global optimality conditions and optimization methods for constrained polynomial programming problems the general constrained polynomial programming problem gpp is considered in this paper problem gpp has a broad range of applications and is proved to be np hard necessary global optimality conditions for problem gpp are established then a new local optimization method for this problem is proposed by exploiting these necessary global optimality conditions a global optimization method is proposed for this problem by combining this local optimization method together with an auxiliary function some numerical examples are also given to illustrate that these approaches are very efficient

distributed control of spatially invariant systems we consider distributed parameter systems where the underlying dynamics are spatially invariant and where the controls and measurements are spatially distributed these systems arise in many applications such as the control of vehicular platoons flow control microelectromechanical systems mems smart structures and systems described by partial differential equations with constant coefficients and distributed controls and measurements for fully actuated distributed control problems involving quadratic criteria such as linear quadratic regulator lqr h sub 2 and h sub spl infin optimal controllers can be obtained by solving a parameterized family of standard finite dimensional problems we show that optimal controllers have an inherent degree of decentralization and this provides a practical distributed controller architecture we also prove a general result that applies to partially distributed control and a variety of performance criteria stating that optimal controllers inherit the spatial invariance structure of the plant connections of this work to that on systems over rings and systems with dynamical symmetries are discussed

global error bounds for piecewise convex polynomials in this paper by examining the recession properties of convex polynomials we provide a necessary and sufficient condition for a piecewise convex polynomial to have a holder type global error bound with an explicit holder exponent our result extends the corresponding results of li siam j control optim 33 5 1510 1529 1995 from piecewise convex quadratic functions to piecewise convex polynomials

a comparison between lqr control for a long string of siso systems and lqr control of the infinite spatially invariant version in this paper we consider a long string of siso systems which in the limit becomes a scalar infinite spatially invariant system we compare the lqr control for long but finite strings with the lqr control for the corresponding infinite strings we give analytical and numerical examples where these are quite different and we investigate the cause in addition we obtain sufficient conditions for the lqr solutions to be similar as the length of the string increases

fractional programming with convex quadratic forms and functions this article is concerned with two global optimization problems p1 and p2 each of these problems is a fractional programming problem involving the maximization of a ratio of a convex function to a convex function where at least one of the convex functions is a quadratic form first the article presents and validates a number of theoretical properties of these problems included among these properties is the result that under a mild assumption any globally optimal solution for problem p1 must belong to the boundary of its feasible region also among these properties is a result that shows that problem p2 can be reformulated as a convex maximization problem second the article presents for the first time an algorithm for globally solving problem p2 the algorithm is a branch and bound algorithm in which the main computational effort involves solving a sequence of convex programming problems convergence properties of the algorithm are presented and computational issues that arise in implementing the algorithm are discussed preliminary indications are that the algorithm can be expected to provide a practical approach for solving problem p2 provided that the number of variables is not too large

eigenvalue bounds versus semidefinite relaxations for the quadratic assignment problem it was recently demonstrated that a well known eigenvalue bound for the quadratic assignment problem qap actually corresponds to a semidefinite programming sdp relaxation however for this bound to be computationally useful the assignment constraints of the qap must first be eliminated and the bound then applied to a lower dimensional problem the resulting projected eigenvalue bound is one of the best available bounds for the qap especially when considering the quality of bounds relative to the complexity of obtaining them in this paper we show that the projected eigenvalue bound is also related to an sdp relaxation of the original qap this implicit sdp relaxation is similar to sdp relaxations of the qap proposed by lin and saigal on solving large scale semidefinite programming problems a case study of quadratic assignment problem department of industrial engineering and operations research university of michigan 1997 and zhao et al j combin optim 2 1998 pp 71 109

