vertex operator for the non autonomous ultradiscrete kp equation we propose an ultradiscrete analogue of the vertex operator in the case of the ultradiscrete kp equation and several other ultradiscrete equations which maps n soliton solutions to n 1 soliton ones

the cauchy problem for a generalized two component short pulse system with high order nonlinearities abstract this paper is concerned with the cauchy problem of a generalized short pulse system in the periodic setting local well posedness of this system is established in the besov space with s max 3 2 1 1 p it is then shown that the data to solution map is not uniformly continuous from b l r s b l r s to c 0 t b l r s c 0 t b l r s finally we provide a blow up result of the solution

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

propagation of ultra short optical pulses in cubic nonlinear media abstract we derive a partial differential equation that approximates solutions of maxwell s equations describing the propagation of ultra short optical pulses in nonlinear media and which extends the prior analysis of alterman and rauch phys lett a 264 2000 390 diffractive nonlinear geometric optics for short pulses preprint 2002 we discuss non rigorously conditions under which this approximation should be valid but the main contributions of this paper are 1 an emphasis on the fact that the model equation for short pulse propagation may depend on the details of the optical susceptibility in the wavelength regime under consideration 2 a numerical comparison of solutions of this model equation with solutions of the full nonlinear partial differential equation 3 a local well posedness result for the model equation and 4 a proof that in contrast to the nonlinear schrodinger equation which models slowly varying wavetrains this equation has no smooth pulse solutions which propagate with fixed shape and speed

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

pfaffianization of the two dimensional toda lattice pfaffianization procedure due to hirota and ohta is applied to the two dimensional toda lattice as a result a pfaffianized version of the two dimensional toda lattice is found

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

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

the cauchy problem for the fokas olver rosenau qiao equation abstract it is shown that the initial value problem for the fokas olver rosenau qiao equation forq is well posed in sobolev spaces h s s 5 2 in the sense of hadamard furthermore it is proved that the dependence on initial data is sharp i e the data to solution map is continuous but not uniformly continuous also peakon travelling wave solutions are derived on both the circle and the line and are used to prove that the solution map is not uniformly continuous in h s for s 3 2

the short pulse hierarchy we study a new hierarchy of equations containing the short pulse equation which describes the evolution of very short pulses in nonlinear media and the elastic beam equation which describes nonlinear transverse oscillations of elastic beams under tension we show that the hierarchy of equations is integrable we obtain the two compatible hamiltonian structures we construct an infinite series of both local and nonlocal conserved charges a lax description is presented for both systems for the elastic beam equations we also obtain a nonstandard lax representation

complex matrix decomposition and quadratic programming this paper studies the possibilities of the linear matrix inequality characterization of the matrix cones formed by nonnegative complex hermitian quadratic functions over specific domains in the complex space in its real case analog such studies were conducted in sturm and zhang sturm j f s zhang 2003 on cones of nonnegative quadratic functions math oper res 28 246 267 in this paper it is shown that stronger results can be obtained for the complex hermitian case in particular we show that the matrix rank one decomposition result of sturm and zhang sturm j f s zhang 2003 on cones of nonnegative quadratic functions math oper res 28 246 267 can be strengthened for the complex hermitian matrices as a consequence it is possible to characterize several new matrix co positive cones over specific domains by means of linear matrix inequality as examples of the potential application of the new rank one decomposition result we present an upper bound on the lowest rank among all the optimal solutions for a standard complex semidefinite programming sdp problem and offer alternative proofs for a result of hausdorff hausdorff f 1919 der wertvorrat einer bilinearform mathematische zeitschrift3 314 316 and a result of brickman brickman l 1961 on the field of values of a matrix proc amer math soc 12 61 66 on the joint numerical range

