on an inequality by n trudinger and j moser and related elliptic equations it has been shown by trudinger and moser that for normalized functions u of the sobolev space w 1 n whereis a bounded domain in r n one has r exp njuj n n 1 dx cn where n is an explicit constant depending only on n and cn is a constant depending only on n and carleson and chang proved that there exists a corresponding extremal function in the case that is the unit ball in r n in this paper we give a new proof a generalization and a new interpretation of this result in particular we give an explicit sequence that is maximizing for the above integral among all normalized concentrating sequences as an application the existence of a nontrivial solution for a related elliptic equation with trudinger moser growth is proved c 2002 john wiley sons inc

antenna array pattern synthesis via convex optimization we show that a variety of antenna array pattern synthesis problems can be expressed as convex optimization problems which can be numerically solved with great efficiency by recently developed interior point methods the synthesis problems involve arrays with arbitrary geometry and element directivity constraints on far and near field patterns over narrow or broad frequency bandwidth and some important robustness constraints we show several numerical simulations for the particular problem of constraining the beampattern level of a simple array for adaptive and broadband arrays

cryptanalysis of a diophantine equation oriented public key cryptosystem c h lin et al 1995 proposed a new public key cipher system whose security is based upon the diophantine equations in this note we show that their scheme is insecure under the ciphertext only attack

primal dual interior point methods for semidefinite programming convergence rates stability and numerical results primal dual interior point path following methods for semidefinite programming are considered several variants are discussed based on newton s method applied to three equations primal feasibility dual feasibility and some form of centering condition the focus is on three such algorithms called the xz xz zx and q methods for the xz zx and q algorithms the newton system is well defined and its jacobian is nonsingular at the solution under nondegeneracy assumptions the associated schur complement matrix has an unbounded condition number on the central path under the nondegeneracy assumptions and an additional rank assumption practical aspects are discussed including mehrotra predictor corrector variants and issues of numerical stability compared to the other methods considered the xz zx method is more robust with respect to its ability to step close to the boundary converges more rapidly and achieves higher accuracy

a krylov subspace method for large scale second order cone linear complementarity problem in this paper we first show that the second order cone linear complementarity problem soclcp can be solved by finding a positive zero s in mathbb r of a particular rational function h s and we then propose a krylov subspace method to reduce h s to h ell s as in the model reduction the zero s of h s can be accurately approximated by that of h ell s 0 which itself can be cast as a small eigenvalue problem the new method is made possible by a complete characterization of the curve of h s and it has several advantages over the bisection newton sf bn iteration recently proposed by l h zhang and w h yang math comp 83 2013 pp 1701 1720 and shown to be very efficient for small to medium size problems the method is tested and compared against the sf bn iteration and two other state of the art packages sf sdpt3 and sf sedumi our numerical results show that the method is very efficient for both small to medium dense problems and large scale ones

a new primal dual path following interior point algorithm for semidefinite optimization abstract in this paper we present a new primal dual path following interior point algorithm for semidefinite optimization the algorithm is based on a new technique for finding the search direction and the strategy of the central path at each iteration we use only full nesterov todd step moreover we obtain the currently best known iteration bound for the algorithm with small update method namely o n log n which is as good as the linear analogue

a primal dual interior point algorithm for second order cone optimization with full nesterov todd step in this paper we propose a primal dual path following interior point algorithm for second order cone optimization the algorithm is based on a new technique for finding the search directions and the strategy of the central path at each iteration we use only full nesterov todd step moreover we derive the currently best known iteration bound for the algorithm with small update method namely email protected where n denotes the number of second order cones in the problem formulation and e the desired accuracy

polynomial convergence of primal dual algorithms for semidefinite programming based on the monteiro and zhang family of directions this paper establishes the polynomial convergence of the class of primal dual feasible interior point algorithms for semidefinite programming sdp based on the monteiro and zhang family of search directions in contrast to monteiro and zhang s work math programming 81 1998 pp 281 299 here no condition is imposed on the scaling matrix that determines the search direction we show that the polynomial iteration complexity bounds of two well known algorithms for linear programming namely the short step path following algorithm of kojima mizuno and yoshise math programming 44 1989 pp 1 26 and monteiro and adler math programming 44 1989 pp 27 41 and pp 43 66 and the predictor corrector algorithm of mizuno todd and ye math oper res 18 1993 pp 945 981 carry over into the context of sdp since the monteiro and zhang family of directions includes the alizadeh haeberly and overton direction we establish for the first time the polynomial convergence of algorithms based on this search direction

an estimate for the morse index of a stokes wave stokes waves are steady periodic water waves on the free surface of an infinitely deep irrotational two dimensional flow under gravity without surface tension they can be described in terms of solutions of the euler lagrange equation of a certain functional this allows one to define the morse index of a stokes wave it is well known that if the morse indices of the elements of a set of non singular stokes waves are bounded then none of them is close to a singular one the paper presents a quantitative variant of this result

cryptanalysis of a knapsack based probabilistic encryption scheme wang et al b wang q wu y hu a knapsack based probabilistic encryption scheme information sciences 177 19 2007 3981 3994 proposed a high density knapsack based probabilistic encryption scheme with non binary coefficients in this paper we present a heuristic attack that can be used to recover the private key parameters from the known public key parameters in particular we show that the restrictions imposed on the system parameters allow the attacker to recover a short list of candidates for the first half of the public key the second half of the public key can then be recovered using an attack based on lattice basis reduction finally by encrypting an arbitrary plaintext using the known public key then decrypting the resulting ciphertext using these estimated candidate solutions the right private key can be uniquely determined

