dc programming and dca thirty years of developments the year 2015 marks the 30th birthday of dc difference of convex functions programming and dca dc algorithms which constitute the backbone of nonconvex programming and global optimization in this article we offer a short survey on thirty years of developments of these theoretical and algorithmic tools the survey is comprised of three parts in the first part we present a brief history of the field while in the second we summarize the state of the art results and recent advances we focus on main theoretical results and dca solvers for important classes of difficult nonconvex optimization problems and then give an overview of real world applications whose solution methods are based on dca the third part is devoted to new trends and important open issues as well as suggestions for future developments

combinatorial interior point methods for generalized network flow problems we present combinatorial interior point methods for the generalized minimum cost flow and the generalized circulation problems based on wallacher and zimmermann s combinatorial interior point method for the minimum cost network flow problem the algorithms have features of both a combinatorial algorithm and an interior point method they work towards optimality by iteratively reducing the value of a potential function while maintaining interior point solutions at each iteration flow is augmented along a generalized circulation which is computed by solving a tvpi two variables per inequality system the algorithms run in time where m and n are respectively the number of arcs and nodes in the graph and l is the length of the input data

linear programming in linear time when the dimension is fixed it is demonstrated that the linear programming problem in d variables and n constraints can be solved in o n time when d is fixed this bound follows from a multidimensional search technique which is applicable for quadratic programming as well there is also developed an algorithm that is polynomial in both n and d provided d is bounded by a certain slowly growing function of n

some group actions on homotopy spheres of dimension seven and fifteen 0 introduction this paper is based on the simple observation that every 4 plane bundle over s4 admits a natural action of 50 3 by bundle maps and that every 8 plane bundle over 58 admits a natural action of the compact lie group g2 by bundle maps these actions are easy to see once one remembers that so 3 is the group of automorphisms of the quaternions and that g2 is the group of automorphisms of the cayley numbers the study of these actions is closely connected to several well known phenomena in differential topology and compact transformation groups the most obvious connection is to milnor s original construction of exotic 7 spheres as 3 sphere bundles over 54 26 milnor proved that if the euler class of such a bundle is a generator of h4 s4 z then its total space is homeomorphic to 57 he also defined a numerical invariant of the diffeomorphism type and used it to detect an exotic differential structure on some of these sphere bundles subsequently eells and kuiper 12 introduced a refinement of this invariant called the it invariant using the u invariant they proved that the sphere bundle m7 milnor s notation is a generator for the group of homotopy 7 spheres these constructions also work for 7 sphere bundles over 58 32 and the manifold m15 is a generator for bp16 the group of homotopy 15 spheres which bound wn manifolds it is a routine matter to check that milnor s arguments and their subsequent refinements work g equivariantly where g 5so 3 or g2 and we shall do this in sections 2 and 3 in particular each sphere bundle with the correct euler class is g homeomorphic to an orthogonal action on s2 1 where 2n 1 7 or 15 moreover distinct sphere bundles have distinct oriented g diffeomorphism types the proof of the second fact uses an equivariant version of the it invariant as originally defined this invariant takes values in q z however it is well known that in the presence of a g action with s c g its value in q is well defined using

complexity of bezout s theorem v polynomial time we show that there are algorithms which find an approximate zero of a system of polynomial equations and which function in polynomial time on the average the number of arithmetic operations is cn 4s where n is the input size and c a universal constant

integer programs for logic constraint satisfaction abstract logic constraint satisfaction problems are in general np hard and a general deterministic polynomial time algorithm is not known since several logic constraint problems can be reduced in polynomial time to the satisfaction of a conjunctive normal form cnf sat this case is very important we present here a technique to transform a cnf sat problem in an integer optimization problem that can be solved by linear programming the size of the obtained integer program has a polynomial growth in comparison with the original problem size

a two resource allocation problem solvable in linear time the allocation problem discussed here is as follows the processing time of a task is a linear function t ax by of amounts x y of resources allocated to it where t a b are positive constants given the amounts x and y of available resources we wish to allocate them among n independent tasks so as to minimize the maximal completion time we solve this problem in linear time

weighted pseudo almost automorphic mild solutions to semilinear fractional differential equations this paper is concerned with the existence and uniqueness of a weighted pseudo almost automorphic mild solution to the semilinear fractional equation d t au t au t d t a 1f t u t t r 1 a 2 in complex banach spaces with s p weighted pseudo almost automorphic coefficients where a is a linear densely defined operator of sectorial type on a complex banach space x moreover we present an application to a fractional wave equation

computing over the reals with addition and order abstract we provide a fairly complete picture of the complexity theory of additive real machines this model of computation is a restriction of the real turing machine of blum et al 1989 since addition and subtraction are the only legal arithmetic operations removing the order relation r yields an even weaker class of machines which is also studied our main results are characterization of the classes of recognizable boolean languages equivalence of real and digital nondeterminism a simpler proof of meer s plin nplin result