implementation of a variance reduction based lower bound in a branch and bound algorithm for the quadratic assignment problem the efficient implementation of a branch and bound algorithm for the quadratic assignment problem qap incorporating the lower bound based on variance reduction of li pardalos ramakrishnan and resende 1994 is presented a new data structure for efficient implementation of branch and bound algorithms for the qap is introduced computational experiments with the branch and bound algorithm on different classes of qap test problems are reported the branch and bound algorithm using the new lower bounds is compared with the same algorithm utilizing the commonly applied gilmore lawler lower bound both implementations use a greedy randomized adaptive search procedure for obtaining initial upper bounds the algorithms report all optimal permutations optimal solutions for previously unsolved instances from the literature of dimensions n 16 and n 20 have been found with the new algorithm in addition the new algorithm has been tested on a class of large data variance problems requiring the examination of much fewer nodes of the branch and bound tree than the same algorithm using the gilmore lawler lower bound

lower bounds for the quadratic assignment problem based upon a dual formulation a new bounding procedure for the quadratic assignment problem qap is described which extends the hungarian method for the linear assignment problem lap to qaps operating on the four dimensional cost array of the qap objective function the qap is iteratively transformed in a series of equivalent qaps leading to an increasing sequence of lower bounds for the original problem to this end two classes of operations which transform the four dimensional cost array are defined these have the property that the values of the transformed objective function z are the corresponding values of the old objective function z shifted by some amount c in the case that all entries of the transformed cost array are nonnegative then c is a lower bound for the initial qap if moreover there exists a feasible solution u to the qap such that its value in the transformed problem is zero then c is the optimal value of z and u is an optimal solution for the original qap the transformations are iteratively applied until no significant increase in constant c as above is found resulting in the so called dual procedure dp several strategies are listed for appropriately determining c or equivalently transforming the cost array the goal is the modification of the elements in the cost array to obtain new equivalent problems that bring the qap closer to solution in some cases the qap is actually solved though solution is not guaranteed the close relationship between the dp and the linear programming formulation of adams and johnson is presented the dp attempts to solve adams and johnson s clp a continuous relaxation of a linearization of the qap this explains why the dp produces bounds close to the optimum values for clp calculated by johnson in her dissertation and by resende et al in their interior point algorithm for linear programming the benefit of using dp within a branch and bound algorithm is described then two versions of dp are tested on the nugent test instances from size 5 to size 30 as well as several other test instances from qaplib these compare favorably with earlier bounding methods

a new bound for the quadratic assignment problem based on convex quadratic programming we describe a new convex quadratic programming bound for the quadratic assignment problem qap the construction of the bound uses a semidefinite programming representation of a basic eigenvalue bound for qap the new bound dominates the well known projected eigenvalue bound and appears to be competitive with existing bounds in the trade off between bound quality and computational effort

optimality conditions for the minimization of quadratic 0 1 problems in this paper we present some explicit local and global optimality conditions for unconstrained binary quadratic problems local optimality conditions are categorized by the neighborhood including 1 flip 2 flip and k flip neighborhoods after defining an evaluation vector and an evaluation matrix we propose algorithms that make it possible for us to obtain 1 flip and or 2 flip local solutions in polynomial time we provide a key insight into the relationship between the local optimality conditions and the global optimality conditions a discussion of global geometric optimality conditions is also presented in this paper

non convex quadratic minimization problems with quadratic constraints global optimality conditions in this paper we first examine how global optimality of non convex constrained optimization problems is related to lagrange multiplier conditions we then establish lagrange multiplier conditions for global optimality of general quadratic minimization problems with quadratic constraints we also obtain necessary global optimality conditions which are different from the lagrange multiplier conditions for special classes of quadratic optimization problems these classes include weighted least squares with ellipsoidal constraints and quadratic minimization with binary constraints we discuss examples which demonstrate that our optimality conditions can effectively be used for identifying global minimizers of certain multi extremal non convex quadratic optimization problems