associative and jordan algebras and polynomial time interior point algorithms for symmetric cones we present a general framework whereby analysis of interior point algorithms for semidefinite programming can be extended verbatim to optimization problems over all classes of symmetric cones derivable from associative algebras in particular such analyses are extendible to the cone of positive semidefinite hermitian matrices with complex and quaternion entries and to the lorentz cone we prove the case of the lorentz cone by using the embedding of its associated jordan algebra in the clifford algebra as an example of such extensions we take monterio s polynomial time complexity analysis of the family of similarly scaled directions introduced by monteiro and zhang 1998 and generalize it to cone lp over all representable symmetric cones

a knapsack based probabilistic encryption scheme knapsack based cryptosystems had been viewed as the most attractive and the most promising asymmetric cryptographic algorithms for a long time due to their np completeness nature and high speed in encryption decryption unfortunately most of them are broken for the low density feature of the underlying knapsack problems in this paper we investigate a new easy compact knapsack problem and propose a novel knapsack based probabilistic public key cryptosystem in which the cipher text is non linear with the plaintext for properly chosen parameters the underlying knapsack problem enjoys a high density larger than 1 06 in the worst case hence it is secure against the low density subset sum attacks our scheme can also defeat other potential attacks such as the brute force attacks and the simultaneous diophantine approximation attack compared with previous knapsack based cryptosystems our scheme is efficient and practical

a knapsack public key cryptosystem based on elliptic curve discrete logarithm there is no question that knapsack systems still warrant continuous researches as a result of the np completeness nature the faster speed and a desire to have a wide variety of available cryptosystems this paper presents a new knapsack type public key cryptosystem based on a novel application of the elliptic curve logarithm problem the difficulty can be made sufficient to frustrate attacks against the survey major knapsack public key cryptosystems developed in the past more than 20 years and hope fully shed some light on the future of the knapsack cryptosystems the proposed new cryptosystem can be found some theoretic support on confidentiality

an improved geometric inequality via vanishing moments with applications to singular liouville equations we consider a class of singular liouville equations on compact surfaces motivated by the study of electroweak and self dual chern simons theories the gaussian curvature prescription with conical singularities and onsager s description of turbulence we analyse the problem of existence variationally and show how the angular distribution of the conformal volume near the singularities may lead to improvements in the moser trudinger inequality and in turn to lower bounds on the euler lagrange functional we then discuss existence and non existence results

an interior point cutting plane method for the convex feasibility problem with second order cone inequalities the convex feasibility problem in general is a problem of finding a point in a convex set that contains a full dimensional ball and is contained in a compact convex set we assume that the outer set is described by second order cone inequalities and propose an analytic center cutting plane technique to solve this problem we discuss primal and dual settings simultaneously two complexity results are reported the complexity of restoration procedure and complexity of the overall algorithm we prove that an approximate analytic center is updated after adding a second order cone cut socc ino 1 newton step and that the analytic center cutting plane method accpm with socc is a fully polynomial algorithm

analysis of the efficiency of the chor rivest cryptosystem implementation in a safe parameter range the chor rivest cryptosystem based on a high density knapsack problem on a finite field f q h was broken by vaudenay for q 200 h 24 and h admitting a factor s verifying a certain condition a new set of parameters q and h which prevent this cryptosystem against vaudenay s attack is presented and the computational aspects of its implementation in the magma computational algebra system are analyzed

second order conic approximation for disassembly line design with joint probabilistic constraints a problem of profit oriented disassembly line design and balancing with possible partial disassembly and presence of hazardous parts is studied the objective is to design a production line providing a maximal revenue with balanced workload task times are assumed to be random variables with known normal probability distributions the cycle time constraints are to be jointly satisfied with at least a predetermined probability level an and or graph is used to model the precedence relationships among tasks several lower and upper bounding schemes are developed using second order cone programming and convex piecewise linear approximation to show the relevance and applicability of the proposed approach a set of instances from the literature are solved to optimality

differentiable exact penalty functions for nonlinear second order cone programs we propose a method for solving nonlinear second order cone programs socps based on a continuously differentiable exact penalty function the construction of the penalty function is given by incorporating a multipliers estimate in the augmented lagrangian for socps under the nondegeneracy assumption and the strong second order sufficient condition we show that a generalized newton method has global and superlinear convergence we also present some preliminary numerical experiments

solving low density subset sum problems the subset sum problem is to decide whether or not the 0 l integer programming problem s n i l a i x i m i x i 0 or 1 has a solution where the a i and m are given positive integers this problem is np complete and the difficulty of solving it is the basis of public key cryptosystems of knapsack type an algorithm is proposed that searches for a solution when given an instance of the subset sum problem this algorithm always halts in polynomial time but does not always find a solution when one exists it converts the problem to one of finding a particular short vector v in a lattice and then uses a lattice basis reduction algorithm due to a k lenstra h w lenstra jr and l lovasz to attempt to find v the performance of the proposed algorithm is analyzed let the density d of a subset sum problem be defined by d n log 2 max i a i then for almost all problems of density d d n it is proved that the lattice basis reduction algorithm locates v extensive computational tests of the algorithm suggest that it works for densities d d c n where d c n is a cutoff value that is substantially larger than 1 n this method gives a polynomial time attack on knapsack public key cryptosystems that can be expected to break them if they transmit information at rates below d c n as n

