probability of directional errors with disordinal qualitative interaction in a factorial design with two or more factors there is nonzero interaction when the differences among the levels of one factor vary with levels of other factors the interaction is disordinal or qualitative with respect to a specific factor saya if the difference between at least two levels ofa is positive for some and negative for some levels of the other factors using standard methods of analysis there is a potentially large probability of drawing incorrect conclusions about the signs of differences in the presence of disordinal interaction the maximum probability of such incorrect conclusions or directional errors is derived for two factor designs in which the factor of interest has two levels and the number of levels of the other factor varies

hidden conic quadratic representation of some nonconvex quadratic optimization problems the problem of minimizing a quadratic objective function subject to one or two quadratic constraints is known to have a hidden convexity property even when the quadratic forms are indefinite the equivalent convex problem is a semidefinite one and the equivalence is based on the celebrated s lemma in this paper we show that when the quadratic forms are simultaneously diagonalizable sd it is possible to derive an equivalent convex problem which is a conic quadratic cq one and as such is significantly more tractable than a semidefinite problem the sd condition holds for free for many problems arising in applications in particular when deriving robust counterparts of quadratic or conic quadratic constraints affected by implementation error the proof of the hidden cq property is constructive and does not rely on the s lemma this fact may be significant in discovering hidden convexity in some nonquadratic problems

an exact test for multiple inequality and equality constraints in the linear regression model abstract in this article we consider the linear regression model y x e where e is n 0 2i in this context we derive exact tests of the form h r r versus k rk for the case in which 2 is unknown we extend these results to consider hypothesis tests of the form h r1 r1 and r2 r2 versus k rk for each of these hypotheses tests we derive several equivalent forms of the test statistics using the duality theory of the quadratic programming for both tests we derive their exact distribution as a weighted sum of snedecor s f distributions normalized by the numerator degrees of freedom of each f distribution of the sum a methodology for computing critical values as well as probability values for the tests is discussed the relationship between this testing framework and the multivariate one sided hypothesis testing literature is also discussed in this context we show that for any size of the hypothesis test h 0 versus k rk the test statistic and critical value obtained a

quadratically constrained least squares and quadratic problems we consider the following problem compute a vectorx such that ax b 2 min subject to the constraint x 2 a new approach to this problem based on gauss quadrature is given the method is especially well suited when the dimensions ofa are large and the matrix is sparse it is also possible to extend this technique to a constrained quadratic form for a symmetric matrixa we consider the minimization ofx t a x 2b t x subject to the constraint x 2 some numerical examples are given

a note on polynomial solvability of the cdt problem recent research has focused on the complexity of extensions of the classical trust region subproblem which addresses the minimization of a quadratic function over a unit ball in mathbb r n

two new tests associated with analysis of variance summary two new tests are described in connexion with analysis of variance the first is for the presence of qualitative interaction between a treatment and an intrinsic factor both with qualitatively defined levels such an interaction is said to be present if there are two levels of the treatment factor and two levels of the intrinsic factor such that the treatment difference has opposite signs at the two levels of the intrinsic factor the second test is for the presence of a treatment effect which can be tested via any of a hierarchical sequence u1 u1 u2 u1 ur of sums of squares an allowance for selection is calculated

on affine scaling algorithms for nonconvex quadratic programming we investigate the use of interior algorithms especially the affine scaling algorithm to solve nonconvex indefinite or negative definite quadratic programming qp problems although the nonconvex qp with a polytope constraint is a hard problem we show that the problem with an ellipsoidal constraint is easy when the hard qp is solved by successively solving the easy qp the sequence of points monotonically converge to a feasible point satisfying both the first and the second order optimality conditions

inventories of unavoidable languages and the word extension conjecture abstract a language x on an alphabet a is unavoidable iff all but finitely many words in a have a factor in x in this paper i prove that the inventory of unavoidable languages of n words can be explicitly made for every n that the reduced unavoidable languages of given cardinality are finite in number an unavoidable language is minimal if no proper subset is unavoidable it is reduced if it is minimal and if whenever a word is replaced by a proper factor the resulting unavoidable language is not minimal and i give a counterexample to the word extension conjecture which said that in every unavoidable language there is a word w and a letter a such that the language where w is replaced by wa is still unavoidable

an approximation scheme for distributionally robust nonlinear optimization we consider distributionally robust optimization problems drops with nonlinear and nonconcave dependence on uncertain parameters the drop can be written as a nonsmooth nonlinear program with a

