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

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

positive semidefinite matrix completion universal rigidity and the strong arnold property this paper addresses the following three topics positive semidenite psd matrix completions universal rigidity of frameworks and the strong arnold property sap we show some strong connections among these topics using semidenite programming as unifying theme our main contribution is a sufcient condition for constructing partial psd matrices which admit a unique completion to a full psd matrix such partial matrices are an essential tool in the study of the gram dimension gd g of a graph g a recently studied graph parameter related to the low psd matrix completion problem additionally we derive an elementary proof of connelly s sucient condition for universal rigidity of tensegrity frameworks and we investigate the links between these two sucient conditions we also give a geometric characterization of psd matrices satisfying the strong arnold property in terms of nondegeneracy of an associated semidenite program which we use to establish some links between the gram dimension gd and the colin de verdi ere type graph parameter

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

application of a mixed simulated annealing genetic algorithm heuristic for the two dimensional orthogonal packing problem abstract in this paper a pure meta heuristic genetic algorithm and a mixed meta heuristic simulated annealing genetic algorithm were applied to two dimensional orthogonal packing problems and the results were compared the major motivation for applying a modified genetic algorithm is as an attempt to alleviate the problem of pre mature convergence we found that in the long run the mixed heuristic produces better results while the pure heuristic produces only good results but produces them faster

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

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

nondegeneracy and quantitative stability of parameterized optimization problems with multiple solutions this paper presents some results on quantitative stability of optimal solutions of parameterized optimization problems having nonisolated optima the analysis is based on the introduced concept of nondegeneracy which may be of independent interest examples of nonlinear and semidefinite programming are discussed

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

semidefinite programming in combinatorial optimization we discuss the use of semidefinite programming for combinatorial optimization problems the main topics covered include i the lovasz theta function and its applications to stable sets perfect graphs and coding theory ii the automatic generation of strong valid inequalities iii the maximum cut problem and related problems and iv the embedding of finite metric spaces and its relationship to the sparsest cut problem

an exact two dimensional non guillotine cutting tree search procedure we consider the two dimensional cutting problem of cutting a number of rectangular pieces from a single large rectangle so as to maximize the value of the pieces cut we develop a lagrangean relaxation of a zero one integer programming formulation of the problem and use it as a bound in a tree search procedure subgradient optimization is used to optimize the bound derived from the lagrangean relaxation problem reduction tests derived from both the original problem and the lagrangean relaxation are given incorporating the bound and the reduction tests into a tree search procedure enables moderately sized problems to be solved

quasi similarity solutions for one phase stefan problems with time dependent boundary conditions we consider the one dimensional one phase stefan problems with temperature or thermal flux boundary conditions exact and closed form solutions for these two free boundary problems are very limited and highly restricted to particular initial and boundary conditions and so are the similarity solutions in general only numerical methods allow one to completely solve such nonlinear problems in this paper the quasi similarity solutions are defined and for each problem an analytical approximation of the interfacial position is given as function of the time dependent boundary datum

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

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

on eigenvalue optimization in this paper we study optimization problems involving eigenvalues of symmetric matrices one of the difficulties with numerical analysis of such problems is that the eigenvalues considered as functions of a symmetric matrix are not differentiable at those points where they coalesce we present a general framework for a smooth differentiable approach to such problems it is based on the concept of transversality borrowed from differential geometry in that framework we discuss first and second order optimality conditions and rates of convergence of the corresponding second order algorithms finally we present some results on the sensitivity analysis of such problems

a reduction approach for solving the rectangle packing area minimization problem abstract in the rectangle packing area minimization problem rpamp we are given a set of rectangles with known dimensions we have to determine an arrangement of all rectangles without overlapping inside an enveloping rectangle of minimum area the paper presents a generic approach for solving the rpamp that is based on two algorithms one for the 2d knapsack problem kp and the other for the 2d strip packing problem spp in this way solving an instance of the rpamp is reduced to solving multiple spp and kp instances a fast constructive heuristic is used as spp algorithm while the kp algorithm is instantiated by a tree search method and a genetic algorithm alternatively all these spp and kp methods have been published previously finally the best variants of the resulting rpamp heuristics are combined within one procedure the guillotine cutting condition is always observed as an additional constraint the approach was tested on 15 well known rpamp instances above all mcnc and gsrc instances and new best solutions were obtained for 10 instances the computational effort remains acceptable moreover 24 new benchmark instances are introduced and promising results are reported

