improvement and comparison of heuristics for solving the uncapacitated multisource weber problem the multisource weber problem is to locate simultaneouslym facilities in the euclidean plane to minimize the total transportation cost for satisfying the demand ofn fixed users each supplied from its closest facility many heuristics have been proposed for this problem as well as a few exact algorithms heuristics are needed to solve quickly large problems and to provide good initial solutions for exact algorithms we compare various heuristics i e alternative location allocation cooper 1964 projection bongartz et al 1994 tabu search brimberg and mladenovic 1996a p median plus weber hansen et al 1996 genetic search and several versions of variable neighbourhood search based on empirical tests that are reported it is found that most traditional and some recent heuristics give poor results when the number of facilities to locate is large and that variable neighbourhood search gives consistently best results on average in moderate computing time

matroid representations and free arrangements we show that terao s conjecture freeness of the module of logarithmic forms at a hyperplane arrangement is determined by its abstract matroid holds over fields with at most four elements however an example demonstrates that the field characteristic has to be fixed for this

variable neighborhood search for the vertex weighted k cardinality tree problem this paper presents some new heuristics based on variable neighborhood search to solve the vertex weighted k cardinality tree problem an efficient local search procedure is also developed for use within these heuristics our computational results demonstrate that the new heuristics substantially outperform the state of the art methodologies including a tabu search and genetic algorithm recently proposed in the literature we also show that a decomposition approach is best for larger problem sizes than previously investigated thus our findings advance in a significant way the capacity to solve this important class of problems

the trust region subproblem with non intersecting linear constraints this paper studies an extended trust region subproblem etrs in which the trust region intersects the unit ball with m m linear inequality constraints when m 0 m 1 m 0 m 1 or m 2 m 2 and the linear constraints are parallel it is known that the etrs optimal value equals the optimal value of a particular convex relaxation which is solvable in polynomial time however it is also known that when m ge 2 m 2 and at least two of the linear constraints intersect within the ball i e some feasible point of the etrs satisfies both linear constraints at equality then the same convex relaxation may admit a gap with etrs this paper shows that the convex relaxation has no gap for arbitrary m m as long as the linear constraints are non intersecting

relations models and a memetic approach for three degree dependent spanning tree problems abstract in this paper we take into account three different spanning tree problems with degree dependent objective functions the main application of these problems is in the field of optical network design in particular we propose the classical minimum leaves spanning tree problem as a relevant problem in this field and show its relations with the minimum branch vertices and the minimum degree sum problems we present a unified memetic algorithm for the three problems and show its effectiveness on a wide range of test instances

semidefinite relaxations for quadratically constrained quadratic programming a review and comparisons at the intersection of nonlinear and combinatorial optimization quadratic programming has attracted significant interest over the past several decades a variety of relaxations for quadratically constrained quadratic programming qcqp can be formulated as semidefinite programs sdps the primary purpose of this paper is to present a systematic comparison of sdp relaxations for qcqp using theoretical analysis it is shown that the recently developed doubly nonnegative relaxation is equivalent to the shor relaxation when the latter is enhanced with a partial first order relaxation linearization technique these two relaxations are shown to theoretically dominate six other sdp relaxations a computational comparison reveals that the two dominant relaxations require three orders of magnitude more computational time than the weaker relaxations while providing relaxation gaps averaging 3 as opposed to gaps of up to 19 for weaker relaxations on 700 randomly generated problems with up to 60 variables an sdp relaxation derived from lagrangian relaxation after the addition of redundant nonlinear constraints to the primal achieves gaps averaging 13 in a few cpu seconds

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

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

theory of cones abstract this survey deals with the aspects of archimedian partially ordered finite dimensional real vector spaces and order preserving linear maps which do not involve spectral theory the first section sketches some of the background of entrywise nonnegative matrices and of systems of inequalities which motivate much of the current investigations the study of inequalities resulted in the definition of a polyhedral cone k and its face lattice f k in section ii a the face lattice of a not necessarily polyhedral cone k in a vector space v is investigated in particular the interplay between the lattice properties of f k and geometric properties of k is emphasized section ii b turns to the cones k in the space of linear maps on v recall that k is the cone of all order preserving linear maps of particular interest are the algebraic structure of k as a semiring and the nature of the group aut k of nonsingular elements a k for which a 1 k as well in a short final section the cone p n of n n positive semidefinite matrices is discussed a characterization of the set of completely positive linear maps is stated the proofs will appear in a forthcoming paper

