a newton cg augmented lagrangian method for semidefinite programming we consider a newton cg augmented lagrangian method for solving semidefinite programming sdp problems from the perspective of approximate semismooth newton methods in order to analyze the rate of convergence of our proposed method we characterize the lipschitz continuity of the corresponding solution mapping at the origin for the inner problems we show that the positive definiteness of the generalized hessian of the objective function in these inner problems a key property for ensuring the efficiency of using an inexact semismooth newton cg method to solve the inner problems is equivalent to the constraint nondegeneracy of the corresponding dual problems numerical experiments on a variety of large scale sdp problems with the matrix dimension n up to 4 110 and the number of equality constraints m up to 2 156 544 show that the proposed method is very efficient we are also able to solve the sdp problem fap36 with n 4 110 and m 1 154 467 in the seventh dimacs implementation challenge much more accurately than in previous attempts

on the moreau yosida regularization of the vector k norm related functions matrix optimization problems mops involving the ky fan k norm arise frequently from many applications in order to design algorithms to solve large scale mops involving the ky fan k norm we need to understand the first and second order properties of the moreau yosida regularization of the ky fan k norm function and the indicator function of the ky fan k norm ball according to the general theory on spectral functions in this paper we shall conduct a thorough study on the moreau yosida regularization of the vector k norm function and the indicator function of the vector k norm ball in particular we show that the proximal mappings associated with these two vector k norm related functions both admit fast and analytically computable solutions moreover we propose algorithms of low computational cost to compute the directional derivatives of these two proximal mappings and then completely characterize their frechet differentiability the work here thus builds the fundamental tools needed

solving lift and project relaxations of binary integer programs we propose a method for optimizing the lift and project relaxations of binary integer programs introduced by lovasz and schrijver in particular we study both linear and semidefinite relaxations the key idea is a restructuring of the relaxations which isolates the complicating constraints and allows for a lagrangian approach we detail an enhanced subgradient method and discuss its efficient implementation computational results illustrate that our algorithm produces tight bounds more quickly than state of the art linear and semidefinite solvers

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

the inverse mean curvature flow and the riemannian penrose inequality let m be an asymptotically flat 3 manifold of nonnegative scalar curvature the riemannian penrose inequality states that the area of an outermost minimal surface n in m is bounded by the adm mass m according to the formula n 16 m 2 we develop a theory of weak solutions of the inverse mean curvature flow and employ it to prove this inequality for each connected component of n using geroch s monotonicity formula for the adm mass our method also proves positivity of bartnik s gravitational capacity by computing a positive lower bound for the mass purely in terms of local geometry

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

a multibody factorization method for independently moving objects the structure from motion problem has been extensively studied in the field of computer vision yet the bulk of the existing work assumes that the scene contains only a single moving object the more realistic case where an unknown number of objects move in the scene has received little attention especially for its theoretical treatment in this paper we present a new method for separating and recovering the motion and shape of multiple independently moving objects in a sequence of images the method does not require prior knowledge of the number of objects nor is dependent on any grouping of features into an object at the image level for this purpose we introduce a mathematical construct of object shapes called the shape interaction matrix which is invariant to both the object motions and the selection of coordinate systems this invariant structure is computable solely from the observed trajectories of image features without grouping them into individual objects once the matrix is computed it allows for segmenting features into objects by the process of transforming it into a canonical form as well as recovering the shape and motion of each object the theory works under a broad set of projection models scaled orthography paraperspective and affine but they must be linear so it excludes projective cameras

on the stable solution of large scale problems over the doubly nonnegative cone the recent approach of solving large scale semidefinite programs with a first order method by minimizing an augmented primal dual function is extended to doubly nonnegative programs a key point governing the convergence of this approach are regularity properties of the underlying problem regularity of the augmented primal dual function is established under the condition of uniqueness and strict complementarity the application to the doubly nonnegative cone is motivated by the fact that the cost per iteration does not increase by adding nonnegativity constraints numerical experiments indicate that a two phase approach based on the augmented primal dual function results in a stable method for solving large scale problems