cryptanalysis of a quadratic compact knapsack public key cryptosystem recently wang and hu have proposed a high density quadratic compact knapsack public key cryptosystem using the chinese remainder theorem to disguise two secret cargo vectors the system is claimed to be secure against certain known attacks however it has not been demonstrated to fulfill any provable security goals in this work we show that this system is not secure exploiting the special structure of system parameters we first show that a candidate list for the secret modulus can be obtained by solving linear equations with small solutions next we show that with this candidate list all other secrets can be recovered in succession with lattice based methods by solving certain modular linear equations with small solutions as a result recovering a private key can be done in about 11 h for the proposed system parameter n 100 we also discuss a method to thwart the proposed attack

sharp sobolev inequalities on the sphere and the moser trudinger inequality where de denotes normalized surface measure v is the conformal gradient and q 2n n 2 a modern folklore theorem is that by taking the infinitedimensional limit of this inequality one obtains the gross logarithmic sobolev inequality for gaussian measure which determines nelson s hypercontractive estimates for the hermite semigroup see 8 one observes using conformal invariance that the above inequality is equivalent to the sharp sobolev inequality on rn for which boundedness and extremal functions can be easily calculated using dilation invariance and geometric symmetrization the roots here go back to hardy and littlewood the advantage of casting the problem on the sphere is that the role of the constants is evident and one is led immediately to the conjecture that this inequality should hold whenever possible for example 2 q 0o if n 2 this is in fact true and will be demonstrated in section 2 a clear question at this point is what is the situation in dimension 2 two important arguments 25 26 27 dealt with this issue both motivated by geometric variational problems because q goes to infinity for dimension 2 the appropriate function space is the exponential class responding in part

product form cholesky factorization in interior point methods for second order cone programming second order cone programming socp problems are typically solved by interior point methods as in linear programming lp interior point methods can in theory solve socps in polynomial time and can in practice exploit sparsity in the problem data specifically when cones of large dimension are present the density that results in the normal equations that are solved at each iteration can be remedied in a manner similar to the treatment of dense columns in an lp here we propose a product form cholesky factorization pfcf approach and show that it is more numerically stable than the alternative sherman morrison woodbury approach we derive several pfcf variants and compare their theoretical perfomance finally we prove that the elements of l in the cholesky factorizations ldlt that arise in interior point methods for socp are uniformly bounded as the duality gap tends to zero as long as the iterates remain is some conic neighborhood of the cental path

moser trudinger trace inequalities sharp constants in exponential inequalities involving a general class of measures in domains are exhibited in the limiting case of the sobolev embedding theorem a comprehensive approach is presented yielding as special instances trace inequalities on o on smooth submanifolds of o of arbitrary dimension and also on fractal subsets of o and recovering in particular the classical moser trudinger inequality

extension of primal dual interior point algorithms to symmetric cones in this paper we show that the so called commutative class of primal dual interior point algorithms which were designed by monteiro and zhang for semidefinite programming extends word for word to optimization problems over all symmetric cones the machinery of euclidean jordan algebras is used to carry out this extension unlike some non commutative algorithms such as the xs sx method this class of extensions does not use concepts outside of the euclidean jordan algebras in particular no assumption is made about representability of the underlying jordan algebra as a special case we prove polynomial iteration complexities for variants of the short semi long and long step path following algorithms using the nesterov todd xs or sx directions

an inexact primal dual path following algorithm for convex quadratic sdp we propose primal dual path following mehrotra type predictor corrector methods for solving convex quadratic semidefinite programming qsdp problems of the form min x frac 1 2 x bullet mathcal q x c bullet x mathcal a x b x succeq 0 where mathcal q is a self adjoint positive semidefinite linear operator on mathcal s n b r m and mathcal a is a linear map from mathcal s n to r m at each interior point iteration the search direction is computed from a dense symmetric indefinite linear system called the augmented equation of dimension m n n 1 2 such linear systems are typically very large and can only be solved by iterative methods we propose three classes of preconditioners for the augmented equation and show that the corresponding preconditioned matrices have favorable asymptotic eigenvalue distributions for fast convergence under suitable nondegeneracy assumptions numerical experiments on a variety of qsdps with n up to 1600 are performed and the computational results show that our methods are efficient and robust

sobolev inequalities for the symmetric gradient in arbitrary domains abstract a form of sobolev inequalities for the symmetric gradient of vector valued functions is proposed which allows for arbitrary ground domains in r n in the relevant inequalities boundary regularity of domains is replaced with information on boundary traces of trial functions the inequalities so obtained exhibit the same exponents as in classical inequalities for the full gradient of sobolev functions in regular domains furthermore they involve constants independent of the geometry of the domain and hence yield novel results yet for smooth domains our approach relies upon a pointwise estimate for the functions in question via a riesz potential of their symmetric gradient and an unconventional potential depending on their boundary trace

sharp moser trudinger inequalities for the laplacian without boundary conditions we derive a sharp moser trudinger inequality for the borderline sobolev imbedding of w2 n 2 bn into the exponential class where bn is the unit ball of rn the corresponding sharp results for the spaces w0d n d are well known for general domains and are due to moser and adams when the zero boundary condition is removed the only known results are for d 1 and are due to chang yang cianchi and leckband the proof of our result is based on a new integral representation formula for the canonical solution of the poisson equation on the ball that is the unique solution of the equation u f which is orthogonal to the harmonic functions on the ball the main technical difficulty of the paper is to establish an asymptotically sharp growth estimate for the kernel of such representation expressed in terms of its distribution function

