard combinatorial optimization problems unconstrained 0 1 quadratic optimization heaviest k subgraph and graph bisection

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

lifting and separation procedures for the cut polytope the max cut problem and the associated cut polytope on complete graphs have been extensively studied over the last 25 years however in comparison only little research has been conducted for the cut polytope on arbitrary graphs in particular separation algorithms have received only little attention in this study we describe new separation and lifting procedures for the cut polytope on general graphs these procedures exploit algorithmic and structural results known for the cut polytope on complete graphs to generate valid and sometimes facet defining inequalities for the cut polytope on arbitrary graphs in a cutting plane framework we report computational results on a set of well established benchmark problems

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

computational experience with a bundle approach for semidefinite cutting plane relaxations of max cut and equipartition we propose a dynamic version of the bundle method to get approximate solutions to semidefinite programs with a nearly arbitrary number of linear inequalities our approach is based on lagrangian duality where the inequalities are dualized and only a basic set of constraints is maintained explicitly this leads to function evaluations requiring to solve a relatively simple semidefinite program our approach provides accurate solutions to semidefinite relaxations of the max cut and the equipartition problem which are not achievable by direct approaches based only on interior point methods

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

computational experience with stable set relaxations we investigate relaxations for the maximum stable set problem based on the lovasz number vartheta g as an initial upper bound we strengthen this relaxation by adding two classes of cutting planes odd circuit and triangle inequalities we present computational results using this tighter model on many classes of graphs

an ϵ sensitivity analysis for semidefinite programming abstract we extend the concept of sensitivity analysis developed for linear programming to that for semidefinite programming first the notion of optimality for a given semidefinite programming problem is defined and then a generic sensitivity analysis for semidefinite programming is introduced based on the definitions we develop an implementation of the generic sensitivity analysis under perturbations of either the cost parameters or the right hand side

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

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

connections between semidefinite relaxations of the max cut and stable set problems we describe links between a recently introduced semidefinite relaxation for the max cut problem and the well known semidefinite relaxation for the stable set problem underlying the lovasz s theta function it turns out that the connection between the convex bodies defining the semidefinite relaxations mimics the connection existing between the corresponding polyhedra we also show how the semidefinite relaxations can be combined with the classical linear relaxations in order to obtain tighter relaxations

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

semidefinite relaxations of ordering problems ordering problems assign weights to each ordering and ask to find an ordering of maximum weight we consider problems where the cost function is either linear or quadratic in the first case there is a given profit if the element u is before v in the ordering in the second case the profit depends on whether u is before v and r is before s the linear ordering problem is well studied with exact solution methods based on polyhedral relaxations the quadratic ordering problem does not seem to have attracted similar attention we present a systematic investigation of semidefinite optimization based relaxations for the quadratic ordering problem extending and improving existing approaches we show the efficiency of our relaxations by providing computational experience on a variety of problem classes

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

strong converse exponents for a quantum channel discrimination problem and quantum feedback assisted communication this paper studies the difficulty of discriminating between an arbitrary quantum channel and a replacer channel that discards its input and replaces it with a fixed state the results obtained here generalize those known in the theory of quantum hypothesis testing for binary state discrimination we show that in this particular setting the most general adaptive discrimination strategies provide no asymptotic advantage over non adaptive tensor power strategies this conclusion follows by proving a quantum stein s lemma for this channel discrimination setting showing that a constant bound on the type i error leads to the type ii error decreasing to zero exponentially quickly at a rate determined by the maximum relative entropy registered between the channels the strong converse part of the lemma states that any attempt to make the type ii error decay to zero at a rate faster than the channel relative entropy implies that the type i error necessarily converges to one we then refine this latter result by identifying the optimal strong converse exponent for this task as a consequence of these results we can establish a strong converse theorem for the quantum feedback assisted capacity of a channel sharpening a result due to bowen furthermore our channel discrimination result demonstrates the asymptotic optimality of a non adaptive tensor power strategy in the setting of quantum illumination as was used in prior work on the topic the sandwiched renyi relative entropy is a key tool in our analysis finally by combining our results with recent results of hayashi and tomamichel we find a novel operational interpretation of the mutual information of a quantum channel mathcal n as the optimal type ii error exponent when discriminating between a large number of independent instances of mathcal n and an arbitrary worst case replacer channel chosen from the set of all replacer channels

