needed reduction and spine strategies for the lambda calculus abstract a redex r in a lambda term m is called needed if in every reduction of m to normal form some residual of r is contracted among others the following results are proved 1 r is needed in m iff r is contracted in the leftmost reduction path of m 2 let r m0 m1 m2 reduce redexes ri mi mi 1 and have the property that i j i rj is needed in mj then r is normalising i e if m0 has a normal form then r is finite and terminates at that normal form 3 neededness is an undecidable property but has several efficiently decidable approximations various versions of the so called spine redexes

on the evaluation complexity of composite function minimization with applications to nonconvex nonlinear programming we estimate the worst case complexity of minimizing an unconstrained nonconvex composite objective with a structured nonsmooth term by means of some first order methods we find that it is unaffected by the nonsmoothness of the objective in that a first order trust region or quadratic regularization method applied to it takes at most mathcal o epsilon 2 function evaluations to reduce the size of a first order criticality measure below epsilon specializing this result to the case when the composite objective is an exact penalty function allows us to consider the objective and constraint evaluation worst case complexity of nonconvex equality constrained optimization when the solution is computed using a first order exact penalty method we obtain that in the reasonable case when the penalty parameters are bounded the complexity of reaching within epsilon of a kkt point is at most mathcal o epsilon 2 problem evaluations which is the same in order as the function evaluation complexity of steepest descent methods applied to unconstrained nonconvex smooth optimization

weak linearization of the lambda calculus we identify a restricted class of terms of the lambda calculus here called weak linear that includes the linear lambda terms keeping their good properties of strong normalization non duplicating reductions and typability in polynomial time the advantage of this class over the linear lambda calculus is the possibility of transforming general terms into weak linear terms with the same normal form we present such transformation and prove its correctness by showing that it preserves normal forms

black box complexity of local minimization the complexity of local minimization in the black box model that is the model in which the objective function and its gradient are available as external subroutines is studied the black box model is used for example in all the optimization algorithms in dennis and schnabel numerical methods for unconstrained optimization and nonlinear equations prentice hall englewood cliffs nj 1983 the first main result is that the complexity grows polynomially with the number of variables n in contrast to other related black box problems global minimization and brouwer fixed points for which the worst case complexity is exponential in n the second contribution is the construction of a family of functions that are bad cases for all possible black box local optimization algorithms

on strong normalization and type inference in the intersection type discipline we introduce a new unification procedure for the type inference problem in the intersection type discipline it is well known that type inference in this case should succeed exactly for the strongly normalizing expressions we give a proof for the strong normalization result in the intersection type discipline which we obtain by putting together some well known results and proof techniques our proof uses a variant of klop s extended l calculus for which it is shown that strong normalization is equivalent to weak normalization this is proved here by means of a finiteness of developments theorem obtained following de vrijer s combinatory technique the main property of this extended calculus is uniformity i e weak and strong normalizabilities coincide the strong normalizability result is therefore a consequence of the fact first established by coppo and dezani for the l calculus that typability implies weak normalizability we then show that the unification process which is the basis for type inference exactly corresponds to reduction in the extended l calculus finally we show that our notion of unification allows us to compute a principal typing for any typable l expression

on the longest perpetual reductions in orthogonal expression reduction systems we study perpetual reductions in orthogonal or conflict free fully extended expression reduction systems oers ers is a formalism for rewriting that subsumes term rewriting systems trss and the l calculus we design a strategy that for any given term t in a fully extended oers constructs a longest reduction starting from t if t is strongly normalizing and otherwise constructs an infinite reduction we call this strategy a limit strategy for a large class of oerss a limit strategy is computable the conservation theorem for fully extended oerss follows easily from the properties of the strategy we develop a method for computing the lengths of longest developments in oerss copyright 2001 elsevier science b v

