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

spectra and eigenvectors of scale free networks we study the spectra and eigenvectors of the adjacency matrices of scale free networks when bidirectional interaction is allowed so that the adjacency matrix is real and symmetric the spectral density shows an exponential decay around the center followed by power law long tails at both spectrum edges the largest eigenvalue ensuremath lambda 1 depends on system size n as ensuremath lambda 1 ensuremath sim n 1 4 for large n and the corresponding eigenfunction is strongly localized at the hub the vertex with largest degree the component of the normalized eigenfunction at the hub is of order unity we also find that the mass gap scales as n ensuremath 0 68

on the quadratic eigenvalue complementarity problem over a general convex cone the solution of the conic quadratic eigenvalue complementarity problem cqeicp is first investigated without assuming symmetry on the matrices defining the problem a new sufficient condition for existence of solutions of cqeicp is presented extending to arbitrary pointed closed and convex cones a condition known to hold when the cone is the nonnegative orthant we also address the symmetric cqeicp where all its defining matrices are symmetric we show that assuming that two of its defining matrices are positive definite this symmetric cqeicp reduces to the computation of a stationary point of an appropriate merit function on a convex set furthermore we discuss the use of the so called spectral projected gradient spg algorithm for solving cqeicp when the cone of interest is the second order cone socqeicp a new algorithm is designed for the computation of the projections required by the spg method to deal with socqeicp numerical results are included to illustrate the efficiency of the spg method and the new projection technique in practice

decreasing the spectral radius of a graph by link removals the decrease of the spectral radius an important characterizer of network dynamics by removing links is investigated the minimization of the spectral radius by removing m links is shown to be an np complete problem which suggests considering heuristic strategies several greedy strategies are compared and several bounds on the decrease of the spectral radius are derived the strategy that removes that link l i j with largest product x1 i x1 j of the components of the eigenvector x1 belonging to the largest adjacency eigenvalue is shown to be superior to other strategies in most cases furthermore a scaling law where the decrease in spectral radius is inversely proportional to the number of nodes n in the graph is deduced another sublinear scaling law of the decrease in spectral radius versus the number m of removed links is conjectured

controllability and linearized regulation a nonlinear controllable plant under mild technical conditions admits a precompensator with the following property along control trajectories joining pairs of states the composite system precompensator plus plant is up to first order isomorphic to a parallel connection of integrators

epidemic threshold in directed networks epidemics have so far been mostly studied in undirected networks however many real world networks such as the online social network twitter and the world wide web on which information emotion or malware spreads are directed networks composed of both unidirectional links and bidirectional links we define the directionality as the percentage of unidirectional links the epidemic threshold c for the susceptible infected susceptible sis epidemic is lower bounded by 1 1 in directed networks where 1 also called the spectral radius is the largest eigenvalue of the adjacency matrix in this work we propose two algorithms to generate directed networks with a given directionality the effect of on the spectral radius 1 principal eigenvector x 1 spectral gap 1 2 and algebraic connectivity n 1 is studied important findings are that the spectral radius 1 decreases with the directionality whereas the spectral gap and the algebraic connectivity increase with the directionality the extent of the decrease of the spectral radius depends on both the degree distribution and the degree degree correlation d hence in directed networks the epidemic threshold is larger and a random walk converges to its steady state faster than that in undirected networks with the same degree distribution

complementary eigenvalues of graphs abstract in this paper we study the eigenvalue complementarity problem eicp when its matrix a belongs to the class s g a a i j a i j a j i 0 iff i j e where g v e is a connected graph it is shown that if all nondiagonal elements of a s g are nonpositive then a has a unique complementary eigenvalue which is the smallest eigenvalue of a in particular zero is the unique complementary eigenvalue of the laplacian and the normalized laplacian matrices of a connected graph the number c g of complementary eigenvalues of the adjacency matrix of a connected graph g is shown to be bounded above by the number b g of induced nonisomorphic connected subgraphs of g furthermore c g b g if the perron roots of the adjacency matrices of these subgraphs are all distinct finally the maximum number of complementary eigenvalues for the adjacency matrices of graphs is shown to grow faster than any polynomial on the number of vertices

spectral analysis of coupled linear complementarity problems abstract this note deals with the so called cone constrained bivariate eigenvalue problem the equilibrium model under consideration is a system of linear complementarity problems p x ax by x p q y cx dy y q involving two closed convex cones and their corresponding duals we study the set of pairs r 2 for which this system has a nontrivial solution x y r n m we discuss also the link between the cone constrained version and the unconstrained one