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

solving max cut to optimality by intersecting semidefinite and polyhedral relaxations we present a method for finding exact solutions of max cut the problem of finding a cut of maximum weight in a weighted graph we use a branch and bound setting that applies a dynamic version of the bundle method as bounding procedure this approach uses lagrangian duality to obtain a nearly optimal solution of the basic semidefinite max cut relaxation strengthened by triangle inequalities the expensive part of our bounding procedure is solving the basic semidefinite relaxation of the max cut problem which has to be done several times during the bounding process we review other solution approaches and compare the numerical results with our method we also extend our experiments to instances of unconstrained quadratic 0 1 optimization and to instances of the graph equipartition problem the experiments show that our method nearly always outperforms all other approaches in particular for dense graphs where linear programming based methods fail our method performs very well exact solutions are obtained in a reasonable time for any instance of size up to n 100 independent of the density for some problems of special structure we can solve even larger problem classes we could prove optimality for several problems of the literature where to the best of our knowledge no other method is able to do so

universal binary semidefinite relaxation for ml signal detection semidefinite relaxation sdr provides a computationally efficient polynomial time approximation of the maximum likelihood detector however most of the existing works mainly focus on particular signal constellations in this paper we propose a universal binary semidefinite relaxation scheme that can handle arbitrary signal constellations in polynomial time the proposed scheme first binarizes the original signal space to a linearly constrained binary space and then solves the detection problem through sdr colorblack a specialized dual barrier method is provided to solve the sdr more efficiently in addition we propose to apply on the fly decision feedback to further reduce the computational complexity and improve the detection performance the proposed binary sdr together with on the fly decision feedback scheme can provide comparable or better solutions compared to existing sdr methods specialized to specific constellations such as 16 qam and 8 psk in terms of computational complexity and symbol error rate furthermore the proposed scheme is universal and can solve any other constellations such as 12 qam 32 qam or m psk

solving quadratic 0 1 problems by semidefinite programs and cutting planes we present computational experiments for solving quadratic 0 1 problems our approach combines a semidefinite relaxation with a cutting plane technique and is applied in a branch and bound setting our experiments indicate that this type of approach is very robust and allows to solve many moderately sized problems having say less than 100 binary variables in a routine manner 1998 the mathematical programming society inc published by elsevier science b v

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

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

a novel signaling scheme for blind unique identification of alamouti space time block coded channel in this paper we present a simple signaling scheme to blindly and uniquely identify the alamouti space time block coded channel first under a noise free environment and then under a complex gaussian noise environment in which pth and qth order statistics p and q are coprime of the received signals are available in both cases closed form solutions to determine the channel coefficients are obtained by exploiting specific properties of the alamouti space time block code stbc and the linear diophantine equation theory under gaussian noise when the length of the received data is finite we propose to use the semidefinite relaxation sdr algorithm to approximate maximum likelihood ml detection so that the joint estimation of the channel and symbols can be efficiently implemented simulation results show that while other existing blind methods fail our signaling scheme works well

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

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

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 version of the bundle idea for minimizing a nonsmooth function conceptual idea convergence analysis numerical results during recent years various proposals for the minimization of a nonsmooth functional have been made amongst these the bundle concept turned out to be an especially fruitful idea based on this concept a number of authors have developed codes that can successfully deal with nonsmooth problems the aim of the paper is to show that by adding some features of the trust region philosophy to the bundle concept the end result is a distinguished member of the bundle family with a more stable behaviour than some other bundle versions the reliability and efficiency of this code is demonstrated on the standard academic test examples and on some real life problems

low rank exploitation in semidefinite programming for control many control related problems can be cast as semidefinite programs even though there exist polynomial time algorithms and excellent publicly available solvers the time it takes to solve these problems can be excessive what many of these problems have in common in particular in control is that some of the variables enter as matrix valued variables this leads to a low rank structure in the basis matrices which can be exploited when forming the newton equations in this article we describe how this can be done and show how our code called strul can be used in conjunction with the semidefinite programming solver sdpt3 the idea behind the structure exploitation is classical and is implemented in lmi lab but we show that when using a modern semidefinite programming framework such as sdpt3 the computational time can be significantly reduced finally we describe how the modelling language yalmip has been changed in such a way that our code which can be freely downloaded can be interfaced using sta