optimality condition and complexity analysis for linearly constrained optimization without differentiability on the boundary in this paper we consider the minimization of a continuous function that is potentially not differentiable or not twice differentiable on the boundary of the feasible region by exploiting an interior point technique we present first and second order optimality conditions for this problem that reduces to classical ones when the derivative on the boundary is available for this type of problems existing necessary conditions often rely on the notion of subdifferential or become non trivially weaker than the kkt condition in the twice differentiable counterpart problems in contrast this paper presents a new set of first and second order necessary conditions that are derived without the use of subdifferential and reduce to exactly the kkt condition when twice differentiability holds as a result these conditions are stronger than some existing ones considered for the discussed minimization problem when only non negativity constraints are present to solve for these optimality conditions in the special but important case of linearly constrained problems we present two novel interior point trust region algorithms and show that their worst case computational efficiency in achieving the potentially stronger optimality conditions match the best known complexity bounds since this work considers a more general problem than those in the literature our results also indicate that best known existing complexity bounds are actually held for a wider class of nonlinear programming problems this new development is significant since optimality conditions play a fundamental role in computational optimization and more and more nonlinear and nonconvex problems need to be solved in practice

on the evaluation complexity of cubic regularization methods for potentially rank deficient nonlinear least squares problems and its relevance to constrained nonlinear optimization we propose a new termination criterion suitable for potentially singular zero or nonzero residual least squares problems with which cubic regularization variants take at most mathcal o epsilon 3 2 residual and jacobian evaluations to drive either the euclidean norm of the residual or its gradient below epsilon this is the best known bound for potentially rank deficient nonlinear least squares problems we then apply the new optimality measure and cubic regularization steps to a family of least squares merit functions in the context of a target following algorithm for nonlinear equality constrained problems this approach yields the first evaluation complexity bound of order epsilon 3 2 for nonconvexly constrained problems when higher accuracy is required for primal feasibility than for dual first order criticality

adaptive cubic regularisation methods for unconstrained optimization part ii worst case function and derivative evaluation complexity an adaptive regularisation framework using cubics arc was proposed for unconstrained optimization and analysed in cartis gould and toint part i math program doi 10 1007 s10107 009 0286 5 2009 generalizing at the same time an unpublished method due to griewank technical report na 12 1981 damtp university of cambridge an algorithm by nesterov and polyak math program 108 1 177 205 2006 and a proposal by weiser deuflhard and erdmann optim methods softw 22 3 413 431 2007 in this companion paper we further the analysis by providing worst case global iteration complexity bounds for arc and a second order variant to achieve approximate first order and for the latter second order criticality of the iterates in particular the second order arc algorithm requires at most mathcal o epsilon 3 2 iterations or equivalently function and gradient evaluations to drive the norm of the gradient of the objective below the desired accuracy epsilon and mathcal o epsilon 3 iterations to reach approximate nonnegative curvature in a subspace the orders of these bounds match those proved for algorithm 3 3 of nesterov and polyak which minimizes the cubic model globally on each iteration our approach is more general in that it allows the cubic model to be solved only approximately and may employ approximate hessians

umbral presentations for polynomial sequences abstract using random variables as motivation this paper presents an exposition of formalisms developed in 1 2 for the classical umbral calculus a new construction and a variety of examples are presented culminating in several descriptions of sequences of binomial type in terms of umbral polynomials

series with general exponents abstract we define the artinian and noetherian algebra which consist of formal series involving exponents which are not necessarily integers all of the usual operations are defined here and characterized as an application we compute the algebra of symmetric functions with nonnegative real exponents the applications to logarithmic series and the umbral calculus are deferred to another paper 9

bck combinators and linear l terms have types abstract a term l is linear when each abstract x m in it has at most one free x in m and each free variable of l occurs free only once bck combinators correspond exactly to linear terms without free variables the main theorem says that all terms of each kind are stratified i e are typable this result is not deep it merely confirms an intuition that non stratification is caused partly by multiple occurrences of variables for bck combinators it was stated as theorem 1 by bunder and meyer in 1 but the proof in 1 seems to contain a gap it uses induction on a definition of level of a combinator but omits to show that the level concept is defined for all bck combinators the proof below proceeds by translating to linear terms for which the induction becomes simply one on length in fact the proof is essentially just an amalgamation of elements from an unpublished proof of m coppo and m dezani ciancaglini and elements from 1 the type schemes of linear terms form a very primitive linear logic and linear logic is of current interest see girard 3 so we shall set out the basic facts about linear terms and bck combinators in some detail these basic facts are not new some appearin komori 6 for example and some are implicit in curry and feys 2