on conic qpccs conic qcqps and completely positive programs this paper studies several classes of nonconvex optimization problems defined over convex cones establishing connections between them and demonstrating that they can be equivalently formulated as convex completely positive programs the problems being studied include a conic quadratically constrained quadratic program qcqp a conic quadratic program with complementarity constraints qpcc and rank constrained semidefinite programs our results do not make any boundedness assumptions on the feasible regions of the various problems considered the first stage in the reformulation is to cast the problem as a conic qcqp with just one nonconvex constraint q x le 0 q x 0 where q x is nonnegative over the convex conic and linear constraints so the problem fails the slater constraint qualification a conic qpcc has such a structure we prove the converse namely that any conic qcqp satisfying a constraint qualification can be expressed as an equivalent conic qpcc the second stage of the reformulation lifts the problem to a completely positive program and exploits and generalizes a result of burer we also show that a frank wolfe type result holds for a subclass of this class of conic qcqps further we derive necessary and sufficient optimality conditions for nonlinear programs where the only nonconvex constraint is a quadratic constraint of the structure considered elsewhere in the paper

skewed vns enclosing second order algorithm for the degree constrained minimum spanning tree problem we develop ideas to enhance the performance of the variable neighborhood search vns by guiding the search in the shaking phase and by employing the skewed strategy for this purpose second order algorithms and skewed functions expressing structural differences are embedded in an efficient vns proposed in the literature for the degree constrained minimum spanning tree problem given an undirected graph with weights associated with its edges the degree constrained minimum spanning tree problem consists in finding a minimum spanning tree of the given graph subject to constraints on node degrees computational experiments are conducted on the largest and hardest benchmark instances found in the literature to date we report results showing that the vns with the proposed strategies improved the best known solutions for all the hardest benchmark instances moreover these new best solutions significantly reduced the gaps with respect to tight lower bounds reported in the literature

a variational approach to copositive matrices this work surveys essential properties of the so called copositive matrices the study of which has been spread over more than fifty five years special emphasis is given to variational aspects related to the concept of copositivity in addition some new results on the geometry of the cone of copositive matrices are presented here for the first time

quadratic programs with hollows let mathcal f be a quadratically constrained possibly nonconvex bounded set and let mathcal e 1 ldots mathcal e l denote ellipsoids contained in mathcal f with non intersecting interiors we prove that minimizing an arbitrary quadratic q cdot over mathcal g mathcal f setminus cup k 1 ell mathrm int mathcal e k is no more difficult than minimizing q cdot over mathcal f in the following sense if a given semidefinite programming sdp relaxation for min q x x in mathcal f is tight then the addition of l linear constraints derived from mathcal e 1 ldots mathcal e l yields a tight sdp relaxation for min q x x in mathcal g we also prove that the convex hull of x xx t x in mathcal g equals the intersection of the convex hull of x xx t x in mathcal f with the same l linear constraints inspired by these results we resolve a related question in a seemingly unrelated area mixed integer nonconvex quadratic programming

trust region problems with linear inequality constraints exact sdp relaxation global optimality and robust optimization the trust region problem which minimizes a nonconvex quadratic function over a ball is a key subproblem in trust region methods for solving nonlinear optimization problems it enjoys many attractive properties such as an exact semi definite linear programming relaxation sdp relaxation and strong duality unfortunately such properties do not in general hold for an extended trust region problem having extra linear constraints this paper shows that two useful and powerful features of the classical trust region problem continue to hold for an extended trust region problem with linear inequality constraints under a new dimension condition first we establish that the class of extended trust region problems has an exact sdp relaxation which holds without the slater constraint qualification this is achieved by proving that a system of quadratic and affine functions involved in the model satisfies a range convexity whenever the dimension condition is fulfilled second we show that the dimension condition together with the slater condition ensures that a set of combined first and second order lagrange multiplier conditions is necessary and sufficient for global optimality of the extended trust region problem and consequently for strong duality through simple examples we also provide an insightful account of our development from sdp relaxation to strong duality finally we show that the dimension condition is easily satisfied for the extended trust region model that arises from the reformulation of a robust least squares problem lsp as well as a robust second order cone programming model problem socp as an equivalent semi definite linear programming problem this leads us to conclude that under mild assumptions solving a robust lsp or socp under matrix norm uncertainty or polyhedral uncertainty is equivalent to solving a semi definite linear programming problem and so their solutions can be validated in polynomial time