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

a recurrent neural network for real time semidefinite programming the semidefinite programming problem is an important optimization problem that has been extensively investigated a real time solution method for solving such a problem however is still not yet available the paper proposes a recurrent neural network for this purpose first an auxiliary cost function is introduced to minimize the duality gap between the admissible points of the primal problem and the corresponding dual problem then a dynamical system is constructed to drive the duality gap to zero exponentially along any trajectory by modifying the gradient of the auxiliary cost function furthermore a subsystem is developed to circumvent the computation of matrix inverse so that the resulting overall dynamical system can be realized using a recurrent neural network the architecture of the resulting neural network is discussed the operating characteristics and performance of the proposed approach are demonstrated by means of simulation results

semidefinite relaxation approaches to soft mimo demodulation for higher order qam signaling the soft demodulator is a key component of receivers that employ iterative demodulation and decoding for multiple input multiple output systems with binary or qpsk signaling the semidefinite relaxation sdr approach to list based soft demodulation provides a desirable balance between the performance and computational cost in this paper that approach is extended to systems with higher order qam signaling for the class of list sdr demodulators the key to the extension is a quadratic approximation of the a priori information and for the more computationally efficient class of single sdr demodulators the key to the extension is an algebraic expression derived herein for the probability that the sdr randomization procedure generates each candidate symbol to implement these demodulators customized primal dual and dual scaling algorithms are developed to solve the underlying semidefinite programs the performance of the proposed demodulators is compared with that of some existing demodulators by using error rate simulations the empirical distribution of the computational cost and exit chart analyses in our comparisons the proposed single sdr demodulator provides performance that comes close to that of the list sequential demodulator at a computational cost that is close to that of the minimum mean square error parallel soft interference canceller

semidefinite programs new search directions smoothing type methods and numerical results motivated by some results for linear programs and complementarity problems this paper gives some new characterizations of the central path conditions for semidefinite programs exploiting these characterizations some smoothing type methods for the solution of semidefinite programs are derived the search directions generated by these methods are automatically symmetric and the overall methods are shown to be globally and locally superlinearly convergent under suitable assumptions some numerical results are also included which indicate that the proposed methods are very promising and comparable to several interior point methods moreover the current method seems to be superior to the smoothing method recently proposed by chen and tseng non interior continuation methods for solving semidefinite complementarity problems technical report department of mathematics university of washington seattle 1999

maximum margin clustering made practical motivated by the success of large margin methods in supervised learning maximum margin clustering mmc is a recent approach that aims at extending large margin methods to unsupervised learning however its optimization problem is nonconvex and existing mmc methods all rely on reformulating and relaxing the nonconvex optimization problem as semidefinite programs sdp though sdp is convex and standard solvers are available they are computationally very expensive and only small data sets can be handled to make mmc more practical we avoid sdp relaxations and propose in this paper an efficient approach that performs alternating optimization directly on the original nonconvex problem a key step to avoid premature convergence in the resultant iterative procedure is to change the loss function from the hinge loss to the laplacian square loss so that overconfident predictions are penalized experiments on a number of synthetic and real world data sets demonstrate that the proposed approach is more accurate much faster hundreds to tens of thousands of times faster and can handle data sets that are hundreds of times larger than the largest data set reported in the mmc literature

some new search directions for primal dual interior point methods in semidefinite programming search directions for primal dual path following methods for semidefinite programming sdp are proposed these directions have the properties that 1 under certain nondegeneracy and strict complementarity assumptions the jacobian matrix of the associated symmetrized newton equation has a bounded condition number along the central path in the limit as the barrier parameter mu tends to zero and 2 the schur complement matrix of the symmetrized newton equation is symmetric and the cost for computing this matrix is 2mn3 0 5m2n2 flops where n and m are the dimension of the matrix and vector variables of the semidefinite program respectively these two properties imply that a path following method using the proposed directions can achieve the high accuracy typically attained by methods employing the direction proposed by alizadeh haeberly and overton currently the best search direction in terms of accuracy but each iteration requires at most half the amount of flops to leading order