strict constraint qualifications and sequential optimality conditions for constrained optimization sequential optimality conditions for constrained optimization are necessarily satisfied by local minimizers independently of the fulfillment of constraint qualifications these conditions support the employment of different stopping criteria for practical optimization algorithms on the other hand when an appropriate property on the constraints holds at a point that satisfies a sequential optimality condition such a point also satisfies the karush kuhn tucker conditions those properties will be called strict constraint qualifications in this paper as a consequence for each sequential optimality condition it is natural to ask for its weakest strict associated constraint qualification this problem has been solved in a recent paper for the approximate karush kuhn tucker sequential optimality condition in the present paper we characterize the weakest strict constraint qualifications associated with other sequential optimality conditions that are useful for defining stopping criteria of algorithms i

proof nets and the call by value λ calculus this paper gives a detailed account of the relationship between a variant of the call by value lambda calculus and linear logic proof nets the presentation is carefully tuned in order to realize an isomorphism between the two systems every single rewriting step on the calculus maps to a single step on proof nets and vice versa in this way we obtain an algebraic reformulation of proof nets moreover we provide a simple correctness criterion for our proof nets which employ boxes in an unusual way and identify a subcalculus that is shown to be as expressive as the full calculus

the barendregt cube with definitions and generalised reduction in this paper we propose to extend the barendregt cube by generalisings reduction and by adding definition mechanisms generalised reduction allows contracting more visible redexes than usual and definitions are an important tool to allow for a more flexible typing system we show that this extension satisfies most of the original properties of the cube including church rosser subject reduction and strong normalisation

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

umbral calculus and special functions developpement du calcul symbolique des puissances et de la methode des coalgebres et des comodules associee application a l etude des fonctions speciales classiques hypergeometriques de laguerre sonine d hermite weber de legendre et de bessel

evaluation complexity for nonlinear constrained optimization using unscaled kkt conditions and high order models the evaluation complexity of general nonlinear possibly nonconvex constrained optimization is analyzed it is shown that under suitable smoothness conditions an epsilon approximate first order critical point of the problem can be computed in order o epsilon 1 2 p 1 p evaluations of the problem s functions and their first p derivatives this is achieved by using a two phase algorithm inspired by cartis gould and toint siam j optim 21 2011 pp 1721 1739 siam j optim 23 2013 pp 1553 1574 it is also shown that strong guarantees in terms of handling degeneracies on the possible limit points of the sequence of iterates generated by this algorithm can be obtained at the cost of increased complexity at variance with previous results the epsilon approximate first order criticality is defined by satisfying a version of the kkt conditions with an accuracy that does not depend on the size of the lagrange multipliers

cubic regularization of newton method and its global performance in this paper we provide theoretical analysis for a cubic regularization of newton method as applied to unconstrained minimization problem for this scheme we prove general local convergence results however the main contribution of the paper is related to global worst case complexity bounds for different problem classes including some nonconvex cases it is shown that the search direction can be computed by standard linear algebra technique

strong normalization from weak normalization in typedλ calculi for some typed calculi it is easier to prove weak normalization than strong normalization techniques to infer the latter from the former have been invented over the last twenty years by nederpelt klop khasidashvili karr de groote and kfoury and wells however these techniques infer strong normalization of one notion of reduction from weak normalization of amore complicatednotion of reduction this paper presents a new technique to infer strong normalization of a notion of reduction in a typed calculus from weak normalization of thesamenotion of reduction the technique is demonstrated to work on some well known systems including second order calculus and the system of positive recursive types it gives hope for a positive answer to the barendregt geuvers conjecture stating that every pure type system which is weakly normalizing is also strongly normalizing the paper also analyzes the relationship between the techniques mentioned above and reviews in less detail other techniques for proving strong normalization

