the fundamental group of a hopf linear category abstract we define the fundamental group of a hopf algebra over a field for this purpose we first consider gradings of hopf algebras and galois coverings the latter are given by linear categories with new additional structure which we call hopf linear categories over a finite group we compare this invariant to the fundamental group of the underlying linear category and we compute those groups for families of examples

an interior proximal method in vector optimization abstract this paper studies the vector optimization problem of finding weakly efficient points for maps from r n to r m with respect to the partial order induced by a closed convex and pointed cone c r m with nonempty interior we develop for this problem an extension of the proximal point method for scalar valued convex optimization problem with a modified convergence sensing condition that allows us to construct an interior proximal method for solving vop on nonpolyhedral set

bundle level type methods uniformly optimal for smooth and nonsmooth convex optimization the main goal of this paper is to develop uniformly optimal first order methods for convex programming cp by uniform optimality we mean that the first order methods themselves do not require the input of any problem parameters but can still achieve the best possible iteration complexity bounds by incorporating a multi step acceleration scheme into the well known bundle level method we develop an accelerated bundle level method and show that it can achieve the optimal complexity for solving a general class of black box cp problems without requiring the input of any smoothness information such as whether the problem is smooth nonsmooth or weakly smooth as well as the specific values of lipschitz constant and smoothness level we then develop a more practical restricted memory version of this method namely the accelerated prox level apl method we investigate the generalization of the apl method for solving certain composite cp problems and an important class of saddle point problems recently studied by nesterov math program 103 127 152 2005 we present promising numerical results for these new bundle level methods applied to solve certain classes of semidefinite programming and stochastic programming problems

from racks to pointed hopf algebras abstract a fundamental step in the classification of finite dimensional complex pointed hopf algebras is the determination of all finite dimensional nichols algebras of braided vector spaces arising from groups the most important class of braided vector spaces arising from groups is the class of braided vector spaces c x c q where x is a rack and q is a 2 cocycle on x with values in c racks and cohomology of racks appeared also in the work of topologists this leads us to the study of the structure of racks their cohomology groups and the corresponding nichols algebras we will show advances in these three directions we classify simple racks in group theoretical terms we describe projections of racks in terms of general cocycles we introduce a general cohomology theory of racks containing properly the existing ones we introduce a fourier transform on racks of certain type finally we compute some new examples of finite dimensional nichols algebras

new large rank nichols algebras over nonabelian groups with commutator subgroup z2 abstract in this article we explicitly construct new finite dimensional indecomposable nichols algebras with dynkin diagrams of type a n c n d n e 6 7 8 f 4 over any group g of order 2 l with commutator subgroup isomorphic to z 2 the construction is generic in the sense that the type just depends on the rank and center of g and thus positively answers for all groups of this class a question raised by susan montgomery in 1995 22 4 our construction uses the new notion of a covering nichols algebra as a special case of a covering hopf algebra 18 and produces non faithful nichols algebras we give faithful examples of doi twists for type a 3 c 3 d 4 c 4 f 4 over several nonabelian groups of order 16 and 32 these are hence the first known examples of nondiagonal finite dimensional indecomposable nichols algebras of rank 2 over nonabelian groups

smooth optimization approach for sparse covariance selection in this paper we first study a smooth optimization approach for solving a class of nonsmooth strictly concave maximization problems whose objective functions admit smooth convex minimization reformulations in particular we apply nesterov s smooth optimization technique y e nesterov dokl akad nauk sssr 269 1983 pp 543 547 y e nesterov math programming 103 2005 pp 127 152 to their dual counterparts that are smooth convex problems it is shown that the resulting approach has cal o 1 sqrt epsilon iteration complexity for finding an epsilon optimal solution to both primal and dual problems we then discuss the application of this approach to sparse covariance selection that is approximately solved as an l 1 norm penalized maximum likelihood estimation problem and also propose a variant of this approach which has substantially outperformed the latter one in our computational experiments we finally compare the performance of these approaches with other first order methods namely nesterov s cal o 1 epsilon smooth approximation scheme and block coordinate descent method studied in a d aspremont o banerjee and l el ghaoui siam j matrix anal appl 30 2008 pp 56 66 j friedman t hastie and r tibshirani biostatistics 9 2008 pp 432 441 for sparse covariance selection on a set of randomly generated instances it shows that our smooth optimization approach substantially outperforms the first method above and moreover its variant substantially outperforms both methods above