the algebraic degree of semidefinite programming given a generic semidefinite program specified by matrices with rational entries each coordinate of its optimal solution is an algebraic number we study the degree of the minimal polynomials of these algebraic numbers geometrically this degree counts the critical points attained by a linear functional on a fixed rank locus in a linear space of symmetric matrices we determine this degree using methods from complex algebraic geometry such as projective duality determinantal varieties and their chern classes

an inexact primal dual path following algorithm for convex quadratic sdp we propose primal dual path following mehrotra type predictor corrector methods for solving convex quadratic semidefinite programming qsdp problems of the form min x frac 1 2 x bullet mathcal q x c bullet x mathcal a x b x succeq 0 where mathcal q is a self adjoint positive semidefinite linear operator on mathcal s n b r m and mathcal a is a linear map from mathcal s n to r m at each interior point iteration the search direction is computed from a dense symmetric indefinite linear system called the augmented equation of dimension m n n 1 2 such linear systems are typically very large and can only be solved by iterative methods we propose three classes of preconditioners for the augmented equation and show that the corresponding preconditioned matrices have favorable asymptotic eigenvalue distributions for fast convergence under suitable nondegeneracy assumptions numerical experiments on a variety of qsdps with n up to 1600 are performed and the computational results show that our methods are efficient and robust

constraint nondegeneracy strong regularity and nonsingularity in semidefinite programming it is known that the karush kuhn tucker kkt conditions of semidefinite programming can be reformulated as a nonsmooth system via the metric projector over the cone of symmetric and positive semidefinite matrices we show in this paper that the primal and dual constraint nondegeneracies the strong regularity the nonsingularity of the b subdifferential of this nonsmooth system and the nonsingularity of the corresponding clarke s generalized jacobian at a kkt point are all equivalent moreover we prove the equivalence between each of these conditions and the nonsingularity of clarke s generalized jacobian of the smoothed counterpart of this nonsmooth system used in several globally convergent smoothing newton methods in particular we establish the quadratic convergence of these methods under the primal and dual constraint nondegeneracies but without the strict complementarity

genericity results in linear conic programming a tour d horizon this paper is concerned with so called generic properties of general linear conic programs many results have been obtained on this subject during the last two decades for example it is known that uniqueness strict complementarity and nondegeneracy of optimal solutions hold for almost all problem instances strong duality holds generically in a stronger sense i e it holds for a generic subset of problem instances in this paper we survey known results and present new ones in particular we give an easy proof of the fact that slater s condition holds generically in linear conic programming we further discuss the problem of stability of uniqueness nondegeneracy and strict complementarity we also comment on the fact that in general a conic program cannot be treated as a smooth problem and that techniques from nonsmooth geometric measure theory are needed

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

duality gap estimation of linear equality constrained binary quadratic programming we investigate in this paper the lagrangian duality properties of linear equality constrained binary quadratic programming we derive an underestimation of the duality gap between the primal problem and its lagrangian dual or sdp relaxation using the distance from the set of binary integer points to certain affine subspace while the computation of this distance can be achieved by the cell enumeration of hyperplane arrangement alternative lagrangian dual schemes via the exact penalty and the squared norm constraint reformulations are also discussed

the u lagrangian of the maximum eigenvalue function in this paper we apply the u lagrangian theory to the maximum eigenvalue function la and to its precomposition with affine matrix valued mappings we first give geometrical interpretations of the u objects that we introduce we also show that the u lagrangian of la has a hessian which can be explicitly computed the second order development of the u lagrangian provides a second order development of la along a characteristic smooth manifold the set of symmetric matrices whose maximal eigenvalues have a fixed multiplicity the same results can be obtained when we precompose la with an affine matrix valued mapping a provided that this mapping satisfies a regularity condition em transversality condition we show that the hessian of the u lagrangian of la circ a coincides with the reduced hessian encountered in sequential quadratic programming finally we use the u lagrangian to derive second order algorithms for minimizing la circ a

an empirical investigation of meta heuristic and heuristic algorithms for a 2d packing problem abstract in this paper we consider the two dimensional 2d rectangular packing problem where a fixed set of items have to be allocated on a single object two heuristics which belong to the class of packing procedures that preserve bottom left bl stability are hybridised with three meta heuristic algorithms genetic algorithms ga simulated annealing sa naive evolution ne and local search heuristic hill climbing this study compares the hybrid algorithms in terms of solution quality and computation time on a number of packing problems of different size in order to show the effectiveness of the design of the different algorithms their performance is compared to random search rs and heuristic packing routines