strong duality in nonconvex quadratic optimization with two quadratic constraints we consider the problem of minimizing an indefinite quadratic function subject to two quadratic inequality constraints when the problem is defined over the complex plane we show that strong duality holds and obtain necessary and sufficient optimality conditions we then develop a connection between the image of the real and complex spaces under a quadratic mapping which together with the results in the complex case lead to a condition that ensures strong duality in the real setting preliminary numerical simulations suggest that for random instances of the extended trust region subproblem the sufficient condition is satisfied with a high probability furthermore we show that the sufficient condition is always satisfied in two classes of nonconvex quadratic problems finally we discuss an application of our results to robust least squares problems

quadratically constrained attitude control via semidefinite programming we consider the attitude control problem augmented with an arbitrary number of nonconvex quadratic constraints by utilizing the implicit magnitude constraint of the state vector i e quaternion we are able to reduce this problem to a convex quadratically constrained quadratic program or a semidefinite program this result leads to an efficient solution strategy for general quadratically constrained attitude control problems an example demonstrates the proposed methodology

an efficient algorithm for globally minimizing a quadratic function under convex quadratic constraints in this paper we investigate two approaches to minimizing a quadratic form subject to the intersection of finitely many ellipsoids the first approach is the d c difference of convex functions optimization algorithm abbr dca whose main tools are the proximal point algorithm and or the projection subgradient method in convex minimization the second is a branch and bound scheme using lagrangian duality for bounding and ellipsoidal bisection in branching the dca was first introduced by pham dinh in 1986 for a general d c program and later developed by our various work is a local method but from a good starting point it provides often a global solution this motivates us to combine the dca and our branch and bound algorithm in order to obtain a good initial point for the dca and to prove the globality of the dca in both approaches we attempt to use the ellipsoidal constrained quadratic programs as the main subproblems the idea is based upon the fact that these programs can be efficiently solved by some available polynomial and nonpolynomial time algorithms among them the dca with restarting procedure recently proposed by pham dinh and le thi has been shown to be the most robust and fast for large scale problems several numerical experiments with dimension up to 200 are given which show the effectiveness and the robustness of the dca and the combined dca branch and bound algorithm

detecting qualitative interaction a bayesian approach differences in treatment effects between centers in a multi center trial may be important these differences represent treatment by subgroup interaction peto defines qualitative interaction qi to occur when the simple treatment effect in one subgroup has a different sign than in another subgroup this interaction is important interaction where the treatment effects are of the same sign in all subgroups is called quantitative and is often not important because the treatment recommendation is identical in all cases a hierarchical model is used here with exchangeable mean responses to each treatment between subgroups the posterior probability of qi and the corresponding bayes factor are proposed as a diagnostic and as a test statistic the model is motivated by two multi center trials with binary responses the frequentist power and size of the test using the bayes factor are examined and compared with two other commonly used tests the impact of imbalance between the sample sizes in each subgroup on power is examined and the test based on the bayes factor typically has better power for unbalanced designs especially for small sample sizes an exact test based on the bayes factor is also suggested assuming the hierarchical model the bayes factor provides a concise summary of the evidence for or against qi it is shown by example that it is easily adapted to summarize the evidence for clinically meaningful qi defined as the simple effects being of opposite signs and larger in absolute value than a minimal clinically meaningful effect

new results on hermitian matrix rank one decomposition in this paper we present several new rank one decomposition theorems for hermitian positive semidefinite matrices which generalize our previous results in huang and zhang math oper res 32 3 758 768 2007 ai and zhang siam j optim 19 4 1735 1756 2009 the new matrix rank one decomposition theorems appear to have wide applications in theory as well as in practice on the theoretical side for example we show how to further extend some of the classical results including a lemma due to yuan math program 47 53 63 1990 the classical results on the convexity of the joint numerical ranges pang and zhang in unpublished manuscript 2004 au yeung and poon in southeast asian bull math 3 85 92 1979 and the so called finsler s lemma bohnenblust in unpublished manuscript au yeung and poon in southeast asian bull math 3 85 92 1979 on the practical side we show that the new results can be applied to solve two typical problems in signal processing and communication one for radar code optimization and the other for robust beamforming the new matrix decomposition theorems are proven by construction in this paper and we demonstrate that the constructive procedures can be implemented efficiently stably and accurately the url of our matlab programs is given in this paper we strongly believe that the new decomposition procedures as a means to solve non convex quadratic optimization with a few quadratic constraints are useful for many other potential engineering applications

