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

blind channel equalization and spl epsiv approximation algorithms we show that a blind equalizer can be obtained without estimating any statistics based on the received data if the constellation alphabet is known the idea is to formulate the blind channel equalization problem into a quadratic optimization with binary constraints then efficient spl epsiv approximation algorithms are presented and applied to find the equalizer coefficients the proposed approach blindly equalizes the channel at symbol baud rates does not rely on estimating any statistics applies to nonminimum phase channels and converges to an spl epsiv neighborhood of a global optimum simulation results support the theoretical analysis

robust optimization for unconstrained simulation based problems in engineering design an optimized solution often turns out to be suboptimal when errors are encountered although the theory of robust convex optimization has taken significant strides over the past decade all approaches fail if the underlying cost function is not explicitly given it is even worse if the cost function is nonconvex in this work we present a robust optimization method that is suited for unconstrained problems with a nonconvex cost function as well as for problems based on simulations such as large partial differential equations pde solver response surface and kriging metamodels moreover this technique can be employed for most real world problems because it operates directly on the response surface and does not assume any specific structure of the problem we present this algorithm along with the application to an actual engineering problem in electromagnetic multiple scattering of aperiodically arranged dielectrics relevant to nanophotonic design the corresponding objective function is highly nonconvex and resides in a 100 dimensional design space starting from an optimized design we report a robust solution with a significantly lower worst case cost while maintaining optimality we further generalize this algorithm to address a nonconvex optimization problem under both implementation errors and parameter uncertainties

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

when a local hamiltonian must be frustration free a broad range of quantum optimization problems can be phrased as the question of whether a specific system has a ground state at zero energy i e whether its hamiltonian is frustration free frustration free hamiltonians in turn play a central role for constructing and understanding new phases of matter in quantum many body physics unfortunately determining whether this is the case is known to be a complexity theoretically intractable problem this makes it highly desirable to search for efficient heuristics and algorithms to at least partially answer this question here we prove a general criterion a sufficient condition under which a local hamiltonian is guaranteed to be frustration free by lifting shearer s theorem from classical probability theory to the quantum world remarkably evaluating this condition proceeds via a fully classical analysis of a hardcore lattice gas at negative fugacity on the hamiltonian s interaction graph which as a statistical mechanics problem is of interest in its own right we concretely apply this criterion to local hamiltonians on various regular lattices while bringing to bear the tools of spin glass physics that permit us to obtain new bounds on the satisfiable to unsatisfiable transition in random quantum satisfiability we are then led to natural conjectures for when such bounds will be tight as well as to a novel notion of universality for these computer science problems besides providing concrete algorithms leading to detailed and quantitative insights this work underscores the power of marrying classical statistical mechanics with quantum computation and complexity theory

a boundary condition with memory for kirchhoff plates equations in this paper we study the stability of solutions for kirchhoff plates equations with a memory condition working at the boundary we show that such dissipation is strong enough to produce exponential decay to the solution provided the relaxation functions also decays exponentially when the relaxation functions decays polynomially we show that the solution decays polynomially and with the same rate

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

convex relaxations for nonconvex quadratically constrained quadratic programming matrix cone decomposition and polyhedral approximation we present a decomposition approximation method for generating convex relaxations for nonconvex quadratically constrained quadratic programming qcqp we first develop a general conic program relaxation for qcqp based on a matrix decomposition scheme and polyhedral piecewise linear underestimation by employing suitable matrix cones we then show that the convex conic relaxation can be reduced to a semidefinite programming sdp problem in particular we investigate polyhedral underestimations for several classes of matrix cones including the cones of rank 1 and rank 2 matrices the cone generated by the coefficient matrices the cone of positive semidefinite matrices and the cones induced by rank 2 semidefinite inequalities we demonstrate that in general the new sdp relaxations can generate lower bounds at least as tight as the best known sdp relaxations for qcqp moreover we give examples for which tighter lower bounds can be generated by the new sdp relaxations we also report comparison results of different convex relaxation schemes for nonconvex qcqp with convex quadratic linear constraints nonconvex quadratic constraints and 0 1 constraints