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

on bar frameworks stress matrices and semidefinite programming a bar framework g p in r dimensional euclidean space is a graph g on the vertices 1 2 n where each vertex i is located at point p i in mathbb r r given a framework g p in mathbb r r a problem of great interest is that of determining whether or not there exists another framework g q not obtained from g p by a rigid motion such that q i q j 2 p i p j 2 for each edge i j of g this problem is known as either the global rigidity problem or the universal rigidity problem depending on whether such a framework g q is restricted to be in the same r dimensional space or not the stress matrix s of a bar framework g p plays a key role in these and other related problems in this paper semidefinite programming sdp theory is used to address in a unified manner several problems concerning universal rigidity new results are presented as well as new proofs of previously known theorems in particular we use the notion of sdp non degeneracy to obtain a sufficient condition for universal rigidity and we show that this condition yields the previously known sufficient condition for generic universal rigidity we present new results concerning positive semidefinite stress matrices and we use a semidefinite version of farkas lemma to characterize bar frameworks that admit a nonzero positive semidefinite stress matrix s

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

an effective quasi human based heuristic for solving the rectangle packing problem abstract in this paper we introduce an effective deterministic heuristic less flexibility first for solving the classical np complete rectangle packing problem many effective heuristics implemented for this problem are cpu intensive and non deterministic in nature others including the polynomial approximation methodology j assoc comput mach 32 1 1985 130 are too laborious for practical problem sizes the technique we propose is inspired and developed by enhancing some rule of thumb guidelines resulting from the generation long work experience of human professionals in ancient days although the less flexibility first heuristic is a deterministic algorithm the results are very encouraging this algorithm can consistently produce packing densities of around 99 on most randomly generated large examples as compared with the recent results of a well known simulated annealing based rectangle packing rp algorithm ieee trans computer aided design integrated circuits systems 17 1 1998 60 the results are much better both in less dead space 2 4 vs 6 7 and much less cpu time 9 57 vs 331 78 seconds experimenting our heuristics on a public rectangle packing data set covering instances of 16 97 rectangles the average unpack ratio is quite satisfactory 0 92 for bounding boxes limited to be optimum and 2 68 for the completed packing while most cases spend only a few minutes in cpu time

universal duality in conic convex optimization given a primal dual pair of linear programs it is well known that if their optimal values are viewed as lying on the extended real line then the duality gap is zero unless both problems are infeasible in which case the optimal values are and in contrast for optimization problems over nonpolyhedral convex cones a nonzero duality gap can exist when either the primal or the dual is feasible for a pair of dual conic convex programs we provide simple conditions on the constraint matrices and cone under which the duality gap is zero for every choice of linear objective function and constraint right hand side we refer to this property as universal duality our conditions possess the following properties i they are necessary and sufficient in the sense that if and only if they do not hold the duality gap is nonzero for some linear objective function and constraint right hand side ii they are metrically and topologically generic and iii they can be verified by solving a single conic convex program we relate to universal duality the fact that the feasible sets of a primal convex program and its dual cannot both be bounded unless they are both empty finally we illustrate our theory on a class of semidefinite programs that appear in control theory applications

interior point trajectories in semidefinite programming in this paper we study interior point trajectories in semidefinite programming sdp including the central path of an sdp this work was inspired by the seminal work of megiddo on linear programming trajectories progress in math programming interior point algorithms and related methods n megiddo ed springer verlag berlin 1989 pp 131 158 under an assumption of primal and dual strict feasibility we show that the primal and dual central paths exist and converge to the analytic centers of the optimal faces of respectively the primal and the dual problems we consider a class of trajectories that are similar to the central path but can be constructed to pass through any given interior feasible or infeasible point and study their convergence finally we study the derivatives of these trajectories and their convergence

positive semidefinite matrix completions on chordal graphs and constraint nondegeneracy in semidefinite programming let g v e be a graph in matrix completion theory it is known that the following two conditions are equivalent i g is a chordal graph ii every g partial positive semidefinite matrix has a positive semidefinite matrix completion in this paper we relate these two conditions to constraint nondegeneracy condition in semidefinite programming and prove that they are each equivalent to iii for any g partial positive definite matrix that has a positive semidefinite completion constraint nondegeneracy is satisfied at each of its positive semidefinite matrix completions