an optimal method for stochastic composite optimization this paper considers an important class of convex programming cp problems namely the stochastic composite optimization sco whose objective function is given by the summation of general nonsmooth and smooth stochastic components since sco covers non smooth smooth and stochastic cp as certain special cases a valid lower bound on the rate of convergence for solving these problems is known from the classic complexity theory of convex programming note however that the optimization algorithms that can achieve this lower bound had never been developed in this paper we show that the simple mirror descent stochastic approximation method exhibits the best known rate of convergence for solving these problems our major contribution is to introduce the accelerated stochastic approximation ac sa algorithm based on nesterov s optimal method for smooth cp nesterov in doklady an sssr 269 543 547 1983 nesterov in math program 103 127 152 2005 and show that the ac sa algorithm can achieve the aforementioned lower bound on the rate of convergence for sco to the best of our knowledge it is also the first universally optimal algorithm in the literature for solving non smooth smooth and stochastic cp problems we illustrate the significant advantages of the ac sa algorithm over existing methods in the context of solving a special but broad class of stochastic programming problems

stochastic block mirror descent methods for nonsmooth and stochastic optimization in this paper we present a new stochastic algorithm namely the stochastic block mirror descent sbmd method for solving large scale nonsmooth and stochastic optimization problems the basic idea of this algorithm is to incorporate block coordinate decomposition and an incremental block averaging scheme into the classic stochastic mirror descent method in order to significantly reduce the cost per iteration of the latter algorithm we establish the rate of convergence of the sbmd method along with its associated large deviation results for solving general nonsmooth and stochastic optimization problems we also introduce variants of this method and establish their rate of convergence for solving strongly convex smooth and composite optimization problems as well as certain nonconvex optimization problems to the best of our knowledge all these developments related to the sbmd methods are new in the stochastic optimization literature moreover some of our results seem to be new for block coordinate descent methods for deterministic optimization

interior proximal methods and central paths for convex second order cone programming we make a unified analysis of interior proximal methods of solving convex second order cone programming problems these methods use a proximal distance with respect to second order cones which can be produced with an appropriate closed proper univariate function in three ways under some mild conditions the sequence generated is bounded with each limit point being a solution and global rates of convergence estimates are obtained in terms of objective values a class of regularized proximal distances is also constructed which can guarantee the global convergence of the sequence to an optimal solution these results are illustrated with some examples in addition we also study the central paths associated with these distance like functions and for the linear socp we discuss their relations with the sequence generated by the interior proximal methods from this we obtain improved convergence results for the sequence for the interior proximal methods using a proximal distance continuous at the boundary of second order cones

a strictly contractive peaceman rachford splitting method with logarithmic quadratic proximal regularization for convex programming recently a strictly contractive peaceman rachford splitting method prsm was proposed for a separable convex minimization model whose variables are subject to some linear constraints and two additional generic constraints in general the strictly contractive prsm requires to solve two constrained minimization subproblems at each iteration in this paper we consider the case where the additional constraints on variables are positive orthants and apply the well developed logarithmic quadratic proximal lqp regularization to regularize the subproblems of the strictly contractive prsm a new algorithm in combination with the strictly contractive prsm and the lqp regularization is thus proposed the new algorithm only needs to solve two unconstrained subproblems at each iteration an inexact version allowing the unconstrained subproblems to be solved approximately subject to certain inexactness criterion is also studied we prove the global convergence and establish a worst case convergence rate measured