direct projection decoding algorithm for sigma delta modulated signals usually sigma delta modulators spl sigma spl delta m are modeled by replacing the quantizer with an additive white noise source based on this linear model linear decoder structures can be designed however the spl sigma spl delta m is a nonlinear system and corresponding nonlinear decoder structures may be able to achieve a better signal to quantization noise ratio sqnr performance in this paper a new nonlinear decoding algorithm is presented it is based on the projection onto complex sets pocs algorithm developed by hein and zakhor our algorithm reduces the decoding problem to the solution of a single quadratic programming problem based on the state equations of the modulator and the condition that the modulator input signal is bandlimited the bandlimitation constraint is directly applied to the state equations thus the resulting quadratic programming problem needs to be solved only once

separation of complexity classes in koiran s weak model abstract we continue the study of complexity classes over the weak model introduced by p koiran in particular we provide several separations of complexity classes the most remarkable being the strict inclusion of p in np other separations concern classes defined by weak polynomial time over parallel or alternating machines as well as over nondeterministic machines whose guesses are required to be 0 or 1

nonuniform proof systems a new framework to describe nonuniform and probabilistic complexity classes the concept of non uniform proof systems is introduced this notion allows a uniform description of non uniform complexity classes 10 probabilistic classes e g bpp 8 15 27 32 am 2 3 and language classes defined by simultaneous non uniform and nondeterministic time bounds non uniform proof systems provide a better understanding of many results concerning these classes particularly their connections to uniform complexity measures we give an uniform approach to lowness results 19 20 for various complexity classes for instance we show that co np poly np is contained in the third level of the low hierarchy and that np c np co np poly implies that the polynomial time hierarchy collapses to its second level see also 1 finally some evidence is given that the low hierarchy cannot be extended beyond its third level by the current techniques

weighted pseudo almost automorphic mild solutions for two term fractional order differential equations using the notion of weyl fractional derivative we study the existence and uniqueness of weighted pseudo almost automorphic mild solutions for a class of two term fractional order abstract differential equation in the form d t u t d t u t a u t d t f t u t t 0 0 1 0 where a is a closed linear operator defined in a banach space x and the forcing term f is sp weighted pseudo almost automorphic

stepanov like almost automorphic functions and monotone evolution equations abstract in this paper we are concerned with a new class of stepanov like almost automorphic s p a a functions with values in a banach space x this class contains the space aa x of all bochner almost automorphic functions we use the results obtained to prove the existence and uniqueness of a weak s p a a solution to the parabolic equation u t a t u f t in a reflexive banach space assuming some appropriate conditions of monotonicity coercitivity of the operators a t and s p almost automorphy of the forced term f t this result extends a known result in the case of almost periodicity an application is also given

stepanov like pseudo almost periodic mild solutions to perturbed nonautonomous evolution equations with infinite delay abstract in this paper we prove the invariance of stepanov like pseudo almost periodic functions under bounded linear operators furthermore we obtain existence and uniqueness theorems of pseudo almost periodic mild solutions to evolution equations u t a t u t h t and u t a t u t f t b u t t c t s u s d s f t on r assuming that a t satisfy acquistapace terreni conditions that the evolution family generated by a t has exponential dichotomy that r 0 a is almost periodic that b c t s t s are bounded linear operators that f is lipschitz with respect to the second argument uniformly in the first argument and that h f f are stepanov like pseudo almost periodic for p 1 and continuous to illustrate our abstract result a concrete example is given

an algorithm of internal feasible directions for linear integer programming abstract an approximate algorithm for solving pure integer problems is examined the algorithm belongs to the class of component algorithms using feasible integer directions the basic characteristics of the algorithm are analyzed testing on an ibm pc at personal computer is carried out the proposed algorithm is compared with other well known approximate algorithms

a strongly polynomial algorithm to solve combinatorial linear programs khachiyan and recently karmarkar gave polynomial algorithms to solve the linear programming problem these algorithms have a small theoretical drawback namely the number of arithmetic steps depends on the size of the input numbers we present a polynomial linear programming algorithm whose number of arithmetic steps depends only on the size of the numbers in the constraint matrix but is independent of the size of the numbers in the right hand side and objective vectors in particular it gives a polynomial algorithm for the minimum cost flow and multicommodity flow problems in which the number of arithmetic steps is independent of the size of the costs and capacities the algorithm makes use of an existing polynomial linear programming algorithm the problem of whether any algorithm has a running time that is independent even of the size of the numbers in the constraint matrix remains open

computation of the output of a function with fuzzy inputs based on a low rank tensor approximation we apply a derivative free optimization method based on novel low rank tensor methods to the problem of propagating fuzzy uncertainty through a continuous real valued function adhering to zadeh s extension principle such a problem can be reformulated as a sequence of optimization problems over nested search spaces the optimization method we use is based on a low rank tensor approximation of the function sampled on a grid and a search for the minimal and maximal entries in this low rank tensor in contrast to classical fuzzy uncertainty propagation methods such as the vertex method and the transformation method the method we propose does not exhibit an inherent exponential scaling for increasing dimension of the search space obviously no derivative free optimization algorithm can exist which shows sub exponential scaling with the dimension for all possible continuous functions the algorithm that we present here however can exploit a specific type of structure and regularity beyond continuity that is often present in real world optimization problems we illustrate this with some high dimensional numerical examples where the presented method clearly outperforms some established derivative free optimization codes