one sided complete stable minimal surfaces we prove that there are no complete one sided stable minimal surfaces in the euclidean 3 space we classify least area surfaces in the quotient of mathbb r 3 by one or two linearly independent translations and we give sharp upper bounds of the genus of compact two sided index one minimal surfaces in non negatively curved ambient spaces finally we estimate from below the index of complete minimal surfaces in flat spaces in terms of the topology of the surface

a modified alternating direction method for convex quadratically constrained quadratic semidefinite programs we propose a modified alternating direction method for solving convex quadratically constrained quadratic semidefinite optimization problems the method is a first order method therefore requires much less computational effort per iteration than the second order approaches such as the interior point methods or the smoothing newton methods in fact only a single inexact metric projection onto the positive semidefinite cone is required at each iteration we prove global convergence and provide numerical evidence to show the effectiveness of this method

iteration complexity of block decomposition algorithms and the alternating direction method of multipliers in this paper we consider the monotone inclusion problem consisting of the sum of a continuous monotone map and a point to set maximal monotone operator with a separable two block structure and introduce a framework of block decomposition prox type algorithms for solving it which allows for each one of the single block proximal subproblems to be solved in an approximate sense moreover by showing that any method in this framework is also a special instance of the hybrid proximal extragradient hpe method introduced by solodov and svaiter we derive corresponding convergence rate results we also describe some instances of the framework based on specific and inexpensive schemes for solving the single block proximal subproblems finally we consider some applications of our methodology to establish for the first time i the iteration complexity of an algorithm for finding a zero of the sum of two arbitrary maximal monotone operators and as a consequence the ergodic iteration complexity of the douglas

compactness of minimal hypersurfaces with bounded index we prove a compactness result for minimal hypersurfaces with bounded index and volume which can be thought of as an extension of the compactness theorem of choi schoen 4 to higher dimensions

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

an embedded minimal surface with no symmetries we construct embedded minimal surfaces of finite total curvature in euclidean space by gluing catenoids and planes we use weierstrass representation and we solve the period problem using the implicit function theorem as a corollary we obtain the existence of minimal surfaces with no symmetries

solving k cluster problems to optimality with semidefinite programming this paper deals with the computation of exact solutions of a classical np hard problem in combinatorial optimization the k cluster problem this problem consists in finding a heaviest subgraph with k nodes in an edge weighted graph we present a branch and bound algorithm that applies a novel bounding procedure based on recent semidefinite programming techniques we use new semidefinite bounds that are less tight than the standard semidefinite bounds but cheaper to get the experiments show that this approach is competitive with the best existing ones

the space of embedded minimal surfaces of fixed genus in a 3 manifold v fixed genus this paper is the fifth and final in a series on embedded minimal surfaces following our earlier papers on disks we prove here two main structure theorems for itnonsimply connected embedded minimal surfaces of any given fixed genus the first of these asserts that any such surface without small necks can be obtained by gluing together two oppositely oriented double spiral staircases the second gives a pair of pants decomposition of any such surface when there are small necks cutting the surface along a collection of short curves after the cutting we are left with graphical pieces that are defined over a disk with either one or two sub disks removed a topological disk with two sub disks removed is called a pair of pants both of these structures occur as different extremes in the two parameter family of minimal surfaces known as the riemann examples

large scale semidefinite programming via a saddle point mirror prox algorithm in this paper we first demonstrate that positive semidefiniteness of a large well structured sparse symmetric matrix can be represented via positive semidefiniteness of a bunch of smaller matrices linked in a linear fashion to the matrix we derive also the dual counterpart of the outlined representation which expresses the possibility of positive semidefinite completion of a well structured partially defined symmetric matrix in terms of positive semidefiniteness of a specific bunch of fully defined submatrices of the matrix using the representations we then reformulate well structured large scale semidefinite problems into smooth convex concave saddle point problems which can be solved by a prox method developed in 6 with efficiency mathcal o epsilon 1 implementations and some numerical results for large scale lovasz capacity and maxcut problems are finally presented