necessary global optimality conditions for nonlinear programming problems with polynomial constraints in this paper we develop necessary conditions for global optimality that apply to non linear programming problems with polynomial constraints which cover a broad range of optimization problems that arise in applications of continuous as well as discrete optimization in particular we show that our optimality conditions readily apply to problems where the objective function is the difference of polynomial and convex functions over polynomial constraints and to classes of fractional programming problems our necessary conditions become also sufficient for global optimality for polynomial programming problems our approach makes use of polynomial over estimators and a polynomial version of a theorem of the alternative which is a variant of the positivstellensatz in semi algebraic geometry we discuss numerical examples to illustrate the significance of our optimality conditions

transposition theorems and qualification free optimality conditions new theorems of the alternative for polynomial constraints based on the positivstellensatz from real algebraic geometry and for linear constraints generalizing the transposition theorems of motzkin and tucker are proved based on these two karush john optimality conditions holding without any constraint qualification are proved for single or multiobjective constrained optimization problems the first condition applies to polynomial optimization problems only and gives for the first time necessary and sufficient global optimality conditions for polynomial problems the second condition applies to smooth local optimization problems and strengthens known local conditions if some linear or concave constraints are present the new version reduces the number of constraints for which a constraint qualification is needed to get the kuhn tucker conditions

strong duality for a trust region type relaxation of the quadratic assignment problem abstract lagrangian duality underlies many efficient algorithms for convex minimization problems a key ingredient is strong duality lagrangian relaxation also provides lower bounds for non convex problems where the quality of the lower bound depends on the duality gap quadratically constrained quadratic programs qqps provide important examples of non convex programs for the simple case of one quadratic constraint the trust region subproblem strong duality holds in addition necessary and sufficient strengthened second order optimality conditions exist however these duality results already fail for the two trust region sub problem surprisingly there are classes of more complex non convex qqps where strong duality holds one example is the special case of orthogonality constraints which arise naturally in relaxations for the quadratic assignment problem qap in this paper we show that strong duality also holds for a relaxation of qap where the orthogonality constraint is replaced by a semidefinite inequality constraint using this strong duality result and semidefinite duality we develop new trust region type necessary and sufficient optimality conditions for these problems our proof of strong duality introduces and uses a generalization of the hoffman wielandt inequality

optimality conditions and optimization methods for quartic polynomial optimization abstract in this paper multivariate quartic polynomial optimization program qpop is considered quartic optimization problems arise in various practical applications and are proved to be np hard we discuss necessary global optimality conditions for quartic problem qpop and then we present a new strongly or e strongly local optimization method according to necessary global optimality conditions which may escape and improve some kkt points finally we design a global optimization method for problem qpop by combining the new strongly or e strongly local optimization method and an auxiliary function numerical examples show that our algorithms are efficient and stable

a survey for the quadratic assignment problem the quadratic assignment problem qap one of the most difficult problems in the np hard class models many real life problems in several areas such as facilities location parallel and distributed computing and combinatorial data analysis combinatorial optimization problems such as the traveling salesman problem maximal clique and graph partitioning can be formulated as a qap in this paper we present some of the most important qap formulations and classify them according to their mathematical sources we also present a discussion on the theoretical resources used to define lower bounds for exact and heuristic algorithms we then give a detailed discussion of the progress made in both exact and heuristic solution methods including those formulated according to metaheuristic strategies finally we analyze the contributions brought about by the study of different approaches

a note on the augmented hessian when the reduced hessian is semidefinite certain matrix relationships play an important role in optimality conditions and algorithms for nonlinear and semidefinite programming let h be an n n symmetric matrix a an m n matrix and z a basis for the null space of a in a typical optimization context h is the hessian of a smooth function and a is the jacobian of a set of constraints when the reduced hessian zthz is positive definite augmented lagrangian methods rely on the known existence of a finite bar rho ge 0 such that for all rho bar rho the augmented hessian h rho ata is positive definite in this note we analyze the case when zthz is positive semidefinite i e singularity is allowed and show that the situation is more complicated in particular we give a simple necessary and sufficient condition for the existence of a finite bar rho so that h rho ata is positive semidefinite for rho ge bar rho a corollary of our result is that if h is nonsingular and indefinite while zthz is positive semidefinite and singular no such bar rho exists

a new lower bound via projection for the quadratic assignment problem new lower bounds for the quadratic assignment problem qap are presented these bounds are based on the orthogonal relaxation of qap the additional improvement is obtained by making efficient use of a tractable representation of orthogonal matrices having constant row and column sums the new bound is easy to implement and often provides high quality bounds under an acceptable computational effort