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

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

a population heuristic for constrained two dimensional non guillotine cutting abstract in this paper we present a heuristic algorithm for the constrained two dimensional non guillotine cutting problem this is the problem of cutting a number of rectangular pieces from a single large rectangle so as to maximise the value of the pieces cut in addition the number of pieces of each type that are cut must lie within prescribed limits our heuristic algorithm is a population heuristic where a population of solutions to the problem are progressively evolved this heuristic is based on a new non linear formulation of the problem computational results are presented for a number of standard test problems taken from the literature and for a number of large randomly generated problems

strengthening the lovasz bound for graph coloring the lovasz number is a way to approximate the independence number of a graph but also its chromatic number we express the lovasz bound as the continuous relaxation of a discrete lovasz number which we derive from karger et al s formulation and which is equal to the chromatic number we also give another relaxation a la schrijver mceliece which is better than the lovasz number

solving log determinant optimization problems by a newton cg primal proximal point algorithm we propose a newton cg primal proximal point algorithm ppa for solving large scale log determinant optimization problems our algorithm employs the essential ideas of ppa the newton method and the preconditioned cg solver when applying the newton method to solve the inner subproblem we find that the log determinant term plays the role of a smoothing term as in the traditional smoothing newton technique focusing on the problem of maximum likelihood sparse estimation of a gaussian graphical model we demonstrate that our algorithm performs favorably compared to existing state of the art algorithms and is much preferred when a high quality solution is required for problems with many equality constraints

a family of simple non parametric kernel learning algorithms previous studies of non parametric kernel learning npkl usually formulate the learning task as a semi definite programming sdp problem that is often solved by some general purpose sdp solvers however for n data examples the time complexity of npkl using a standard interior point sdp solver could be as high as o n6 5 which prohibits npkl methods applicable to real applications even for data sets of moderate size in this paper we present a family of efficient npkl algorithms termed simplenpkl which can learn non parametric kernels from a large set of pairwise constraints efficiently in particular we propose two efficient simplenpkl algorithms one is simplenpkl algorithm with linear loss which enjoys a closed form solution that can be efficiently computed by the lanczos sparse eigen decomposition technique another one is simplenpkl algorithm with other loss functions including square hinge loss hinge loss square loss that can be re formulated as a saddle point optimization problem which can be further resolved by a fast iterative algorithm in contrast to the previous npkl approaches our empirical results show that the proposed new technique maintaining the same accuracy is significantly more efficient and scalable finally we also demonstrate that the proposed new technique is also applicable to speed up many kernel learning tasks including colored maximum variance unfolding minimum volume embedding and structure preserving embedding

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

a squared smoothing newton method for nonsmooth matrix equations and its applications in semidefinite optimization problems we study a smoothing newton method for solving a nonsmooth matrix equation that includes semidefinite programming and the semidefinite complementarity problem as special cases this method if specialized for solving semidefinite programs needs to solve only one linear system per iteration and achieves quadratic convergence under strict complementarity and nondegeneracy we also establish quadratic convergence of this method applied to the semidefinite complementarity problem under the assumption that the jacobian of the problem is positive definite on the affine hull of the critical cone at the solution these results are based on the strong semismoothness and complete characterization of the b subdifferential of a corresponding squared smoothing matrix function which are of general theoretical interest

inverse optimal control for discrete time finite horizon linear quadratic regulators abstract in this paper we consider the inverse optimal control problem for discrete time linear quadratic regulators lqr over finite time horizons given observations of the optimal trajectories or optimal control inputs to a linear time invariant system the goal is to infer the parameters that define the quadratic cost function the well posedness of the inverse optimal control problem is first justified in the noiseless case when these observations are exact we analyze the identifiability of the problem and provide sufficient conditions for uniqueness of the solution in the noisy case when the observations are corrupted by additive zero mean noise we formulate the problem as an optimization problem and prove that the solution to this problem is statistically consistent the performance of the proposed method is illustrated through numerical examples