a decomposition method for minlps with lipschitz continuous nonlinearities many mixed integer optimization problems are constrained by nonlinear functions that do not possess desirable analytical properties like convexity or factorability or cannot even be evaluated exactly this is e g the case for many problems constrained by differential equations or for models that rely on black box simulation runs for these problem classes we present analyze and test algorithms that solve mixed integer problems with lipschitz continuous nonlinearities our theoretical results depend on the assumptions made on the in exactness of function evaluations and on the knowledge of lipschitz constants if lipschitz constants are known we prove finite termination at approximate globally optimal points both for the case of exact and inexact function evaluations if only approximate lipschitz constants are known we prove finite termination and derive additional conditions under which infeasibility can be detected a computational study for gas transport problems and an academic case study show the applicability of our algorithms to real world problems and how different assumptions on the constraint functions up or downgrade the practical performance of the methods

a polynomial combinatorial algorithm for generalized minimum cost flow we propose the first combinatorial solution to the generalized minimum cost flow problem flow with losses and gains despite a rich history dating back to kantorovich and dantzig until now the only known way to solve the problem in polynomial time was via general purpose linear programming techniques polynomial combinatorial algorithms were previously known only for the version of our problem without costs we design the first such algorithms for the version with costs our algorithms also find provably good solutions faster than optimal ones providing the first strongly polynomial approximation schemes for the problem our techniques extend to optimize linear programs with two variables per inequality polynomial combinatorial algorithms were previously developed for testing the feasibility of such linear programs we propose the first such methods for the optimization version

testing the necklace condition for shortest tours and optimal factors in the plane a tour of a finite set p of points is a necklace tour if there are disks with the points in p as centers such that two disks intersect if and only if their centers are adjacent in it has been observed by sanders that a necklace tour is an optimal traveling salesman tour

a simpler and faster strongly polynomial algorithm for generalized flow maximization we present a new strongly polynomial algorithm for generalized flow maximization that is significantly simpler and faster than the previous strongly polynomial algorithm 34 for the uncapacitated problem formulation the complexity bound o mn m n log n log n2 m improves on the previous estimate by almost a factor o n2 even for small numerical parameter values our running time bound is comparable to the best weakly polynomial algorithms the key new technical idea is relaxing the primal feasibility conditions this allows us to work almost exclusively with integral flows in contrast to all previous algorithms for the problem

on some interior point algorithms for nonconvex quadratic optimization recently interior point algorithms have been applied to nonlinear and nonconvex optimization most of these algorithms are either primal dual path following or affine scaling in nature and some of them are conjectured to converge to a local minimum we give several examples to show that this may be untrue and we suggest some strategies for overcoming this difficulty

on the complexity of quadratic programming in real number models of computation abstract the complexity of linearly constrained nonconvex quadratic programming is analyzed within the framework of real number models namely the one of blum shub and smale and its modification recently introduced by koiran weak bss model in particular we show that this problem is not np complete in the koiran setting applications to the full bss model are discussed

towards a genuinely polynomial algorithm for linear programming a linear programming algorithm is called genuinely polynomial if it requires no more than p m n arithmetic operations to solve problems of order m times n where p is a polynomial it is not known whether such an algorithm exists we present a genuinely polynomial algorithm for the simpler problem of solving linear inequalities with at most two variables per inequality the number of operations required is o mn 3 log text m the technique used was developed in a previous paper where a novel binary search idea was introduced

range assignment of base stations maximizing coverage area without interference abstract we study the problem of assigning non overlapping geometric objects centered at a given set of points such that the sum of area covered by them is maximized the problem remains open since 2002 as mentioned in a lecture slide of david eppstein in this paper we have performed an exhaustive study on the problem we show that if the points are placed in r 2 then the problem is np hard even for simplest type of covering objects like disks or squares in contrast eppstein 2017 10 proposed a polynomial time algorithm for maximizing the sum of radii or perimeter of non overlapping disks when the points are arbitrarily placed in r 2 we show that eppstein s algorithm for maximizing sum of perimeter of the disks in r 2 gives a 2 approximation solution for the sum of area maximization problem we also propose a ptas for the same problem our results can be extended in higher dimensions as well as for a class of centrally symmetric convex objects

on the complexity of approximating a kkt point of quadratic programming we present a potential reduction algorithm to approximate a karush kuhn tucker kkt point of general quadratic programming qp we show that the algorithm is a fully polynomial time approximation scheme and its running time dependency on accuracy 0 1 is o l log l log log l compared to the previously best known result o l 2 furthermore the limit of the kkt point satisfies the second order necessary optimality condition of being a local minimizer 1998 the mathematical programming society inc published by elsevier science b v

on the oracle complexity of first order and derivative free algorithms for smooth nonconvex minimization the optimal function gradient evaluations worst case complexity analysis available for the adaptive regularization algorithms with cubics arc for nonconvex smooth unconstrained optimization is extended to finite difference versions of this algorithm yielding complexity bounds for first order and derivative free methods applied on the same problem class a comparison with the results obtained for derivative free methods by vicente worst case complexity of direct search technical report preprint 10 17 department of mathematics university of coimbra coimbra portugal 2010 is also discussed giving some theoretical insight into the relative merits of various methods in this popular class of algorithms