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

the simplest semidefinite programs are trivial we consider optimization problems of the following type min left mbox tr cx colon ax b x quad mbox positive semidefinite right here tr denotes the trace operator c and x are symmetric n n matrices b is a symmetric m m matrix and a denotes a linear operator such problems are called semidefinite programs and have recently become the object of considerable interest due to important connections with max min eigenvalue problems and with new bounds for integer programming in the context of symmetric matrices the simplest linear operators have the following form ax mxm t where m is an arbitrary m n matrix in this paper we show that for such linear operators the optimization problem is trivial in the sense that an explicit solution can be given

nonpolyhedral relaxations of graph bisection problems we study the problem of finding the minimum bisection of a graph into two parts of prescribed sizes we formulate two lower bounds on the problem by relaxing node and edge incidence vectors of cu

a semidefinite framework for trust region subproblems with applications to large scale minimization primal dual pairs of semidefinite programs provide a general framework for the theory and algorithms for the trust region subproblem trs this latter problem consists in minimizing a general quadratic function subject to a convex quadratic constraint and therefore it is a generalization of the minimum eigenvalue problem the importance of trs is due to the fact that it provides the step in trust region minimization algorithms the semidefinite framework is studied as an interesting instance of semidefinite programming as well as a tool for viewing known algorithms and deriving new algorithms for trs in particular a dual simplex type method is studied that solves trs as a parametric eigenvalue problem this method uses the lanczos algorithm for the smallest eigenvalue as a black box therefore the essential cost of the algorithm is the matrix vector multiplication and thus sparsity can be exploited a primal simplex type method provides steps for the so called hard case extensive numerical tests for large sparse problems are discussed these tests show that the cost of the algorithm is 1 n times the cost of finding a minimum eigenvalue using the lanczos algorithm where 0 n 1 is a fraction which decreases as the dimension increases

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

symmetric indefinite systems for interior point methods we present a unified framework for solving linear and convex quadratic programs via interior point methods at each iteration this method solves an indefinite system whose matrix is left begin array 20 c d 2 a t a 0 end array right instead of reducing to obtain the usualad 2 a t system this methodology affords two advantages 1 it avoids the fill created by explicitly forming the productad 2 a t whena has dense columns and 2 it can easily be used to solve nonseparable quadratic programs since it requires only thatd be symmetric we also present a procedure for converting nonseparable quadratic programs to separable ones which yields computational savings when the matrix of quadratic coefficients is dense

a novel neural network for solving semidefinite programming problems with some applications abstract in this paper we present a neural network model for solving semidefinite programming we show that the equilibrium point of the proposed neural network and the kkt point of the semidefinite programming are equal by employing lyapunov function approach it is also shown that the suggested neural network model is stable in the sense of lyapunov and globally convergent to an exact optimal solution of the original problem some applications of the proposed scheme are stated by modeling of some important problems as semidefinite programming problems numerical simulations verify the obtained theoretical 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

quasi maximum likelihood multiuser detection using semi definite relaxation with application to synchronous cdma the maximum likelihood ml multiuser detector is well known to exhibit better bit error rate ber performance than many other multiuser detectors unfortunately ml detection mld is a nondeterministic polynomial time hard np hard problem for which there is no known algorithm that can find the optimal solution with polynomial time complexity in the number of users in this paper a polynomial time approximation method called semi definite sd relaxation is applied to the mld problem with antipodal data transmission sd relaxation is an accurate approximation method for certain np hard problems the sd relaxation ml sdr ml detector is efficient in that its complexity is of the order of k3 5 where k is the number of users we illustrate the potential of the sdr ml detector by showing that some existing detectors such as the decorrelator and the linear minimum mean square error detector can be interpreted as degenerate forms of the sdr ml detector simulation results indicate that the ber performance of the sdr ml detector is better than that of these existing detectors and is close to that of the true ml detector even when the cross correlations between users are strong or the near far effect is significant

the bounded smooth reformulation and a trust region algorithm for semidefinite complementarity problems in this paper we present a new merit function for the semidefinite complementarity problem sdcp by extending the bounded smooth reformulation for variational inequality problems we prove that the merit function has a bounded level set and any stationary point of it is a global minimizer without the assumption of monotonicity moreover we present a trust region algorithm for solving the minimization problem with semidefinite constraints the trust region subproblem is solved by the truncated conjugate gradient method and the global convergence is established even without requiring the existence of an accumulation point of the generated sequence

a computational study of graph partitioning letg n e be an edge weighted undirected graph the graph partitioning problem is the problem of partitioning the node setn intok disjoint subsets of specified sizes so as to minimize the total weight of the edges connecting nodes in distinct subsets of the partition we present a numerical study on the use of eigenvalue based techniques to find upper and lower bounds for this problem results for bisecting graphs with up to several thousand nodes are given and for small graphs some trisection results are presented we show that the techniques are very robust and consistently produce upper and lower bounds having a relative gap of typically a few percentage points

higher order predictor corrector interior point methods with application to quadratic objectives in this paper the authors explore the full utility of mehrotra s predictor corrector method in the context of linear and convex quadratic programs they describe a procedure for doing multiple corrections at each iteration and implement it within the framework of ob1 computational results are provided for the multiple correcting procedure using several strategies for determining the number of corrections in a given iteration the results indicate that iteration counts can be significantly reduced by allowing higher order corrections but at the the cost of extra work per iteration the procedure is shown to be a level m composite newton interior point method where m is the number of corrections performed in an iteration