a first eigenvalue estimate for minimal hypersurfaces on obtient une borne inferieure pour la premiere valeur propre d une hypersurface orientable compacte plongee de facon minimale dans une variete orientable compacte a courbure de ricci positive

intrinsic constraints in space time filtering a new approach to representing uncertainty in low level vision describes how in the process of extracting the optical flow through space time filtering one has to consider the constraints associated with the motion uncertainty as well as the spatial and temporal sampling rates of the sequence of images the motion uncertainty satisfies the cramer rao cr inequality which is shown to be a function of the filter parameters on the other hand the spatial and temporal sampling rates have lower bounds which depend on the motion uncertainty the maximum support in the frequency domain and the optical flow these lower bounds on the sampling rates and on the motion uncertainty are constraints that constitute an intrinsic part of the computational structure of space time filtering the author shows that if he uses these constraints simultaneously the filter parameters cannot be arbitrarily determined but instead have to satisfy consistency constraints by using explicit representations of uncertainties in extracting visual attributes one can constrain the range of values assumed by the filter parameters

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

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

complete embedded minimal surfaces of finite total curvature we will survey what is known about minimal surfaces s r 3 which are complete embedded and have finite total curvature int s k da infty the only classically known examples of such surfaces were the plane and the catenoid the discovery by costa 14 15 early in the last decade of a new example that proved to be embedded sparked a great deal of research in this area many new examples have been found even families of them as will be described below the central question has been transformed from whether or not there are any examples except surfaces of rotation to one of understanding the structure of the space of examples

regularization methods for sdp relaxations in large scale polynomial optimization we study how to solve semidefinite programming sdp relaxations for large scale polynomial optimization when interior point methods are used typically only small or moderately large problems could be solved this paper studies regularization methods for solving polynomial optimization problems we describe these methods for semidefinite optimization with block structures and then apply them to solve large scale polynomial optimization problems the performance is tested on various numerical examples with regularization methods significantly bigger problems could be solved on a regular computer which is almost impossible with interior point methods

fast alternating linearization methods for minimizing the sum of two convex functions we present in this paper alternating linearization algorithms based on an alternating direction augmented lagrangian approach for minimizing the sum of two convex functions our basic methods require at most o 1 epsilon iterations to obtain an epsilon optimal solution while our accelerated i e fast versions of them require at most o 1 sqrt epsilon iterations with little change in the computational effort required at each iteration for both types of methods we present one algorithm that requires both functions to be smooth with lipschitz continuous gradients and one algorithm that needs only one of the functions to be so algorithms in this paper are gauss seidel type methods in contrast to the ones proposed by goldfarb and ma in fast multiple splitting algorithms for convex optimization columbia university 2009 where the algorithms are jacobi type methods numerical results are reported to support our theoretical conclusions and demonstrate the practical potential of our algorithms

minimal hypersurfaces with bounded index we prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal hypersurfaces with uniformly bounded index and volume if the ambient dimension is greater than three in a riemannian manifold m n g 3 le n le 7 can degenerate loosely speaking our results show that embedded minimal hypersurfaces with bounded index behave qualitatively like embedded stable minimal hypersurfaces up to controlled errors several compactness finiteness theorems follow from our local picture

an inexact accelerated proximal gradient method for large scale linearly constrained convex sdp the accelerated proximal gradient apg method first proposed by nesterov for minimizing smooth convex functions later extended by beck and teboulle to composite convex objective functions and studied in a unifying manner by tseng has proven to be highly efficient in solving some classes of large scale structured convex optimization possibly nonsmooth problems including nuclear norm minimization problems in matrix completion and l 1 minimization problems in compressed sensing the method has superior worst case iteration complexity over the classical projected gradient method and usually has good practical performance on problems with appropriate structures in this paper we extend the apg method to the inexact setting where the subproblem in each iteration is solved only approximately and show that it enjoys the same worst case iteration complexity as the exact counterpart if the subproblems are progressively solved to sufficient accuracy we apply our inexact apg method to solve large scale