on a subproblem of trust region algorithms for constrained optimization we study a subproblem that arises in some trust region algorithms for equality constrained optimization it is the minimization of a general quadratic function with two special quadratic constraints properties of such subproblems are given it is proved that the hessian of the lagrangian has at most one negative eigenvalue and an example is presented to show that the hessian may have a negative eigenvalue when one constraint is inactive at the solution

an exact algorithm for nonconvex quadratic integer minimization using ellipsoidal relaxations we propose a branch and bound algorithm for minimizing a not necessarily convex quadratic function over integer variables the algorithm is based on lower bounds computed as continuous minima of the objective function over appropriate ellipsoids in the nonconvex case we use ellipsoids enclosing the feasible region of the problem in spite of the nonconvexity these minima can be computed quickly the corresponding optimization problems are equivalent to trust region subproblems we present several ideas that allow us to accelerate the solution of the continuous relaxation within a branch and bound scheme and examine the performance of the overall algorithm by computational experiments good computational performance is shown especially for ternary instances

a linear time algorithm for trust region problems we consider the fundamental problem of minimizing a general quadratic function over an ellipsoidal domain also known as the trust region sub problem we give the first provable linear time in the number of non zero entries of the input algorithm for approximately solving this problem specifically our algorithm returns an epsilon approximate solution in runtime of order n sqrt epsilon where n is the number of non zero entries in the input this matches the runtime of nesterov s accelerated gradient descent suitable for the special case in which the quadratic objective is convex and the runtime of the lanczos method which is applicable when the problem is purely quadratic

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

on the classical necessary second order optimality conditions in the presence of equality and inequality constraints for nonconvex optimization problems in rn a counterexample was recently given by anitescu siam j optim 10 2000 pp 1116 1135 that showed that the mangasarian fromovitz constraint qualification mfcq is not sufficient for the classical necessary second order optimality conditions to hold we prove these optimality conditions with the assumptions that the set of lagrange multipliers is a bounded line segment and a relaxed strict complementary slackness scs condition holds a new constraint qualification is presented in this paper we assume that the mfcq holds and the active constraint rank deficiency is 1 this assumption relaxes the linear independence constraint qualification and enforces the mfcq in addition the set of lagrange multipliers is a bounded line segment

tests against qualitative interaction exact critical values and robust tests summary consider a study to evaluate treatment a with a placebo in two or more groups of patients if treatment a is beneficial to one group of patients and harmful to another then we say that there is qualitative interaction or crossover interaction between patient groups and the treatments gail and simon 1985 biometrics41 361 372 developed a large sample procedure for this testing problem their test has received favorable coverage in the literature in this article we obtain corresponding exact finite sample results for normal error distribution and provide a table of critical values the test statistic is similar to the familiar f ratio and its p value is equal to a weighted sum of tail areas of f distributions the computations to implement this are simple a simulation study shows that the exact critical values provided here for normal error distribution are preferable to the asymptotic critical values for a wide range of error distributions we also develop tests that are power robust against long tailed error distributions our robust test uses m estimators instead of the least squares estimators we show that the efficiency robustness of the m estimator translates to power robustness of the corresponding test therefore our robust tests are better if outliers are expected a simulation study illustrates the substantial power advantages of our robust tests

uniformly more powerful tests for hypotheses concerning linear inequalities and normal means abstract this article considers some hypothesis testing problems regarding normal means in these problems the hypotheses are defined by linear inequalities on the means we show that in certain problems the likelihood ratio test lrt is not very powerful we describe a test that has the same size as the lrt and is uniformly more powerful the test is easily implemented since its critical values are standard normal percentiles the increase in power with the new test can be substantial for example the new test s power is 1 2 times bigger 10 times bigger for 05 than the lrt s power for some parameter points in a simple example specifically let x x 1 xp p 2 be a multivariate normal random vector with unknown mean 1 p and known nonsingular covariance matrix we consider testing the null hypothesis h 0 b i 0 for some i 1 k versus the alternative hypothesis h 1 b i 0 for all i 1 k here b 1 b k k 2 are specified p dimensional ve

on the solution of a two ball trust region subproblem in this paper we investigate the structure of a two ball trust region subproblem arising frequently in nonlinear parameter identification problems and propose a method for its solution the method decomposes the subproblem and allows the application of efficient well studied methods for the solution of trust region subproblems arising in unconstrained optimization in the discussion of the structure we focus on the case where both constraints are active and on the treatment of the unconstrained problem

