further relaxations of the semidefinite programming approach to sensor network localization recently a semidefinite programming sdp relaxation approach has been proposed to solve the sensor network localization problem although it achieves high accuracy in estimating the sensor locations the speed of the sdp approach is not satisfactory for practical applications in this paper we propose methods to further relax the sdp relaxation more precisely to relax the single semidefinite matrix cone into a set of small size semidefinite submatrix cones which we call a sub sdp ssdp approach we present two such relaxations although they are weaker than the original sdp relaxation they retain the key theoretical property and numerical experiments show that they are both efficient and accurate the speed of the ssdp is even faster than that of other approaches based on weaker relaxations the ssdp approach may also pave a way to efficiently solving general sdp problems without sacrificing the solution quality

a nonlinear programming algorithm for solving semidefinite programs via low rank factorization in this paper we present a nonlinear programming algorithm for solving semidefinite programs sdps in standard form the algorithm s distinguishing feature is a change of variables that replaces the symmetric positive semidefinite variable x of the sdp with a rectangular variable r according to the factorization x rr t the rank of the factorization i e the number of columns of r is chosen minimally so as to enhance computational speed while maintaining equivalence with the sdp fundamental results concerning the convergence of the algorithm are derived and encouraging computational results on some large scale test problems are also presented

an implementation of karmarkar s algorithm for linear programming this paper describes the implementation of power series dual affine scaling variants of karmarkar s algorithm for linear programming based on a continuous version of karmarkar s algorithm two variants resulting from first and second order approximations of the continuous trajectory are implemented and tested linear programs are expressed in an inequality form which allows for the inexact computation of the algorithm s direction of improvement resulting in a significant computational advantage implementation issues particular to this family of algorithms such as treatment of dense columns are discussed the code is tested on several standard linear programming problems and compares favorably with the simplex codeminos 4 0

a two stage method for the design of near field broadband beamformer in this paper the design of near field broadband beamformer is considered the idea is to design the beamformer filter coefficients such that the error between the actual response and the desired response is minimized this problem can be formulated as a minimax optimization problem and can be transformed into a linear semi definite programming problem interior point algorithm can then be applied after discretization however if the dimension of the problem is high the number of discrete points of the specified region and consequently the dimension of the resultant constraint matrix is high this high dimension provides a difficulty when directly employing interior point algorithm since it results in a large size problem this problem is considered in this paper to reduce the computational complexity and memory usage a two stage method has been proposed in the first stage by using an infinite length filter formulation an infinite length limit for the original problem is obtained then based on that limit a reduced problem with a finite filter length is found the computational complexity of this method is derived it provides a significant reduction compared to a direct solution of the original problem it is shown that as the filter length increases the approximative problem comes close to the optimal infinite solution furthermore we demonstrate with examples the close correspondence between proposed design and optimal design also the superiority of the suggested method is illustrated in a three dimensional beamformer design where the direct method can only design broadband beamformer of very short filter lengths

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

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

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

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

cones of matrices and set functions and 0 1 optimization it has been recognized recently that to represent a polyhedron as the projection of a higher dimensional but simpler polyhedron is a powerful tool in polyhedral combinatorics a general method is developed to construct higher dimensional polyhedra or in some cases convex sets whose projection approximates the convex hull of 0 1 valued solutions of a system of linear inequalities an important feature of these approximations is that one can optimize any linear objective function over them in polynomial time in the special case of the vertex packing polytope a sequence of systems of inequalities is obtained such that the first system already includes clique odd hole odd antihole wheel and orthogonality constraints in particular for perfect and many other graphs this first system gives the vertex packing polytope for various classes of graphs including t perfect graphs it follows that the stable set polytope is the projection of a polytope with a polynomial number of facets an extension o

robust optimization of sums of piecewise linear functions with application to inventory problems robust optimization is a methodology that has gained a lot of attention in the recent years this is mainly due to the simplicity of the modeling process and ease of resolution even for large scale models unfortunately the second property is usually lost when the cost function that needs to be robustified is not concave or linear with respect to the perturbing parameters in this paper we study robust optimization of sums of piecewise linear functions over polyhedral uncertainty set given that these problems are known to be intractable we propose a new scheme for constructing conservative approximations based on the relaxation of an embedded mixed integer linear program and relate this scheme to methods that are based on exploiting affine decision rules our new scheme gives rise to two tractable models that respectively take the shape of a linear program and a semidefinite program with the latter having the potential to provide solutions of better quality than the former at the price of heavie

a matrix generation approach for eigenvalue optimization we study the extension of a column generation technique to nonpolyhedral models in particular we study the problem of minimizing the maximum eigenvalue of an affine combination of symmetric matrices at each step of the algorithm a restricted master problem in the primal space corresponding to the relaxed dual original problem is formed a query point is obtained as an approximate analytic center of a bounded set that contains the optimal solution of the dual problem the original objective function is evaluated at the query point and depending on its differentiability a column or a matrix is added to the restricted master problem we discuss the issues of recovering feasibility after the restricted master problem is updated by a column or a matrix the computational experience of implementing the algorithm on randomly generated problems are reported and the cpu time of the matrix generation algorithm is compared with that of the primal dual interior point methods on dense and sparse problems using the software sdpt3 our numerical results illustrate that the matrix generation algorithm outperforms primal dual interior point methods on dense problems with no structure and also on a class of sparse problems