the tracial moment problem and trace optimization of polynomials the main topic addressed in this paper is trace optimization of polynomials in noncommuting nc variables given an nc polynomial f what is the smallest trace f underline a can attain for a tuple of matrices underline a a relaxation using semidefinite programming sdp based on sums of hermitian squares and commutators is proposed while this relaxation is not always exact it gives effectively computable bounds on the optima to test for exactness the solution of the dual sdp is investigated if it satisfies a certain condition called flatness then the relaxation is exact in this case it is shown how to extract global trace optimizers with a procedure based on two ingredients the first is the solution to the truncated tracial moment problem and the other crucial component is the numerical implementation of the artin wedderburn theorem for matrix algebras due to murota kanno kojima kojima and maehara trace optimization of nc polynomials is a nontrivial extension of polynomial optimization in commuting variables on one side and eigenvalue optimization of nc polynomials on the other side two topics with many applications the most prominent being to linear systems engineering and quantum physics the optimization problems discussed here facilitate new possibilities for applications e g in operator algebras and statistical physics

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

analyzing looming motion components from their spatiotemporal spectral signature this paper addresses the use of spatio temporal transform methods applied to the analysis of dynamic image sequences and the characterization of image motion the image motion including a divergent component resulting from a looming camera component is analyzed in the spatio temporal mellin transform mt domain resulting in the separation of the spectrum into two parts a structural term corresponding to the spatial mt of the static image and a kinematic term depending on time to collision a motion support we examine potential applications of this property for the recovery of image motion from integral image brightness measurements and the computation of time to collision using spatio temporal mt analysis

solving large scale semidefinite programs via an iterative solver on the augmented systems the search directions in an interior point method for large scale semidefinite programming sdp can be computed by applying a krylov iterative method to either the schur complement equation sce or the augmented equation both methods suffer from slow convergence as interior point iterates approach optimality numerical experiments have shown that a diagonally preconditioned conjugate residual method on the sce typically takes a huge number of steps to converge however it is difficult to incorporate cheap and effective preconditioners into the sce this paper proposes to apply the preconditioned symmetric quasi minimal residual psqmr method to a reduced augmented equation that is derived from the augmented equation by utilizing the eigenvalue structure of the interior point iterates numerical experiments on sdp problems arising from maximum clique and selected sdplib sdp library problems show that moderately accurate solutions can be obtained with a modest number of psqmr steps using the proposed preconditioned reduced augmented equation an sdp problem with 127600 constraints is solved in about 6 5 hours to an accuracy of 10 6 in relative duality gap

semidefinite programming relaxations for graph coloring and maximal clique problems the semidefinite programming formulation of the lovasz theta number does not only give one of the best polynomial simultaneous bounds on the chromatic number g or the clique number g of a graph but also leads to heuristics for graph coloring and extracting large cliques this semidefinite programming formulation can be tightened toward either g or g by adding several types of cutting planes we explore several such strengthenings and show that some of them can be computed with the same effort as the theta number we also investigate computational simplifications for graphs with rich automorphism groups

a local regularity theorem for mean curvature flow this paper proves curvature bounds for mean curvature flows and other related flows in regions of spacetime where the gaussian densities are close to 1

regularization methods for semidefinite programming we introduce a new class of algorithms for solving linear semidefinite programming sdp problems our approach is based on classical tools from convex optimization such as quadratic regularization and augmented lagrangian techniques we study the theoretical properties and we show that practical implementations behave very well on some instances of sdp having a large number of constraints we also show that the boundary point method from povh rendl and wiegele computing 78 2006 pp 277 286 is an instance of this class