a cyclic block coordinate descent method with generalized gradient projections the aim of this paper is to present the convergence analysis of a very general class of gradient projection methods for smooth constrained possibly nonconvex optimization the key features of these methods are the armijo linesearch along a suitable descent direction and the non euclidean metric employed to compute the gradient projection we develop a very general framework from the point of view of block coordinate descent methods which are useful when the constraints are separable in our numerical experiments we consider a large scale image restoration problem to illustrate the impact of the metric choice on the practical performances of the corresponding algorithm

proving properties of continuous systems qualitative simulation and temporal logic we demonstrate an automated method for proving temporal logic statements about solutions to ordinary differential equations odes even in the face of an incomplete specification of the ode the method combines an implemented on the fly model checking algorithm for statements in the temporal logic ctl 3 7 8 with the output of the qualitative simulation algorithm qsim 13 16 based on the qsim guaranteed coverage theorem we prove that for certain ctl statements if is true for the temporal structure produced by qsim then a corresponding temporal statement holds for the solution of any ode consistent with the qualitative differential equation qde that qsim used to generate the temporal structure

the algorithmic analysis of hybrid systems we present a general framework for the formal specification and algorithmic analysis of hybrid systems a hybrid system consists of a discrete program with an analog environment we model hybrid systems as finite automata equipped with variables that evolve continuously with time according to dynamical laws for verification purposes we restrict ourselves to linear hybrid systems where all variables follow piecewise linear trajectories we provide decidability and undecidability results for classes of linear hybrid systems and we show that standard program analysis techniques can be adapted to linear hybrid systems in particular we consider symbolic model checking and minimization procedures that are based on the reachability analysis of an infinite state space the procedures iteratively compute state sets that are definable as unions of convex polyhedra in multidimensional real space we also present approximation techniques for dealing with systems for which the iterative procedures do not converge

primal dual first order methods with mathcal o 1 epsilon iteration complexity for cone programming in this paper we consider the general cone programming problem and propose primal dual convex smooth and or nonsmooth minimization reformulations for it we then discuss first order methods suitable for solving these reformulations namely nesterov s optimal method nesterov in doklady an sssr 269 543 547 1983 math program 103 127 152 2005 nesterov s smooth approximation scheme nesterov in math program 103 127 152 2005 and nemirovski s prox method nemirovski in siam j opt 15 229 251 2005 and propose a variant of nesterov s optimal method which has outperformed the latter one in our computational experiments we also derive iteration complexity bounds for these first order methods applied to the proposed primal dual reformulations of the cone programming problem the performance of these methods is then compared using a set of randomly generated linear programming and semidefinite programming instances we also compare the approach based on the variant of nesterov s optimal method with the low rank method proposed by burer and monteiro math program ser b 95 329 357 2003 math program 103 427 444 2005 for solving a set of randomly generated sdp instances

system identification via sparse multiple kernel based regularization using sequential convex optimization techniques model estimation and structure detection with short data records are two issues that receive increasing interests in system identification in this paper a multiple kernel based regularization method is proposed to handle those issues multiple kernels are conic combinations of fixed kernels suitable for impulse response estimation and equip the kernel based regularization method with three features first multiple kernels can better capture complicated dynamics than single kernels second the estimation of their weights by maximizing the marginal likelihood favors sparse optimal weights which enables this method to tackle various structure detection problems e g the sparse dynamic network identification and the segmentation of linear systems third the marginal likelihood maximization problem is a difference of convex programming problem it is thus possible to find a locally optimal solution efficiently by using a majorization minimization algorithm and an interior point method where the cost of a single interior point iteration grows linearly in the number of fixed kernels monte carlo simulations show that the locally optimal solutions lead to good performance for randomly generated starting points

a fully automated framework for control of linear systems from temporal logic specifications we consider the following problem given a linear system and a linear temporal logic ltl formula over a set of linear predicates in its state variables find a feedback control law with polyhedral bounds and a set of initial states so that all trajectories of the closed loop system satisfy the formula our solution to this problem consists of three main steps first we partition the state space in accordance with the predicates in the formula and construct a transition system over the partition quotient which captures our capability of designing controllers second using a procedure resembling model checking we determine runs of the transition system satisfying the formula third we generate the control strategy illustrative examples are included