low rank matrix approximation in the infinity norm abstract the low rank matrix approximation problem with respect to the entry wise l norm is the following given a matrix m and a factorization rank r find a matrix x whose rank is at most r and that minimizes max i j m i j x i j in this paper we prove that the decision variant of this problem for r 1 is np complete using a reduction from the problem not all equal 3sat we also analyze several cases when the problem can be solved in polynomial time and propose a simple practical heuristic algorithm which we apply on the problem of the recovery of a quantized low rank matrix that is to the problem of recovering a low rank matrix whose entries have been rounded up to some accuracy

computational complexity of real structured singular value in l sub p setting this paper studies a generalized real structured singular value spl mu problem where uncertain parameters are bounded by an l sub p norm two results are presented the first one shows that this generalized spl mu problem is np hard for any given rational number p spl isin 1 spl infin the np hardness holds as long as le the size of the largest repeated block exceeds one this result generalizes the known np hardness result for the conventional spl mu problem with p spl infin the second result which strengthens the first one considers the approximability problem of the generalized spl mu we show that the problem of obtaining an estimate for the generalized spl mu with some guaranteed bounds on the relative error remains to be np hard regardless how large this bound is

a dynamical trajectory based methodology for systematically computing multiple optimal solutions of general nonlinear programming problems in this paper a novel dynamical trajectory based methodology is developed for systematically computing multiple local optimal solutions of general nonlinear programming problems with disconnected feasible components satisfying nonlinear equality inequality constraints the proposed methodology deterministic in nature exploits trajectories of two different nonlinear dynamical systems to find multiple local optimal solutions the methodology consists of two phases phase i starts from an arbitrary infeasible initial point and finds systematically multiple or all the disconnected feasible components phase ii finds an adjacent local optimal solution from a local optimum via a decomposition point thereby systematically locating multiple local optimal solutions which lie within each feasible component found in phase i by alternating between these two phases the methodology locates multiple or all the local optimal solutions which lie in all the disconnected feasible components a theoretical foundation for the proposed methodology is also developed the methodology is illustrated with a numerical example with promising results

a storm of feasibility pumps for nonconvex minlp one of the foremost difficulties in solving mixed integer nonlinear programs either with exact or heuristic methods is to find a feasible point we address this issue with a new feasibility pump algorithm tailored for nonconvex mixed integer nonlinear programs feasibility pumps are algorithms that iterate between solving a continuous relaxation and a mixed integer relaxation of the original problems such approaches currently exist in the literature for mixed integer linear programs and convex mixed integer nonlinear programs both cases exhibit the distinctive property that the continuous relaxation can be solved in polynomial time in nonconvex mixed integer nonlinear programming such a property does not hold and therefore special care has to be exercised in order to allow feasibility pump algorithms to rely only on local optima of the continuous relaxation based on a new high level view of feasibility pump algorithms as a special case of the well known successive projection method we show that many possible different variants of the approach can be developed depending on how several different orthogonal implementation choices are taken a remarkable twist of feasibility pump algorithms is that unlike most previous successive projection methods from the literature projection is naturally taken in two different norms in the two different subproblems to cope with this issue while retaining the local convergence properties of standard successive projection methods we propose the introduction of appropriate norm constraints in the subproblems these actually seem to significantly improve the practical performance of the approach we present extensive computational results on the minlplib showing the effectiveness and efficiency of our algorithm

stepanov like pseudo almost automorphic mild solutions to nonautonomous evolution equations abstract we obtain new existence and uniqueness theorems of pseudo almost automorphic mild solutions to several nonautonomous evolution equations with stepanov like pseudo almost automorphic terms two explanation examples are given

new algorithms for generalized network flows this paper of which a preliminary version appeared in istcs 92 is concerned with generalized network flow problems in a generalized network each edgee u v has a positive flow multiplier a e associated with it the interpretation is that if a flow ofx e enters the edge at nodeu then a flow ofa e x e exits the edge atv the uncapacitated generalized transshipment problem ugt is defined on a generalized network where demands and supplies real numbers are associated with the vertices and costs real numbers are associated with the edges the goal is to find a flow such that the excess or deficit at each vertex equals the desired value of the supply or demand and the sum over the edges of the product of the cost and the flow is minimized adler and cosares operations research 39 1991 955 960 reduced the restricted uncapacitated generalized transshipment problem where only demand nodes are present to a system of linear inequalities with two variables per inequality the algorithms presented by the authors in siam journal on computing to appear result in a faster algorithm for restricted ugt generalized circulation is defined on a generalized network with demands at the nodes and capacity constraints on the edges i e upper bounds on the amount of flow the goal is to find a flow such that the excesses at the nodes are proportional to the demands and maximized we present a new algorithm that solves the capacitated generalized flow problem by iteratively solving instances of ugt the algorithm can be used to find an optimal flow or an approximation thereof when used to find a constant factor approximation the algorithm is not only more efficient than previous algorithms but also strongly polynomial it is believed to be the first strongly polynomial approximation algorithm for generalized circulation the existence of such an approximation algorithm is interesting since it is not known whether the exact problem has a strongly polynomial algorithm