perturbation analysis of second order cone programming problems we discuss first and second order optimality conditions for nonlinear second order cone programming problems and their relation with semidefinite programming problems for doing this we extend in an abstract setting the notion of optimal partition then we state a characterization of strong regularity in terms of second order optimality conditions this is the first time such a characterization is given for a nonpolyhedral conic problem

second order cone programming second order cone programming socp problems are convex optimization problems in which a linear function is minimized over the intersection of an affine linear manifold with the cartesian product of second order lorentz cones linear programs convex quadratic programs and quadratically constrained convex quadratic programs can all be formulated as socp problems as can many other problems that do not fall into these three categories these latter problems model applications from a broad range of fields from engineering control and finance to robust optimization and combinatorial optimization on the other hand semidefinite programming sdp that is the optimization problem over the intersection of an affine set and the cone of positive semidefinite matrices includes socp as a special case therefore socp falls between linear lp and quadratic qp programming and sdp like lp qp and sdp problems socp problems can be solved in polynomial time by interior point methods the computational effort per iteration required by these methods to solve socp problems is greater than that required to solve lp and qp problems but less than that required to solve sdp s of similar size and structure because the set of feasible solutions for an socp problem is not polyhedral as it is for lp and qp problems it is not readily apparent how to develop a simplex or simplex like method for socp while socp problems can be solved as sdp problems doing so is not advisable both on numerical grounds and computational complexity concerns for instance many of the problems presented in the survey paper of vandenberghe and boyd vb96 as examples of sdps can in fact be formulated as socps and should be solved as such in 2 3 below we give socp formulations for four of these examples the convex quadratically constrained quadratic programming qcqp problem problems involving fractional quadratic functions rutcor rutgers university e mail alizadeh rutcor rutgers edu research supported in part by the u s national science foundation grant ccr 9901991 ieor columbia university e mail gold ieor columbia edu research supported in part by the department of energy grant de fg02 92er25126 national science foundation grants dms 94 14438 cda 97 26385 and dms 01 04282

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

generic minimizing behavior in semialgebraic optimization we present a theorem of sard type for semialgebraic set valued mappings whose graphs have dimension no larger than that of their range space the inverse of such a mapping admits a single valued analytic localization around any pair in the graph for a generic value parameter this simple result yields a transparent and unified treatment of generic properties of semialgebraic optimization problems typical semialgebraic problems have finitely many critical points around each of which they admit a unique active manifold analogue of an active set in nonlinear optimization moreover such critical points satisfy strict complementarity and second order sufficient conditions for optimality are indeed necessary

a newton like method for solving rank constrained linear matrix inequalities this paper presents a newton like algorithm for solving systems of rank constrained linear matrix inequalities though local quadratic convergence of the algorithm is not a priori guaranteed or observed in all cases numerical experiments including application to an output feedback stabilization problem show the effectiveness of the algorithm

a method for solving moving boundary problems a new method for solving moving boundary problems or stefan problems is proposed two associated heat conduction problems which have fixed domains and have the same boundary and initial conditions as the original problem are introduced each of these includes a moving source as a supplementary condition the new problem has no moving boundary and can be studied by many known methods and techniques one dimensional problems are analyzed in detail for exploratory purposes two examples one of which is the neumann problem are discussed

a dual optimization approach to inverse quadratic eigenvalue problems with partial eigenstructure the inverse quadratic eigenvalue problem iqep arises in the field of structural dynamics it aims to find three symmetric matrices known as the mass the damping and the stiffness matrices such that they are closest to the given analytical matrices and satisfy the measured data the difficulty of this problem lies in the fact that in applications the mass matrix should be positive definite and the stiffness matrix positive semidefinite based on an equivalent dual optimization version of the iqep we present a quadratically convergent newton type method our numerical experiments confirm the high efficiency of the proposed method