the iterated logarithmic algebra ii sheffer sequences abstract an extension of the theory of the iterated logarithmic algebra 1 gives the logarithmic analog of a sheffer or appell sequence of polynomials this leads to several examples including stirling s formula and a logarithmic version of the euler maclaurin summation formula

perpetuality and uniform normalization in orthogonal rewrite systems we study perpetuality of reduction steps as well as perpetuality of redexes in orthogonal rewrite systems a perpetual step is a reduction step which retains the possibility of infinite reductions a perpetual redex is a redex which when put into an arbitrary context yields a perpetual step we generalize and refine existing criteria for the perpetuality of reduction steps and redexes in orthogonal term rewriting systems and the calculus due to bergstra and klop and others we first introduce context sensitive conditional expression reduction systems ccerss and define a concept of orthogonality which implies confluence for them in particular several important calculi and their extensions and restrictions can naturally be embedded into orthogonal ccerss we then define a perpetual reduction strategy which enables one to construct minimal w r t levy s permutation ordering on reductions infinite reductions in orthogonal fully extended ccerss using the properties of the minimal perpetual strategy we prove 1 perpetuality of any reduction step that does not erase potentially infinite arguments which are arguments that may become via substitution infinite after a number of outside steps and 2 perpetuality in every context of any safe redex which is a redex whose substitution instances may discard infinite arguments only when the corresponding contracta remain infinite we prove both these perpetuality criteria for orthogonal fully extended ccerss and then specialize and apply them to restricted calculi demonstrating their usefulness in particular we prove the equivalence of weak and strong normalization whose equivalence is here called uniform normalization for various restricted calculi most of which cannot be derived from previously known perpetuality criteria

on the complexity of steepest descent newton s and regularized newton s methods for nonconvex unconstrained optimization problems it is shown that the steepest descent and newton s methods for unconstrained nonconvex optimization under standard assumptions may both require a number of iterations and function evaluations arbitrarily close to o epsilon 2 to drive the norm of the gradient below epsilon this shows that the upper bound of o epsilon 2 evaluations known for the steepest descent is tight and that newton s method may be as slow as the steepest descent method in the worst case the improved evaluation complexity bound of o epsilon 3 2 evaluations known for cubically regularized newton s methods is also shown to be tight

an equivalence between lambda terms resume dans ce papier on definit une relation d equivalence sur les lambda termes identifiant les termes qui ne different que par des permutations de radicaux la equivalence on demontre qu aucun des criteres operationnels standards de classification du lambda calcul e g la longueur de la plus longue normalisation ne permet de distinguer deux termes equivalents enfin la equivalence est utilisee pour demontrer une generalisation du theoreme de la strategie perpetuelle

on the complexity of finding first order critical points in constrained nonlinear optimization the complexity of finding epsilon approximate first order critical points for the general smooth constrained optimization problem is shown to be no worse that o epsilon 2 in terms of function and constraints evaluations this result is obtained by analyzing the worst case behaviour of a first order short step homotopy algorithm consisting of a feasibility phase followed by an optimization phase and requires minimal assumptions on the objective function since a bound of the same order is known to be valid for the unconstrained case this leads to the conclusion that the presence of possibly nonlinear nonconvex inequality equality constraints is irrelevant for this bound to apply

a type assignment for λ calculus complete both for fptime and strong normalization one of the aims of implicit computational complexity is the design of programming languages with bounded computational complexity one of the most promising approaches to this aim is based on the use of lambda calculus as paradigmatic programming language and the design of type assignment systems for lambda terms where types guarantee both the functional correctness and the complexity bound along this line we propose a system of stratified types inspired by intersection types where intersection is a non associative operator this system is correct and complete for polynomial time when a uniform coding scheme is considered moreover all the strongly normalizing terms are typed in it thus increasing the typing power with respect to the previous proposals based on light logics

two identities in the theory of polynomials of binomial type demonstration de deux identites generalisant celles etablies par hurwitz 1902 application aux cycles de permutations

