benchmarking optimization software with performance profiles we propose performance profiles distribution functions for a performance metric as a tool for benchmarking and comparing optimization software we show that performance profiles combine the best features of other tools for performance evaluation

relaxing nonconvex quadratic functions by multiple adaptive diagonal perturbations the current bottleneck of globally solving mixed integer nonconvex quadratically constrained programming problems miqcps is still to construct strong but computationally cheap convex relaxations especially when dense quadratic functions are present we propose a cutting surface method based on multiple diagonal perturbations to derive convex quadratic relaxations for nonconvex quadratic problems with separable constraints our relaxations can also be realized as outer approximations of a semi infinite convex formulation of a semidefinite relaxation proposed by buchheim and wiegele the corresponding separation problem is a highly structured semidefinite program sdp with a convex but nonsmooth objective function we propose solving this separation problem with a specialized barrier coordinate minimization algorithm numerical experiments on randomly generated instances show that our approach is very promising we also discuss how to apply our method to more general cases of miqcps

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 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

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

localization in equivariant intersection theory and the bott residue formula we prove the localization theorem for torus actions in equivariant intersection theory using the theorem we give another proof of the bott residue formula for chern numbers of bundles on smooth complete varieties in addition our techniques allow us to obtain residue formulas for bundles on a certain class of singular schemes which admit torus actions this class is rather special but it includes some interesting examples such as complete intersections and schubert varieties 1 introduction the purpose of this paper is to prove the localization theorem for torus actions in equivariant intersection theory using the theorem we give another proof of the bott residue formula for chern numbers of bundles on smooth complete varieties in addition our techniques allow us to obtain residue formulas for bundles on a certain class of singular schemes which admit torus actions this class is rather special but it includes some interesting examples such as complete intersections cf bfq and schubert varieties let t be a split torus acting on a scheme x the 7 equivariant chow groups of x are a module over rt sym t where f is the character group of t the localization theorem states that up to torsion the equivariant chow groups of the fixed locus xt are isomorphic to those of x such a theorem is a hallmark of any equivariant theory the earliest version for equivariant cohomology is due to borel bo subsequently it theory versions were proved by segal se in topological a theory quart qu for actions of a cyclic group and thomason th2 for algebraic theory th2 for equivariant chow groups the localization isomorphism is given by the equivariant push forward induced by the inclusion of xt to x an interesting aspect of this theory is that the push forward is naturally defined on the level of cycles even in the singular case the closest topological analogue of this is equivariant borel moore homology see e g3 for a definition and a similar proof establishes localization in that theory for smooth spaces the inverse to the equivariant push forward can be written explicitly it was realized independently by several authors i n a b b v

on the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues we derive some basic results on the geometry of semidefinite programming sdp and eigenvalue optimization i e the minimization of the sum of the k largest eigenvalues of a smooth matrix valued function we provide upper bounds on the rank of extreme matrices in sdps and the first theoretically solid explanation of a phenomenon of intrinsic interest in eigenvalue optimization in the spectrum of an optimal matrix the kth and k 1 st largest eigenvalues tend to be equal and frequently have multiplicity greater than two this clustering is intuitively plausible and has been observed as early as 1975 when the matrix valued function is affine we prove that clustering must occur at extreme points of the set of optimal solutions if the number of variables is sufficiently large we also give a lower bound on the multiplicity of the critical eigenvalue these results generalize to the case of a general matrix valued function under appropriate conditions

an unconstrained minimization method for solving low rank sdp relaxations of the maxcut problem in this paper we consider low rank semidefinite programming lrsdp relaxations of combinatorial quadratic problems that are equivalent to the maxcut problem using the gramian representation of a positive semidefinite matrix the lrsdp problem can be formulated as the nonconvex nonlinear programming problem of minimizing a quadratic function with quadratic equality constraints for the solution of this problem we propose a continuously differentiable exact merit function that exploits the special structure of the constraints and we use this function to define an efficient and globally convergent algorithm finally we test our code on an extended set of instances of the maxcut problem and we report comparisons with other existing codes

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

prefiltering for improved correlation detection of bandlimited transient signals prefiltering or limiting the passband of a received signal can be used to improve ordinary correlation threshold detector performance for an unknown source model for the known source model the cross correlation detector is equivalent to matched filtering and intrinsically contains prefiltering prefiltering also improves higher order correlation threshold detector performance but often with more advantage than seen in the ordinary correlation detector this is true for both the unknown and known source models geometric interpretations are given to provide insight into the origin of potential higher order advantage eight energy signals each with three different fourier magnitude based filters are used to test the cross correlation bicorrelation and tricorrelation detectors by monte carlo simulation and hypothesis testing significant signal to noise ratio snr gains are evident for both the known and unknown source models with the tricorrelation exhibiting the largest gains the tricorrelation detector has superior performance versus the cross correlation detector for all the eight test signals in the unknown source case and was superior for seven of the eight test signals in the known source case for the narrow pulse test signal a probability of detection of 0 5 at probability of false alarm 0 001 was achieved at a power snr 23 57 db in the known source model and at snr 20 28 db in the unknown source model some other test signals have lower snrs in the unknown source model higher order detectors can be limited to have equivalent information to that of the ordinary correlation detector an example illustrating this case is given