similarity and other spectral relations for symmetric cones abstract the similarity relations that are derived in this paper reduce to well known results in the special case of symmetric matrices in particular for two positive definite matrices x and y the square of the spectral geometric mean is known to be similar to the matrix product xy it is shown in this paper that this property carries over to symmetric cones more elementary similarity relations such as xy 2 x yx 2 y are generalized as well we also extend the result that the eigenvalues of a matrix product xy are less dispersed than the eigenvalues of the jordan product xy yx 2 the paper further contains a number of inequalities on norms and spectral values this type of inequality is often used in the analysis of interior point methods in optimization we also derive an extension of stein s theorem to symmetric cones

quadratic convergence of a nonsmooth newton type method for semidefinite programs without strict complementarity we consider a newton type method for the solution of semidefinite programs this newton type method is based on a semismooth reformulation of the semidefinite program as a nonsmooth system of equations we establish local quadratic convergence of this method under a linear independence assumption and a weak nondegeneracy condition that implies uniqueness of the optimal solution but does not imply strict complementarity

on the closedness of the linear image of a closed convex cone when is the linear image of a closed convex cone closed we present very simple and intuitive necessary conditions that 1 unify and generalize seemingly disparate classical sufficientconditions such as polyhedrality of the cone and slater type conditions 2 are necessary and sufficient when the dual cone belongs to a class that we call nice cones nice cones subsume all cones amenable to treatment by efficient optimization algorithms for instance polyhedral semidefinite and p cones and 3 provide similarly attractive conditions for an equivalent problem the closedness of the sum of two closed convex cones

an exact algorithm for general orthogonal two dimensional knapsack problems abstract we present a new exact tree search procedure for solving two dimensional knapsack problems in which a number of small rectangular pieces each of a given size and value are required to be cut from a large rectangular stock plate the objective is to maximise the value of pieces cut or minimise the wastage we consider the case where there is a maximum number of times that a piece may be used in a cutting pattern the algorithm limits the size of the tree search by using a bound derived from a langrangean relaxation of a 0 1 integer programming formulaton of the problem subgradient optimisation is used to optimise this bound reduction tests derived from both the original problem and the lagrangean relaxation produce substantial computational gains the computational performance of the algorithm indicates that it is an effective procedure capable of solving optimally practical two dimensional cutting problems of medium size

second order sufficiency and quadratic growth for nonisolated minima for standard nonlinear programming problems the weak second order sufficient condition is equivalent to the quadratic growth condition as far as the set of minima consists of isolated points and some qualification hypothesis holds this kind of condition is instrumental in the study of numerical algorithms and sensitivity analysis the arm of the paper is to study the relations between various types of sufficient conditions and quadratic growth in cases when the set of minima may have nonisolated points

application of standard and refined heat balance integral methods to one dimensional stefan problems the work in this paper concerns the study of conventional and refined heat balance integral methods for a number of phase change problems these include standard test problems both with one and two phase changes which have exact solutions that enable us to test the accuracy of the approximate solutions we also consider situations where no analytical solution is available and compare these to numerical solutions it is popular to use a quadratic profile as an approximation of the temperature but we show that a cubic profile seldom considered in the literature is far more accurate in most circumstances in addition the refined integral method can give greater improvement still and we develop a variation on this method which turns out to be optimal in some cases we assess which integral method is better for various problems showing that it is largely dependent on the specified boundary conditions

new and improved level heuristics for the rectangular strip packing and variable sized bin packing problems we consider two types of orthogonal oriented rectangular two dimensional packing problems the first is the strip packing problem for which four new and improved level packing algorithms are presented two of these algorithms guarantee a packing that may be disentangled by guillotine cuts these are combined with a two stage heuristic designed to find a solution to the variable sized bin packing problem where the aim is to pack all items into bins so as to minimise the packing area this heuristic packs the levels of a solution to the strip packing problem into large bins and then attempts to repack the items in those bins into smaller bins in order to reduce wasted space the results of the algorithms are compared to those of seven level packing heuristics from the literature by means of a large number of strip packing benchmark instances it is found that the new algorithms are an improvement over known level packing heuristics for the strip packing problem the advancements made by the new and improved algorithms are limited in terms of utilised space when applied to the variable sized bin packing problem however they do provide results faster than many existing algorithms