recursive matrices and umbral calculus two major objectives can be seen to guide much recent work in enumeration 1 to single out a limited variety of recurrences for numerical sequences which will encompass counting problems of wide enough type 2 to recover from empirical data an underlying set theoretic structure which would reveal the source of the given recursion we are here concerned with the first of these objectives though the eventual understanding of the second is tacitly present if only as a goal we noticed the coincidence of several computations which similar as they are in retrospect had failed to realize their kinship on leafing through the unique assembly of recursively solvable combinatorial problems in comtet s and sloane s invaluable collections one is struck by the repeated occurrence of one and the same kind of double recursion more strikingly the same recursion is seen to occur in the polynomial sequences of the umbra1 calculus of roman and rota see 25 everywhere the lagrange inversion formula for power series plays a pivotal role much work is nowadays going into the unraveling of the everdeeper layers of combinatorial significance of this formula both in the ordinary case and in its as yet partially worked out noncommutative and qanalogs andrews foata garsia gessel joni raney reiner schtitzenberger to name but a few whatever their origins the identities abutting lagrange inversion are expressed by integers alone this suggests not only a hidden set theoretic layer but a characteristic free generalization as well this generalization is the central theme of our work we define a monoid of infinite matrices recursive matrices for short the entries of these matrices give the sought out recursion for example that for coefficients of binomial and sheffer polynomials and factor sequences as well as that of the special sequences recently introduced by roman in 26 546 002 l 8693 82 040546 28 02 00 0

on the convergence of successive linear quadratic programming algorithms the global convergence properties of a class of penalty methods for nonlinear programming are analyzed these methods include successive linear programming approaches and more specifically the successive linear quadratic programming approach presented by byrd et al math program 100 2004 pp 27 48 every iteration requires the solution of two trust region subproblems involving piecewise linear and quadratic models respectively it is shown that for a fixed penalty parameter the sequence of iterates approaches stationarity of the penalty function a procedure for dynamically adjusting the penalty parameter is described and global convergence results for it are established

perpetual reductions in λ calculus this paper surveys a part of the theory ofs reduction in calculus which might aptly be calledperpetual reductions the theory is concerned withperpetual reduction strategies i e reduction strategies that compute infinite reduction paths from terms when possible and withperpetual redexes i e redexes whose contraction in terms preserves the possibility when present of infinite reduction paths the survey not only recasts classical theorems in a unified setting but also offers new results proofs and techniques as well as a number of applications to problems in calculus and type theory

conservation and uniform normalization in lambda calculi with erasing reductions for a notion of reduction in a calculus one can ask whether a term satisfies conservation and uniform normalization conservation means that single step reductions of the term preserve infinite reduction paths from the term uniform normalization means that either the term will have no reduction paths leading to a normal form or all reduction paths will lead to a normal form in the classical conservation theorem for i the distinction between the two notions is not clear uniform normalization implies conservation and conservation also implies uniform normalization the reason for this is that i is closed under reduction due to the fact that reductions never erase terms in i more generally for nonerasing reductions the two notions are equivalent on a set closed under reduction however when turning to erasing reductions the distinction becomes important as conservation no longer implies uniform normalization this paper presents a new technique for finding uniformly normalizing subsets of a calculus this is done by combining a syntactic and a semantic criterion the technique is demonstrated by several applications the technique is used to present a new uniformly normalizing subset of the pure calculus this subset is a superset of i and thus contains erasing k redexes the technique is also used to prove strong normalization from weak normalization of the simply typed calculus extended with pairs this is an extension of techniques developed recently by sorensen and xi before presenting the technique the paper presents a simple proof of a slightly weaker form of the characterization of perpetual redexes by bergstra and klop this is a step for the later applications of the technique

principal types of bck lambda terms abstract bck terms are the terms in which each variable occurs at most once the principal type of a term is the most general type of the term in this paper we prove that if two bck terms in normal form have the same principal type then they are identical this solves the following problem y komori 1987 in more general form if two closed normal form bck terms are assigned to the same minimal bck formula are they identical a minimal bck formula is the most general formula among bck provable formulas with respect to substitutions for type variables to analyze type assignment the notion of connection is introduced a connection is a series of occurrences of a type in a type assignment figure connected occurrences of a type have the same meaning the occurrences of the type in distinct connection classes can be rewritten separately as a result we have more general type assignment by formulae as types correspondence the result implies the uniqueness of the normal proof figure for principal bck formulas the result is valid for bci logic or implicational fragment of linear logic as well