affine scaling algorithm fails for semidefinite programming in this paper we introduce an affine scaling algorithm for semidefinite programming sdp and give an example of a semidefinite program such that the affine scaling algorithm converges to a non optimal point both our program and its dual have interior feasible solutions and unique optimal solutions which satisfy strict complementarity and they are non degenerate everywhere

numerical evaluation of sbmethod we report numerical results for sbmethod a publically available implementation of the spectral bundle method applied to the 7 th dimacs challenge test sets that are semidefinite relaxations of combinatorial optimization problems the performance of the code is heavily influenced by parameters that control bundle update and eigenvalue computation unfortunately no mathematically sound guidelines for setting them are known based on our experience with sbmethod we propose heuristics for dynamically updating the parameters as well as a heuristc for improving the starting point these are now the default settings of sbmethod version 1 1 we compare their performance on the dimacs instances to our previous best choices for version 1 0 sbmethod version 1 1 is also part of the independent dimacs benchmark by h mittelmann based on these results we try to analyze strengths and weaknesses of our approach in comparison to other codes for large scale semidefinite programming

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

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

solving large scale sparse semidefinite programs for combinatorial optimization we present a dual scaling interior point algorithm and show how it exploits the structure and sparsity of some large scale problems we solve the positive semidefinite relaxation of combinatorial and quadratic optimization problems subject to boolean constraints we report the first computational results of interior point algorithms for approximating maximum cut semidefinite programs with dimension up to 3 000

a feasible method for optimization with orthogonality constraints minimization with orthogonality constraints e g x top x i and or spherical constraints e g vert x vert 2 1 has wide applications in polynomial optimization combinatorial optimization eigenvalue problems sparse pca p harmonic flows 1 bit compressive sensing matrix rank minimization etc these problems are difficult because the constraints are not only non convex but numerically expensive to preserve during iterations to deal with these difficulties we apply the cayley transform a crank nicolson like update scheme to preserve the constraints and based on it develop curvilinear search algorithms with lower flops compared to those based on projections and geodesics the efficiency of the proposed algorithms is demonstrated on a variety of test problems in particular for the maxcut problem it exactly solves a decomposition formulation for the sdp relaxation for polynomial optimization nearest correlation matrix estimation and extreme eigenvalue problems the proposed algorithms run very fast and return solutions no worse than those from their state of the art algorithms for the quadratic assignment problem a gap 0 842 to the best known solution on the largest problem tai256c in qaplib can be reached in 5 min on a typical laptop

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

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

rotation invariant hog descriptors using fourier analysis in polar and spherical coordinates the histogram of oriented gradients hog is widely used for image description and proves to be very effective in many vision problems rotation invariant analysis is necessary or preferred popular solutions are mainly based on pose normalization or learning neglecting some intrinsic properties of rotations this paper presents a method to build rotation invariant hog descriptors using fourier analysis in polar spherical coordinates which are closely related to the irreducible representation of the 2d 3d rotation groups this is achieved by considering a gradient histogram as a continuous angular signal which can be well represented by the fourier basis 2d or spherical harmonics 3d as rotation invariance is established in an analytical way we can avoid discretization artifacts and create a continuous mapping from the image to the feature space in the experiments we first show that our method outperforms the state of the art in a public dataset for a car detection task in aerial images we further use the princeton shape benchmark and the shrec 2009 generic shape benchmark to demonstrate the high performance of our method for similarity measures of 3d shapes finally we show an application on microscopic volumetric data

the analytic center cutting plane method with semidefinite cuts we analyze an analytic center cutting plane algorithm for convex feasibility problems with semidefinite cuts the problem of interest seeks a feasible point in a bounded convex set which contains a full dimensional ball with varepsilon 0 is a condition number of the field of cuts

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

dual versus primal dual interior point methods for linear and conic programming we observe a curious property of dual versus primal dual path following interior point methods when applied to unbounded linear or conic programming problems in dual form while primal dual methods can be viewed as implicitly following a central path to detect primal infeasibility and dual unboundedness dual methods can sometimes implicitly move away from the analytic center of the set of infeasibility unboundedness detectors

the approximability of max csp with fixed value constraints in the maximum constraint satisfaction problem max csp one is given a finite collection of possibly weighted constraints on overlapping sets of variables and the goal is to assign values from a given finite domain to the variables so as to maximize the number or the total weight for the weighted case of satisfied constraints this problem is np hard in general and therefore it is natural to study how restricting the allowed types of constraints affects the approximability of the problem in this article we show that any max csp problem with a finite set of allowed constraint types which includes all fixed value constraints i e constraints of the form x e a is either solvable exactly in polynomial time or else is apx complete even if the number of occurrences of variables in instances is bounded moreover we present a simple description of all polynomial time solvable cases of our problem this description relies on the well known algebraic combinatorial property of supermodularity

robust stability analysis of sparsely interconnected uncertain systems in this paper we consider robust stability analysis of large scale sparsely interconnected uncertain systems by modeling the interconnections among the subsystems with integral quadratic constraints we show that robust stability analysis of such systems can be performed by solving a set of sparse linear matrix inequalities we also show that a sparse formulation of the analysis problem is equivalent to the classical formulation of the robustness analysis problem and hence does not introduce any additional conservativeness the sparse formulation of the analysis problem allows us to apply methods that rely on efficient sparse factorization techniques and our numerical results illustrate the effectiveness of this approach compared to methods that are based on the standard formulation of the analysis problem

approximating quadratic programming with bound and quadratic constraints we consider the problem of approximating the global maximum of a quadratic program qp subject to bound and simple quadratic constraints based on several early results we show that a 4 7 approximate solution can be obtained in polynomial time