diffusion tensors for processing sheared and rotated rectangles image restoration and simplification methods that respect important features such as edges play a fundamental role in digital image processing however known edge preserving methods like common nonlinear diffusion methods tend to round vertices for large diffusion times in this paper we adapt the diffusion tensor for anisotropic diffusion to avoid this effects in images containing rotated and sheared rectangles respectively in this context we propose a new method for estimating rotation angles and shear parameters based on the so called structure tensor further we show how the knowledge of appropriate diffusion tensors can be used in variational models numerical examples including orientation estimation denoising and segmentation demonstrate the good performance of our methods

primal dual path following algorithms for semidefinite programming this paper deals with a class of primal dual interior point algorithms for semidefinite programming sdp which was recently introduced by kojima shindoh and hara siam j optim 7 1997 pp 86 125 these authors proposed a family of primal dual search directions that generalizes the one used in algorithms for linear programming based on the scaling matrix x1 2s 1 2 they study three primal dual algorithms based on this family of search directions a short step path following method a feasible potential reduction method and an infeasible potential reduction method however they were not able to provide an algorithm which generalizes the long step path following algorithm introduced by kojima mizuno and yoshise progress in mathematical programming interior point and related methods n megiddor ed springer verlag berlin new york 1989 pp 29 47 in this paper we characterize two search directions within their family as being unique solutions of systems of linear equations in symmetric variables based on this characterization we present a simplified polynomial convergence proof for one of their short step path following algorithms and for the first time a polynomially convergent long step path following algorithm for sdp which requires an extra sqrt n factor in its iteration complexity order as compared to its linear programming counterpart where n is the number of rows or columns of the matrices involved

on the convergence of the method of analytic centers when applied to convex quadratic programs this work examines the method of analytic centers of sonnevend when applied to solve generalized convex quadratic programs where also the constraints are given by convex quadratic functions we establish the existence of a two sided ellipsoidal approximation for the set of feasible points around its center and show that a simple zero order algorithm starting from an initial center of the feasible set generates a sequence of strictly feasible points whose objective function values converge to the optimal value concerning the speed of convergence it is shown that an upper bound for the gap in between the objective function value and the optimal value is reduced by a factor ofe with o sqrt m left ln varepsilon right iterations wherem is the number of inequality constraints here each iteration involves the computation of one newton step the bound of o sqrt m left ln varepsilon right newton iterations to guarantee an error reduction by a factor ofe in the objective function is as good as the one currently given forlinear programs however the algorithm considered here is of theoretical interest only full efficiency of the method can only be obtained when accelerating it by some higher order extrapolation scheme see e g the work of jarre sonnevend and stoer

a generalized newton method for absolute value equations associated with second order cones in this paper we introduce the absolute value equations associated with second order cones socave in short which is a generalization of the absolute value equations discussed recently in the literature it is proved that the socave is equivalent to a class of second order cone linear complementarity problems soclcp in short in particular we propose a generalized newton method for solving the socave and show that the proposed method is globally linearly and locally quadratically convergent under suitable assumptions we also report some preliminary numerical results of the proposed method for solving the socave and the soclcp which show the efficiency of the proposed method

a thermodynamical formalism for monge ampere equations moser trudinger inequalities and kahler einstein metrics we develop a variational calculus for a certain free energy functional on the space of all probability measures on a kahler manifold x this functional can be seen as a generalization of mabuchi s k energy functional and its twisted versions to more singular situations applications to monge ampere equations of mean field type twisted kahler einstein metrics and moser trudinger type inequalities on miller manifolds are given tian s alpha invariant is generalized to singular measures allowing in particular a proof of the existence of kahler einstein metrics with positive ricci curvature that are singular along a given anti canonical divisor which combined with very recent developments concerning miller metrics with conical singularities confirms a recent conjecture of donaldson as another application we show that if the calabi flow in the anti canonical class exists for all times then it converges to a kahler einstein metric when a unique one exists which is in line with a well known conjecture c 2013 elsevier inc all rights reserved

a jordan algebraic approach to potential reduction algorithms we consider the linear monotone complementarity problem for domains obtained as the intersection of an affine subspace and the cartesian product of symmetric cones a primal dual potential reduction algorithm is described and its complexity estimates are established with the help of the jordan algebraic technique

extremal functions for moser s inequality let q be a bounded smooth domain in ri and u x a c1 function with compact support in q moser s inequality states that there is a constant co depending only on the dimension n such that 1 wn1 n j ee n u dx co where iqi is the lebesgue measure of q and wn 1 the surface area of the unit ball in rn we prove in this paper that there are extremal functions for this inequality in other words we show that the sup j nw u dx u e wj 11vuh1n 1 is attained earlier results include carleson chang 1986 q is a ball in any dimension and flucher 1992 q is any domain in 2 dimensions

complementarity and nondegeneracy in semidefinite programming primal and dual nondegeneracy conditions are defined for semidefinite programming given the existence of primal and dual solutions it is shown that primal nondegeneracy implies a unique dual solution and that dual nondegeneracy implies a unique primal solution the converses hold if strict complementarity is assumed primal and dual nondegeneracy assumptions do not imply strict complementarity as they do in lp the primal and dual nondegeneracy assumptions imply a range of possible ranks for primal and dual solutions x and z this is in contrast with lp where nondegeneracy assumptions exactly determine the number of variables which are zero it is shown that primal and dual nondegeneracy and strict complementarity all hold generically numerical experiments suggest probability distributions for the ranks of x and z which are consistent with the nondegeneracy conditions