a short proof that adding some permutation rules to β preserves sn i show that if a term is strongly normalizing sn for beta it remains sn when some permutation rules are added

the interated logarithmic algebra abstract we generalize the umbral calculus of g c rota adv in math 27 1978 95 188 by studying not only sequences of polynomials and inverse power series or even the logarithms studied by d loeb and g c rota adv in math 75 1989 1 118 but instead we study sequences of formal expressions involving the iterated logarithms and x to an arbitrary real power using a theory of formal power series with real exponents and a more general definition of factorial binomial coefficient and stirling numbers to all the real numbers we define the iterated logarithmic algebra j its elements are the formal representations of the asymptotic expansions of a large class of real functions and we define the harmonic logarithm basis of j which will be interpreted as a generalization of the powers xn since it behaves nicely with respect to the derivative we classify all operators over j which commute with the derivative classically there are known as shift invariant operators and formulate several equivalent definitions of a sequence of binomial type we then derive many formulas useful towards the calculation of these sequences including the recurrence formula the transfer formula and the lagrange inversion formula in the sequel 22 we study sheffer sequences and give many examples

on the evaluation complexity of constrained nonlinear least squares and general constrained nonlinear optimization using second order methods when solving the general smooth nonlinear and possibly nonconvex optimization problem involving equality and or inequality constraints an approximate first order critical point of accuracy epsilon can be obtained by a second order method using cubic regularization in at most o epsilon 3 2 evaluations of problem functions the same order bound as in the unconstrained case this result is obtained by first showing that the same result holds for inequality constrained nonlinear least squares as a consequence the presence of possibly nonconvex equality inequality constraints does not affect the complexity of finding approximate first order critical points in nonconvex optimization this result improves on the best known o epsilon 2 evaluation complexity bound for solving general nonconvexly constrained optimization problems

on principal types of combinators in this paper we study in some cases the relationship between the combinatory completeness of a set of typable combinators with simple types for a system of calculus and the axiomatic completeness under substitution and modus ponens of the respective set of principal types for the corresponding logical system we show that combinatory completeness is a necessary but not sufficient condition for axiomatic completeness in the k and in the i calculus while the two problems become equivalent for the bck as well as for the bci calculus furthermore we present an algorithm which whenever b is a principal pair for some normal bck term m reconstructs m up to conversion and which fails if there is no normal bck term for which b is a principal pair from the correctness proof of the algorithm we also obtain another proof for the fact that each bck term in normal form is completely determined by its principal pairs

an exact correspondence between a typed pi calculus and polarised proof nets this paper presents an exact correspondence in typing and dynamics between polarised linear logic and a typed p calculus based on io typing the respective incremental constraints one on geometric structures of proof nets and one based on types precisely correspond to each other leading to the exact correspondence of the respective formalisms as they appear in olivier laurent 2003 27 for proof nets and kohei honda et al 2004 24 for the p calculus

filter models with polymorphic types abstract using ideas and results from barendrecht 1983 and coppo 1984 on intersection types a comparable theory is developed for second order polymorphic types the set of filters constructed with polymorphic type forms with inclusion a continuous lattice which yields a model of what we call expansion i e the value of a term increases under reduction but not of conversion combining intersection with polymorphic types does give filter models but the two standard ways of interpreting terms do not coincide

godel s system t revisited the linear lambda calculus where variables are restricted to occur in terms exactly once has a very weak expressive power in particular all functions terminate in linear time in this paper we consider a simple extension with natural numbers and a restricted iterator only closed linear functions can be iterated we show properties of this linear version of godel s tusing a closed reduction strategy and study the class of functions that can be represented surprisingly this linear calculus offers a huge increase in expressive power over previous linear versions of t which are closed at construction rather than closed at reduction we show that a linear twith closed reduction is as powerful as t