discretization and localization in successive convex relaxation methods for nonconvex quadratic optimization based on the authors previous work which established theoretical foundations of two conceptual successive convex relaxation methods i e the ssdp successive semidefinite programming relaxation method and the ssilp successive semi infinite linear programming relaxation method this paper proposes their implementable variants for general quadratic optimization problems these problems have a linear objective function c t x to be maximized over a nonconvex compact feasible region f described by a finite number of quadratic inequalities we introduce two new techniques discretization and localization into the ssdp and ssilp relaxation methods the discretization technique makes it possible to approximate an infinite number of semi infinite sdps or semi infinite lps which appeared at each iteration of the original methods by a finite number of standard sdps or standard lps with a finite number of linear inequality constraints we establish given any open convex set u containing f there is an implementable discretization of the ssdp or ssilp relaxation method which generates a compact convex set c such that f c u in a finite number of iterations the localization technique is for the cases where we are only interested in upper bounds on the optimal objective value for a fixed objective function vector c but not in a global approximation of the convex hull of f this technique allows us to generate a convex relaxation of f that is accurate only in certain directions in a neighborhood of the objective direction c this cuts off redundant work to make the convex relaxation accurate in unnecessary directions we establish given any positive number e there is an implementable localization discretization of the ssdp or ssilp relaxation method which generates an upper bound of the objective value within e of its maximum in a finite number of iterations

approximation bounds for quadratic optimization with homogeneous quadratic constraints we consider the np hard problem of finding a minimum norm vector in n dimensional real or complex euclidean space subject to m concave homogeneous quadratic constraints we show that a semidefinite programming sdp relaxation for this nonconvex quadratically constrained quadratic program qp provides an o m 2 approximation in the real case and an o m approximation in the complex case moreover we show that these bounds are tight up to a constant factor when the hessian of each constraint function is of rank 1 namely outer products of some given so called steering vectors and the phase spread of the entries of these steering vectors are bounded away from pi 2 we establish a certain constant factor approximation depending on the phase spread but independent of m and n for both the sdp relaxation and a convex qp restriction of the original np hard problem finally we consider a related problem of finding a maximum norm vector subject to m convex homogeneous quadratic constraints we show that an sdp relaxation for this nonconvex qp provides an o 1 ln m approximation which is analogous to a result of nemirovski e math program 86 1999 pp 463 473 for the real case

a nonlinear semidefinite optimization relaxation for the worst case linear optimization under uncertainties in this paper we consider the so called worst case linear optimization wclo with uncertainties in the right hand side of the constraints such a problem arises from numerous applications such as systemic risk estimate in finance and stochastic optimization we first show the problem is np hard and present a coarse semidefinite relaxation sdr for wclo an iterative procedure is introduced to sequentially refine the relaxation model based on the solution of the current relaxation model by simply changing some parameters in the coarse sdr we show that the sequence of the proposed sdrs will converge to some nonlinear semidefinite optimization problem sdo a bi section search algorithm is proposed to solve the resulting nonlinear sdo our preliminary experimental results illustrate that the nonlinear sdr can provide very tight bounds for the original wclo and is able to locate the exact global solution in most cases within a few iterations on average

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

on approximation of max vertex cover abstract we consider the max vertex cover mvc problem i e find k vertices from an undirected and edge weighted graph g v e where v n k such that the total edge weight covered by the k vertices is maximized there is a 3 4 approximation algorithm for mvc based on a linear programming relaxation we show that the guaranteed ratio can be improved by a simple greedy algorithm for k 3 4 n and a simple randomized algorithm for k 1 2 n furthermore we study a semidefinite programming sdp relaxation based approximation algorithms for mvc we show that for a range of k our sdp based algorithm achieves the best performance guarantee among the four types of algorithms mentioned in this paper

constrained optimal input signal design for data centric estimation methods this technical note examines the design of constrained input signals for data centric estimation methods which systematically generate a local function approximation from a database of regressors at a current operating point the proposed method addresses the optimal distribution of regressor vectors under constraints for a linear time invariant lti system the resulting nonconvex optimization problems are solved using semidefinite relaxation methods numerical examples illustrate the benefits and usefulness of the proposed input signal design formulations