strong duality for the cdt subproblem a necessary and sufficient condition in this paper we consider the problem of minimizing a nonconvex quadratic function subject to two quadratic inequality constraints as an application such a quadratic program plays an important role in the trust region method for nonlinear optimization such a problem is known as the celis dennis and tapia cdt subproblem in the literature the lagrangian dual of the cdt subproblem is a semidefinite program sdp hence convex and solvable however a positive duality gap may exist between the cdt subproblem and its lagrangian dual because the cdt subproblem itself is nonconvex in this paper we present a necessary and sufficient condition to characterize when the cdt subproblem and its lagrangian dual admits no duality gap i e the strong duality holds this necessary and sufficient condition is easy verifiable and involves only one any optimal solution of the sdp relaxation for the cdt subproblem moreover the condition reveals that it is actually rare to render a positive duality gap for the cdt subproblems in general moreover if the strong duality holds then an optimal solution for the cdt problem can be retrieved from an optimal solution of the sdp relaxation by means of a matrix rank one decomposition procedure the same analysis is extended to the framework where the necessary and sufficient condition is presented in terms of the lagrangian multipliers at a kkt point furthermore we show that the condition is numerically easy to work with approximatively

narrowing the difficulty gap for the celis dennis tapia problem we study the celis dennis tapia cdt problem minimize a non convex quadratic function over the intersection of two ellipsoids in contrast to the well studied trust region problem where the feasible set is just one ellipsoid the cdt problem is not yet fully understood our main objective in this paper is to narrow the difficulty gap that occurs when the hessian of the lagrangian is indefinite at all karush kuhn tucker points we prove new sufficient and necessary conditions both for local and global optimality based on copositivity giving a complete characterization in the degenerate case

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

box and ball system as a realization of ultradiscrete nonautonomous kp equation cellular automata which are realized by dynamics of several kinds of balls in an infinite array of boxes are investigated they show soliton patterns even in the case when each box has arbitrary capacity the analytical expression for the soliton patterns are obtained using ultradiscretization of the nonautonomous discrete kp equation

optimality conditions for the minimization of a quadratic with two quadratic constraints the trust region method has been proven to be very successful in both unconstrained and constrained optimization it requires the global minimum of a general quadratic function subject to ellipsoid constraints in this paper we generalize the trust region subproblem by allowing two general quadratic constraints conditions and properties of its solution are discussed

periodic solutions of the short pulse model equation we develop a systematic procedure for constructing periodic solutions of the short pulse sp model equation that describes the propagation of ultrashort pulses in nonlinear media we first summarize a novel exact method of solution that consists of a hodograph transformation to convert the sp equation into the sine gordon sg equation we then exemplify some one and two phase periodic solutions of the sg equation for which the system of linear partial differential equations governing the inverse mapping can be integrated analytically to obtain periodic solutions of the sp equation in the form of the parametric representation we obtain a new class of two phase periodic solutions and investigate their properties in detail of particular interest is a nonsingular periodic solution that reduces to the breather solution in the long wave limit it may play an important role in studying the propagation of ultrashort pulses in a finite length optical system

ultradiscrete plucker relation specialized for soliton solutions we propose an ultradiscrete analog of the plucker relation specialized for soliton solutions it is expressed by an ultradiscrete permanent which is obtained by ultradiscretizing the permanent that is the signature free determinant using this relation we also show soliton solutions to the ultradiscrete kadomtsev petviashvili equation and the ultradiscrete two dimensional toda lattice equation respectively

an integrable coupled short pulse equation an integrable coupled short pulse csp equation is proposed for the propagation of ultra short pulses in optical fibers based on two sets of bilinear equations to a two dimensional toda lattice linked by a backlund transformation and an appropriate hodograph transformation the proposed csp equation is derived meanwhile its n soliton solutions are given by the casorati determinant in a parametric form the properties of one and two soliton solutions are investigated in detail same as the short pulse equation the two soliton solution turns out to be a breather type if the wave numbers are complex conjugate we also illustrate an example of soliton breather interaction

on local solutions of the celis dennis tapia subproblem we discuss the distribution of the local solutions of the celis dennis tapia cdt subproblem which appears in some trust region algorithms for nonlinear optimization we also give some examples to show the differences between the cdt subproblem and the single ball constraint subproblem these results show that the complexity of the cdt subproblem does not depend on the complexity of the structure of the dual plane thus they provide the possibility to search for the global minimizer in the dual plane