true random number generator based on mouse movement and chaotic hash function a true random number generator trng makes use of a non deterministic source to produce randomness it is considered as more secure than a pseudorandom number generator as the degree of randomness is higher in this letter we propose a novel trng which generates random bits by moving the mouse casually it is convenient universal and low cost for the personal computer pc platform to eliminate the regular patterns of mouse movements caused by the habit of the same user we propose three trngs based on chaotic hash functions experiments are conducted to evaluate the speed diffusion and randomness performance of these trngs the results show that two of them possess satisfactory performance and can be implemented on common pc platform

a polynomial time algorithm for breaking the basic merkle hellman cryptosystem the merkle hellman cryptosystem is one of the two major public key cryptosystems proposed so far it is shown that the basic variant of this cryptosystem in which the elements of the public key are modular multiples of a superincreasing sequence is breakable in polynomial time

an active set method for second order conic constrained quadratic programming we consider the minimization of a convex quadratic objective subject to second order cone constraints this problem generalizes the well studied bound constrained quadratic programming qp problem we propose a new two phase method in the first phase a projected gradient method is used to quickly identify the active set of cones and in the second phase newton s method is applied to rapidly converge given the subsystem of active cones computational experiments confirm that the conically constrained qp is solved more efficiently by our method than by a specialized conic optimization solver and more robustly than by general nonlinear programming solvers

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

a new approach to sharp moser trudinger and adams type inequalities a rearrangement free argument abstract the main purpose of this paper is two fold on the one hand we will develop a new approach to establish sharp singular moser trudinger and adams type inequalities in unbounded domains of euclidean spaces without using the standard symmetrization on the other hand we will prove the sharp singular adams type inequality on high order sobolev spaces w m n m r n of arbitrary integer order m theorem 1 1 which improves the results of ruf and sani 2013 48 where sharp adams inequalities were established for even m and those of the authors lam and lu 2012 28 29 for odd m but with different and more restricted norms we first establish the sharp local singular adams inequality on domains in r n of finite measure theorem 1 4 we take a perspective that any function in the high order sobolev spaces w m n m r n can be represented as a bessel potential thus we can fully use the tools from harmonic analysis and the kernel properties of the polyharmonic operators i m 2 once we have established this sharp local adams inequality then we can adapt the rearrangement free method we will develop in this paper to derive a global sharp adams inequality from a local one our argument substantially simplifies those in ruf and sani 2013 48 and lam and lu 2012 28 29 and avoids the use of rather deep and complicated comparison principle of solutions to polyharmonic operators used in ruf and sani 2013 48 lam and lu 2012 28 29 moreover our theorem holds on sobolev spaces w n r n of any positive fractional order n theorems 1 6 and 1 7 as consequences we can also establish sharp adams inequalities with less restrictions on the sobolev norms see theorems 1 2 1 3 and 1 5 our approach is surprisingly simple and general and can be easily applied to scenarios such as riemannian and sub riemannian manifolds where symmetrization argument does not work see e g on the heisenberg group lam and lu 2012 30

conformal deformation of metrics on s 2 soit d un domaine lisse de s 2 et k une fonction lisse definie sur d alors pour que le probleme aux limites de neumann u k exp 2u 1 sur d u n 0 sur d ait une solution u on a a quand aire d max x d k x 0

a one parametric class of merit functions for the symmetric cone complementarity problem abstract in this paper we extend the one parametric class of merit functions proposed by kanzow and kleinmichel c kanzow h kleinmichel a new class of semismooth newton type methods for nonlinear complementarity problems comput optim appl 11 1998 227 251 for the nonnegative orthant complementarity problem to the general symmetric cone complementarity problem sccp we show that the class of merit functions is continuously differentiable everywhere and has a globally lipschitz continuous gradient mapping from this we particularly obtain the smoothness of the fischer burmeister merit function associated with symmetric cones and the lipschitz continuity of its gradient in addition we also consider a regularized formulation for the class of merit functions which is actually an extension of one of the ncp function classes studied by c kanzow y yamashita m fukushima new ncp functions and their properties j optim theory appl 97 1997 115 135 to the sccp by exploiting the cartesian p properties for a nonlinear transformation we show that the class of regularized merit functions provides a global error bound for the solution of the sccp and moreover has bounded level sets under a rather weak condition which can be satisfied by the monotone sccp with a strictly feasible point or the sccp with the joint cartesian r 02 property all of these results generalize some recent important works in j s chen p tseng an unconstrained smooth minimization reformulation of the second order cone complementarity problem math program 104 2005 293 327 c k sim j sun d ralph a note on the lipschitz continuity of the gradient of the squared norm of the matrix valued fischer burmeister function math program 107 2006 547 553 p tseng merit function for semidefinite complementarity problems math program 83 1998 159 185 under a unified framework

a new public key cipher system based upon the diophantine equations a new public key two key cipher scheme is proposed in this paper in our scheme keys can be easily generated in addition both encryption and decryption procedures are simple to encrypt a message the sender needs to conduct a vector product of the message being sent and the enciphering key on the other hand the receiver can easily decrypt it by conducting several multiplication operations and modulus operations for security analysis we also examine some possible attacks on the presented scheme

sharp exponential integrability for critical riesz potentials and fractional laplacians on rn abstract we derive sharp adams inequalities for the riesz and more general riesz like potentials on the whole of r n as a consequence we obtain sharp moser trudinger inequalities for the critical sobolev spaces w n r n 0 n these inequalities involve fractional laplacians higher order gradients general homogeneous elliptic operators with constant coefficients and general trace type borel measures