constrained polynomial optimization problems with noncommuting variables in this paper we study constrained eigenvalue optimization of noncommutative nc polynomials focusing on the polydisc and the ball our three main results are as follows 1 an nc polynomial is nonnegative if and only if it admits a weighted sum of hermitian squares decomposition 2 eigenvalue optima for nc polynomials can be computed using a single semidefinite program sdp this sharply contrasts with the commutative case where sequences of sdps are needed 3 the dual solution to this single sdp can be exploited to extract eigenvalue optimizers with an algorithm based on two ingredients solution to a truncated nc moment problem via flat extensions and gelfand naimark segal construction the implementation of these procedures in our computer algebra system ncsostools is presented and several examples pertaining to matrix inequalities are given to illustrate our results

alternating direction methods for latent variable gaussian graphical model selection chandrasekaran parrilo and willsky 2012 proposed a convex optimization problem for graphical model selection in the presence of unobserved variables this convex optimization problem aims to estimate an inverse covariance matrix that can be decomposed into a sparse matrix minus a low rank matrix from sample data solving this convex optimization problem is very challenging especially for large problems in this letter we propose two alternating direction methods for solving this problem the first method is to apply the classic alternating direction method of multipliers to solve the problem as a consensus problem the second method is a proximal gradient based alternating direction method of multipliers our methods take advantage of the special structure of the problem and thus can solve large problems very efficiently a global convergence result is established for the proposed methods numerical results on both synthetic data and gene expression data show that our methods usually solve problems with 1 million variables in 1 to 2 minutes and are usually 5 to 35i times faster than a state of the art newton cg proximal point algorithm

the space of minimal embeddings of a surface into a three dimensional manifold of positive ricci curvature in this paper we obtain a curvature estimate for embedded minimal surfaces in a three dimensional manifold of positive ricci curvature in terms of the geometry of the ambient manifold and the genus of the minimal surface it should be mentioned that there are two main points in our result one is the absence of a stability assumption and the other is the requirement of being embedded most known curvature estimates require the stability assumption and once the stability assumption is dropped many of these known results cease to be valid see ss and an for another example of a curvature estimate without the assumption of stability the embeddedness condition is rather subtle because of the way it enters in our proof our proof depends on the eigenvalue estimate and the area bound due to the first author and wang cw which require embeddedness in an essential way present knowledge indicates that closed embedded minimal surfaces in s 3 are rare while immersed surfaces are more plentiful for example only a finite number of minimal embeddings of a given genus are known on the other hand otsuki o constructed infinitely many immersed minimal tori with arbitrarily large area we obtain the curvature estimate indirectly by proving the smooth compactness theorem theorem 1 theorem 1 has many interesting consequences it shows that the set of conformal structures that can be realized on a minimal embedding in s 3 is a compact subset of the moduli space this is in contrast with the result of bryant b who showed that every riemann surface is conformally and minimally immersed in s 4 in view of our compactness result and the scarcity of examples it is very tempting to conjecture that there are only finitely many embedded minimal surfaces up to rigid motion in s 3 for each fixed genus throughout this paper manifold means manifold without boundary unless explictly stated otherwise when we say a sequence m i of surfaces converges to a

chordal decomposition in operator splitting methods for sparse semidefinite programs we employ chordal decomposition to reformulate a large and sparse semidefinite program sdp either in primal or dual standard form into an equivalent sdp with smaller positive semidefinite psd constraints in contrast to previous approaches the decomposed sdp is suitable for the application of first order operator splitting methods enabling the development of efficient and scalable algorithms in particular we apply the alternating direction method of multipliers admm to solve decomposed primal and dual standard form sdps each iteration of such admm algorithms requires a projection onto an affine subspace and a set of projections onto small psd cones that can be computed in parallel we also formulate the homogeneous self dual embedding hsde of a primal dual pair of decomposed sdps and extend a recent admm based algorithm to exploit the structure of our hsde the resulting hsde algorithm has the same leading order computational cost as those for the primal or dual problems only with the advantage of being able to identify infeasible problems and produce an infeasibility certificate all algorithms are implemented in the open source matlab solver cdcs numerical experiments on a range of large scale sdps demonstrate the computational advantages of the proposed methods compared to common state of the art solvers