local duality of nonlinear semidefinite programming recently chan and sun chan z x d sun constraint nondegeneracy strong regularity and nonsingularity in semidefinite programming siam j optim 19 370 376 reported for semidefinite programming sdp that the primal dual constraint nondegeneracy is equivalent to the dual primal strong second order sufficient condition ssosc this result is responsible for a number of important results in stability analysis of sdp in this paper we study duality of this type in nonlinear semidefinite programming nsdp we introduce the dual ssosc at a karush kuhn tucker kkt triple of nsdp and study its various characterizations and relationships to the primal nondegeneracy although the dual ssosc is nothing but the ssosc for the wolfe dual of the nsdp it suggests new information for the primal nsdp for example it ensures that the inverse of the hessian of the lagrangian function exists at the kkt triple and the inverse is positive definite on some normal space it also ensures the primal nondegeneracy some of our results generalize the corresponding classical duality results in nonlinear programming studied by fujiwara et al fujiwara o s p han o l mangasarian 1984 local duality of nonlinear programs siam j control optim 22 162 169 for the convex quadratic sdp qsdp we have complete characterizations for the primal and dual ssosc our results reveal that the nearest correlation matrix problem satisfies not only the primal and dual ssosc but also the primal and dual nondegeneracy at its solution suggesting that it is a well conditioned qsdp

holder and lipschitz stability of solution sets in programs with probabilistic constraints we study perturbations of a stochastic program with a probabilistic constraint and r concave original probability distribution first we improve our earlier results substantially and provide conditions implying holder continuity properties of the solution sets w r t the kolmogorov distance of probability distributions secondly we derive an upper lipschitz continuity property for solution sets under more restrictive conditions on the original program and on the perturbed probability measures the latter analysis applies to linear quadratic models and is based on work by bonnans and shapiro the stability results are illustrated by numerical tests showing the different asymptotic behaviour of parametric and nonparametric estimates in a program with a normal probabilistic constraint

an and or graph approach to the solution of two dimensional non guillotine cutting problems this work is concerned with unconstrained two dimensional non guillotine cutting problems where a rectangular plate is cut into a number of rectangular pieces in such a way as to optimize an objective function we reduce the state space of the problem by considering non guillotine cutting patterns which are combinations of guillotine and simple non guillotine cuts these cutting patterns are represented as complete paths in an and or graph and a branch and bound method is described we provide rules and heuristics that reduce the graph search and present computational results from randomly generated examples as well as from the literature

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

on the reduction of duality gap in box constrained nonconvex quadratic program in this paper we investigate in this paper the reduction of the duality gap between box constrained nonconvex quadratic programming and its semidefinite programming sdp relaxation or lagrangian dual characterizing the zero duality gap by a set of saddle point type conditions we propose a parameterized distance measure between a polyhedral set c and a perturbed nonconvex set to measure the dissatisfaction degree of the optimality conditions for zero duality gap an underestimation of the duality gap is then derived which leads to a reduction of the duality gap proportional to 2 for the identified best parameter this reduction of duality gap can be extended to the cases with both box and linear equality constraints we demonstrate that the computation of can be reduced to the cell enumeration of hyperplane arrangement in discrete geometry in particular we show that the reduction of duality gap can be achieved in polynomial time for a fixed degeneracy degree of the modified

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

a tabu search algorithm for a two dimensional non guillotine cutting problem in this paper we study a two dimensional non guillotine cutting problem the problem of cutting rectangular pieces from a large stock rectangle so as to maximize the total value of the pieces cut the problem has many industrial applications whenever small pieces have to be cut from or packed into a large stock sheet we propose a tabu search algorithm several moves based on reducing and inserting blocks of pieces have been defined intensification and diversification procedures based on long term memory have been included the computational results on large sets of test instances show that the algorithm is very efficient for a wide range of packing and cutting problems

optimization problems with perturbations a guided tour this paper presents an overview of some recent and significant progress in the theory of optimization problems with perturbations we put the emphasis on methods based on upper and lower estimates of the objective function of the perturbed problems these methods allow one to compute expansions of the optimal value function and approximate optimal solutions in situations where the set of lagrange multipliers is not a singleton may be unbounded or is even empty we give rather complete results for nonlinear programming problems and describe some extensions of the method to more general problems we illustrate the results by computing the equilibrium position of a chain that is almost vertical or horizontal