polynomial algorithms for lp over a subring of the algebraic integers with applications to lp with circulant matrices we show that a modified variant of the interior point method can solve linear programs lps whose coefficients are real numbers from a subring of the algebraic integers by defining the encoding size of such numbers to be the bit size of the integers that represent them in the subring we prove the modified algorithm runs in time polynomial in the encoding size of the input coefficients the dimension of the problem and the order of the subring we then extend the tardos scheme to our case obtaining a running time which is independent of the objective and right hand side data as a consequence of these results we are able to show that lps with real circulant coefficient matrices can be solved in strongly polynomial time finally we show how the algorithm can be applied to lps whose coefficients belong to the extension of the integers by a fixed set of square roots

on the complexity of a practical interior point method the theory of self concordance in convex optimization has been used to analyze the complexity of interior point methods based on newton s method for large problems it may be impractical to use newton s method here we analyze a truncated newton method in which an approximation to the newton search direction is used in addition practical interior point methods often include enhancements such as extrapolation that are absent from the theoretical algorithms analyzed previously we derive theoretical results that apply to such an algorithm one similar to a sophisticated computer implementation of a barrier method the results for a single barrier subproblem are a satisfying extension of the results for newton s method when extrapolation is used in the overall barrier method however our results are more limited we indicate by both theoretical arguments and examples why more elaborate results may be difficult to obtain

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 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

deciding linear inequalities by computing loop residues v r pratt has shown that the real and integer feastbdlty of sets of linear mequallues of the form x y c can be decided quickly by examining the loops m certain graphs pratt s method is generahzed first to real feaslbdlty of mequahues m two variables and arbitrary coefficients and ultimately to real feaslbdlty of arbitrary sets of hnear mequahtles the method is well suited to apphcatlons m program verification

a unifying geometric solution framework and complexity analysis for variational inequalities in this paper we propose a concept of polynomiality for variational inequality problems and show how to find a near optimal solution of variational inequality problems in a polynomial number of iterations to establish this result we build upon insights from several algorithms for linear and nonlinear programs the ellipsoid algorithm the method of centers of gravity the method of inscribed ellipsoids and vaidya s algorithm to develop a unifying geometric framework for solving variational inequality problems the analysis rests upon the assumption of strong f monotonicity which is weaker than strict and strong monotonicity since linear programs satisfy this assumption the general framework applies to linear programs

a strongly polynomial algorithm for a special class of linear programs we extend the list of linear programming problems that are known to be solvable in strongly polynomial time to include a class of lps which contains special cases of the generalized transshipment problem the result is facilitated by exploiting some special properties associated with leontief substitution systems and observing that a feasible solution to the system ax b x 0 in which no variable appears in more than two equations can be found in strongly polynomial time for b belonging to some set

stepanov like almost automorphic solutions of nonautonomous semilinear evolution equations with delay abstract we consider the existence and uniqueness of a stepanov like almost automorphic solution to the nonautonomous semilinear evolution equations with a constant delay u t a t u t f t u t h t r where a t t r generates an exponentially stable evolution family u t s and f r x x satisfies a lipschitz condition with respect to the second argument

quadratic programming and combinatorial minimum weight product problems we present a fully polynomial time approximation scheme fptas for minimizing an objective a t x b t x under linear constraints a x d examples of such problems are combinatorial minimum weight product problems such as the following given a graph g v e and two edge weights a b e rightarrow mathbb r find an s t path p that minimizes a p b p the product of its edge weights relative to a and b

a strongly polynomial algorithm for generalized flow maximization a strongly polynomial algorithm is given for the generalized flow maximization problem it uses a new variant of the scaling technique called continuous scaling the main measure of progress is that within a strongly polynomial number of steps an arc can be identified that must be tight in every dual optimal solution and thus can be contracted as a consequence of the result we also obtain a strongly polynomial algorithm for the linear feasibility problem with at most two nonzero entries per column in the constraint matrix

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

tight bounds and 2 approximation algorithms for integer programs with two variables per inequality the problem of integer programming in bounded variables over constraints with no more than two variables in each constraint is np complete even when all variables are binary this paper deals with integer linear minimization problems inn variables subject tom linear constraints with at most two variables per inequality and with all variables bounded between 0 andu for such systems a 2 approximation algorithm is presented that runs in time o mnu2 log un2m so it is polynomial in the input size if the upper boundu is polynomially bounded the algorithm works by finding first a super optimal feasible solution that consists of integer multiples of 1 2 that solution gives a tight bound on the value of the minimum it furthermore has an identifiable subset of integer components that retain their value in an integer optimal solution of the problem these properties are a generalization of the properties of the vertex cover problem the algorithm described is in particular a 2 approximation algorithm for the problem of minimizing the total weight of true variables among all truth assignments to the 2 satisfiability problem