indefinite trust region subproblems and nonsymmetric eigenvalue perturbations this paper extends the theory of trust region subproblems in two ways i it allows indefinite inner products in the quadratic constraint and ii it uses a two sided upper and lower bound quadratic constraint characterizations of optimality are presented that have no gap between necessity and sufficiency conditions for the existence of solutions are given in terms of the definiteness of a matrix pencil a simple dual program is introduced that involves the maximization of a strictly concave function on an interval this dual program simplifies the theory and algorithms for trust region subproblems it also illustrates that the trust region subproblems are implicit convex programming problems and thus explains why they are so tractable the duality theory also provides connections to eigenvalue perturbation theory trust region subproblems with zero linear term in the objective function correspond to eigenvalue problems and adding a linear term in the objective function is seen to correspond to a p

computing a celis dennis tapia trust region step for equality constrained optimization we study an approach for minimizing a convex quadratic function subject to two quadratic constraints this problem stems from computing a trust region step for an sqp algorithm proposed by celis dennis and tapia 1985 for equality constrained optimization our approach is to reformulate the problem into a univariate nonlinear equation 0 where the function is continuous at least piecewise differentiable and monotone well established methods then can be readily applied we also consider an extension of our approach to a class of non convex quadratic functions and show that our approach is applicable to reduced hessian sqp algorithms numerical results are presented indicating that our algorithm is reliable robust and has the potential to be used as a building block to construct trust region algorithms for small sized problems in constrained optimization

convex hull of two quadratic or a conic quadratic and a quadratic inequality in this paper we consider an aggregation technique introduced by yildiran j math control inf 26 417 450 2009 to study the convex hull of regions defined by two quadratic inequalities or by a conic quadratic and a quadratic inequality yildiran 2009 shows how to characterize the convex hull of open sets defined by two strict quadratic inequalities using linear matrix inequalities we show how this aggregation technique can be easily extended to yield valid conic quadratic inequalities for the convex hull of open sets defined by two strict quadratic inequalities or by a strict conic quadratic and a strict quadratic inequality we also show that for sets defined by a strict conic quadratic and a strict quadratic inequality under one additional containment assumption these valid inequalities characterize the convex hull exactly we also show that under certain topological assumptions the results from the open setting can be extended to characterize the closed convex hull of sets defined with non strict conic and quadratic inequalities

globally solving the trust region subproblem using simple first order methods we consider the trust region subproblem which is given by a minimization of a quadratic not necessarily convex function over the euclidean ball based on the well known second order necessary and

s lemma with equality and its applications let f x x tax 2a tx c f x xtax 2atx c and h x x tbx 2b tx d h x xtbx 2btx d be two quadratic functions having symmetric matrices a a and b b the s lemma with equality asks when the unsolvability of the system f x 0 h x 0 f x 0 h x 0 implies the existence of a real number mu such that f x mu h x ge 0 forall x in mathbb r n f x h x 0 x rn the problem is much harder than the inequality version which asserts that under slater condition f x 0 h x le 0 f x 0 h x 0 is unsolvable if and only if f x mu h x ge 0 forall x in mathbb r n f x h x 0 x rn for some mu ge 0 0 in this paper we show that the s lemma with equality does not hold only when the matrix a a has exactly one negative eigenvalue and h x h x is a non constant linear function b 0 b not 0 b 0 b 0 as an application we can globally solve inf f x h x 0 inf f x h x 0 as well as the two sided generalized trust region subproblem inf f x l le h x le u inf f x l h x u without any condition moreover the convexity of the joint numerical range f x h 1 x ldots h p x x in mathbb r n f x h1 x hp x x rn where f f is a possibly non convex quadratic function and h 1 x ldots h p x h1 x hp x are affine functions can be characterized using the newly developed s lemma with equality

on regularity of context free languages this paper considers conditions under which a context free language is regular and conditions which imposed on productions of a rewriting system generating a context free language will guarantee that the generated language is regular in particular 1 1 necessary and sufficient conditions on productions of a unitary grammar are given that guarantee the generated language to be regular a unitary grammar is a semi thue system in which the left hand of each production is the empty word and 2 2 it is proved that commutativity of a linear language implies its regularity to obtain the former result we give a generalization of the myhill nerode characterization of the regular languages in terms of well quasi orders along with a generalization of higman s well quasi order result concerning the subsequence embedding relation on in obtaining the latter results we introduce the class of periodic languages and demonstrate how they can be used to characterize the commutative regular languages here we also utilize the theory of well quasi orders