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

robust convex optimization we study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set u yet the constraints must hold for all possible values of the data from u the ensuing optimization problem is called robust optimization in this paper we lay the foundation of robust convex optimization in the main part of the paper we show that if u is an ellipsoidal uncertainty set then for some of the most important generic convex optimization problems linear programming quadratically constrained programming semidefinite programming and others the corresponding robust convex program is either exactly or approximately a tractable problem which lends itself to efficientalgorithms such as polynomial time interior point methods

semidefinite programming approach for range difference based source localization a common technique for passive source localization is to utilize the range difference rd measurements between the source and several spatially separated sensors the rd information defines a set of hyperbolic equations from which the source position can be calculated with the knowledge of the sensor positions under the standard assumption of gaussian distributed rd measurement errors it is well known that the maximum likelihood ml position estimation is achieved by minimizing a multimodal cost function which corresponds to a difficult task in this correspondence we propose to approximate the nonconvex ml optimization by relaxing it to a convex optimization problem using semidefinite programming a semidefinite relaxation rd based positioning algorithm which makes use of the admissible source position information is proposed and its estimation performance is contrasted with the two step weighted least squares method and nonlinear least squares estimator as well as cramer rao lower bound

on the truncated conjugate gradient method in this paper we consider the truncated conjugate gradient method for minimizing a convex quadratic function subject to a ball trust region constraint it is shown that the reduction in the objective function by the solution obtained by the truncated cg method is at least half of the reduction by the global minimizer in the trust region

inhomogeneous polynomial optimization over a convex set an approximation approach in this paper we consider computational methods for optimizing a multivariate inhomogeneous polynomial function over a general convex set the focus is on the design and analysis of polynomial time approximation algorithms the methods are able to deal with optimization models with polynomial objective functions in any fixed degrees in particular we first study the problem of maximizing an inhomogeneous polynomial over the euclidean ball a polynomial time approximation algorithm is proposed for this problem with an assured relative worst case performance ratio which is dependent only on the dimensions of the model the method and approximation ratio are then generalized to optimize an inhomogeneous polynomial over the intersection of a finite number of co centered ellipsoids furthermore the constraint set is extended to a general convex compact set specifically we propose a polynomial time approximation algorithm with a relative worst case performance ratio for polynomial optimization over some convex compact sets e g a polytope finally numerical results are reported revealing good practical performance of the proposed algorithms for solving some randomly generated instances

injecting problem dependent knowledge to improve evolutionary optimization search ability the flexibility introduced by evolutionary algorithms eas has allowed the use of virtually arbitrary objective functions and constraints even when evaluations require as for real world problems running complex mathematical and or procedural simulations of the systems under analysis even so eas are not a panacea traditionally the solution search process has been totally oblivious of the specific problem being solved and optimization processes have been applied regardless of the size complexity and domain of the problem in this paper we justify our claim that far reaching benefits may be obtained from more directly influencing how searches are performed we propose using data mining techniques as a step for dynamically generating knowledge that can be used to improve the efficiency of solution search processes in this paper we use kohonen soms and show an application for a well known benchmark problem in the water distribution system design literature the result crystallizes the conceptual rules for the ea to apply at certain stages of the evolution which reduces the search space and accelerates convergence

generating moment matching scenarios using optimization techniques an optimization based method is proposed to generate moment matching scenarios for numerical integration and its use in stochastic programming the main advantage of the method is its flexibility it can generate scenarios matching any prescribed set of moments of the underlying distribution rather than matching all moments up to a certain order and the distribution can be defined over an arbitrary set this allows for a reduction in the number of scenarios and allows the scenarios to be better tailored to the problem at hand the method is based on a semi infinite linear programming formulation of the problem that is shown to be solvable with polynomial iteration complexity a practical column generation method is implemented the column generation subproblems are polynomial optimization problems however they need not be solved to optimality it is found that the columns in the column generation approach can be efficiently generated by random sampling the number of scenarios generated matches a lower

applications of second order cone programming in a second order cone program socp a linear function is minimized over the intersection of an affine set and the product of second order quadratic cones socps are nonlinear convex problems that include linear and convex quadratic programs as special cases but are less general than semidefinite programs sdps several efficient primaldual interior point methods for socp have been developed in the last few years after reviewing the basic theory of socps we describe general families of problems that tan be recast as socps these include robust linear programming and robust leastsquares problems problems involving sums or maxima of norms or with convex hyperbolic constraints we discuss a variety of engineering applications such as filter design antenna array weight design truss design and grasping forte optimization in robotics we describe an efficient primaldual interior point method for solving socps which shares many of the features of primaldual interior point methods for linear program