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

strong duality for semidefinite programming it is well known that the duality theory for linear programming lp is powerful and elegant and lies behind algorithms such as simplex and interior point methods however the standard lagrangian for nonlinear programs requires constraint qualifications to avoid duality gaps semidefinite linear programming sdp is a generalization of lp where the nonnegativity constraints are replaced by a semidefiniteness constraint on the matrix variables there are many applications e g in systems and control theory and combinatorial optimization however the lagrangian dual for sdp can have a duality gap we discuss the relationships among various duals and give a unified treatment for strong duality in semidefinite programming these duals guarantee strong duality i e a zero duality gap and dual attainment this paper is motivated by the recent paper by ramana where one of these duals is introduced

large scale optimization of eigenvalues optimization problems involving eigenvalues arise in many applications let x be a vector of real parameters and let a x be a continuously differentiable symmetric matrix function of x we consi

global stochastic stability analysis for stochastic neural networks with infinite delay and markovian switching we consider a novel stochastic nns model the method is the lyapunov method stochastic analysis technique and m matrix theory we derive sufficient criteria for three kinds of stochastic stability numerical examples demonstrate the applicability and effectiveness of the theoretical theorems this paper focuses on the stochastic stability for a novel kind of stochastic neural networks with infinite delay and markovian switching by using the lyapunov method stochastic analysis technique and m matrix theory some simple and easily testable sufficient conditions are presented to ensure the trivial solution is stochastically stable stochastically asymptotically stable and globally stochastically asymptotically stable respectively as a subsequent result we develop the conditions that guarantee global stochastic asymptotic stability for stochastic neural networks with infinite delay as well as global asymptotic stability for neural networks with infinite delay from the perspective of theory the derived stability criteria include some existing ones as its special cases and are thus less conservative finally two examples are given to demonstrate the applicability and effectiveness of the theoretical theorems

an interior point method with a primal dual quadratic barrier penalty function for nonlinear semidefinite programming in this paper we consider an interior point method for nonlinear semidefinite programming yamashita yabe and harada presented a primal dual interior point method in which a nondifferentiable merit function was used by using shifted barrier kkt conditions we propose a differentiable primal dual merit function within the framework of the line search strategy and prove its global convergence property under weaker assumptions than the method of yamashita yabe and harada

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

an interior point method for minimizing the maximum eigenvalue of a linear combination of matrices we consider the problem of finding lambda opt mathop inf limits x in r m lambda max a 0 sum limits i 1 m x i a i 1 and an optimal solution xopt if it exists here the a i are given symmetric matrices in ir n n for 0 i m and max a is the largest eigenvalue of a matrix a problem 1 can be rewritten as a convex differentiable problem with a positive definite constraint positive definiteness can easily be verified numerically via cholesky decomposition moreover the cone of positive definite matrices a is characterized by the domain of the smooth convex barrier function log det a

designing low complexity near optimal multiple symbol detectors for impulse radio uwb systems multiple symbol detection has recently caught attention in ultrawideband uwb communications because of its high performance without requiring explicit channel estimation in this context the generalized likelihood ratio test glrt for multiple symbol detection problem is developed to jointly detect multiple symbols which exhibits considerable error performance improvement over transmitted reference transmissions methods unfortunately the glrt is a boolean quadratic programming bqp problem which is generally nondeterministic polynomial hard np hard in this paper we propose two near optimal detectors with polynomial complexity the first detector performs semidefinite relaxation to approximately solve the bqp problem called sdp msd the second detector is based on a sphere based relaxation of the bqp problem we refer to this as modified unconstrained relaxation multiple symbol detector mur msd both detectors achieve near optimal performance while the sdp msd performs slightly better than the mur msd at the price of higher computational complexity furthermore the mur msd can be treated as a further nontight relaxation of sdp msd simulations are utilized to validate our findings and to demonstrate performance robustness to multiaccess interference