optimal stochastic approximation algorithms for strongly convex stochastic composite optimization i a generic algorithmic framework in this paper we present a generic algorithmic framework namely the accelerated stochastic approximation ac sa algorithm for solving strongly convex stochastic composite optimization sco problems while the classical stochastic approximation algorithms are asymptotically optimal for solving differentiable and strongly convex problems the ac sa algorithm when employed with proper stepsize policies can achieve optimal or nearly optimal rates of convergence for solving different classes of sco problems during a given number of iterations moreover we investigate these ac sa algorithms in more detail such as by establishing the large deviation results associated with the convergence rates and introducing an efficient validation procedure to check the accuracy of the generated solutions

graphs with relations coverings and group graded algebras the paper studies the interrelationship between coverings of finite directed graphs and gradings of the path algebras associated to the directed graphs to include gradings of all basic finite dimensional algebras over an algebraically closed field a theory of coverings of graphs with relations is introduced the object of this paper is to relate group gradings on algebras to coverings of a graph which is associated to the algebra the linking of the theories allows one to relate purely algebraic questions to questions in algebraic topology group theory or combinatorics in the representation theory of artin algebras the association to each algebra of a finite directed graph called the quiver of the algebra has been a useful tool the reason that the quiver of such an algebra is of interest is that there is a natural definition of representations of the quiver so that the category of representations of the quiver satisfying certain relations is equivalent to the category of finitely generated modules over the algebra 1 gives a slight extension of these concepts to finitely generated algebras the main emphasis of the paper is to show that the theory of coverings of graphs with relations introduced by c riedtmann 9 and expanded by p gabriel 2 and the theory of group graded algebras are essentially the same although the original connection between coverings and gradings was inspired by the similarity of results for z graded artin algebras 3 4 and p gabriel s announced results 2 the context of this paper is more general and deals with all finitely generated algebras over a field we associate to each such algebra a finite directed graph which we still call a quiver of the algebra we show that for each regular covering t of a quiver t0 of an algebra a with certain prescribed restrictions we get a g grading of the algebra a where g is the automorphism group of the covering y over ro conversely given a certain type of g grading of a where g is a group we construct a regular covering y of the quiver t0 of the algebra such that g is isomorphic to the automorphism group of t over ro furthermore if y is a regular covering of t0 with automorphism group g we show that the category of representations of y satisfying a certain set of relations is equivalent to the category of finite dimensional graded g modules we

a simplified view of first order methods for optimization we discuss the foundational role of the proximal framework in the development and analysis of some iconic first order optimization algorithms with a focus on non euclidean proximal distances of bregman type which are central to the analysis of many other fundamental first order minimization relatives we stress simplification and unification by highlighting self contained elementary proof patterns to obtain convergence rate and global convergence both in the convex and the nonconvex settings which in turn also allows to present some novel results

finite dimensional pointed or copointed hopf algebras over affine racks abstract we study the pointed or copointed liftings of nichols algebras associated to affine racks and constant cocycles for any finite group admitting a principal yd realization of these racks in the copointed case we complete the classification for the six affine racks whose nichols algebra is known to be of finite dimension in the pointed case our method allows us to finish four of them in all of the cases the hopf algebras obtained turn out to be cocycle deformations of their associated graded hopf algebras all of them are new examples of finite dimensional copointed or pointed hopf algebras over non abelian groups