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

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

robust discrete optimization and network flows we propose an approach to address data uncertainty for discrete optimization and network flow problems that allows controlling the degree of conservatism of the solution and is computationally tractable both practically and theoretically in particular when both the cost coefficients and the data in the constraints of an integer programming problem are subject to uncertainty we propose a robust integer programming problem of moderately larger size that allows controlling the degree of conservatism of the solution in terms of probabilistic bounds on constraint violation when only the cost coefficients are subject to uncertainty and the problem is a 0 1 discrete optimization problem on n variables then we solve the robust counterpart by solving at most n 1 instances of the original problem thus the robust counterpart of a polynomially solvable 0 1 discrete optimization problem remains polynomially solvable in particular robust matching spanning tree shortest path matroid intersection etc are polynomially solvable we also show that the robust counterpart of an np hard approximable 0 1 discrete optimization problem remains approximable finally we propose an algorithm for robust network flows that solves the robust counterpart by solving a polynomial number of nominal minimum cost flow problems in a modified network

blind source separation of signals with known alphabets using spl epsi approximation algorithms we show that blind separation of signals in given alphabets can be formulated into a quadratic optimization problem with integer constraints then efficient spl epsi approximation algorithms are applied to directly estimate the transmitted signals the proposed approach does not require any high order statistics moreover the algorithms converge to an spl epsi neighborhood of the global optimum with polynomial computational complexity simulations show that the algorithm achieves satisfactory performance using a short length of data

brief paper convex relaxations for mixed integer predictive control the main objective in this work is to compare different convex relaxations for model predictive control mpc problems with mixed real valued and binary valued control signals in the problem description considered the objective function is quadratic the dynamics are linear and the inequality constraints on states and control signals are all linear the relaxations are related theoretically and the quality of the bounds and the computational complexities are compared in numerical experiments the investigated relaxations include the quadratic programming qp relaxation the standard semidefinite programming sdp relaxation and an equality constrained sdp relaxation the equality constrained sdp relaxation appears to be new in the context of hybrid mpc and the result presented in this work indicates that it can be useful as an alternative relaxation which is less computationally demanding than the ordinary sdp relaxation and which often gives a better bound than the bound from the qp relaxation furthermore it is discussed how the result from the sdp relaxations can be used to generate suboptimal solutions to the control problem moreover it is also shown that the equality constrained sdp relaxation is equivalent to a qp in an important special case

tractable approximations to robust conic optimization problems in earlier proposals the robust counterpart of conic optimization problems exhibits a lateral increase in complexity i e robust linear programming problems lps become second order cone problems socps robust socps become semidefinite programming problems sdps and robust sdps become np hard we propose a relaxed robust counterpart for general conic optimization problems that a preserves the computational tractability of the nominal problem specifically the robust conic optimization problem retains its original structure i e robust lps remain lps robust socps remain socps and robust sdps remain sdps and b allows us to provide a guarantee on the probability that the robust solution is feasible when the uncertain coefficients obey independent and identically distributed normal distributions

error estimates in the optimization of degree two polynomials on a discrete hypercube this paper considers the distribution of values q x x in 1 1 n where q is a quadratic form in n variables with real coefficients error estimates are established for approximations of the maximum and minimum values of q on 1 1 n which can be obtained by semidefinite programming bounds are given involving the sum of the absolute values of the off diagonal entries other bounds are given which are useful in the case of extreme skewness used in conjunction with earlier bounds of nesterov in optim methods softw 9 1998 pp 141 160 these new bounds lead to improvements on the bound given by the trace the trigonometric description of the maximum and minimum given in optim methods softw 9 1998 pp 141 160 which is based on the rounding argument introduced by goemans and williamson in j assoc comput mach 6 1995 pp 1115 1145 is a major tool in obtaining these bounds

solving large scale sparse semidefinite programs for combinatorial optimization we present a dual scaling interior point algorithm and show how it exploits the structure and sparsity of some large scale problems we solve the positive semidefinite relaxation of combinatorial and quadratic optimization problems subject to boolean constraints we report the first computational results of interior point algorithms for approximating maximum cut semidefinite programs with dimension up to 3 000

robust design of structures using optimization methods the robust design of structures with stochastic parameters is studied using optimization techniques the first two statistical moments of the stochastic parameters including design variables are considered in conjunction with the second order perturbation method for the approximation of mean value and variance of the structural response in this framework the sensitivities of the mean values and variances of the structural performance function with respect to the design variables are calculated for use in the optimization task the robust design of structures is formulated as a multi criteria optimization problem in which both the expected value and the standard deviation of the objective function are to be minimized the robustness of the feasibility is also taken into account by involving the variability of the structural response in the constraints the two criteria optimization problem is converted into a scalar one and is then solved by a gradient based optimization algorithm to demonstrate the applicability of the presented method numerical examples are given involving static and dynamic response

connectivity of quadratic hypersurfaces and its applications in optimization part i general theory in this paper we establish a basic theory on the connectivity of quadratic hypersurfaces in addition we examine how both the connectivity and the hidden convexity of quadratic functions are related to each other and prove several connectivity theorems on quadratic functions that combine and generalize dines s and brickman s results on the hidden convexity of quadratic mappings in real field many classical results in quadratic optimization are reproved by the unified approach and more properties involving the matrix pencils are given