ultra discrete permanent and the consistency of max plus linear equations abstract in this paper we investigate the consistency conditions for three classes of the max plus linear equations including the case corresponding to the convex polyhedra which was the subject of the previous paper the necessary and sufficient conditions for the existence of finite solutions are expressed in the form of single equations by using the ultra discrete permanent

moment inequalities for sums of random matrices and their applications in optimization in this paper we consider various moment inequalities for sums of random matrices which are well studied in the functional analysis and probability theory literature and demonstrate how they can be used to obtain the best known performance guarantees for several problems in optimization first we show that the validity of a recent conjecture of nemirovski is actually a direct consequence of the so called non commutative khintchine s inequality in functional analysis using this result we show that an sdp based algorithm of nemirovski which is developed for solving a class of quadratic optimization problems with orthogonality constraints has a logarithmic approximation guarantee this improves upon the polynomial approximation guarantee established earlier by nemirovski furthermore we obtain improved safe tractable approximations of a certain class of chance constrained linear matrix inequalities secondly we consider a recent result of delage and ye on the so called data driven distributionally robust stochastic programming problem one of the assumptions in the delage ye result is that the underlying probability distribution has bounded support however using a suitable moment inequality we show that the result in fact holds for a much larger class of probability distributions given the close connection between the behavior of sums of random matrices and the theoretical properties of various optimization problems we expect that the moment inequalities discussed in this paper will find further applications in optimization

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

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

ising field theory quadratic difference equations for the n point green s functions on the lattice for the two dimensional ising model at arbitrary temperature a system of quadratic difference equations is presented involving the correlations of n order variables and that of n ensuremath 2 order and two disorder variables with suitable boundary conditions these equations specify the correlations

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

continuous properties of the data to solution map for a generalized μ camassa holm integrable equation this work studies a generalized type integrable equation with both quadratic and cubic nonlinearities the camassa holm and modified camassa holm equations are members of this family of equations it has been shown that the cauchy problem for this generalized camassa holm integrable equation is locally well posed for initial data u0 hs s 5 2 in this work we further investigate the continuity properties to this equation it is proved in this work that the data to solution map of the proposed equation is not uniformly continuous it is also found that the solution map is holder continuous in the hr topology when 0 r 5 2 in this work we further investigate the continuity properties to this equation it is proved in this work that the data to solution map of the proposed equation is not uniformly continuous it is also found that the solution map is holder continuous in the hr topology when 0 r s with holder exponent depending on both s and r

well posedness and continuity properties of the fornberg whitham equation in besov spaces abstract in this paper we prove well posedness of the fornberg whitham equation in besov spaces b 2 r s in both the periodic and non periodic cases this will imply the existence and uniqueness of solutions in the aforementioned spaces along with the continuity of the data to solution map provided that the initial data belongs to b 2 r s we also establish sharpness of continuity on the data to solution map by showing that it is not uniformly continuous from any bounded subset of b 2 r s to c t t b 2 r s furthermore we prove a cauchy kowalevski type theorem for this equation that establishes the existence and uniqueness of real analytic solutions and also provide blow up criterion for solutions

application of a multiplier method to uncertain lur e like systems abstract this paper investigates the conditions under which an abstract matrix multiplier method can be applied to determine guaranteeing cost controls for systems containing nonlinear uncertain elements via linear matrix inequalities lmis quadratically constrained uncertainties and nonlinearities are considered which comprehend the cases of norm bounded positive real and sector bounded uncertainties nonlinearities both the discrete time and the continuous time cases are discussed necessary and sufficient conditions are formulated in the case of unstructured uncertainty the conditions are sufficient in the structured case the cost guaranteeing controls can be determined by solving lmis numerical examples illustrate the effectiveness of the proposed method

local minimizers of quadratic functions on euclidean balls and spheres in this paper a characterization of the local nonglobal minimizes of a quadratic function defined on a euclidean ball or a sphere is given it is proven that there exists at most one local nonglobal minimizes and that the lagrange multiplier that corresponds to this minimizes is the largest solution of a nonlinear scalar equation an algorithm is proposed for computing the local nonglobal minimizes