on the copositive representation of binary and continuous nonconvex quadratic programs in this paper we model any nonconvex quadratic program having a mix of binary and continuous variables as a linear program over the dual of the cone of copositive matrices this result can be viewed as an extension of earlier separate results which have established the copositive representation of a small collection of np hard problems a simplification which reduces the dimension of the linear conic program and an extension to complementarity constraints are established and computational issues are discussed

on cones of nonnegative quadratic functions we derive linear matrix inequality lmi characterizations and dual decomposition algorithms for certain matrix cones which are generated by a given set using generalized co positivity these matrix cones are in fact cones of nonconvex quadratic functions that are nonnegative on a certain domain as a domain we consider for instance the intersection of a upper level set of a quadratic function and a half plane consequently we arrive at a generalization of yakubovich s s procedure result although the primary concern of this paper is to characterize the matrix cones by lmis we show as an application of our results that optimizing a general quadratic function over the intersection of an ellipsoid and a half plane can be formulated as semidefinite programming sdp thus proving the polynomiality of this class of optimization problems which arise e g from the application of the trust region method for nonlinear programming other applications are in control theory and robust optimization

on cone of nonsymmetric positive semidefinite matrices abstract in this paper we analyze and characterize the cone of nonsymmetric positive semidefinite matrices ns psd firstly we study basic properties of the geometry of the ns psd cone and show that it is a hyperbolic but not homogeneous cone secondly we prove that the ns psd cone is a maximal convex subcone of p 0 matrix cone which is not convex but the interior of the ns psd cone is not a maximal convex subcone of p matrix cone as the byproducts some new sufficient and necessary conditions for a nonsymmetric matrix to be positive semidefinite are given finally we present some properties of metric projection onto the ns psd cone

second order cone constraints for extended trust region subproblems the classical trust region subproblem trs minimizes a nonconvex quadratic objective over the unit ball in this paper we consider extensions of trs having extra constraints when two parallel cuts are added to trs we show that the resulting nonconvex problem has an exact representation as a semidenite program with additional linear and second order cone constraints for the case where an additional ellipsoidal constraint is added to trs resulting in the two trust region subproblem ttrs we provide a new relaxation including second order cone constraints that strengthens the usual sdp relaxation

an exact duality theory for semidefinite programming and its complexity implications in this paper we present a new and more complete duality for semidefinite programming sdp with the following features begin itemize item this dual is an explicit semidefinite program whose number of variables and the coefficient bitlengths are polynomial in those of the primal item if the primal is feasible then it is bounded if and only if the dual is feasible item the duality gap ie the difference between the primal and the dual objective function values is zero whenever the primal is feasible and bounded also in this case the dual attains its optimum item it yields a precise farkas lemma for semidefinite feasibility systems ie characterization of the it infeasibility of a semidefinite inequality in terms of the it feasibility of another polynomial size semidefinite inequality end itemize note that the standard duality for linear programming satisfies all of the above features but no such duality theory was previously known for sdp without slater like conditions being assumed then we apply the dual to derive certain complexity results for semidefinite programming problems the decision problem of semidefinite feasibility sdfp ie that of determining if a given semidefinite inequality system is feasible is the central problem of interest the complexity of sdfp is unknown but we show the following 1 in the turing machine model sdfp is not np complete unless np co np 2 in the real number model of blum shub and smale cite bss sdfp is in np cap co np we then give polynomial reductions from the following problems to sdfp 1 checking whether an sdp is bounded 2 checking whether a feasible and bounded sdp attains the optimum 3 checking the optimality of a feasible solution

stable zero duality gaps in convex programming complete dual characterisations with applications to semidefinite programs the zero duality gap that underpins the duality theory is one of the central ingredients in optimisation in convex programming it means that the optimal values of a given convex program and its associated dual program are equal it allows in particular the development of efficient numerical schemes however the zero duality gap property does not always hold even for finite dimensional problems and it frequently fails for problems with non polyhedral constraints such as the ones in semidefinite programming problems over the years various criteria have been developed ensuring zero duality gaps for convex programming problems in the present work we take a broader view of the zero duality gap property by allowing it to hold for each choice of linear perturbation of the objective function of the given problem globalising the property in this way permits us to obtain complete geometric dual characterisations of a stable zero duality gap in terms of epigraphs and conjugate functions for convex semidefinite programs we establish necessary and sufficient dual conditions for stable zero duality gaps as well as for a universal zero duality gap in the sense that the zero duality gap property holds for each choice of constraint right hand side and convex objective function zero duality gap results for second order cone programming problems are also given our approach makes use of elegant conjugate analysis and fenchel s duality