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

a finite branch and bound algorithm for nonconvex quadratic programming via semidefinite relaxations existing global optimization techniques for nonconvex quadratic programming qp branch by recursively partitioning the convex feasible set and thus generate an infinite number of branch and bound nodes an open question of theoretical interest is how to develop a finite branch and bound algorithm for nonconvex qp one idea which guarantees a finite number of branching decisions is to enforce the first order karush kuhn tucker kkt conditions through branching in addition such an approach naturally yields linear programming lp relaxations at each node however the lp relaxations are unbounded a fact that precludes their use in this paper we propose and study semidefinite programming relaxations which are bounded and hence suitable for use with finite kkt branching computational results demonstrate the practical effectiveness of the method with a particular highlight being that only a small number of nodes are required

exactness of semidefinite relaxations for nonlinear optimization problems with underlying graph structure this work is concerned with finding a global optimization technique for a broad class of nonlinear optimization problems including quadratic and polynomial optimization problems the main objective of this paper is to investigate how the hidden structure of a given real complex valued optimization problem makes it easy to solve to this end three conic relaxations are proposed necessary and sufficient conditions are derived for the exactness of each of these relaxations and it is shown that these conditions are satisfied if the optimization problem is highly structured more precisely the structure of the optimization problem is mapped into a generalized weighted graph where each edge is associated with a weight set extracted from the coefficients of the optimization problem in the real valued case it is shown that the relaxations are all exact if each weight set is sign definite and in addition a condition is satisfied for each cycle of the graph it is also proved that if some of these conditi

an approach to identification for robust control given measured data generated by a discrete time linear system we propose a model consisting of a linear time invariant system affected by norm bounded perturbation under mild assumptions the plants belonging to the resulting uncertain family form a convex set the approach depends on two key parameters an a priori given bound of the perturbation and the input used to generate the data it turns out that the size of the uncertain family can be reduced by intersecting the model families obtained by making use of different inputs the model validation problem in this identification scheme is analyzed for a given energy level the invalidation problem yields the family of those models which can never be invalidated for any possible input of fixed energy and any possible perturbation this leads to the intersection of all uncertain families a consequence of the invalidation problem is that for finite length measurements not all models can be invalidated using fixed energy inputs

global existence and stability for wave equation of kirchhoff type with memory condition at the boundary abstract we consider a nonlinear wave equation of kirchhoff type with memory condition at the boundary and we study the asymptotic behavior of the corresponding solutions we proved that the energy decay with the same rate of decay of the relaxation function that is the energy decays exponentially when the relaxation function decay exponentially and polynomially when the relaxation function decay polynomially

new relations between the monomer dimer and the yang lee models a hamiltonian approach to the monomer dimer model is derived by a special truncation of the spin operators in the yang lee model at its critical point the negative fugacity dimer problem is proved to belong to the universality class of the yang lee singularity as previously conjectured

existence uniqueness and uniform decay for the nonlinear beam degenerate equation with weak damping in this paper we prove global existence and uniqueness of weak solutions of the problem for the nonlinear beam degenerate equationk x t u email protected email protected w u 2dx email protected u email protected 0 whereq is a cylindrical domain of r n 1 n 1 with the lateral boundary s and k x t is a real function defined in q k x t 0 for all x t email protected 0 which satisfies some appropriate conditions m l is a real function such that m l email protected 0 b l 1 l 1 is the first eigenvalue of the operator d 2 moreover the uniform decay rates of the energy are obtained as time goes to infinity

three dimensional perfectly matched layer for the absorption of electromagnetic waves the perfectly matched layer is a technique of free space simulation developed for solving unbounded electromagnetic problems with the finite difference time domain method referred to as pml this technique has been described in a previous paper for two dimensional problems the present paper is devoted to the three dimensional case the theory of the perfectly matched layer is generalized to three dimensions and some numerical experiments are shown to illustrate the efficiency of this new method of free space simulation