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

prediction of signal to noise ratio gain for passive higher order correlation detection of energy transients in general higher order correlation detectors perform well in passive detection for signals of high third and fourth order moments previous studies by the authors have shown that the normalized third and fourth order signal moments are reliable indicators of higher order correlation detector performance pflug et al 1992b for a deterministic energy transient of known moments through fourth order it is possible to predict theoretically the amount of gain over an ordinary cross correlation detector for a bicorrelation or tricorrelation detector applied in a noise environment of known variance in this paper formulas that predict detector performance for passive detection at the minimum detectable level are derived the noise is assumed to be stationary and zero mean with gaussian correlation central ordinate probability density functions to test the formulas snr detection and gain curves are generated using hypothesis testing and monte carlo simulations on a set of test signals the test signal

detection of non gaussian signals in non gaussian noise using the bispectrum the problem of detecting a non gaussian time series in the presence of additive gaussian or non gaussian noise is cast into a classical hypothesis testing framework using the sample bispectrum as the test statistic the power of the test is demonstrated as a function of signal to noise ratio the degree of skewness of the signal and processing parameters the results are compared to the power of a classical energy detection test it is concluded that the bispectrum can be used effectively to detect non gaussian signals in the presence of interfering noise and that it may perform better depending on the degree of non gaussianity than energy detection

detection of oscillatory and impulsive transients using higher order correlations and spectra higher order cross and ordinary correlation detectors are applied to four deterministic transients contaminated by uncorrelated gaussian noise only histograms and moments are used to examine the properties of the signals and their effect on detector performance receiver operating characteristic roc curve analysis and limiting signal to noise ratios for good detection provide comparative measures for different detectors probability density functions of detection ordinate values of signal present and noise only correlations are used to explain roc curve behavior using a known source the cross correlation detector performs better than the higher order correlation detectors for each transient studied however for an unknown narrow pulse source signal the bicorrelation and tricorrelation detectors outperform the cross correlation detector in contrast the bicorrelation detector performs very poorly for low frequency narrow band signals with a small third moment embedded in uncorrelated gaussian noise rectification as part of the detection process improves the performance of the bicorrelation detector and also places the peak of maximum magnitude at the origin this eliminates the problem in detection or time delay estimation that the existence of multiple peaks due to symmetries in the auto bicorrelation or the bicorrelation of repeated signals may cause the tricorrelation detector also performs better with rectification than without for an unknown source the bicorrelation and tricorrelation detectors with rectification perform on a level comparable to the cross correlation detector for certain signals comparisons are made between repeating a known source and repeating noisy received signals in the bicorrelation

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

speedp an algorithm to compute sdp bounds for very large max cut instances we consider low rank semidefinite programming lrsdp relaxations of unconstrained 1 1 quadratic problems or equivalently of max cut problems that can be formulated as the non convex nonlinear programming problem of minimizing a quadratic function subject to separable quadratic equality constraints we prove the equivalence of the lrsdp problem with the unconstrained minimization of a new merit function and we define an efficient and globally convergent algorithm called speedp for finding critical points of the lrsdp problem we provide evidence of the effectiveness of speedp by comparing it with other existing codes on an extended set of instances of the max cut problem we further include speedp within a simply modified version of the goemans williamson algorithm and we show that the corresponding heuristic speedp mc can generate high quality cuts for very large sparse graphs of size up to a million nodes in a few hours

low rank optimization on the cone of positive semidefinite matrices we propose an algorithm for solving optimization problems defined on a subset of the cone of symmetric positive semidefinite matrices this algorithm relies on the factorization x yy t where the number of columns of y fixes an upper bound on the rank of the positive semidefinite matrix x it is thus very effective for solving problems that have a low rank solution the factorization x yy t leads to a reformulation of the original problem as an optimization on a particular quotient manifold the present paper discusses the geometry of that manifold and derives a second order optimization method with guaranteed quadratic convergence it furthermore provides some conditions on the rank of the factorization to ensure equivalence with the original problem in contrast to existing methods the proposed algorithm converges monotonically to the sought solution its numerical efficiency is evaluated on two applications the maximal cut of a graph and the problem of sparse principal component analysis

sampling requirements and aliasing for higher order correlations while sampling at a nyquist frequency equal to the highest frequency present in the data critical sampling is sufficient to prevent aliasing in both the data and the autocorrelation of a bandlimited energy signal the sampling requirements for the avoidance of aliasing in higher order correlations and spectra are not the same also there is a difference in aliasing effects depending on whether one samples the original continuous time signal and calculates the autocorrelation or one samples the continuous time autocorrelation this distinction between sampling procedures must be made for correlations of higher order as well for which not only the type of aliasing but also the sampling requirements to prevent aliasing differ in particular if one samples the continuous time autobicorrelation or autotricorrelation critical sampling is sufficient to prevent aliasing in practice however it is not usually the continuous time autobicorrelation or autotricorrelation that is sampled generally it is the