an improved column generation algorithm for minimum sum of squares clustering given a set of entities associated with points in euclidean space minimum sum of squares clustering mssc consists in partitioning this set into clusters such that the sum of squared distances from each point to the centroid of its cluster is minimized a column generation algorithm for mssc was given by du merle et al in siam journal scientific computing 21 1485 1505 the bottleneck of that algorithm is the resolution of the auxiliary problem of finding a column with negative reduced cost we propose a new way to solve this auxiliary problem based on geometric arguments this greatly improves the efficiency of the whole algorithm and leads to exact solution of instances with over 2 300 entities i e more than 10 times as much as previously done

a signal dependent time frequency representation fast algorithm for optimal kernel design a time frequency representation based on an optimal signal dependent kernel has been previously been proposed in an attempt to overcome one of the primary limitations of bilinear time frequency distributions that the best kernel and distribution depend on the signal to be analyzed the optimization formulation for the signal dependent kernel results in a linear program with a unique feature a tree structure that summarizes a set of constraints on the kernel the authors present a fast algorithm based on sorting to solve a special class of linear programs that includes the problem of interest for a kernel with q variables the running time of the algorithm is o q log q which is several orders of magnitude less than any other known method for solving this class of linear program this efficiency enables the computation of the signal dependent optimal kernel time frequency representation at a cost that is on the same order as a fixed kernel distribution an important property of the optimal kernel is that it takes on essentially only the values of 1 and 0

necessary and sufficient condition for local minima of a class of nonconvex quadratic programs the author 1992 1993 earlier studied the equivalence of a class of 0 1 quadratic programs and their relaxed problems thus a class of combinatorial optimization problems can be solved by solving a class of nonconvex quadratic programs in this paper a necessary and sufficient condition for local minima of this class of nonconvex quadratic programs is given this will be the foundation for study of algorithms

fast suboptimal predictive control with guaranteed stability efficient algorithms for constrained minimization of infinite horizon predictive control costs are presented control laws with guaranteed stability and asymptotic tracking are derived on the basis of appropriately chosen one dimensional and ellipsoidal constraint set approximations these achieve comparable performance with existing qp based control laws with significant reductions in computational burden

stepanov like pseudo almost periodic mild solutions to nonautonomous neutral partial evolution equations abstract we obtain new existence and uniqueness theorems of pseudo almost periodic mild solutions to nonautonomous neutral partial evolution equations d d t u t f t u t a t u t f t u t g t u t t r d d t u t f t b u t a t u t f t b u t g t c u t t r assuming that a t satisfy acquistapace terreni conditions the evolution family generated by a t has exponential dichotomy r 0 a is almost periodic b c are densely defined closed linear operators f g are lipschitz with respect to the second argument uniformly in the first argument f is pseudo almost periodic in the first argument g is stepanov like pseudo almost periodic in the first argument for p 1 and jointly continuous to illustrate our abstract results two examples are given

nonlinear kernel based statistical pattern analysis the eigenstructure of the second order statistics of a multivariate random population can be inferred from the matrix of pairwise combinations of inner products of the samples therefore it can be also efficiently obtained in the implicit high dimensional feature spaces defined by kernel functions we elaborate on this property to obtain general expressions for immediate derivation of nonlinear counterparts of a number of standard pattern analysis algorithms including principal component analysis data compression and denoising and fisher s discriminant the connection between kernel methods and nonparametric density estimation is also illustrated using these results we introduce the kernel version of mahalanobis distance which originates nonparametric models with unexpected and interesting properties and also propose a kernel version of the minimum squared error mse linear discriminant function this learning machine is particularly simple and includes a number of generalized linear models such as the potential functions method or the radial basis function rbf network our results shed some light on the relative merit of feature spaces and inductive bias in the remarkable generalization properties of the support vector machine svm although in most situations the svm obtains the lowest error rates exhaustive experiments with synthetic and natural data show that simple kernel machines based on pseudoinversion are competitive in problems with appreciable class overlapping

random walks totally unimodular matrices and a randomised dual simplex algorithm we discuss the application of random walks to generating a random basis of a totally unimodular matrix and to solving a linear program with such a constraint matrix we also derive polynomial upper bounds on the combinatorial diameter of an associated polyhedron

fast fair and efficient flows in networks we study the problem of minimizing the maximum latency of flows in networks with congestion we show that this problem is np hard even when all arc latency functions are linear and there is a single source and sink still an optimal flow and an equilibrium flow share a desirable property in this situation all flow carrying paths have the same length i e these solutions are fair which is in general not true for optimal flows in networks with nonlinear latency functions in addition the maximum latency of the nash equilibrium which can be computed efficiently is within a constant factor of that of an optimal solution that is the so called price of anarchy is bounded in contrast we present a family of instances with multiple sources and a single sink for which the price of anarchy is unbounded even in networks with linear latencies furthermore we show that an s t flow that is optimal with respect to the average latency objective is near optimal for the maximum latency objective and it is close to being fair conversely the average latency of a flow minimizing the maximum latency is also within a constant factor of that of a flow minimizing the average latency