complex quadratic optimization and semidefinite programming in this paper we study the approximation algorithms for a class of discrete quadratic optimization problems in the hermitian complex form a special case of the problem that we study corresponds to the max 3 cut model used in a recent paper of goemans and williamson j comput system sci 68 2004 pp 442 470 we first develop a closed form formula to compute the probability of a complex valued normally distributed bivariate random vector to be in a given angular region this formula allows us to compute the expected value of a randomized with a specific rounding rule solution based on the optimal solution of the complex semidefinite programming relaxation problem in particular we present an m 2 1 cos frac 2 pi m 8 pi approximation algorithm and then study the limit of that model in which the problem remains np hard we show that if the objective is to maximize a positive semidefinite hermitian form then the randomization rounding procedure guarantees a worst case performance ratio of pi 4 approx 0 7854 which is better than the ratio of 2 pi approx 0 6366 for its counterpart in the real case due to nesterov furthermore if the objective matrix is real valued positive semidefinite with nonpositive off diagonal elements then the performance ratio improves to 0 9349

scenarios and policy aggregation in optimization under uncertainty a common approach in coping with multiperiod optimization problems under uncertainty where statistical information is not really enough to support a stochastic programming model has been to set up and analyze a number of scenarios the aim then is to identify trends and essential features on which a robust decision policy can be based this paper develops for the first time a rigorous algorithmic procedure for determining such a policy in response to any weighting of the scenarios the scenarios are bundled at various levels to reflect the availability of information and iterative adjustments are made to the decision policy to adapt to this structure and remove the dependence on hindsight

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

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

alternative theorems for quadratic inequality systems and global quadratic optimization we establish alternative theorems for quadratic inequality systems consequently we obtain lagrange multiplier characterizations of global optimality for classes of nonconvex quadratic optimization problems we present a generalization of dine s theorem to a system of two homogeneous quadratic functions with a regular cone the class of regular cones are cones k for which k cup k is a subspace as a consequence we establish a generalization of the powerful s lemma which paves the way to obtain a complete characterization of global optimality for a general quadratic optimization model problem involving a system of equality constraints in addition to a single quadratic inequality constraint we then present an alternative theorem for a system of three nonhomogeneous inequalities by way of establishing the convexity of the joint range of three homogeneous quadratic functions using a regular cone this yields lagrange multiplier characterizations of global optimality for classes of trust region type problems with two inequality constraints finally we establish an alternative theorem for systems involving an arbitrary finite number of quadratic inequalities involving z matrices which are matrices with nonpositive off diagonal elements and present necessary and sufficient conditions for global optimality for classes of nonconvex inequality constrained quadratic optimization problems

spanning forests and the q state potts model in the limit q 0 we study the q state potts model with nearest neighbor coupling v e j 1 in the limit q v 0 with the ratio w v q held fixed combinatorially this limit gives rise to the generating polynomial of spanning forests physically it provides information about the potts model phase diagram in the neighborhood of q v 0 0 we have studied this model on the square and triangular lattices using a transfer matrix approach at both real and complex values of w for both lattices we have computed the symbolic transfer matrices for cylindrical strips of widths 2 l 10 as well as the limiting curves b of partition function zeros in the complex w plane for real w we find two distinct phases separated by a transition point w w0 where w0 1 4 resp w0 0 1753 0 0002 for the square resp triangular lattice for w w0 we find a non critical disordered phase that is compatible with the predicted asymptotic freedom as w for w w0 our results are compatible with a massless berker kadanoff phase with central charge c 2 and leading thermal scaling dimension xt 1 2 marginally irrelevant operator at w w0 we find a first order critical point the first derivative of the free energy is discontinuous at w0 while the correlation length diverges as w w0 and is infinite at w w0 the critical ehavior at w w0 seems to be the same for both lattices and it differs from that of the berker kadanoff phase our results suggest that the central charge is c 1 the leading thermal scaling dimension is xt 1 0 and the critical exponents are 1 d 1 2 and 1

lmi based robust mixed integer model predictive control for hybrid systems this paper proposes a linear matrix inequality lmi approach to mixed integer model predictive control mpc of uncertain hybrid systems with binary and real valued control inputs the stability c

on the range of validity of the 6 e expansion for percolation in 6 epsilon dimensions the critical properties of percolation can be computed starting from a phi 3 theory having the same symmetry of the one state potts model the authors argue that this fixed point must be unstable with respect to a phi 4 interaction in two dimensions therefore a phi 4 perturbation is relevant this suggestion is confirmed by an explicit two loops computation