accelerated schemes for a class of variational inequalities we propose a novel stochastic method namely the stochastic accelerated mirror prox samp method for solving a class of monotone stochastic variational inequalities svi the main idea of the proposed algorithm is to incorporate a multi step acceleration scheme into the stochastic mirror prox method the developed samp method computes weak solutions with the optimal iteration complexity for svis in particular if the operator in svi consists of the stochastic gradient of a smooth function the iteration complexity of the samp method can be accelerated in terms of their dependence on the lipschitz constant of the smooth function for svis with bounded feasible sets the bound of the iteration complexity of the samp method depends on the diameter of the feasible set for unbounded svis we adopt the modified gap function introduced by monteiro and svaiter for solving monotone inclusion and show that the iteration complexity of the samp method depends on the distance from the initial point to the set of strong solutions it is worth noting that our study also significantly improves a few existing complexity results for solving deterministic variational inequality problems we demonstrate the advantages of the samp method over some existing algorithms through our preliminary numerical experiments

optimal primal dual methods for a class of saddle point problems we present a novel accelerated primal dual apd method for solving a class of deterministic and stochastic saddle point problems spps the basic idea of this algorithm is to incorporate a multistep acceleration scheme into the primal dual method without smoothing the objective function for deterministic spp the apd method achieves the same optimal rate of convergence as nesterov s smoothing technique our stochastic apd method exhibits an optimal rate of convergence for stochastic spp not only in terms of its dependence on the number of the iteration but also on a variety of problem parameters to the best of our knowledge this is the first time that such an optimal algorithm has been developed for stochastic spp in the literature furthermore for both deterministic and stochastic spp the developed apd algorithms can deal with the situation when the feasible region is unbounded as long as a saddle point exists in the unbounded case we incorporate the modified termination criterion introduced b

osga a fast subgradient algorithm with optimal complexity this paper presents an algorithm for approximately minimizing a convex function in simple not necessarily bounded convex finite dimensional domains assuming only that function values and subgradients are available no global information about the objective function is needed apart from a strong convexity parameter which can be put to zero if only convexity is known the worst case number of iterations needed to achieve a given accuracy is independent of the dimension and apart from a constant factor best possible under a variety of smoothness assumptions on the objective function

coverings of graded pointed hopf algebras we introduce the concept of a covering of a graded pointed hopf algebra the theory developed shows that coverings of a bosonized nichols algebra can be concretely expressed by biproducts using a quotient of the universal coalgebra covering group of the nichols algebra if there are enough quadratic relations the universal coalgebra covering is given by the bosonization by the enveloping group of the underlying rack

inexact proximal point methods for variational inequality problems we present a new family of proximal point methods for solving monotone variational inequalities our algorithm has a relative error tolerance criterion in solving the proximal subproblems our convergence analysis covers a wide family of regularization functions including double regularizations recently introduced by silva eckstein and humes jr siam j optim 12 2001 pp 238 261 and the bregman distance induced by h x sum i 1 n x i log x i we do not use in our analysis the assumption of paramonotonicity which is standard in proving convergence of bregman based proximal methods

projected subgradient methods with non euclidean distances for non differentiable convex minimization and variational inequalities we study subgradient projection type methods for solving non differentiable convex minimization problems and monotone variational inequalities the methods can be viewed as a natural extension of subgradient projection type algorithms and are based on using non euclidean projection like maps which generate interior trajectories the resulting algorithms are easy to implement and rely on a single projection per iteration we prove several convergence results and establish rate of convergence estimates under various and mild assumptions on the problem s data and the corresponding step sizes

optimal stochastic approximation algorithms for strongly convex stochastic composite optimization ii shrinking procedures and optimal algorithms in this paper we study new stochastic approximation sa type algorithms namely the accelerated sa ac sa for solving strongly convex stochastic composite optimization sco problems specifically by introducing a domain shrinking procedure we significantly improve the large deviation results associated with the convergence rate of a nearly optimal ac sa algorithm presented by ghadimi and lan in siam j optim 22 2012 pp 1469 1492 moreover we introduce a multistage ac sa algorithm which possesses an optimal rate of convergence for solving strongly convex sco problems in terms of the dependence on not only the target accuracy but also a number of problem parameters and the selection of initial points to the best of our knowledge this is the first time that such an optimal method has been presented in the literature from our computational results these ac sa algorithms can substantially outperform the classical sa and some other sa type algorithms for solving certain classes of strongly