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

computing lower bounds for the quadratic assignment problem with an interior point algorithm for linear programming an example of the quadratic assignment problem qap is the facility location problem in which n facilities are assigned at minimum cost to n sites between each pair of facilities there is a given amount of flow contributing a cost equal to the product of the flow and the distance between sites to which the facilities are assigned proving optimality of qaps has been limited to instances having fewer than 20 facilities largely because known lower bounds are weak we compute lower bounds for a wide range of qaps using a linear programming based lower bound studied by z drezner 1995 on the majority of qaps tested a new best known lower bound is computed on 87 of the instances we produced the best known lower bound on several instances including some having more the 20 facilities the lower bound is tight the linear programs which can be large even for small qaps are solved with an interior point code that uses a preconditioned conjugate gradient algorithm to compute the interior point directions attempts to solve these instances using the cplex primal simplex algorithm as well as the cplex barrier primal dual interior point method were successful only for the smallest instances

minimizing polynomials via sum of squares over the gradient ideal a method is proposed for finding the global minimum of a multivariate polynomial via sum of squares sos relaxation over its gradient variety that variety consists of all points where the gradient is zero and it need not be finite a polynomial which is nonnegative on its gradient variety is shown to be sos modulo its gradient ideal provided the gradient ideal is radical or the polynomial is strictly positive on the real gradient variety this opens up the possibility of solving previously intractable polynomial optimization problems the related problem of constrained minimization is also considered and numerical examples are discussed experiments show that our method using the gradient variety outperforms prior sos methods

an approximation theorem for the algebraic riccati equation for an infinite dimensional linear quadratic control problem in hilbert space approximation of the solution of the algebraic riccati operator equation in the strong operator topology is considered under conditions weaker than uniform exponential stability of the approximating systems as an application strong onvergence of the approximating riccati operators in case of a previously developed spline approximation scheme for delay systems is established finally convergence of the transfer functions of the approximating systems is investigated

recent advances in the solution of quadratic assignment problems the quadratic assignment problem qap is notoriously difficult for exact solution methods in the past few years a number of long open qaps including those posed by steinberg 1961 nugent et al 1968 and krarup 1972 were solved to optimality for the first time the solution of these problems has utilized both new algorithms and novel computing structures we describe these developments as well as recent work which is likely to result in the solution of even more difficult instances

quadratic assignment problems abstract this paper surveys quadratic assignment problems qap at first several applications of this problem class are described and mathematical formulations of qaps are given then some exact solution methods and good heuristics are outlined their computational behaviour is illustrated by numerical results further recent results on the asymptotic probabilistic behaviour of qaps are outlined

applications of parametric programming and eigenvalue maximization to the quadratic assignment problem we investigate new bounding strategies based on different relaxations of the quadratic assignment problem in particular we improve the lower bound found by using an eigenvalue decomposition of the quadratic part and by solving a linear program for the linear part the improvement is accomplished by applying a steepest ascent algorithm to the sum of the two bounds

feasible partition problem in reverse convex and convex mixed integer programming abstract in this paper we consider the consistent partition problem in reverse convex and convex mixed integer programming in particular we will show that for the considered classes of convex functions both integer and relaxed systems can be partitioned into two disjoint subsystems each of which is consistent and defines an unbounded region the polynomial time algorithm to generate the partition will be proposed and the algorithm for a maximal partition will also be provided