incorporating nonmonotone strategies into the trust region method for unconstrained optimization this paper concerns a nonmonotone line search technique and its application to the trust region method for unconstrained optimization problems in our line search technique the current nonmonotone term is a convex combination of the previous nonmonotone term and the current objective function value instead of an average of the successive objective function values that was introduced by zhang and hager h zhang w w hager a nonmonotone line search technique and its application to unconstrained optimization siam j optim 14 4 2004 1043 1056 we incorporate this nonmonotone scheme into the traditional trust region method such that the new algorithm possesses nonmonotonicity unlike the traditional trust region method our algorithm performs a nonmonotone line search to find a new iteration point if a trial step is not accepted instead of resolving the subproblem under mild conditions we prove that the algorithm is global and superlinear convergence holds primary numerical results are reported

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

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

randomized algorithms for optimal solutions of double sided qcqp with applications in signal processing quadratically constrained quadratic programming qcqp with double sided constraints has plenty of applications in signal processing as have been addressed in recent years qcqp problems are hard to solve in general and they are typically approached by solving a semidefinite programming sdp relaxation followed by a postprocessing procedure existing postprocessing schemes include gaussian randomization to generate an approximate solution rank reduction procedure the so called purification and some specific rank one matrix decomposition techniques to yield a globally optimal solution in this paper we propose several randomized postprocessing methods to output not an approximate solution but a globally optimal solution for some solvable instances of the double sided qcqp i e instances with a small number of constraints we illustrate their applicability in robust receive beamforming radar optimal code design and broadcast beamforming for multiuser communications as a byproduct we derive an alternative shorter proof for the sturm zhang rank one matrix decomposition theorem

unavoidable sets we obtain new results on minimum lengths of words in an unavoidable set of words of cardinality n before introducing the notion of aperiodic unavoidable sets a natural extension of unavoidable sets

the generalized trust region subproblem solution complexity and convex hull results we consider the generalized trust region subproblem gtrs of minimizing a nonconvex quadratic objective over a nonconvex quadratic constraint a lifting of this problem recasts the gtrs as minimizing a linear objective subject to two nonconvex quadratic constraints our first main contribution is structural we give an explicit description of the convex hull of this nonconvex set in terms of the generalized eigenvalues of an associated matrix pencil this result may be of interest in building relaxations for nonconvex quadratic programs moreover this result allows us to reformulate the gtrs as the minimization of two convex quadratic functions in the original space our next set of contributions is algorithmic we present an algorithm for solving the gtrs up to an epsilon additive error based on this reformulation we carefully handle numerical issues that arise from inexact generalized eigenvalue and eigenvector computations and establish explicit running time guarantees for these algorithms notably our algorithms run in linear in the size of the input time furthermore our algorithm for computing an epsilon optimal solution has a slightly improved running time dependence on epsilon over the state of the art algorithm our analysis shows that the dominant cost in solving the gtrs lies in solving a generalized eigenvalue problem establishing a natural connection between these problems finally generalizations of our convex hull results allow us to apply our algorithms and their theoretical guarantees directly to equality interval and hollow constrained variants of the gtrs this gives the first linear time algorithm in the literature for these variants of the gtrs

a semidefinite framework for trust region subproblems with applications to large scale minimization primal dual pairs of semidefinite programs provide a general framework for the theory and algorithms for the trust region subproblem trs this latter problem consists in minimizing a general quadratic function subject to a convex quadratic constraint and therefore it is a generalization of the minimum eigenvalue problem the importance of trs is due to the fact that it provides the step in trust region minimization algorithms the semidefinite framework is studied as an interesting instance of semidefinite programming as well as a tool for viewing known algorithms and deriving new algorithms for trs in particular a dual simplex type method is studied that solves trs as a parametric eigenvalue problem this method uses the lanczos algorithm for the smallest eigenvalue as a black box therefore the essential cost of the algorithm is the matrix vector multiplication and thus sparsity can be exploited a primal simplex type method provides steps for the so called hard case extensive numerical tests for large sparse problems are discussed these tests show that the cost of the algorithm is 1 n times the cost of finding a minimum eigenvalue using the lanczos algorithm where 0 n 1 is a fraction which decreases as the dimension increases