a cutting plane algorithm for semi definite programming problems with applications to failure discriminant analysis we will propose a new cutting plane algorithm for solving a class of semi definite programming problems sdp with a small number of variables and a large number of constraints problems of this type appear when we try to classify a large number of multi dimensional data into two groups by a hyper ellipsoidal surface among such examples are cancer diagnosis failure discrimination of enterprises also a certain class of option pricing problems can be formulated as this type of problem we will show that the cutting plane algorithm is much more efficient than the standard interior point algorithms for solving sdp

soft quasi maximum likelihood detection for multiple antenna wireless channels the paper addresses soft maximum likelihood ml detection for multiple antenna wireless communication channels we propose a soft quasi ml detector that maximizes the log likelihood function by deploying a semi definite relaxation sdr given perfect channel state information at the receiver the quasi ml sdr detector closely approximates the performance of the optimal ml detector in both coded and uncoded multiple input multiple output mimo channels with quadrature phase shift keying qpsk modulation and frequency flat rayleigh fading the complexity of the quasi ml sdr detector is much less than that of the optimal ml detector thus offering more favorable performance complexity characteristics in contrast to the existing sphere decoder the new quasi ml detector enjoys guaranteed polynomial worst case complexity the two detectors exhibit quite comparable performance in a variety of ergodic qpsk mimo channels but the complexity of the quasi ml detector scales better with increasing number of transmit and receive antennas especially in the region of low signal to noise ratio snr

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

sensor selection based on generalized information gain for target tracking in large sensor networks in this paper sensor selection problems for target tracking in large sensor networks with linear equality or inequality constraints are considered first we derive an equivalent kalman filter for sensor selection i e generalized information filter then under a regularity condition we prove that the multistage look ahead policy that minimizes either the final or the average estimation error covariances of next multiple time steps is equivalent to a myopic sensor selection policy that maximizes the trace of the generalized information gain at each time step moreover when the measurement noises are uncorrelated between sensors the optimal solution can be obtained analytically for sensor selection when constraints are temporally separable when constraints are temporally inseparable sensor selections can be obtained by approximately solving a linear programming problem so that the sensor selection problem for a large sensor network can be dealt with quickly although there is no guarantee that the gap between the performance of the chosen subset and the performance bound is always small numerical examples suggest that the algorithm is near optimal in many cases finally when the measurement noises are correlated between sensors the sensor selection problem with temporally inseparable constraints can be relaxed to a boolean quadratic programming problem which can be efficiently solved by a gaussian randomization procedure along with solving a semi definite programming problem numerical examples show that the proposed method is much better than the method that ignores dependence of noises

semidefinite relaxations of robust binary least squares under ellipsoidal uncertainty sets the problem of finding the least squares solution s to a system of equations hs y is considered when s is a vector of binary variables and the coefficient matrix h is unknown but of bounded uncertainty similar to previous approaches to robust binary least squares we explore the potential of a min max design with the aim to provide solutions that are less sensitive to the uncertainty in h we concentrate on the important case of ellipsoidal uncertainty i e the matrix h is assumed to be a deterministic unknown quantity which lies in a given uncertainty ellipsoid the resulting problem is np hard yet amenable to convex approximation techniques starting from a convenient reformulation of the original problem we propose an approximation algorithm based on semidefinite relaxation that explicitly accounts for the ellipsoidal uncertainty in the coefficient matrix next we show that it is possible to construct a tighter relaxation by suitably changing the description of the feasible region of the problem and formulate an approximation algorithm that performs better in practice interestingly both relaxations are derived as lagrange bidual problems corresponding to the two equivalent problem reformulations the strength of the proposed tightened relaxation is demonstrated by pertinent simulations

inexact non interior continuation method for solving large scale monotone sdcp for exact newton method for solving monotone semidefinite complementarity problems sdcp one needs to exactly solve a linear system of equations at each iteration for problems of large size solving the linear system of equations exactly can be very expensive in this paper we propose a new inexact smoothing continuation algorithm for solution of large scale monotone sdcp at each iteration the corresponding linear system of equations is solved only approximately under mild assumptions the algorithm is shown to be both globally and superlinearly convergent

laplacian eigenvalues and the maximum cut problem we introduce and study an eigenvalue upper bound g on the maximum cut mc g of a weighted graph the function g has several interesting properties that resemble the behaviour of mc g the following results are presented we show that is subadditive with respect to amalgam and additive with respect to disjoint sum and 1 sum we prove that g is never worse that 1 131 mc g for a planar or more generally a weakly bipartite graph with nonnegative edge weights we give a dual characterization of g and show that g is computable in polynomial time with an arbitrary precision