bad semidefinite programs they all look the same conic linear programs among them semidefinite programs often behave pathologically the optimal values of the primal and dual programs may differ and may not be attained we present a novel analysis of these pathological behaviors we call a conic linear system cal a x leq k b badly behaved if the value of sup langle c x rangle cal a x leq k b is finite but the dual program has no solution with the same value for some c we describe simple and intuitive geometric characterizations of badly behaved conic linear systems our main motivation is the striking similarity of badly behaved semidefinite systems in the literature we characterize such systems by certain excluded matrices which are easy to spot in all published examples we prove several other structural results about badly behaved semidefinite systems for example we show that they are in cal np cap mbox rm co cal np in the real number model of computing

some np complete problems in quadratic and nonlinear programming in continuous variable smooth nonconvex nonlinear programming we analyze the complexity of checking whether a a given feasible solution is not a local minimum and b the objective function is not bounded below on the set of feasible solutions we construct a special class of indefinite quadratic programs with simple constraints and integer data and show that checking a or b on this class is np complete as a corollary we show that checking whether a given integer square matrix is not copositive is np complete

a conic duality frank wolfe type theorem via exact penalization in quadratic optimization the famous frank wolfe theorem ensures attainability of the optimal value for quadratic objective functions over a possibly unbounded polyhedron if the feasible values are bounded this theorem does not hold in general for conic programs where linear constraints are replaced by more general convex constraints like positive semidefiniteness or copositivity conditions despite the fact that the objective can be even linear this paper studies exact penalizations of classical quadratic programs i e optimization of quadratic functions over a polyhedron and applies the results to establish a frank wolfe type theorem for the primal dual pair of a class of conic programs that frequently arises in applications one result is that uniqueness of the solution of the primal ensures dual attainability i e existence of the solution of the dual

lyapunov type least squares problems over symmetric cones abstract the lyapunov type least squares problem over symmetric cone is to find the least squares solution of the lyapunov equation with a constraint of symmetric cone in the euclidean jordan algebra and it contains the lyapunov type least squares problem over cone of semidefinite matrices as a special case in this paper we first give a detailed analysis for the image of lyapunov operator in the euclidean jordan algebra relying on these properties together with some characterizations of symmetric cone we then establish some necessary and or sufficient conditions for solution existence of the lyapunov type least squares problem finally we study uniqueness of the least squares solution

strong duality for semidefinite programming it is well known that the duality theory for linear programming lp is powerful and elegant and lies behind algorithms such as simplex and interior point methods however the standard lagrangian for nonlinear programs requires constraint qualifications to avoid duality gaps semidefinite linear programming sdp is a generalization of lp where the nonnegativity constraints are replaced by a semidefiniteness constraint on the matrix variables there are many applications e g in systems and control theory and combinatorial optimization however the lagrangian dual for sdp can have a duality gap we discuss the relationships among various duals and give a unified treatment for strong duality in semidefinite programming these duals guarantee strong duality i e a zero duality gap and dual attainment this paper is motivated by the recent paper by ramana where one of these duals is introduced

combinatorial construction of logarithmic differential forms abstract h terao has shown that the structure of the module of rational differential forms with at most logarithmic poles at an arrangement of hyperplanes as defined by k saito is very strongly controlled by the combinatorial structure of the arrangement in this paper we demonstrate how the existence of rational logarithmic forms with poles of high order depends on the existence of highly degenerate special position subarrangements the associated combinatorial structures are studied first a strong version of k saito s preparation lemma for logarithmic forms leads to a new simple proof of terao s celebrated addition deletion theorem for hyperplane arrangements with free module of logarithmic 1 forms free arrangements then structural characterizations are developed in two extreme cases for the generation of the module of logarithmic differential 1 forms this module has a triangular basis iff the arrangement is supersolvable strictly linearly fibered and it is generated by forms of degree 1 iff the arrangement is generic in codimension 3 both conditions on the geometry of the arrangement are combinatorial in a very strong sense determined by restricted data on the lattice of intersections of the hyperplanes however examples show that the cardinality and degree sequence of a minimal set of generators for the module of logarithmic 1 forms are not in general determined by this intersection lattice