least area incompressible surfaces in 3 manifolds let m be a riemannian manifold and let f be a closed surface a map f f m is called least area if the area of f is less than the area of any homotopic map from f to m note that least area maps are always minimal surfaces but that in general minimal surfaces are not least area as they represent only local stationary points for the area function the existence of least area immersions in a homotopy class of maps has been established when the homotopy class satisfies certain injectivity conditions on the fundamental group 18 17 in this paper we shall consider the possible singularities of such immersions our results show that the general philosophy is that least area surfaces intersect least meaning that the intersections and self intersections of least area immersions are as small as their homotopy classes allow when measured correctly one should note that evidence supporting this view had been found by meeks yau in their embedding theorems for minimal disks and 2 spheres 13 143 our main result asserts that if a least area immersion is homotopic to an embedding then it has no self intersections which clearly exemplifies the above philosophy the precise result is the following

an introduction to a class of matrix cone programming in this paper we define a class of linear conic programming which we call matrix cone programming or mcp involving the epigraphs of five commonly used matrix norms and the well studied symmetric cone mcp has recently been found to have many important applications for example in nuclear norm relaxations of affine rank minimization problems in order to make the defined mcp tractable and meaningful we must first understand the structure of these epigraphs so far only the epigraph of the frobenius matrix norm which can be regarded as a second order cone has been well studied here we take an initial step to study several important properties including its closed form solution calm bouligand differentiability and strong semismoothness of the metric projection operator over the epigraph of the l 1 l infty spectral or operator and nuclear matrix norm respectively these properties make it possible to apply augmented lagrangian methods which have recently received a great deal of interests due to their high efficiency in solving large scale semidefinite programming to this class of mcp problems the work done in this paper is far from comprehensive rather it is intended as a starting point to call for more insightful research on mcp so that it can serve as a basic tool to solve more challenging convex matrix optimization problems in years to come

large scale binary quadratic optimization using semidefinite relaxation and applications in computer vision many problems can be formulated as binary quadratic programs bqps which are in general np hard finding a solution when the problem is of large size to be of practical interest typically requires relaxation semidefinite relaxation usually yields tight bounds but its computational complexity is high in this work we present a semidefinite programming sdp formulation for bqps with two desirable properties first it produces similar bounds to the standard sdp formulation second compared with the conventional sdp formulation the proposed sdp formulation leads to a considerably more efficient and scalable dual optimization approach we then propose two solvers namely quasi newton and smoothing newton methods for the simplified dual problem both of them are significantly more efficient than standard interior point methods empirically the smoothing newton solver is faster than the quasi newton solver for dense or medium sized problems while the quasi newton solver is preferable for large sparse structured problems

strong lift and project cutting planes for the stable set problem a great deal of research has been focusing since the early seventies on finding strong relaxations for the stable set problem polyhedral combinatorics techniques have been at first developed to strengthen the natural linear formulation afterward strong semidefinite programming relaxations have been deeply investigated nevertheless the resulting integer programming ip algorithms cannot be regarded as being quite successful in practice as most of the relaxations give rise to one out of two extreme situations either provide weak bounds at low computational cost or give good bounds sometimes excellent but too demanding to compute in this paper we present a method to bridge such a gap in particular a new lift and project relaxation is obtained by a problem specific variant of the lifting operator m k k by lovasz and schrijver combined with benders decomposition this yields strong cutting planes generated by solving a cut generating linear program an extensive computational experience shows that embedding these cuts in a branch and cut framework significantly reduces the size of the enumeration trees as well as the cpu times with respect to state of the art ip algorithms

an application of the lovasz schrijver m k k operator to the stable set problem although the lift and project operators of lovasz and schrijver have been the subject of intense study their m k k operator has received little attention we consider an application of this operator to the stable set problem we begin with an initial linear programming lp relaxation consisting of clique and non negativity inequalities and then apply the operator to obtain a stronger extended lp relaxation we discuss theoretical properties of the resulting relaxation describe the issues that must be overcome to obtain an effective practical implementation and give extensive computational results remarkably the upper bounds obtained are sometimes stronger than those obtained with semidefinite programming techniques