existence uniqueness and uniform decay for the non linear degenerate equation with memory condition at the boundary this paper is concerned with the existence uniqueness and uniform decay for the solution to the non linear degenerate equation with memory condition at the boundary

an improved rounding method and semidefinite programming relaxation for graph partition given an undirected graph g v e with v n and an integer k between 0 and n the maximization graph partition max gp problem is to determine a subset s v of k nodes such that an objective function w s is maximized the max gp problem can be formulated as a binary quadratic program and it is np hard semidefinite programming sdp relaxations of such quadratic programs have been used to design approximation algorithms with guaranteed performance ratios for various max gp problems based on several earlier results we present an improved rounding method using an sdp relaxation and establish improved approximation ratios for several max gp problems including dense subgraph max cut max not cut and max vertex cover

robust optimization methodology and applications robust optimization ro is a modeling methodology combined with computational tools to process optimization problems in which the data are uncertain and is only known to belong to some uncertainty set the paper surveys the main results of ro as applied to uncertain linear conic quadratic and semidefinite programming for these cases computationally tractable robust counterparts of uncertain problems are explicitly obtained or good approximations of these counterparts are proposed making ro a useful tool for real world applications we discuss some of these applications specifically antenna design truss topology design and stability analysis synthesis in uncertain dynamic systems we also describe a case study of 90 lps from the netlib collection the study reveals that the feasibility properties of the usual solutions of real world lps can be severely affected by small perturbations of the data and that the ro methodology can be successfully used to overcome this phenomenon

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

derivative free robust optimization by outer approximations we develop an algorithm for minimax problems that arise in robust optimization in the absence of objective function derivatives the algorithm utilizes an extension of methods for inexact outer approximation in sampling a potentially infinite cardinality uncertainty set clarke stationarity of the algorithm output is established alongside desirable features of the model based trust region subproblems encountered we demonstrate the practical benefits of the algorithm on a new class of test problems

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

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

an approximation technique for robust nonlinear optimization nonlinear equality and inequality constrained optimization problems with uncertain parameters can be addressed by a robust worst case formulation that is however difficult to treat computationally in this paper we propose and investigate an approximate robust formulation that employs a linearization of the uncertainty set in case of any norm bounded parameter uncertainty this formulation leads to penalty terms employing the respective dual norm of first order derivatives of the constraints the main advance of the paper is to present two sparsity preserving ways for efficient computation of these derivatives in the case of large scale problems one similar to the forward mode the other similar to the reverse mode of automatic differentiation we show how to generalize the techniques to optimal control problems and discuss how even infinite dimensional uncertainties can be treated efficiently finally we present optimization results for an example from process engineering a batch distillation

semidefinite relaxation bounds for indefinite homogeneous quadratic optimization this paper studies the relationship between the optimal value of a homogeneous quadratic optimization problem and its semidefinite programming sdp relaxation we consider two quadratic optimization models 1 min x c x mid x a k x ge 1 k 0 1 ldots m x in mathbb f n and 2 max x c x mid x a k x le 1 k 0 1 ldots m x in mathbb f n where mathbb f is either the real field mathbb r or the complex field mathbb c and a k c are symmetric matrices for the minimization model 1 we prove that if the matrix c and all but one of the a k s are positive semidefinite then the ratio between the optimal value of 1 and its sdp relaxation is upper bounded by o m 2 when mathbb f mathbb r and by o m when mathbb f mathbb c moreover when two or more of the a k s are indefinite this ratio can be arbitrarily large for the maximization model 2 we show that if c and at most one of the a k s are indefinite while other a k s are positive semidefinite then the ratio between the optimal value of 2 and its sdp relaxation is bounded from below by o 1 log m for both the real and the complex case this result improves the bound based on the so called approximate s lemma of ben tal nemirovski and roos siam j optim 13 2002 pp 535 560 when two or more of the a k s in 2 are indefinite we derive a general bound in terms of the problem data and the sdp solution for both optimization models we present examples to show that the derived approximation bounds are essentially tight