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

nonmonotone trust region method for solving optimization problems in this paper a trust region tr method with nonmonotone technique for optimization is proposed we construct a new ratio of actual descent and predicted descent which is a simple and natural generalization of the modified armijo line search rule the paper exposes the relationship between the trust region method and line search approach since this method possesses the robust properties of trust region subproblem it is globally convergent although we employ the nonmonotone sequence of function values instead of the monotone sequence in addition the proof of convergence is obviously simpler than one of newton type method with nonmonotone line search finally applications of the nonmonotone tr algorithm to some optimization problems are discussed

an efficient search algorithm for road network optimization in this paper we propose an efficient search algorithm for determining optimal reserve capacity of a road network with toll settings optimization of network reserve capacity with toll settings can be formulated as a mathematical program with equilibrium constraints mpec when user equilibrium flows are taken into account a trust region subgradient projection method is proposed to efficiently solve this network design problem with global convergence numerical calculations are illustrated using an example network and promising results are reported

duality and sensitivity in nonconvex quadratic optimization over an ellipsoid abstract in this paper a duality framework is discussed for the problem of optimizing a nonconvex quadratic function over an ellipsoid additional insight is obtained from the observation that this nonconvex problem is in a sense equivalent to a convex problem of the same type from which known necessary and sufficient conditions for optimality readily follow based on the duality results some existing solution procedures are interpreted as in fact solving the dual the duality relations are also shown to provide a natural framework for sensitivity analysis

second order global optimality conditions for convex composite optimization in recent years second order sufficient conditions of an isolated local minimizer for convex composite optimization problems have been established in this paper second order optimality conditions are obtained of aglobal minimizer for convex composite problems with a non finite valued convex function and a twice strictly differentiable function by introducing a generalized representation condition this result is applied to a minimization problem with a closed convex set constraint which is shown to satisfy the basic constraint qualification in particular second order necessary and sufficient conditions of a solution for a variational inequality problem with convex composite inequality constraints are obtained 1998 the mathematical programming society inc published by elsevier science b v

a simple duality proof in convex quadratic programming with a quadratic constraint and some applications abstract in this paper a simple derivation of duality is presented for convex quadratic programs with a convex quadratic constraint this problem arises in a number of applications including trust region subproblems of nonlinear programming regularized solution of ill posed least squares problems and ridge regression problems in statistical analysis in general the dual problem is a concave maximization problem with a linear equality constraint we apply the duality result to 1 the trust region subproblem 2 the smoothing of empirical functions and 3 to piecewise quadratic trust region subproblems arising in nonlinear robust huber m estimation problems in statistics the results are obtained from a straightforward application of lagrange duality

testing avoidability on sets of partial words is hard we prove that the problem of deciding whether a finite set of partial words is unavoidable is np hard for any alphabet of size larger than or equal to two which is in contrast with the well known feasability results for unavoidability of a set of full words we raise some related questions on avoidability of sets of partial words

solving generalized cdt problems via two parameter eigenvalues we consider solving a nonconvex quadratic minimization problem with two quadratic constraints one of which being convex this problem is a generalization of the celis denis tapia cdt problem and thus we refer to it as gcdt generalized cdt the cdt problem has been widely studied but no polynomial time algorithm was known until bienstock s recent work his algorithm solves the cdt problem in polynomial time with respect to the number of bits in data and log epsilon 1 by admitting an epsilon error in the constraints the algorithm however appears to be difficult to implement in this paper we present another algorithm for gcdt which is guaranteed to find a global solution for almost all gcdt instances and slightly perturbed ones in some exceptionally rare cases in exact arithmetic including eigenvalue computation our algorithm is based on the approach proposed by iwata nakatsukasa and takeda 2015 for computing the signed distance between overlapping ellipsoids our algorithm computes all the lagrange multipliers of gcdt by solving a two parameter linear eigenvalue problem obtains the corresponding kkt points and finds a global solution as the kkt point with the smallest objective value in practice in finite precision arithmetic our algorithm requires o n 6 log log u 1 computational time where n is the number of variables and u is the unit roundoff although we derive our algorithm under the unrealistic assumption that exact eigenvalues can be computed numerical experiments illustrate that our algorithm performs well in finite precision arithmetic