regular graphs with four eigenvalues we study the connected regular graphs with four distinct eigenvalues properties and feasibility conditions of the eigenvalues are found several examples constructions and characterizations are given as well as some uniqueness and nonexistence results

on the reduced signless laplacian spectrum of a degree maximal graph abstract for a simple graph g the signless laplacian of g is the matrix a g d g where a g is the adjacency matrix and d g is the diagonal matrix of vertex degrees of g the reduced signless laplacian of g is the matrix g b g where b g is the reduced adjacency matrix of g and g is the diagonal matrix whose diagonal entries are the common degrees for vertices belonging to the same neighborhood equivalence class of g a graph is said to be degree maximal if it is connected and its degree sequence is not majorized by the degree sequence of any other connected graph for a maximal graph we obtain a formula for the characteristic polynomial of its reduced signless laplacian and use the formula to derive a localization result for its reduced signless laplacian eigenvalues and to compare the signless laplacian spectral radii of two well known maximal graphs we also obtain a necessary condition for a maximal graph to have maximal signless laplacian spectral radius among all connected graphs with given numbers of vertices and edges

on the characterizing properties of the permanental polynomials of graphs abstract let g be a graph and a g the adjacency matrix of g the permanental polynomial of g is defined as g x per x i a g if two graphs g and h have the same permanental polynomial then g is called a per cospectral mate of h a graph g is said to be characterized by its permanental polynomial if all the per cospectral mates of g are isomorphic to g it is shown that complete graphs stars regular complete bipartite graphs and odd cycles are characterized by their permanental polynomials we prove that in general the permanental polynomial cannot characterize the paths and even cycles in particular for each l 1 and m 2 we can find non isomorphic per cospectral mates of p 4 l 3 and c 4 m respectively when we restrict our consideration to connected graphs both the paths and even cycles c 4 l 2 are characterized by their permanental polynomials

on spectral maps induced by convex cones abstract let l e be the vector space of linear endomorphisms on a euclidean space e and k be a closed convex cone in e the spectral map k l e 2 r induced by k is defined as follows to each linear endomorphism a on e it associates the set k a of all r for which the complementarity problem k x a x x k has a nonzero solution x e here k is the dual cone of k and stands for orthogonality the set k a is known as the k spectrum of a the purpose of this work is twofold firstly we wish to compare algebraic properties of k and geometric properties of k secondly we wish to identify classes of linear transformations f l e l e that leave the k spectrum invariant i e such that k f a k a for all a l e the set of linear transformations preserving the k spectrum depends of course on the cone k under consideration

on least eigenvalues and least eigenvectors of real symmetric matrices and graphs abstract we give an upper bound for the least eigenvalue of a principal submatrix of a real symmetric matrix with zero diagonal from which we establish an upper bound for the least eigenvalue of a graph when some vertices are removed using the components of the least eigenvector s we give lower and upper bounds for the least eigenvalue of a graph when some edges are removed we also establish bounds for the components of the least eigenvector s of a real symmetric matrix and a graph

on the finiteness of the cone spectrum of certain linear transformations on euclidean jordan algebras abstract let l be a linear transformation on a finite dimensional real hilbert space h and k be a closed convex cone with dual k in h the cone spectrum of l relative to k is the set of all real for which the linear complementarity problem x k y l x x k and x y 0 admits a nonzero solution x in the setting of a euclidean jordan algebra h and the corresponding symmetric cone k we discuss the finiteness of the cone spectrum for z transformations and quadratic representations on h

a new method for linearizing non linear systems the pseudolinearization consider the single input non linear system x f x u it has been shown that under mild conditions there exist transformations z t x v s x u such that in the z space the linear tangent model is independent of the operating point a constructive procedure to obtain the above transformations is described this new method is computationally attractive and effective in many practical cases

the complement of the path is determined by its spectrum abstract it is proved that a graph whose 0 1 adjacency matrix has the spectrum of p n the complement of the path on n vertices must be p n

copositivity and constrained fractional quadratic problems we provide completely positive and copositive optimization formulations for the constrained fractional quadratic problem cfqp and standard fractional quadratic problem stfqp based on these formulations semidefinite programming relaxations are derived for finding good lower bounds to these fractional programs which can be used in a global optimization branch and bound approach applications of the cfqp and stfqp related with the correction of infeasible linear systems and eigenvalue complementarity problems are also discussed