almost automorphic solutions to abstract volterra equations on the line abstract given a l 1 r and a the generator of an l 1 integrable family of bounded and linear operators defined on a banach space x we prove the existence of an almost automorphic mild solution to the semilinear integral equation u t t a t s a u s f s u s d s for each f r x x s p almost automorphic in t uniformly in x x and satisfying diverse lipschitz type conditions for the scalar linear case we prove that a l 1 r completely monotonic is already sufficient

existence of pseudo almost automorphic solutions to some abstract differential equations with sp pseudo almost automorphic coefficients abstract this paper introduces and studies the notion of s p pseudo almost automorphy or stepanov like pseudo almost automorphy which generalizes in a natural fashion the concept of pseudo almost automorphy that was recently introduced in the literature by liang xiao and zhang we next obtain the existence of pseudo almost automorphic solutions to some differential equations with s p pseudo almost automorphic coefficients

generalized finite automata over real and complex numbers in a recent work gandhi khoussainov and liu 7 introduced and studied a generalized model of finite automata able to work over arbitrary structures as one relevant area of research for this model the authors identify studying such automata over particular structures such as real and algebraically closed fields in this paper we start investigations into this direction we prove several structural results about sets accepted by such automata and analyze decidability as well as complexity of several classical questions about automata in the new framework our results show quite a diverse picture when compared to the well known results for finite automata over finite alphabets

strongly polynomial algorithms for the quadratic transportation problem with a fixed number of sources the transportation problem with a linear objective function is known to be solvable in strongly polynomial time whereas instances with a convex nonquadratic objective function are not even for cases with the integrality constraints relaxed we present a linear time algorithm for the continuous quadratic transportation problem qtp with two source nodes further we show how problem instances with a fixed number of source or destination nodes k could be solved in strongly polynomial time onk 1 using an algebraic tree computation model the algorithms exploit a relationship between the transportation problem and the resource allocation problem the strong polynomiality of the algorithms presented here imply the existence of strongly polynomial algorithms for integer quadratic transportation problems with a fixed number of source nodes as well

complexity analysis of interior point algorithms for non lipschitz and nonconvex minimization we propose a first order interior point algorithm for a class of non lipschitz and nonconvex minimization problems with box constraints which arise from applications in variable selection and regularized optimization the objective functions of these problems are continuously differentiable typically at interior points of the feasible set our first order algorithm is easy to implement and the objective function value is reduced monotonically along the iteration points we show that the worst case iteration complexity for finding an epsilon scaled first order stationary point is o epsilon 2 o 2 furthermore we develop a second order interior point algorithm using the hessian matrix and solve a quadratic program with a ball constraint at each iteration although the second order interior point algorithm costs more computational time than that of the first order algorithm in each iteration its worst case iteration complexity for finding an epsilon scaled second order stationary point is reduced to o epsilon 3 2 o 3 2 note that an epsilon scaled second order stationary point must also be an epsilon scaled first order stationary point

improved algorithms for linear inequalities with two variables per inequality the authors show that a system of m linear inequalities with n variables where each inequality involves at most two variables can be solved in tilde o mn 2 time we denote tilde o f o f polylog n polylog m deterministically and in tilde o n 3 mn expected time using randomization parallel implementations of these algorithms run in tilde o n time where the deterministic algorithm uses tilde o mn processors and the randomized algorithm uses tilde o n 2 m processors the bounds significantly improve over previous algorithms the randomized algorithm is based on novel insights into the structure of the problem

a polyhedral study of nonconvex quadratic programs with box constraints by reformulating quadratic programs using necessary optimality conditions we obtain a linear program with complementarity constraints for the case where the only constraints are bounds on the variables we study a relaxation based on a subset of the optimality conditions by characterizing its convex hull we obtain a large class of valid inequalities these inequalities are used in a branch and cut scheme see 13

a note on the convergence of barrier algorithms to second order necessary points it has long been known that barrier algorithms for constrained optimization can produce a sequence of iterates converging to a critical point satisfying weak second order necessary optimality conditions when their inner iterations ensures that second order necessary conditions hold at each barrier minimizer we show that despite this strong second order necessary conditions may fail to be attained at the limit even if the barrier minimizers satisfy second order sufficient optimality conditions

ellipsoids and ellipsoid shaped fuzzy sets as natural multi variate generalization of intervals and fuzzy numbers how to elicit them from users and how to use them in data processing in this paper we show that ellipsoids are natural multi variate generalization of intervals and ellipsoid shaped fuzzy sets are a natural generalization of fuzzy numbers we explain how to elicit them from users and how to use them in data processing

continuous cubic formulations for cluster detection problems in networks the celebrated motzkin straus formulation for the maximum clique problem provides a nontrivial characterization of the clique number of a graph in terms of the maximum value of a nonconvex quadratic function over a standard simplex it was originally developed as a way of proving turan s theorem in graph theory but was later used to develop competitive algorithms for the maximum clique problem based on continuous optimization clique relaxations such as s defective clique and s plex emerged as attractive more practical alternatives to cliques in network based cluster detection models arising in numerous applications this paper establishes continuous cubic formulations for the maximum s defective clique problem and the maximum s plex problem by generalizing the motzkin straus formulation to the corresponding clique relaxations the formulations are used to extend turan s theorem and other known lower bounds on the clique number to the considered clique relaxations results of preliminary numerical experiments with the conopt solver demonstrate that the proposed formulations can be used to quickly compute high quality solutions for the maximum s defective clique problem and the maximum s plex problem the proposed formulations can also be used to generate interesting test instances for global optimization solvers