resource operators for λ calculus we present a simple term calculus with an explicit control of erasure and duplication of substitutions enjoying a sound and complete correspondence with the intuitionistic fragment of linear logic s proof nets we show the operational behaviour of the calculus and some of its fundamental properties such as confluence preservation of strong normalisation strong normalisation of simply typed terms step by step simulation of b reduction and full composition

a note on preservation of strong normalisation in the λ calculus an auxiliary notion of reduction r on the l terms preserves strong normalisation if all strongly normalising terms for b are also strongly normalising for b r we give a sufficient condition for r to preserve strong normalisation as an example of application we check easily the sufficient condition for regnier s s reduction rules and the assoc reduction rule inspired by calculi with let expressions this gives the simplest proof so far that the union of all these rules preserves strong normalisation

recursive trust region methods for multiscale nonlinear optimization a class of trust region methods is presented for solving unconstrained nonlinear and possibly nonconvex discretized optimization problems like those arising in systems governed by partial differential equations the algorithms in this class make use of the discretization level as a means of speeding up the computation of the step this use is recursive leading to true multilevel multiscale optimization methods reminiscent of multigrid methods in linear algebra and the solution of partial differential equations a simple algorithm of the class is then described and its numerical performance is shown to be numerically promising this observation then motivates a proof of global convergence to first order stationary points on the fine grid that is valid for all algorithms in the class

corrigendum on the complexity of finding first order critical points in constrained nonlinear optimization in a recent paper cartis et al in math prog a 144 2 93 106 2014 the evaluation complexity of an algorithm to find an approximate first order critical point for the general smooth constrained optimization problem was examined unfortunately the proof of lemma 3 5 in that paper uses a result from an earlier paper in an incorrect way and indeed the result of the lemma is false the purpose of this corrigendum is to provide a modification of the previous analysis that allows us to restore the complexity bound for a different scaled measure of first order criticality

majorization minimization procedures and convergence of sqp methods for semi algebraic and tame programs in view of solving nonsmooth and nonconvex problems involving complex constraints like standard nlp problems we study general maximization minimization procedures produced by families of strongly convex subproblems using techniques from semi algebraic geometry and variational analysis in particular ojasiewicz inequality we establish the convergence of sequences generated by these types of schemes to critical points the broad applicability of this process is illustrated in the context of nlp in that case critical points coincide with kkt points when the data are semi algebraic or real analytic our method applies for instance to the study of various sequential quadratic programming sqp schemes the moving balls method the penalized sqp method and the extended sqp method under standard qualification conditions this provides to the best of our knowledge the first general convergence results for general nonlinear programming problems we emphasize the fact that unlike most works on this subjec

dual operators and lagrange inversion in several variables the original lagrange inversion formula which gives explicitly the inverse under composition of a formal series was obtained by lagrange in 1770 through formal computations involving logarithms of inlmite products see lagrange 121 there exists an extensive literature on various versions of the inversion formula obtained by algebraic analytic and combinatorial methods comtet 3 gives a survey and an ample bibliography during the last ten years there have appeared several papers on generalizations and new proofs of the inversion formula in 1974 abhyankar obtained an inversion formula using algebraic methods see bass connell and wright 11 several authors have related lagrange inversion with rota s finite operator calculus 151 for example joni 8 garsia and joni 4 hofhauer 7 and roman and rota 14 joyal and labelle used a combinatorial theory of formal series to obtain inversion formulas see 9 l i and viskov 161 used ideas from lie algebra theory to prove formulas similar to abhyankar s in the present paper we use operator methods to obtain inversion formulas in several variables that generalize the formulas of abhyankar joni and viskov we get our results studying an algebra with involution of linear operators on the algebra of formal laurent series in several indeterminates we also show that by means of an anti isomorphism of operator algebras which we call the operator bore1 transform the study of finite operator calculus can be reduced to the study of certain groups of operators on the algebra of formal laurent series our approach may be considered as an algebraic analogue of the complex variable methods used by good 5 to obtain the lagrange inversion formula as a consequence of the change of variables theorem for the