fundamental solution for the q laplacian and sharp moser trudinger inequality in carnot groups abstract for a general carnot group g with homogeneous dimension q we prove the existence of a fundamental solution of the q laplacian u q and a constant a q 0 such that exp a q u q is a homogeneous norm on g this implies a representation formula for smooth functions on g which is used to prove the sharp carnot group version of the celebrated moser trudinger inequality

polynomial convergence of primal dual algorithms for the second order cone program based on the mz family of directions in this paper we study primal dual path following algorithms for the second order cone programming socp based on a family of directions that is a natural extension of the monteiro zhang mz family for semidefinite programming we show that the polynomial iteration complexity bounds of two well known algorithms for linear programming namely the short step path following algorithm of kojima et al and monteiro and adler and the predictor corrector algorithm of mizuno et al carry over to the context of socp that is they have an o loge 1 iteration complexity to reduce the duality gap by a factor of e where n is the number of second order cones since the mz type family studied in this paper includes an analogue of the alizadeh haeberly and overton pure newton direction we establish for the first time the polynomial convergence of primal dual algorithms for socp based on this search direction

adams inequalities on measure spaces abstract in 1988 adams obtained sharp moser trudinger inequalities on bounded domains of r n the main step was a sharp exponential integral inequality for convolutions with the riesz potential in this paper we extend and improve adams results to functions defined on arbitrary measure spaces with finite measure the riesz fractional integral is replaced by general integral operators whose kernels satisfy suitable and explicit growth conditions given in terms of their distribution functions natural conditions for sharpness are also given most of the known results about moser trudinger inequalities can be easily adapted to our unified scheme we give some new applications of our theorems including sharp higher order moser trudinger trace inequalities sharp adams moser trudinger inequalities for general elliptic differential operators scalar and vector valued for sums of weighted potentials and for operators in the cr setting

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 polyhedral approximations of the second order cone we demonstrate that a conic quadratic problem is polynomially reducible to linear programming we demonstrate this by constructing for every an lp program explicitly given in terms of e and the data of cqp with the following properties i the number dimx dimu of variables and the number dimp of constraints in lp do not exceed ii every feasible solutionx to cqp can be extended to a feasible solution x u to lp iii if x u is feasible for lp thenx satisfies the e relaxed constraints of cqp namely

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

on extending some primal dual interior point algorithms from linear programming to semidefinite programming this work concerns primal dual interior point methods for semidefinite programming sdp that use a search direction originally proposed by helmberg et al siam j optim 6 1996 pp 342 361 and kojima shindoh and hara siam j optim 7 1997 pp 86 125 and recently rediscovered by monteiro siam j optim 7 1997 pp 663 678 in a more explicit form in analyzing these methods a number of basic equalities and inequalities were developed in kojima shindoh and hara and also in monteiro through different means and in different forms in this paper we give a concise derivation of the key equalities and inequalities for complexity analysis along the exact line used in linear programming lp producing basic relationships that have compact forms almost identical to their counterparts in lp we also introduce a new formulation of the central path and variable metric measures of centrality these results provide convenient tools for deriving polynomiality results for primal dual algorithms extended from lp to sdp using the aforementioned and related search directions we present examples of such extensions including the long step infeasible interior point algorithm of zhang siam j optim 4 1994 pp 208 227

secure public key encryption scheme without random oracles since the first practical and secure public key encryption scheme without random oracles proposed by cramer and shoup in 1998 cramer shoup s scheme and its variants remained the only practical and secure public key encryption scheme without random oracles until 2004 in 2004 canetti et al proposed a generic transformation from a selective identity based encryption scheme to a public key encryption by adding a one time strongly signature scheme since then some transformation techniques from a selective identity based encryption scheme to a public key encryption have been proposed to enhance the computational efficiency for example boneh katz s construction and boyen mei waters scheme these transformations have either traded off the publicly verifiable properties or tightness of security reduction in 2007 zhang proposed another generic transformation by adding chameleon hash functions in this paper we introduce another technique from the boneh boyen s selective identity based encryption scheme to a public key encryption which is publicly verifiable and is slightly more efficient than zhang s transformation the proposed public key encryption scheme is based on the decisional bilinear diffie hellman assumption and the target collision resistant hash functions

improved cryptanalysis of a knapsack based probabilistic encryption scheme wang et al b wang q wu y hu information sciences 177 2007 proposed a knapsack based probabilistic encryption scheme with non binary coefficients which enjoys a high density larger than 1 06 in the worst case in this work we successfully attack this scheme by showing that a public key and a restriction on system parameters allow the attacker to recover a secret key in a cubic time complexity using modular equations this approach is much more efficient than the previous attack by youssef a m youssef information sciences 179 2009 in which lattice basis reductions are used recovering secret keys can be done within 4h and 4days when n 100 and 200 respectively a simple modification that helps resist known attacks is also discussed

second order cone programming second order cone programming socp problems are convex optimization problems in which a linear function is minimized over the intersection of an affine linear manifold with the cartesian product of second order lorentz cones linear programs convex quadratic programs and quadratically constrained convex quadratic programs can all be formulated as socp problems as can many other problems that do not fall into these three categories these latter problems model applications from a broad range of fields from engineering control and finance to robust optimization and combinatorial optimization on the other hand semidefinite programming sdp that is the optimization problem over the intersection of an affine set and the cone of positive semidefinite matrices includes socp as a special case therefore socp falls between linear lp and quadratic qp programming and sdp like lp qp and sdp problems socp problems can be solved in polynomial time by interior point methods the computational effort per iteration required by these methods to solve socp problems is greater than that required to solve lp and qp problems but less than that required to solve sdp s of similar size and structure because the set of feasible solutions for an socp problem is not polyhedral as it is for lp and qp problems it is not readily apparent how to develop a simplex or simplex like method for socp while socp problems can be solved as sdp problems doing so is not advisable both on numerical grounds and computational complexity concerns for instance many of the problems presented in the survey paper of vandenberghe and boyd vb96 as examples of sdps can in fact be formulated as socps and should be solved as such in 2 3 below we give socp formulations for four of these examples the convex quadratically constrained quadratic programming qcqp problem problems involving fractional quadratic functions rutcor rutgers university e mail alizadeh rutcor rutgers edu research supported in part by the u s national science foundation grant ccr 9901991 ieor columbia university e mail gold ieor columbia edu research supported in part by the department of energy grant de fg02 92er25126 national science foundation grants dms 94 14438 cda 97 26385 and dms 01 04282