an extension of proximal methods for quasiconvex minimization on the nonnegative orthant abstract in this paper we propose an extension of proximal methods to solve minimization problems with quasiconvex objective functions on the nonnegative orthant assuming that the function is bounded from below and lower semicontinuous and using a general proximal distance it is proved that the iterations given by our algorithm are well defined and stay in the positive orthant if the objective function is quasiconvex we obtain the convergence of the iterates to a certain set which contains the set of optimal solutions and convergence to a kkt point if the function is continuously differentiable and the proximal parameters are bounded furthermore we introduce a sufficient condition on the proximal distance such that the sequence converges to an optimal solution of the problem

mini batch stochastic approximation methods for nonconvex stochastic composite optimization this paper considers a class of constrained stochastic composite optimization problems whose objective function is given by the summation of a differentiable possibly nonconvex component together with a certain non differentiable but convex component in order to solve these problems we propose a randomized stochastic projected gradient rspg algorithm in which proper mini batch of samples are taken at each iteration depending on the total budget of stochastic samples allowed the rspg algorithm also employs a general distance function to allow taking advantage of the geometry of the feasible region complexity of this algorithm is established in a unified setting which shows nearly optimal complexity of the algorithm for convex stochastic programming a post optimization phase is also proposed to significantly reduce the variance of the solutions returned by the algorithm in addition based on the rspg algorithm a stochastic gradient free algorithm which only uses the stochastic zeroth order information has been also discussed some preliminary numerical results are also provided

a descent lemma beyond lipschitz gradient continuity first order methods revisited and applications the proximal gradient and its variants is one of the most attractive first order algorithm for minimizing the sum of two convex functions with one being nonsmooth however it requires the differentiable part of the objective to have a lipschitz continuous gradient thus precluding its use in many applications in this paper we introduce a framework which allows to circumvent the intricate question of lipschitz continuity of gradients by using an elegant and easy to check convexity condition which captures the geometry of the constraints this condition translates into a new descent lemma which in turn leads to a natural derivation of the proximal gradient scheme with bregman distances we then identify a new notion of asymmetry measure for bregman distances which is central in determining the relevant step size these novelties allow to prove a global sublinear rate of convergence and as a by product global pointwise convergence is obtained this provides a new path to a broad spectrum of problems arising in key applications which were until now considered as out of reach via proximal gradient methods we illustrate this potential by showing how our results can be applied to build new and simple schemes for poisson inverse problems

accelerating variance reduced stochastic gradient methods variance reduction is a crucial tool for improving the slow convergence of stochastic gradient descent only a few variance reduced methods however have yet been shown to directly benefit from nesterov s acceleration techniques to match the convergence rates of accelerated gradient methods such approaches rely on negative momentum a technique for further variance reduction that is generally specific to the svrg gradient estimator in this work we show for the first time that negative momentum is unnecessary for acceleration and develop a universal acceleration framework that allows all popular variance reduced methods to achieve accelerated convergence rates the constants appearing in these rates including their dependence on the number of functions n scale with the mean squared error and bias of the gradient estimator in a series of numerical experiments we demonstrate that versions of saga svrg sarah and sarge using our framework significantly outperform non accelerated versions and compare favourably with algorithms using negative momentum

interior gradient and proximal methods for convex and conic optimization interior gradient subgradient and proximal methods for convex constrained minimization have been much studied in particular for optimization problems over the nonnegative octant these methods are using non euclidean projections and proximal distance functions to exploit the geometry of the constraints in this paper we identify a simple mechanism that allows us to derive global convergence results of the produced iterates as well as improved global rates of convergence estimates for a wide class of such methods and with more general convex constraints our results are illustrated with many applications and examples including some new explicit and simple algorithms for conic optimization problems in particular we derive a class of interior gradient algorithms which exhibits an o k 2 global convergence rate estimate