doubly constrained robust capon beamformer with ellipsoidal uncertainty sets the doubly constrained robust dcr capon beamformer with a spherical uncertainty set was introduced and studied by stoica and wang here we consider the generalized dcr problem gdcr in which the uncertainty set is an ellipsoid rather than a sphere although as noted previously by stoica and wang this problem is nonconvex and appears to be intractable we show that it can be solved efficiently in fact we prove that the gdcr beamformer can be obtained as a solution to a convex optimization problem to this end we first derive a strong duality result for nonconvex quadratic programs with two quadratic constraints over the complex domain specializing the results to our context leads to a semidefinite programming formulation of the gdcr beamformer

a note on quadratic forms we extend an interesting theorem of yuan 12 for two quadratic forms to three matrices let c 1 c 2 c 3 be three symmetric matrices in n n if max x t c 1 x x t c 2 x x t c 3 x 0 for all x n it is proved that there exist t i 0 i 1 2 3 such that i 1 3 t i 1 and i 1 3 t i c i has at most one negative eigenvalue

a survey of the s lemma in this survey we review the many faces of the s lemma a result about the correctness of the s procedure the basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization convex geometry and linear algebra as well these were all active research areas but as there was little interaction between researchers in these different areas their results remained mainly isolated here we give a unified analysis of the theory by providing three different proofs for the s lemma and revealing hidden connections with various areas of mathematics we prove some new duality results and present applications from control theory error estimation and computational geometry

on the solution of the tikhonov regularization of the total least squares problem total least squares tls is a method for treating an overdetermined system of linear equations bf a bf x approx bf b where both the matrix bf a and the vector bf b are contaminated by noise tikhonov regularization of the tls trtls leads to an optimization problem of minimizing the sum of fractional quadratic and quadratic functions as such the problem is nonconvex we show how to reduce the problem to a single variable minimization of a function mathcal g over a closed interval computing a value and a derivative of mathcal g consists of solving a single trust region subproblem for the special case of regularization with a squared euclidean norm we show that mathcal g is unimodal and provide an alternative algorithm which requires only one spectral decomposition a numerical example is given to illustrate the effectiveness of our method

an alternative theorem for quadratic forms and extensions abstract this note concerns an alternative theorem for quadratic forms established recently by y yuan we explore some directions in which yuan s result can be extended and we indicate also some important situations where an alternative theorem a la yuan does not hold

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 trust region algorithm for equality constrained optimization a trust region algorithm for equality constrained optimization is proposed that employs a differentiable exact penalty function under certain conditions global convergence and local superlinear convergence results are proved

a global convergence theory for dennis el alem and maciel s class of trust region algorithms for constrained optimization without assuming regularity this work presents a convergence theory for dennis el alem and maciel s class of trust region based algorithms for solving the smooth nonlinear programming problem with equality constraints the results are proved under very mild conditions on the quasi normal and tangential components of the trial steps the lagrange multiplier estimates and the hessian estimates are assumed to be bounded no regularity assumption is made in particular linear independence of the gradients of the constraints is not assumed the theory proves global convergence for the class in particular it shows that a subsequence of the iteration sequence satisfies one of four types of mayer bliss stationary conditions in the limit

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

quadratic matrix programming we introduce and study a special class of nonconvex quadratic problems in which the objective and constraint functions have the form f boldmath x tr boldmath x t boldmath a boldmath x 2 tr boldmath b t boldmath x c boldmath x in real r n times r the latter formulation is termed quadratic matrix programming qmp of order r we construct a specially devised semidefinite relaxation sdr and dual for the qmp problem and show that under some mild conditions strong duality holds for qmp problems with at most r constraints using a result on the equivalence of two characterizations of the nonnegativity property of quadratic functions of the above form we are able to compare the constructed sdr and dual problems to other known sdrs and dual formulations of the problem an application to robust least squares problems is discussed

hidden convexity in some nonconvex quadratically constrained quadratic programming we consider the problem of minimizing an indefinite quadratic objective function subject to twosided indefinite quadratic constraints under a suitable simultaneous diagonalization assumption which trivially holds for trust region type problems we prove that the original problem is equivalent to a convex minimization problem with simple linear constraints we then consider a special problem of minimizing a concave quadratic function subject to finitely many convex quadratic constraints which is also shown to be equivalent to a minimax convex problem in both cases we derive the explicit nonlinear transformations which allow for recovering the optimal solution of the nonconvex problems via their equivalent convex counterparts special cases and applications are also discussed we outline interior point polynomial time algorithms for the solution of the equivalent convex programs

