error bounds for nondegenerate monotone linear complementarity problems error bounds and upper lipschitz continuity results are given for monotone linear complementarity problems with a nondegenerate solution the existence of a nondegenerate solution considerably simplifies the error bounds compared with problems for which all solutions are degenerate thus when a point satisfies the linear inequalities of a nondegenerate complementarity problem the residual that bounds the distance from a solution point consists of the complementarity condition alone whereas for degenerate problems this residual cannot bound the distance to a solution without adding the square root of the complementarity condition to it this and other simplified results are a consequence of the polyhedral characterization of the solution set as the intersection of the feasible region z mz q 0 z 0 with a single linear affine inequality constraint

the classification of n dimensional non lie malcev algebras with n 4 dimensional annihilator abstract in this paper we give a complete classification of all n dimensional non lie malcev algebras with n 4 dimensional annihilator over an algebraically closed field of characteristic 0 we also show that such algebras are special

arbitrary algebras with a multiplicative basis abstract let a be an algebra of arbitrary dimension over an arbitrary base field f and in which any identity on the product is not supposed a basis b e i i i of a is called multiplicative if for any i j i we have that e i e j f e k for some k i we show that if a admits a multiplicative basis then it decomposes as the direct sum a k i k of well described ideals admitting each one a multiplicative basis also the minimality of a is characterized in terms of the multiplicative basis and it is shown that under a mild condition the above direct sum is by means of the family of its minimal ideals admitting a multiplicative basis

the classification of 4 dimensional leibniz algebras abstract this paper is a contribution to the development of the non associative algebras theory more precisely this work deals with the classification of the complex 4 dimensional leibniz algebras note that the classification of 4 dimensional nilpotent complex leibniz algebras was obtained in albeverio et al 2006 1 therefore we will only consider non nilpotent case in this work

on finite convergence of proximal point algorithms for variational inequalities in this paper we first characterize finite convergence of an arbitrary iterative algorithm for solving the variational inequality problem vip where the finite convergence means that the algorithm can find an exact solution of the problem in a finite number of iterations by using this result we obtain that the well known proximal point algorithm possesses finite convergence if the solution set of vip is weakly sharp as an extension we show finite convergence of the inertial proximal method for solving the general variational inequality problem under the condition of weak g sharpness

interior point methods for massive support vector machines we investigate the use of interior point methods for solving quadratic programming problems with a small number of linear constraints where the quadratic term consists of a low rank update to a positive semidefinite matrix several formulations of the support vector machine fit into this category an interesting feature of these particular problems is the volume of data which can lead to quadratic programs with between 10 and 100 million variables and if written explicitly a dense q matrix our code is based on ooqp an object oriented interior point code with the linear algebra specialized for the support vector machine application for the targeted massive problems all of the data is stored out of core and we overlap computation and input output to reduce overhead results are reported for several linear support vector machine formulations demonstrating that the method is reliable and scalable

partial proximal minimization algorithms for convex pprogramming an extension of the proximal minimization algorithm is considered where only some of the minimization variables appear in the quadratic proximal term the resulting iterates are interpreted in terms of the iterates of the standard algorithm and a uniform descent property is shown that holds independently of the proximal terms used this property is used to give simple convergence proofs of parallel algorithms where multiple processors simultaneously execute proximal iterations using different partial proximal terms it is also shown that partial proximal minimization algorithms are dual to multiplier methods with partial elimination of constraints and a relation is established between parallel proximal minimization algorithms and parallel constraint distribution algorithms

solvable leibniz algebras with triangular nilradicals abstract in this paper the description of solvable lie algebras with triangular nilradicals is extended to leibniz algebras it is proven that the matrices of the left and the right operators on the elements of leibniz algebra have the upper triangular forms we establish that solvable leibniz algebra of a maximal possible dimension with a given triangular nilradical is a lie algebra furthermore solvable leibniz algebras with triangular nilradicals of the low dimensions are classified

weak sharp minima for convex infinite optimization problems in normed linear spaces in the present paper we study the characterization issue of the weak sharp minima properties for convex infinite optimization problems in normed linear spaces we develop a new approach to establi

finite termination of the proximal point algorithm in banach spaces abstract in this paper we show that the convex optimization problem can be solved by the proximal point algorithm in a finite number of steps under the assumption that the solution set is a set of weak sharp minima

classification of solvable leibniz algebras with naturally graded filiform nilradical in this paper we show that the method for describing solvable lie algebras with given nilradical by means of non nilpotent outer derivations of the nilradical is also applicable to the case of leibniz algebras using this method we extend the classification of solvable lie algebras with naturally graded filiform lie algebra to the case of leibniz algebras namely the classification of solvable leibniz algebras whose nilradical is a naturally graded filiform leibniz algebra is obtained