a sharp inequality of j moser for higher order derivatives in fact go nwl 1 where wnis the area of the surface of the unit n ball also he notes that if 3 exceeds go then 1 can no longer hold with some co independent of the u satisfying 2 the fact that this result holds for some 3 0 independent of u has been observed now by several authors with a variety of proofs see 24 20 23 22 10 4 1 2 13 14 for example the earliest of these arguments expands the exponential function in a power series then observing that the norm in the imbedding

non injective knapsack public key cryptosystems abstract two public key 0 1 knapsack cryptosystems are proposed that have so high a density and use so weak a modular multiplication as a trapdoor that known attacks can be avoided decryption is fairly slow and may produce more than one decipherment but all alternative decipherments can be found disambiguating protocols are needed to determine the correct decipherment it is suggested to use also redundancy for this purpose in the first system the initial knapsack is constructed from the powers of two which are multiplied by a constant and reduced with respect to a modulus to a specific range thus producing the easy knapsack then weak modular multiplication is used as a trapdoor transformation with respect to another modulus which is typically smaller than some or all of the elements of the easy knapsack the second knapsack is constructed iteratively from modularly injective or nearly injective components decryption of small components is based on look up tables the specific form of the proposal uses also one large non injective component which is generated and decrypted in a way that resembles superincrease

a product form cholesky factorization method for handling dense columns in interior point methods for linear programming cholesky factorization has become the method of choice for solving the symmetric system of linear equations arising in interior point methods ipms for linear programming lp its main advantages being numerical stability and efficiency for sparse systems however in the presence of dense columns there is dramatic loss of efficiency a typical approach to remedy this problem is to apply the sherman morrison woodbury smw update formula to the dense part this approach while being very efficient is not numerically stable here we propose using product form cholesky factorization to handle dense columns the proposed approach is essentially as stable as the original cholesky factorization and nearly as efficient as the smw approach we demonstrate these properties both theoretically and computationally a key part of our theoretical analysis is the proof that the elements of the cholesky factors of the matrices that arise in ipms for lp are uniformly bounded as the duality gap converges to zero

an interior point method for semidefinite programming we propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices

scenario reduction revisited fundamental limits and guarantees the goal of scenario reduction is to approximate a given discrete distribution with another discrete distribution that has fewer atoms we distinguish continuous scenario reduction where the new atoms may be chosen freely and discrete scenario reduction where the new atoms must be chosen from among the existing ones using the wasserstein distance as measure of proximity between distributions we identify those n point distributions on the unit ball that are least susceptible to scenario reduction i e that have maximum wasserstein distance to their closest m point distributions for some prescribed m n we also provide sharp bounds on the added benefit of continuous over discrete scenario reduction finally to our best knowledge we propose the first polynomial time constant factor approximations for both discrete and continuous scenario reduction as well as the first exact exponential time algorithms for continuous scenario reduction

moser inequalities in gauss space the sharp constants in a family of exponential sobolev type inequalities in gauss space are exhibited they constitute the gaussian analogues of the moser inequality in the borderline case of the sobolev embedding in the euclidean space interestingly the gaussian results have features in common with the euclidean ones but also reveal marked diversities

quadratic compact knapsack public key cryptosystem knapsack type cryptosystems were among the first public key cryptographic schemes to be invented their np completeness nature and the high speed in encryption decryption made them very attractive however these cryptosystems were shown to be vulnerable to the low density subset sum attacks or some key recovery attacks in this paper additive knapsack type public key cryptography is reconsidered we propose a knapsack type public key cryptosystem by introducing an easy quadratic compact knapsack problem the system uses the chinese remainder theorem to disguise the easy knapsack sequence the encryption function of the system is nonlinear about the message vector under the relinearization attack model the system enjoys a high density we show that the knapsack cryptosystem is secure against the low density subset sum attacks by observing that the underlying compact knapsack problem has exponentially many solutions it is shown that the proposed cryptosystem is also secure against some brute force attacks and some known key recovery attacks including the simultaneous diophantine approximation attack and the orthogonal lattice attack

unconditionally secure cryptosystems based on quantum cryptography most modern cryptographic studies design cryptosystems and algorithms using mathematical concepts in designing and analyzing cryptosystems and protocols mathematical concepts are critical in supporting the claim that the intended cryptosystem is secure most early cryptographic algorithms are based either on factorization or on discrete logarithm problem such systems generally adopt rather simple mathematics and therefore need extensive secondary index computation this study discusses quantum cryptosystems protection of system security and optimization of system efficiency quantum cryptography detects intrusion and wiretap in quantum mechanics a wiretap is neither external nor passive rather it modifies its entity based on the internal component of the system the status of the quantum system changes once a wiretap is detected hence only the designer of the system can discover the quantum status of the system an eavesdropper can neither determine the quantum state nor duplicate the system the quantum cryptosystem can achieve unconditional security and thus guarantees secure communication