on the problem of interactions in the analysis of variance abstract for the two way analysis of variance without the assumption of zero interactions consider the following recommendation test the zero interactions hypothesis and if the hypothesis is not rejected proceed as if the interactions were zero this recommendation is unwarranted and may give incorrect decisions with a large probability as has been pointed out in several articles beginning in the year 1935 despite such isolated criticism it is a recommendation suggested in most applied books in the field and used in countless applications an apology for the recommendation is that it can be easily improved by considering the power of the test and thus obtaining information on the interactions neglected we consider this improved recommendation for the goals of i an interval estimate of one of the cell expectations ii a simultaneous interval estimate of the cell expectations and iii an estimate of the cell with the largest expectation we show that the improved recommendation leads to method

potpourri of conjectures and open questions in nonlinear analysis and optimization we present a collection of fourteen conjectures and open problems in the fields of nonlinear analysis and optimization these problems can be classified into three groups problems of pure mathematical interest problems motivated by scientific computing and applications and problems whose solutions are known but for which we would like to know better proofs for each problem we provide a succinct presentation a list of appropriate references and a view of the state of the art of the subject

qualitative interactions in multifactor studies in clinical trials qualitative interaction or crossover interaction is said to occur when one treatment is superior for some sets of patients and the alternative treatment is superior for other subsets here we propose a definition of no qualitative interaction with respect to a single continuous covariate which implies that one treatment is superior to the other treatment over a prespecified range of the covariate further in studies involving patients cross classified by two or more prognostic factors we define a marginal qualitative interaction with respect to a treatment factor and a prognostic factor by averaging in some sense over the remaining prognostic factors in some situations the methodology of gail and simon 1985 biometrics 41 361 372 is shown to be appropriate their procedure assumes independent estimates of treatment effect for each subset of patients and this may not be appropriate for the generalizations we propose therefore we generalize the procedure of gail and simon to the case of two correlated estimates of treatment effect providing a table of critical values results for one sided tests for the case of j correlated estimates are also obtained we also present an example illustrating our procedure

socp reformulation for the generalized trust region subproblem via a canonical form of two symmetric matrices we investigate in this paper the generalized trust region subproblem gtrs of minimizing a general quadratic objective function subject to a general quadratic inequality constraint by applying a simultaneous block diagonalization approach we obtain a congruent canonical form for the symmetric matrices in both the objective and constraint functions by exploiting the block separability of the canonical form we show that all gtrss with an optimal value bounded from below are second order cone programming socp representable our result generalizes the recent work of ben tal and den hertog math program 143 1 2 1 29 2014 which establishes the socp representability of the gtrs under the assumption of the simultaneous diagonalizability of the two matrices in the objective and constraint functions we then derive a closed form solution for the gtrs when the two matrices are not simultaneously diagonalizable we further extend our method to two variants of the gtrs in which the inequality constraint is replaced by either an equality constraint or an interval constraint

on the complexity of quadratic programming with two quadratic constraints the complexity of quadratic programming problems with two quadratic constraints is an open problem in this paper we show that when one constraint is a ball constraint and the hessian of the quadratic function defining the other constraint is positive definite then under quite general conditions the problem can be solved in polynomial time in the real number model of computation through an approach based on the analysis of the dual space of the lagrange multipliers however the degree of the polynomial is rather large thus making the result mostly of theoretical interest

a convex optimization approach for minimizing the ratio of indefinite quadratic functions over an ellipsoid we consider the nonconvex problem rq of minimizing the ratio of two nonconvex quadratic functions over a possibly degenerate ellipsoid this formulation is motivated by the so called regularized total least squares problem rtls which is a special case of the problem s class we study we prove that under a certain mild assumption on the problem s data problem rq admits an exact semidefinite programming relaxation we then study a simple iterative procedure which is proven to converge superlinearly to a global solution of rq and show that the dependency of the number of iterations on the optimality tolerance varepsilon grows as o sqrt ln varepsilon 1

a geometric characterization of strong duality in nonconvex quadratic programming with linear and nonconvex quadratic constraints we first establish a relaxed version of dines theorem associated to quadratic minimization problems with finitely many linear equality and a single nonconvex quadratic inequality constraints the case of unbounded optimal valued is also discussed then we characterize geometrically the strong duality and some relationships with the conditions employed in finsler theorem are established furthermore necessary and sufficient optimality conditions with or without the slater assumption are derived our results can be used to situations where none of the results appearing elsewhere are applicable in addition a revisited theorem due to frank and wolfe along with that due to eaves is established for asymptotically linear sets