exact semidefinite formulations for a class of random and non random nonconvex quadratic programs we study a class of quadratically constrained quadratic programs qcqps called diagonal qcqps which contain no off diagonal terms x j x k for j ne k and we provide a sufficient condition on the problem data guaranteeing that the basic shor semidefinite relaxation is exact our condition complements and refines those already present in the literature and can be checked in polynomial time we then extend our analysis from diagonal qcqps to general qcqps i e ones with no particular structure by reformulating a general qcqp into diagonal form we establish new polynomial time checkable sufficient conditions for the semidefinite relaxations of general qcqps to be exact finally these ideas are extended to show that a class of random general qcqps has exact semidefinite relaxations with high probability as long as the number of constraints grows no faster than a fixed polynomial in the number of variables to the best of our knowledge this is the first result establishing the exactness of the semidefinite relaxation for random general qcqps

on the duality operator of a convex cone let c be a convex set in r for each y c the cone of c at y denoted by cone y c is the cone a x y a 0 and x e c if k is a cone in r we shall denote by k its dual cone and by k the lattice of faces of k then the duality operator of k is the mapping dk k k given by dn f span f a cq k properties of the duality operator d k of a closed pointed hill cone k have been studied before in this paper we study da for a general cone k especially m relation to d o c v k where y e k our main result says that for any closed cone k in r the duality operator d k is injective surjective if and only if the duality operator d o u n is injective surjective for each vector y e k k n k in the last part of the paper we obtain some partial results on the problem of constructing a compact convex set c which contains the zero vector such that cone 0 c is equal to a given cone

a second order cone based approach for solving the trust region subproblem and its variants we study the trust region subproblem trs of minimizing a nonconvex quadratic function over the unit ball with additional conic constraints despite having a nonconvex objective it is known that the classical trs and a number of its variants are polynomial time solvable in this paper we follow a second order cone soc based approach to derive an exact convex reformulation of the trs under a structural condition on the conic constraint our structural condition is immediately satisfied when there are no additional conic constraints and it generalizes several such conditions studied in the literature as a result our study highlights an explicit connection between the classical nonconvex trs and smooth convex quadratic minimization which allows for the application of cheap iterative methods such as nesterov s accelerated gradient descent to the trs furthermore under slightly stronger conditions we give a low complexity characterization of the convex hull of the epigraph of the nonconvex quadrati

a fresh cp look at mixed binary qps new formulations and relaxations triggered by burer s seminal characterization from 2009 many copositive reformulations of mixed binary qps have been discussed by now most of them can be used as proper relaxations if the intractable co mpletely positive cones are replaced by tractable approximations while the widely used approximation hierarchies have the disadvantage to use positive semidefinite psd matrices of orders which rapidly increase with the level of approximation alternatives focus on the problem of keeping psd matrix orders small with the aim to avoid memory problems in the interior point algorithms this work continues this approach proposing new reformulations and relaxations moreover we provide a thorough comparison of the respective duals and establish a monotonicity relation among their duality gaps we also identify sufficient conditions for strong duality zero duality gap in some of these formulations and generalize some of our observations to general conic problems

some almost exceptional arrangements abstract we introduce and study a family of real hyperplane arrangements that includes the reflection arrangements of types a and d as well as an infinite family of arrangements e n constructed by orlik solomon and terao related to the exceptional arrangements of type e n n 8 we investigate freeness in the sense of terao of these arrangements disproving the conjecture that the arrangements e n are free in general

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

generalized exponents of a free arrangement of hyperplanes and shepherd todd brieskorn formula we define an n arrangement as a finite family of hyperplanes through the origin in c 1 in 11 and 12 we studied the free arrangement and defined its structure sequence their definitions will be given again in sect 2 in this article we say the generalized exponents instead of the structure sequence let d o d 1 d be the generalized exponents of a free n arrangement x let xj u h our main result is hex

new results on quadratic minimization in this paper we present several new results on minimizing an indefinite quadratic function under quadratic linear constraints the emphasis is placed on the case in which the constraints are two quadratic inequalities this formulation is termed the extended trust region subproblem in this paper to distinguish it from the ordinary trust region subproblem in which the constraint is a single ellipsoid the computational complexity of the extended trust region subproblem in general is still unknown in this paper we consider several interesting cases related to this problem and show that for those cases the corresponding semidefinite programming relaxation admits no gap with the true optimal value and consequently we obtain polynomial time procedures for solving those special cases of quadratic optimization for the extended trust region subproblem itself we introduce a parameterized problem and prove the existence of a trajectory that will lead to an optimal solution combining this with a result obtained in the first part of the paper we propose a polynomial time solution procedure for the extended trust region subproblem arising from solving nonlinear programs with a single equality constraint