weak sharp solutions for variational inequalities in banach spaces abstract in this paper we introduce the notion of a weak sharp set of solutions to a variational inequality problem vip in a reflexive strictly convex and smooth banach space and present its several equivalent conditions we also prove under some continuity and monotonicity assumptions that if any sequence generated by an algorithm for solving vip converges to a weak sharp solution then we can obtain solutions for vip by solving a finite number of convex optimization subproblems with linear objective moreover in order to characterize finite convergence of an iterative algorithm we introduce the notion of a weak subsharp set of solutions to a variational inequality problem vip which is more general than that of weak sharp solutions in hilbert spaces we establish a sufficient and necessary condition for the finite convergence of an algorithm for solving vip which satisfies that the sequence generated by which converges to a weak subsharp solution of vip and show that the proximal point algorithm satisfies this condition as a consequence we prove that the proximal point algorithm possesses finite convergence whenever the sequence generated by which converges to a weak subsharp solution of vip

the linear l 1 estimator and the huber m estimator relationships between a linear l1 estimation problem and the huber m estimator problem can be easily established by their dual formulations the least norm solution of a linear programming problem studied by mangasarian and meyer siam j control optim 17 1979 pp 745 752 provides a key link between the dual problems based on the dual formulations we establish a local linearity property of the huber m estimators with respect to the tuning parameter gamma and prove that the solution set of the huber m estimator problem is lipschitz continuous with respect to perturbations of the tuning parameter gamma as a consequence the set of the linear l1 estimators is the limit of the set of the huber m estimators as gamma to 0 thus the huber m estimator problem has many solutions for small tuning parameter gamma if the linear l1 estimation problem has multiple solutions a recursive version of madsen and nielsen s algorithm siam j optim 3 1993 pp 223 235 based on computation of the huber m estimator is proposed for finding a linear l1 estimator

solvable lie algebras with an mathbb n graded nilradical of maximal nilpotency degree and their invariants the class of solvable lie algebras with an graded nilradical of maximal nilpotency index is classified it is shown that such solvable extensions are unique up to isomorphism the generalized casimir invariants for the graded nilradicals and their associated solvable extensions are computed by the method of moving frames

solvable lie algebras with abelian nilradicals a procedure is presented for classifying solvable lie algebras with abelian nilradicals theorems on the structure of such algebras are proven and their centres are constructed many examples are analysed

generic minimizing behavior in semialgebraic optimization we present a theorem of sard type for semialgebraic set valued mappings whose graphs have dimension no larger than that of their range space the inverse of such a mapping admits a single valued analytic localization around any pair in the graph for a generic value parameter this simple result yields a transparent and unified treatment of generic properties of semialgebraic optimization problems typical semialgebraic problems have finitely many critical points around each of which they admit a unique active manifold analogue of an active set in nonlinear optimization moreover such critical points satisfy strict complementarity and second order sufficient conditions for optimality are indeed necessary

leibniz algebras with small derived ideal abstract a nilpotent leibniz algebra l is called extra special if dim z l dim l 2 1 in this paper the structure the second homology and the capability of extra special leibniz algebras are completely determined

a simple proof of the entropy power inequality this correspondence gives a simple proof of shannon s entropy power inequality epi using the relationship between mutual information and minimum mean square error mmse in gaussian channels

minimizing and stationary sequences of constrained optimization problems we study some fundamental asymptotic properties associated with the well posedness of constrained optimization problems emphasis is placed on the relation between minimizing and stationary sequences and their characterizations in terms of a set of asymptotic karush kuhn tucker kkt optimality conditions unlike the subdifferential approach used by auslender cominetti and crouzeix we use a residual function approach that is closely tied to the theory of error bounds this approach handles constraints explicitly and allows the effective treatment of infeasible stationary sequences the asymptotic kkt conditions provide an asymptotic optimality certificate for inequality constrained programs specializations of the results to convex quadratically constrained convex quadratic spline minimization problems and convex programs with holderian minima are discussed an application of the results to the convergence of a family of newton type iterative descent methods involving singular quasi newton matrices for solving a constrained minimization problem is also presented

invariants of solvable lie algebras with triangular nilradicals and diagonal nilindependent elements the invariants of solvable lie algebras with nilradicals isomorphic to the algebra of strongly upper triangular matrices and diagonal nilindependent elements are studied exhaustively bases of the invariant sets of all such algebras are constructed by an original purely algebraic algorithm based on cartan s method of moving frames

solvable lie algebras with triangular nilradicals all finite dimensional indecomposable solvable lie algebras having the triangular algebra t n as their nilradical are constructed the number of non nilpotent elements f in satisfies and the dimension of the lie algebra is

a class of solvable lie algebras and their casimir invariants a nilpotent lie algebra with an n 1 dimensional abelian ideal is studied all indecomposable solvable lie algebras with as their nilradical are obtained their dimension is at most n 2 the generalized casimir invariants of and of its solvable extensions are calculated for n 4 these algebras figure in the petrov classification of einstein spaces for larger values of n they can be used in a more general classification of riemannian manifolds

finite termination of the proximal point algorithm this paper concerns the notion of a sharp minimum on a set and its relationship to the proximal point algorithm we give several equivalent definitions of the property and use the notion to prove finite termination of the proximal point algorithm

on classification of 5 dimensional solvable leibniz algebras in the paper we describe 5 dimensional solvable leibniz algebras with three dimensional nilradical since those 5 dimensional solvable leibniz algebras whose nilradical is three dimensional heisenberg algebra have been classified before we focus on the rest cases the result of the paper together with heisenberg nilradical case gives complete classification of all 5 dimensional solvable leibniz algebras with three dimensional nilradical