strongly polynomial time and nc algorithms for detecting cycles in periodic graphs this paper is concerned with the problem of recognizing in a graph with rational vector weights associates with the edges the existence of a cycle whose total weight is the zero vector this problem is known to be equivalent to the problem of recognizing the existence of cycles in periodic dynamic graphs and to the validity of systems of recursive formulas it was previously conjectured that combinatorial algorithms exist for the cases of two and three dimensional vector weights it is shown that strongly polynomial algorithms exist for any fixed dimension d moreover these algorithms also establish membership in the class nc on the other hand it is shown that when the dimension of the weights is not fixed the problem is equivalent to the general linear programming problem under strongly polynomial and logspace reductions

on the degree and separability of nonconvexity and applications to optimization problems we study qualitative indications for d c representations of closed sets in and functions on hilbert spaces the first indication is an index of nonconvexity which can be regarded as a measure for the degree of nonconvexity we show that a closed set is weakly closed if this indication is finite using this result we can prove the solvability of nonconvex minimization problems by duality a minimization problem on a feasible set in which this indication is low can be reduced to a quasi concave minimization over a convex set in a low dimensional space the second indication is the separability which can be incorporated in solving dual problems both the index of nonconvexity and the separability can be characteristics to good d c representations for practical computation we present a notion of clouds which enables us to obtain a good d c representation for a class of nonconvex sets using a generalized caratheodory s theorem we present various applications of clouds

on ladner s result for a class of real machines with restricted use of constants we study the question whether there are analogues of email protected s result in the computational model of blum shub and smale it is known that in the complex and the additive bss model a pure analogue holds i e there are non complete problems in email protected p assuming np p in the full real number model only a non uniform version is known we define a new variant which seems relatively close to the full real number model in this variant inputs can be treated as in the full model whereas real machine constants can be used in a restricted way only our main result shows that in this restricted model email protected s result holds our techniques analyze a class p const that has been known previously to be crucial for this kind of results by topological arguments relying on the polyhedral structure of certain sets of machine constants we show that this class coincides with the new restricted version of p r thus implying email protected s result

the d step conjecture and its relatives the d step conjecture arose from an attempt to understand the computational complexity of edge following algorithms for linear programming such as the simplex algorithm it can be stated in terms of diameters of graphs of convex polytopes in terms of the existence of nonrevisiting paths in such graphs in terms of an exchange process for simplicial bases of a vector space and in terms of matrix pivot operations first formulated by w m hirsch in 1957 the conjecture remains unsettled though it has been proved in many special cases and counterexamples have been found for slightly stronger conjectures if the conjecture is false as we believe to be the case then finding a counterexample will be merely a small first step in the line of investigation related to the conjecture this report summarizes what is known about the d step conjecture and its relatives a considerable amount of new material is included but it does not seem to come close to settling the conjecture of special interest is the first example of a polytope that is not vertex decomposable showing that a certain natural approach to the conjecture will not work also significant are the quantitative relations among the lengths of paths associated with various forms of the conjecture

nonconvex phase synchronization we estimate n phases angles from noisy pairwise relative phase measurements the task is modeled as a nonconvex least squares optimization problem it was recently shown that this problem can be solved in polynomial time via convex relaxation under some conditions on the noise in this paper under similar but more restrictive conditions we show that a modified version of the power method converges to the global optimum this is simpler and empirically faster than convex approaches empirically they both succeed in the same regime further analysis shows that in the same noise regime as previously studied second order necessary optimality conditions for this quadratically constrained quadratic program are also sufficient despite nonconvexity

pseudo almost periodicity of some nonautonomous evolution equations with delay abstract this paper is concerned with pseudo almost periodicity of the solutions to the nonautonomous evolution equation with delay u t a t u t f t u t h some sufficient conditions which ensure the existence and uniqueness of pseudo almost periodic mild solutions to the evolution equation with delay are given an example is shown to illustrate our results

on the optimal synthesis of sum or difference patterns of centrosymmetric arrays under arbitrary sidelobe constraints this paper deals with the optimal synthesis of sum or difference patterns radiated by an array of fixed geometry the aim is to establish the most general class of arrays for which such synthesis can be reduced to a linear programming problem extending and completing previous results regarding uniformly spaced linear and planar arrays it is shown that such a class consists of the centrosymmetric arrays whose element patterns are either equal or hermitian symmetric with respect to the center of symmetry the optimal solutions are also hermitian symmetric and the corresponding patterns are real furthermore it is shown that the above class of arrays coincides with that of the arrays capable of radiating real patterns some numerical examples are provided to give an idea of the geometries and the synthesis problems to which the achieved result successfully applies and of the achievable computational advantage