a gentle geometric introduction to copositive optimization this paper illustrates the fundamental connection between nonconvex quadratic optimization and copositive optimization a connection that allows the reformulation of nonconvex quadratic problems as convex ones in a unified way we focus on examples having just a few variables or a few constraints for which the quadratic problem can be formulated as a copositive style problem which itself can be recast in terms of linear second order cone and semidefinite optimization a particular highlight is the role played by the geometry of the feasible set

copositive optimization recent developments and applications due to its versatility copositive optimization receives increasing interest in the operational research community and is a rapidly expanding and fertile field of research it is a special case of conic optimization which consists of minimizing a linear function over a cone subject to linear constraints the diversity of copositive formulations in different domains of optimization is impressive since problem classes both in the continuous and discrete world as well as both deterministic and stochastic models are covered copositivity appears in local and global optimality conditions for quadratic optimization but can also yield tighter bounds for np hard combinatorial optimization problems here some of the recent success stories are told along with principles algorithms and applications

convexifiability of continuous and discrete nonnegative quadratic programs for gap free duality abstract in this paper we show that a convexifiability property of nonconvex quadratic programs with nonnegative variables and quadratic constraints guarantees zero duality gap between the quadratic programs and their semi lagrangian duals more importantly we establish that this convexifiability is hidden in classes of nonnegative homogeneous quadratic programs and discrete quadratic programs such as mixed integer quadratic programs revealing zero duality gaps as an application we prove that robust counterparts of uncertain mixed integer quadratic programs with objective data uncertainty enjoy zero duality gaps under suitable conditions various sufficient conditions for convexifiability are also given

regularizing the abstract convex program abstract characterizations of optimality for the abstract convex program inf p x g x s x p where s is an arbitrary convex cone in a finite dimensional space is a convex set and p and g are respectively convex and s convex on were given in 10 these characterizations hold without any constraint qualification they use the minimal cone s f of p and the cone of directions of constancy d g s f in the faithfully convex case these cones can be used to regularize p i e transform p into an equivalent program p r for which slater s condition holds we present an algorithm that finds both s f and d g s f the main step of the algorithm consists in solving a particular complementarity problem we also present a characterization of optimality for p in terms of the cone of directions of constancy of a convex functional d g rather than d g s f

computable representations for convex hulls of low dimensional quadratic forms let mathcal c be the convex hull of points 1 choose x 1 choose x t x in mathcal f subset re n representing or approximating mathcal c is a fundamental problem for global optimization algorithms based on convex relaxations of products of variables we show that if n 4 and mathcal f is a simplex then mathcal c has a computable representation in terms of matrices x that are doubly nonnegative positive semidefinite and componentwise nonnegative we also prove that if n 2 and mathcal f is a box then mathcal c has a representation that combines semidefiniteness with constraints on product terms obtained from the reformulation linearization technique rlt the simplex result generalizes known representations for the convex hull of x 1 x 2 x 1x 2 x in mathcal f when mathcal f subset re 2 is a triangle while the result for box constraints generalizes the well known fact that in this case the rlt constraints generate the convex hull of x 1 x 2 x 1x 2 x in mathcal f when n 3 and mathcal f is a box we show that a representation for mathcal c can be obtained by utilizing the simplex result for n 4 in conjunction with a triangulation of the 3 cube

copositive relaxation beats lagrangian dual bounds in quadratically and linearly constrained quadratic optimization problems we study nonconvex quadratic minimization problems under possibly nonconvex quadratic and linear constraints characterizing both lagrangian and semi lagrangian dual bounds in terms of conic optimization while the lagrangian dual is equivalent to the sdp relaxation which has been known for quite a while although the presented form incorporating explicitly linear constraints seems to be novel we show that the semi lagrangian dual is equivalent to a natural copositive relaxation and this has apparently not been observed before this way we arrive at conic bounds tighter than the usual lagrangian dual and thus than the sdp bounds any of the known tractable inner approximations of the copositive cone can be used for this tightening but in particular the above mentioned characterization with explicit linear constraints is a natural much cheaper relaxation than the usual zero order approximation by doubly nonnegative dnn matrices and still improves upon the lagrangian dual bounds these app