properties of higher order correlations and spectra for bandlimited deterministic transients higher order correlations and spectra may be used for detection time delay estimation classification and discrimination of signals for these applications a detailed knowledge of their attributes can be highly useful in this paper the properties of the bicorrelation and tricorrelation of bandlimited deterministic transients i e energy signals and their spectra the bispectrum and trispectrum are studied bandlimited transients that contain frequencies down to and including zero and those that have a nonzero lower cutoff frequency are both considered using symmetries inherent in the bispectrum of a signal the entire bispectrum can be mapped from bispectral elements defined in two polygons one for the unaliased and one for the aliased domain each of which is one twelfth the area of its total domain the nonredundant unaliased region of the trispectrum is contained in two principal unaliased polyhedra each replicated 48 times to reproduce the full trispectrum if there is aliasing in the trisp

a semidefinite programming based search strategy for feature selection with mutual information measure feature subset selection as a special case of the general subset selection problem has been the topic of a considerable number of studies due to the growing importance of data mining applications in the feature subset selection problem there are two main issues that need to be addressed i finding an appropriate measure function than can be fairly fast and robustly computed for high dimensional data ii a search strategy to optimize the measure over the subset space in a reasonable amount of time in this article mutual information between features and class labels is considered to be the measure function two series expansions for mutual information are proposed and it is shown that most heuristic criteria suggested in the literature are truncated approximations of these expansions it is well known that searching the whole subset space is an np hard problem here instead of the conventional sequential search algorithms we suggest a parallel search strategy based on semidefinite programming sdp that can search through the subset space in polynomial time by exploiting the similarities between the proposed algorithm and an instance of the maximum cut problem in graph theory the approximation ratio of this algorithm is derived and is compared with the approximation ratio of the backward elimination method the experiments show that it can be misleading to judge the quality of a measure solely based on the classification accuracy without taking the effect of the non optimum search strategy into account

extremal correlation matrices abstract let r n denote the convex compact set of all real n by n positive semidefinite matrices with main diagonal entries equal to 1 we examine the extreme points of r n focusing mainly on their rank the principal result is that r n contains extreme points of rank k if and only if k k 1 2 n

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

semi definite matrix constraints in optimization positive semi definite matrix constraints arise in a number of optimization problems in which some or all of the elements of a matrix are variables such as the educational testing and matrix modification problems the structure of such constraints is developed including expressions for the normal cone feasible directions and their application to optimality conditions a computational framework is given within which these concepts can be exploited and which permits the quantification of second order effects the matrix of lagrange multipliers in this formulation is shown to have an important relationship to the characterization of the normal cone modifications of various iterative schemes in nonlinear programming are considered in order to develop an effective algorithm for the educational testing problem and comparative numerical experiments are described it is shown that a particular choice of the penalty parameter for an l 1 exact penalty function is appropriate for this type of problem the be

solving a class of semidefinite programs via nonlinear programming in this paper we introduce a transformation that converts a class of linear and nonlinear semidefinite programming sdp problems into nonlinear optimization problems for those problems of interest the transformation replaces matrix valued constraints by vector valued ones hence reducing the number of constraints by an order of magnitude the class of transformable problems includes instances of sdp relaxations of combinatorial optimization problems with binary variables as well as other important sdp problems we also derive gradient formulas for the objective function of the resulting nonlinear optimization problem and show that both function and gradient evaluations have affordable complexities that effectively exploit the sparsity of the problem data this transformation together with the efficient gradient formulas enables the solution of very large scale sdp problems by gradient based nonlinear optimization techniques in particular we propose a first order log barrier method designed for solving a class of large scale linear sdp problems this algorithm operates entirely within the space of the transformed problem while still maintaining close ties with both the primal and the dual of the original sdp problem global convergence of the algorithm is established under mild and reasonable assumptions

known source detection predictions for higher order correlators the problem addressed in this paper is whether higher order correlation detectors can perform better in white noise than the cross correlation detector for the detection of a known transient source signal if additional receiver information is included in the higher order correlations while the cross correlation is the optimal linear detector for white noise additional receiver information in the higher order correlations makes them nonlinear in this paper formulas that predict the performance of higher order correlation detectors of energy signals are derived for a known source signal given the first through fourth order signal moments and the noise variance the formulas predict the snr for which the detectors achieve a probability of detection of 0 5 for any level of false alarm when noise at each receiver is independent and identically distributed results show that the performance of the cross correlation bicorrelation and tricorrelation detectors are proportional to the second fourth and s

using svm to combine global heuristics for the standard quadratic problem abstract the standard quadratic problem stqp is an np hard problem with many local minimizers stationary points in the literature heuristics based on unconstrained continuous non convex formulations have been proposed bomze palagi 2005 bomze grippo palagi 2012 but none dominates the other in terms of best value found following cassioli dilorenzo locatelli schoen sciandrone 2012 we propose to use support vector machines svms to define a multistart global strategy which selects the best heuristic we test our method on stqp arising from the maximum clique problem on a graph which is a challenging combinatorial problem we use as benchmark the clique problems in the dimacs challenge