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

a pfaffian version of the modified discrete kp equation we first present casorati determinant and gram type determinant solutions to a modified discrete kp equation and then produce a pfaffianized version of modified discrete kp equations by using hirota and ohta s pfaffianization procedure the solutions to a coupled modified discrete kp equation are expressed by two types of pfaffians

integrable discretizations of the short pulse equation in this paper we propose integrable semi discrete and full discrete analogues of the short pulse sp equation the key construction is the bilinear form and determinant structure of solutions of the sp equation we also give the determinant formulas of n soliton solutions of the semi discrete and full discrete analogues of the sp equations from which the multi loop and multi breather solutions can be generated in the continuous limit the full discrete sp equation converges to the semi discrete sp equation and then to the continuous sp equation based on the semi discrete sp equation an integrable numerical scheme i e a self adaptive moving mesh scheme is proposed and used for the numerical computation of the short pulse equation

a symmetric generalization of linear backlund transformation associated with the hirota bilinear difference equation the hirota bilinear difference equation is generalized to discrete space of arbitrary dimension solutions to the nonlinear difference equations can be obtained via backlund transformation of the corresponding linear problems

discrete and ultradiscrete backlund transformation for kdv equation the backlund transformation for the discrete korteweg de vries equation is introduced in the bilinear form the superposition formula is also derived from the transformation an ultradiscrete analogue of the transformation is presented by means of the ultradiscretization technique this analogue gives the backlund transformation for the box and ball system the ultradiscrete soliton solutions for the system are also discussed with explicit examples

pfaffianization of the discrete kp equation we use the procedure of ohta and hirota to generate an integrable coupled system of discrete equations from the discrete kp equation

gram type pfaffian solution to the coupled discrete kp equation in this paper we first prove the grammian determinant solution to the discrete kp dkp equation by the algebraic identity of pfaffian instead of laplace expansion for determinants then we present the gram type pfaffian solution to the pfaffianized dkp system as an example the n soliton solution for the system is obtained

solitary wave solutions of the short pulse equation an exact nonsingular solitary wave solution of the schafer wayne short pulse equation is derived from the breather solution of the sine gordon equation by means of a transformation between these two integrable equations

a two variable approach to the two trust region subproblem the trust region subproblem minimizes a general quadratic function over an ellipsoid and can be solved in polynomial time using a semidefinite programming sdp relaxation intersecting the feasible set with a second ellipsoid results in the two trust region subproblem ttrs even though ttrs can also be solved in polynomial time existing algorithms do not use sdp in this paper we investigate the use of sdp for ttrs starting from the basic sdp relaxation of ttrs which admits a gap recent research has tightened the basic relaxation using valid second order cone inequalities even still closing the gap requires more for the special case of ttrs in dimension n 2 we fully characterize the remaining valid inequalities which can be viewed as strengthened versions of the second order cone inequalities just mentioned we also demonstrate that these valid inequalities can be used computationally even when n 2 to solve ttrs instances that were previously unsolved using sdp based techniques

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

the backlund transformation equations for the ultradiscrete kp equation the backlund transformation for the ultradiscrete kp equation is proposed an algorithm to eliminate variables from the ultradiscrete linear equations is proposed the consistency condition for the backlund transformation equations is obtained via the algorithm

category of nonlinear evolution equations algebraic structure and r matrix in this paper we deal with the category of nonlinear evolution equations nlees associated with the spectral problem and provide an approach for constructing their algebraic structure and r matrix first we introduce the category of nlees which is composed of various positive order and negative order hierarchies of nlees both integrable and nonintegrable the whole category of nlees possesses a generalized lax representation next we present two different lie algebraic structures of the lax operator one of them is universal in the category i e independent of the hierarchy while the other one is nonuniversal in the hierarchy i e dependent on the underlying hierarchy moreover we find that two kinds of adjoint maps are r matrices under the algebraic structures in particular the virasoro algebraic structures without a central extension of isospectral and nonisospectral lax operators can be viewed as reductions of our algebraic structure finally we give several concrete examples to illustrate

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