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

distance regularity and the spectrum of graphs abstract we deal with the question can one see from the spectrum of a graph whether it is distance regular or not up till now the answer has not been known when has precisely four distinct eigenvalues the diameter 3 case we show that in this case the answer is negative we also give positive answers in some special situations for instance if has the spectrum of a distance regular graph with diameter 3 and 1 then is distance regular our main tools are eigenvalue techniques for partitioned matrices

constructing trees with given eigenvalues and angles abstract let eigenvalues and angles between eigensubspaces and the axes of the corresponding real vector space be given for a tree an algorithm for constructing all trees with given eigenvalues and angles is described

control of systems without drift via generic loops this paper proposes a simple numerical technique for the steering of arbitrary analytic systems with no drift it is based on the generation of nonsingular loops which allow linearized controllability along suitable trajectories once such loops are available it is possible to employ standard newton or steepest descent methods as classically done in numerical control the theoretical justification of the approach relies on results establishing the genericity of nonsingular controls as well as a simple convergence lemma

solving inverse cone constrained eigenvalue problems we compare various algorithms for constructing a matrix of order n whose pareto spectrum contains a prescribed set lambda lambda 1 ldots lambda p of reals in order to avoid overdetermination one assumes that p does not exceed n 2 the inverse pareto eigenvalue problem under consideration is formulated as an underdetermined system of nonlinear equations we also address the issue of computing lorentz spectra and solving inverse lorentz eigenvalue problems

a variational approach to copositive matrices this work surveys essential properties of the so called copositive matrices the study of which has been spread over more than fifty five years special emphasis is given to variational aspects related to the concept of copositivity in addition some new results on the geometry of the cone of copositive matrices are presented here for the first time

complementarity eigenvalue analysis of connected graphs abstract this work concerns the spectral analysis of connected graphs from a non traditional point of view instead of the usual eigenvalues of the adjacency matrix a g of a graph g under consideration we compute and analyze the complementarity eigenvalues of a g the complementarity eigenvalues of a general square matrix are defined in terms of a certain complementarity system relative to the componentwise ordering the complementarity eigenvalues of a g form the so called complementarity spectrum of g in general the structure of a connected graph is better discriminated in terms of its complementarity spectrum than in terms of its usual spectrum this observation is one of the leading motivation behind our work

feedback control of nonlinear systems by extended linearization for single input multiple output nonlinear systems we consider a design method based on the family of linearizations of the system parameterized by constant operating points nonlinear state feedback control laws and observer state feedback control laws are designed such that the eigenvalues of the family of linearized closed loop systems are placed at specified values that are locally invariant with respect to the closed loop operating point the method is illustrated by application to the problem of automatically balancing an inverted pendulum

a block active set algorithm with spectral choice line search for the symmetric eigenvalue complementarity problem in this paper we address the solution of the symmetric eigenvalue complementarity problem eicp by treating an equivalent reformulation of finding a stationary point of a fractional quadratic program on the unit simplex the spectral projected gradient spg method has been recommended to this optimization problem when the dimension of the symmetric eicp is large and the accuracy of the solution is not a very important issue we suggest a new algorithm which combines elements from the spg method and the block active set method where the latter was originally designed for box constrained quadratic programs in the new algorithm the projection onto the unit simplex in the spg method is replaced by the much cheaper projection onto a box this can be of particular advantage for large and sparse symmetric eicps global convergence to a solution of the symmetric eicp is established computational experience with medium and large symmetric eicps is reported to illustrate the efficacy and efficiency of the new algorithm

bounds for the spectral radius of a graph when nodes are removed abstract we present a new type of lower bound for the spectral radius of a graph in which m nodes are removed as a corollary cioab s theorem 4 which states that the maximum normalized principal eigenvector component in any graph never exceeds 1 2 with equality for the star appears as a special case of our more general result

universal nonsingular controls abstract in this note we prove that for analytical systems satisfying the strong accessibility rank condition generic inputs produce trajectories along which the linearized system is controllable applications to the steering of systems without drift are briefly mentioned

tensor eigenvalue complementarity problems this paper studies tensor eigenvalue complementarity problems basic properties of standard and complementarity tensor eigenvalues are discussed we formulate tensor eigenvalue complementarity problems as constrained polynomial optimization when one tensor is strictly copositive the complementarity eigenvalues can be computed by solving polynomial optimization with normalization by strict copositivity when no tensor is strictly copositive we formulate the tensor eigenvalue complementarity problem equivalently as polynomial optimization by a randomization process the complementarity eigenvalues can be computed sequentially the formulated polynomial optimization can be solved by lasserre s hierarchy of semidefinite relaxations we show that it has finite convergence for generic tensors numerical experiments are presented to show the efficiency of proposed methods