similarity and other spectral relations for symmetric cones abstract the similarity relations that are derived in this paper reduce to well known results in the special case of symmetric matrices in particular for two positive definite matrices x and y the square of the spectral geometric mean is known to be similar to the matrix product xy it is shown in this paper that this property carries over to symmetric cones more elementary similarity relations such as xy 2 x yx 2 y are generalized as well we also extend the result that the eigenvalues of a matrix product xy are less dispersed than the eigenvalues of the jordan product xy yx 2 the paper further contains a number of inequalities on norms and spectral values this type of inequality is often used in the analysis of interior point methods in optimization we also derive an extension of stein s theorem to symmetric cones

a method of analytic centers for quadratically constrained convex quadratic programs an interior point method is developed for maximizing a concave quadratic function under convex quadratic constraints the algorithm constructs a sequence of nested convex sets and finds their approximate centers using a partial newton step given the first convex set and its approximate center the total arithmetic operations required to converge to an approximate solution are of order o sqrt m m n n 2 ln varepsilon where m is the number of constraints n is the number of variables and varepsilon is determined by the desired tolerance of the optimal value and the size of the first convex set a method to initialize the algorithm is also proposed so that the algorithm can start from an arbitrary perhaps infeasible point

robust twin support vector machine for pattern classification in this paper we proposed a new robust twin support vector machine called r twsvm via second order cone programming formulations for classification which can deal with data with measurement noise efficiently preliminary experiments confirm the robustness of the proposed method and its superiority to the traditional robust svm in both computation time and classification accuracy remarkably since there are only inner products about inputs in our dual problems this makes us apply kernel trick directly for nonlinear cases simultaneously we does not need to solve the extra inverse of matrices which is totally different with existing twsvms in addition we also show that the twsvms are the special case of our robust model and simultaneously give a new dual form of twsvm by degenerating r twsvm which successfully overcomes the existing shortcomings of twsvm

primal dual interior point algorithm for convex quadratic semi definite optimization abstract in this paper we present a new primal dual interior point algorithm for solving a special case of convex quadratic semi definite optimization based on a parametric kernel function the proposed parametric kernel function is used both for determining the search directions and for measuring the distance between the given iterate and the center for the algorithm these properties enable us to derive the currently best known iteration bounds for the algorithm with large and small update methods namely o n log n log n e and o n log n e respectively which reduce the gap between the practical behavior of the algorithm and its theoretical performance results

on the semidefinite representation of real functions applied to symmetric matrices abstract we present a new semidefinite representation for the trace of a real function f applied to symmetric matrices when a semidefinite representation of the convex or concave function f is known our construction is intuitive and yields a representation that is more compact than the previously known one we also show with the help of matrix geometric means and a riemannian metric over the set of positive definite matrices that for a rational exponent p in the interval 0 1 the matrix x raised to p is the largest element of a set represented by linear matrix inequalities this result further generalizes to the case of the matrix a p b which is the point of coordinate p on the geodesic from a to b we give numerical results for a problem inspired from the theory of experimental designs which show that the new semidefinite programming formulation can yield an important speed up factor

complete characterizations of stable farkas lemma and cone convex programming duality we establish necessary and sufficient conditions for a stable farkas lemma we then derive necessary and sufficient conditions for a stable duality of a cone convex optimization problem where strong duality holds for each linear perturbation of a given convex objective function as an application we obtain stable duality results for convex semi definite programs and convex second order cone programs

linear systems in jordan algebras and primal dual interior point algorithms we discuss a possibility of the extension of a primal dual interior point algorithm suggested recently by alizadeh et al 1994 we consider optimization problems defined on the intersection of a symmetric cone and an affine subspace the question of solvability of a linear system arising in the implementation of the primal dual algorithm is analyzed a nondegeneracy theory for the considered class of problems is developed the jordan algebra technique suggested by faybusovich 1995 plays major role in the present paper

a new full nesterov todd step feasible interior point method for convex quadratic symmetric cone optimization in this paper we generalize the classical primal dual logarithmic barrier method for linear optimization to convex quadratic optimization over symmetric cone by using euclidean jordan algebras the symmetrization of the search directions used in this paper is based on the nesterov todd scaling scheme and only full nesterov todd step is used at each iteration we derive the iteration bound that matches the currently best known iteration bound for small update methods namely o r log r e some preliminary numerical results are provided to demonstrate the computational performance of the proposed algorithm

volumetric barrier decomposition algorithms for stochastic quadratic second order cone programming ariyawansa and zhu 2011 have derived volumetric barrier decomposition algorithms for solving two stage stochastic semidefinite programs and proved polynomial complexity of certain members of the algorithms in this paper we utilize their work to derive volumetric barrier decomposition algorithms for solving two stage stochastic convex quadratic second order cone programming and establish polynomial complexity of certain members of the proposed algorithms

a new non interior continuation method for solving the second order cone complementarity problem abstract in this paper based on a symmetrically perturbed smoothing fischer burmeister function a non interior continuation method is proposed for solving the second order cone complementarity problem soccp the proposed algorithm solves only one linear system of equations and performs only one line search at each iteration under monotonicity it is shown that our algorithm is globally and locally superlinearly convergent without requiring strict complementarity assumption at the soccp solution furthermore the proposed algorithm has local quadratic convergence under mild conditions some numerical results are reported which indicate the effectiveness of the proposed algorithm