further results on approximating nonconvex quadratic optimization by semidefinite programming relaxation we study approximation bounds for the semidefinite programming sdp relaxation of quadratically constrained quadratic optimization min f 0 x subject to f k x le 0 k 1 dots m where fk x xtakx bk tx ck in the special case of ellipsoid constraints with interior feasible solution at 0 we show that the sdp relaxation coupled with a rank 1 decomposition result of sturm and zhang math oper res to appear yields a feasible solution of the original problem with objective value at most 1 gamma 2 sqrt m gamma 2 times the optimal objective value where gamma sqrt smash b max k f k 0 1 for the single trust region problem corresponding to m 1 this yields an exact optimal solution in the general case we extend some bounds derived by nesterov optim methods softw 9 1998 pp 141 160 working paper core universite catholique de louvain louvain la neuve belgium 1998 ye math program 84 1999 pp 219 226 and nesterov wolkowicz and ye in handbook of semidefinite programming h wolkowicz r saigal and l vandenberghe eds kluwer academic publishers dordrecht the netherlands 2000 pp 360 419 for the special case where a k is diagonal and bk 0 for k 1 m we also discuss the generation of approximate solutions with high probability

cones of matrices and successive convex relaxations of nonconvex sets let f be a compact subset of the n dimensional euclidean space rn represented by finitely or infinitely many quadratic inequalities we propose two methods one based on successive semidefinite programming sdp relaxations and the other on successive linear programming lp relaxations each of our methods generates a sequence of compact convex subsets ck k 1 2 of rn such that a the convex hull of f subseteq c k 1 subseteq c k monotonicity b cap k 1 infty c k text the convex hull of f asymptotic convergence our methods are extensions of the corresponding lovasz schrijver lift and project procedures with the use of sdp or lp relaxation applied to general quadratic optimization problems qops with infinitely many quadratic inequality constraints utilizing descriptions of sets based on cones of matrices and their duals we establish the exact equivalence of the sdp relaxation and the semi infinite convex qop relaxation proposed originally by fujie and kojima using this equivalence we investigate some fundamental features of the two methods including a and b above

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 mathsf d induced duality and its applications this paper attempts to extend the notion of duality for convex cones by basing it on a prescribed conic ordering and a fixed bilinear mapping this is an extension of the standard definition of dual cones in the sense that the nonnegativity of the inner product is replaced by a pre specified conic ordering defined by a convex cone mathsf d and the inner product itself is replaced by a general multi dimensional bilinear mapping this new type of duality is termed the mathsf d induced duality in the paper we further introduce the notion of mathsf d induced polar sets within the same framework which can be viewed as a generalization of the mathsf d induced dual cones and is convenient to use for some practical applications properties of the extended duality including the extended bi polar theorem are proven furthermore attention is paid to the computation and approximation of the mathsf d induced dual objects we discuss as examples applications of the newly introduced mathsf d induced duality concepts in robust conic optimization and the duality theory for multi objective conic optimization

approximating quadratic programming with bound and quadratic constraints we consider the problem of approximating the global maximum of a quadratic program qp subject to bound and simple quadratic constraints based on several early results we show that a 4 7 approximate solution can be obtained in polynomial time

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

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

the yang lee edge singularity by the phenomenological renormalisation group the phenomenological renormalisation group is applied to the 1d transverse ising model with longitudinal imaginary field in order to investigate the yang lee edge singularity 1952 of the equivalent 2d ising model the method is extended to calculate directly the exponent sigma of the magnetisation

the complexity of approximating a nonlinear program we consider the problem of finding the maximum of a multivariate polynomial inside a convex polytope we show that there is no polynomial time approximation algorithm for this problem even one with a very poor guarantee unless p np we show that even when the polynomial is quadratic i e quadratic programming there is no polynomial time approximation unless np is contained in quasi polynomial time

approximation algorithms for homogeneous polynomial optimization with quadratic constraints in this paper we consider approximation algorithms for optimizing a generic multi variate homogeneous polynomial function subject to homogeneous quadratic constraints such optimization models have wide applications e g in signal processing magnetic resonance imaging mri data training approximation theory and portfolio selection since polynomial functions are non convex the problems under consideration are all np hard in general in this paper we shall focus on polynomial time approximation algorithms in particular we first study optimization of a multi linear tensor function over the cartesian product of spheres we shall propose approximation algorithms for such problem and derive worst case performance ratios which are shown to be dependent only on the dimensions of the model the methods are then extended to optimize a generic multi variate homogeneous polynomial function with spherical constraint likewise approximation algorithms are proposed with provable approximation performance ratios furthermore the constraint set is relaxed to be an intersection of co centered ellipsoids namely we consider maximization of a homogeneous polynomial over the intersection of ellipsoids centered at the origin and propose polynomial time approximation algorithms with provable worst case performance ratios numerical results are reported illustrating the effectiveness of the approximation algorithms studied