a unified continuous optimization framework for center based clustering methods center based partitioning clustering algorithms rely on minimizing an appropriately formulated objective function and different formulations suggest different possible algorithms in this paper we start with the standard nonconvex and nonsmooth formulation of the partitioning clustering problem we demonstrate that within this elementary formulation convex analysis tools and optimization theory provide a unifying language and framework to design analyze and extend hard and soft center based clustering algorithms through a generic algorithm which retains the computational simplicity of the popular k means scheme we show that several well known and more recent center based clustering algorithms which have been derived either heuristically or and have emerged from intuitive analogies in physics statistical techniques and information theoretic perspectives can be recovered as special cases of the proposed analysis and we streamline their relationships

bisimilar linear systems the notion of bisimulation in theoretical computer science is one of the main complexity reduction methods for the analysis and synthesis of labeled transition systems bisimulations are special quotients of the state space that preserve many important properties expressible in temporal logics and in particular reachability in this paper the framework of bisimilar transition systems is applied to various transition systems that are generated by linear control systems given a discrete time or continuous time linear system and a finite observation map we characterize linear quotient maps that result in quotient transition systems that are bisimilar to the original system interestingly the characterizations for discrete time systems are more restrictive than for continuous time systems due to the existence of an atomic time step we show that computing the coarsest bisimulation which results in maximum complexity reduction corresponds to computing the maximal controlled or reachability invariant subspace inside the kernel of the observations map these results establish strong connections between complexity reduction concepts in control theory and computer science

galois coverings of pointed coalgebras we introduce the concept of a galois covering of a pointed coalgebra the theory developed shows that galois coverings of pointed coalgebras can be concretely expressed by smash coproducts using the coaction of the automorphism group of the covering thus the theory of galois coverings is seen to be equivalent to group gradings of coalgebras an advantageous feature of the coalgebra theory is that neither the grading group nor the quiver is assumed finite in order to obtain a smash product coalgebra

nichols algebras of group type with many quadratic relations abstract we classify nichols algebras of irreducible yetter drinfeld modules over nonabelian groups satisfying an inequality for the dimension of the homogeneous subspace of degree two all such nichols algebras are finite dimensional and all known finite dimensional nichols algebras of nonabelian group type appear in the result of our classification we find a new finite dimensional nichols algebra over fields of characteristic two

an optimal algorithm for constrained differentiable convex optimization we describe two algorithms for solving differentiable convex optimization problems constrained to simple sets in mathbb r n i e sets on which it is easy to project an arbitrary point the algorithms are optimal in the sense that they achieve an absolute precision of varepsilon in relation to the optimal value in o 1 sqrt varepsilon iterations using only first order information this complexity depends on an unknown lipschitz constant l for the function derivatives and on a known strong convexity constant mu geq 0 the algorithms are extensions of well known methods devised by nesterov introductory lectures on convex optimization kluwer academic boston 2004 without the need for estimates of l and including in the second algorithm line searches and an adaptive procedure for estimating a strong convexity constant the algorithms are such that all computed points are feasible and the complexity analysis follows a simple geometric approach numerical tests for

a very simple sqcqp method for a class of smooth convex constrained minimization problems with nice convergence results we introduce a new and very simple algorithm for a class of smooth convex constrained minimization problems which is an iterative scheme related to sequential quadratically constrained quadratic programming methods called sequential simple quadratic method ssqm the computational simplicity of ssqm which uses first order information makes it suitable for large scale problems theoretical results under standard assumptions are given proving that the whole sequence built by the algorithm converges to a solution and becomes feasible after a finite number of iterations when in addition the objective function is strongly convex then asymptotic linear rate of convergence is established

an inexact proximal method for quasiconvex minimization abstract in this paper we propose an inexact proximal point method to solve constrained minimization problems with locally lipschitz quasiconvex objective functions assuming that the function is also bounded from below lower semicontinuous and using proximal distances we show that the sequence generated for the method converges to a stationary point of the problem