a transfer function approach to linear time varying discrete time systems in the first part of the paper a transfer function approach is developed for the class of linear time varying discrete time systems the theory is specified in terms of skew noncommutative rings of polynomials and formal power series both with coefficients in a ring of time functions the transfer function matrix is defined to be a matrix whose entries belong to a skew ring of formal power series it is shown that various system properties such as asymptotic stability can be characterized in terms of the skew ring framework in the last part of the paper the transfer function framework is applied to the study of feedback control new results are obtained on assignability of system dynamics by using dynamic output feedback and dynamic state feedback the results are applied to the control of an armature controlled do motor with a variable loading

a conjecture on the spectral radius of graphs abstract let g be a simple graph of order n and also let g be the spectral radius of the graph g in this paper we prove that g 2 g v k 2 d k 1 for any non isolated vertex v k of degree d k in g thus confirming a conjecture posed by guo et al 2019 3 moreover we characterized all connected graphs for which this bound is attained

crc handbook of combinatorial designs balanced incomplete block designs and t designs2 v k l designs of small orderbibds with small block sizet designs t 3steiner systemssymmetric designsresolvable and near resolvable designslatin squares mols and orthogonal arrayslatin squaresmutually orthogonal latin squares mols incomplete molsorthogonal arrays of index more than oneorthogonal arrays of strength more than twopairwise balanced designspbds and gdds the basicspbds recursive constructionspbd closurepairwise balanced designs as linear spacespbds and gdds of higher indexpbds frames and resolvabilityother combinatorial designsassociation schemesbalanced part ternary designsbalanced tournament designsbhaskar rao designscomplete mappings and sequencings of finite groupsconfigurationscostas arrayscoveringscycle systemsdifference familiesdifference matricesdifference sets abeliandifference sets nonabeliandifference triangle setsdirected designsd optimal matricesembedding partial quasigroupsequidistant permutation arraysfactorial designsfrequency squaresgeneralized quadranglesgraph decompositions and designsgraphical designshadamard matrices and designshall triple systemshowell designsmaximal sets of molsmendelsohn designsthe oberwolfach problemordered designs and perpendicular arraysorthogonal designsorthogonal main effect planspackingspartial geometriespartially balanced incomplete block designsquasigroupsquasi symmetric designs r l designsroom squaresself orthogonal latin squares sols sols with a symmetric orthogonal mate solssom sequences with zero autocorrelationskolem sequencesspherical t designsstarterstrades and defining sets t m s netstuscan squarest wise balanced designsuniformly resolvable designsvector space designsweighing matrices and conference matriceswhist tournamentsyouden designs generalizedyouden squaresapplicationscodescomputer science selected applicationsapplications of designs to cryptographyderandomizationoptimality and efficiency comparing block designsgroup testingscheduling a tournamentwinning the lotteryrelated mathematics and computational methodsfinite groups and designsnumber theory and finite fieldsgraphs and multigraphsfactorizations of graphsstrongly regular graphstwo graphsclassical geometriesprojective planes nondesarguesiancomputational methods in design theoryindex

which graphs are determined by their spectrum for almost all graphs the answer to the question in the title is still unknown here we survey the cases for which the answer is known not only the adjacency matrix but also other types of matrices such as the laplacian matrix are considered

the spectral radius of irregular graphs abstract let 1 be the largest eigenvalue and n the least eigenvalue of the adjacency matrix of a connected graph g of order n we prove that if g is irregular with diameter d maximum degree minimum degree and average degree d then 1 n d 1 d d 2 1 the inequality improves previous bounds of various authors and implies two lower bounds on n which improve previous bounds of nikiforov it also gives some fine tuning of a result of alon and sudakov a similar inequality is also obtained for the laplacian spectral radius of a connected irregular graph

new results on pareto spectra abstract a pareto eigenvalue of a matrix a of order n is a scalar r for which the complementarity problem 0 x a x x 0 admits a nonzero solution x r n pareto eigenvalues are also known as complementarity eigenvalues they have found applications in graph theory cone constrained dynamical systems and mathematical modeling in general in this paper we continue our study on theoretical properties of pareto spectra special attention is paid to classification of pareto eigenvalues and cardinality issues

inverse eigenvalue problems for linear complementarity systems abstract traditionally an inverse eigenvalue problem is about reconstructing a matrix from a given spectral data in this work we study the set of real matrices a of order n such that the linear complementarity system x 0 ax x 0 x ax x 0 holds for prescribed pairs x 1 1 x p p the analysis of this new type of inverse eigenvalue problem differs substantially from the classical one