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

response vector constrained robust lcmv beamforming based on semidefinite programming although linearly constrained minimum variance lcmv beamforming is robust against imprecise target information it usually leads to relatively high sidelobe and distorted mainlobe which would induce a high false alarm probability to circumvent this problem this work devises a novel robust lcmv beamforming approach by utilizing response vector optimization it intends to find the optimal response vector in lieu of the all one response vector in traditional lcmv beamformer the proposed robust beamformer is first formulated as a nonconvex quadratically constrained quadratic programming problem and then transformed into a semidefinite programming problem which can be efficiently and exactly solved the proposed beamformer not only improves the performance in terms of signal to interference plus noise ratio substantially but also possesses low sidelobe and well maintained mainlobe moreover since the response vector is quite small in size the complexity of calculating the optimal response vector is negligible additionally the proposed beamformer is also extended to two dimensional space time adaptive processing simulation results are presented to demonstrate the superiority of the proposed approach

a linear fractional semidefinite relaxation approach to maximum likelihood detection of higher order qam ostbc in unknown channels this paper considers the blind maximum likelihood ml detection problem for orthogonal space time block codes ostbcs in multiple input multiple output flat fading channels while the blind ml detection problem for general space time codes is difficult to solve it has been shown that for ostbcs with constant modulus constellations the blind ml detection problem can be formulated as a discrete quadratic program and then handled by a powerful convex approximation technique known as semidefinite relaxation sdr in this paper we turn our attention to the case of higher order qam ostbcs due to the nonconstant modulus nature of higher order qam signals the blind ml detection problem turns out to be a discrete rayleigh quotient maximization problem and as a result the current sdr technique is no longer directly applicable we propose a linear fractional sdr lfsdr approach to this problem this approach first relaxes the higher order qam blind ml detection problem into a quasi convex problem followed by a simple solution approximation procedure in general quasi convex problems are computationally more complex to solve than convex problems but we show that an optimum solution of our quasi convex problem can be efficiently obtained by solving a convex semidefinite program the approximation accuracy of the proposed approach relative to other possible relaxation approaches is also studied simulation results are presented to demonstrate that the proposed lfsdr based blind ml detector outperforms some existing suboptimal detectors and can yield promising performance even with a small to moderate number of code blocks

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

semidefinite relaxation based multiuser detection for m ary psk multiuser systems because of the powerful symbol error performance of multiuser maximum likelihood ml detection recently there has been much interest in seeking effective ways of approximating multiuser ml detection mld with affordable computational costs it has been illustrated that for the synchronous code division multiple access cdma scenario the so called semidefinite relaxation sdr algorithm can accurately and efficiently approximate multiuser mld this sdr mld algorithm however can only handle binary and quadratic phase shift keying psk symbol constellations in this sequel we propose an extended sdr algorithm for mld with m ary psk mpsk constellations for the synchronous cdma scenario the proposed sdr algorithm provides an attractive polynomial time complexity order of k sup 3 5 where k is the number of users simulation results indicate that the proposed detector provides improved symbol error performance compared with several commonly used multiuser detectors

some applications of optimization in matrix theory abstract we apply a recent characterization of optimality for the abstract convex program with a cone constraint to three matrix theory problems 1 a generalization of farkas s lemma 2 paired duality in linear programming over cones 3 a constrained matrix best approximation problem in particular these results are not restricted to polyhedral or closed cones

the diversity order of the semidefinite relaxation detector in this paper we consider the detection of binary antipodal signals transmitted in a spatially multiplexed fashion over a fading multiple input multiple output mimo channel and where the detection is done by means of semidefinite relaxation sdr the sdr detector is an attractive alternative to maximum likelihood ml detection since the complexity is polynomial rather than exponential assuming that the channel matrix is drawn with independent identically distributed i i d real valued gaussian entries we study the receiver diversity and prove that the sdr detector achieves the maximum possible diversity thus the error probability of the receiver tends to zero at the same rate as the optimal ml receiver in the high signal to noise ratio snr limit this significantly strengthens previous performance guarantees available for the semidefinite relaxation detector additionally it proves that full diversity detection is also possible in certain scenarios when using a noncombinatorial receiver structure