worst case evaluation complexity for unconstrained nonlinear optimization using high order regularized models the worst case evaluation complexity for smooth possibly nonconvex unconstrained optimization is considered it is shown that if one is willing to use derivatives of the objective function up to order p for p ge 1 p 1 and to assume lipschitz continuity of the p th derivative then an epsilon approximate first order critical point can be computed in at most o epsilon p 1 p o p 1 p evaluations of the problem s objective function and its derivatives this generalizes and subsumes results known for p 1 p 1 and p 2 p 2

bisimulation relations for dynamical control and hybrid systems the fundamental notion of bisimulation equivalence for concurrent processes has escaped the world of continuous and subsequently hybrid systems inspired by the categorical framework of joyal nielsen and winskel we develop novel notions of bisimulation equivalence for dynamical systems as well as control systems we prove that these notions can be captured by the abstract notion of bisimulation as developed by joyal nielsen and winskel this is the first unified notion of system equivalence that transcends discrete and continuous systems furthermore this enables the development of a novel and natural notion of bisimulation for hybrid systems which is the final goal of this paper

optimization of lyapunov invariants in verification of software systems the paper proposes a control theoretic framework for verification of numerical software systems and puts forward software verification as an important application of control and systems theory the idea is to transfer lyapunov functions and the associated computational techniques from control systems analysis and convex optimization to verification of various software safety and performance specifications these include but are not limited to absence of overflow absence of division by zero termination in finite time absence of dead code and certain user specified assertions central to this framework are lyapunov invariants these are properly constructed functions of the program variables and satisfy certain properties analogous to those of lyapunov functions along the execution trace the search for the invariants can be formulated as a convex optimization problem if the associated optimization problem is feasible the result is a certificate for the specification

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

mesh adaptive direct search algorithms for constrained optimization this paper addresses the problem of minimization of a nonsmooth function under general nonsmooth constraints when no derivatives of the objective or constraint functions are available we introduce the mesh adaptive direct search mads class of algorithms which extends the generalized pattern search gps class by allowing local exploration called polling in an asymptotically dense set of directions in the space of optimization variables this means that under certain hypotheses including a weak constraint qualification due to rockafellar mads can treat constraints by the extreme barrier approach of setting the objective to infinity for infeasible points and treating the problem as unconstrained the main gps convergence result is to identify limit points hat x where the clarke generalized derivatives are nonnegative in a finite set of directions called refining directions although in the unconstrained case nonnegative combinations of these directions span the whole space the fact that there can only be finitely many gps refining directions limits rigorous justification of the barrier approach to finitely many linear constraints for gps the main result of this paper is that the general mads framework is flexible enough to allow the generation of an asymptotically dense set of refining directions along which the clarke derivatives are nonnegative we propose an instance of mads for which the refining directions are dense in the hypertangent cone at hat x with probability 1 whenever the iterates associated with the refining directions converge to a single hat x the instance of mads is compared to versions of gps on some test problems we also illustrate the limitation of our results with examples

a relaxed constant positive linear dependence constraint qualification and applications in this work we introduce a relaxed version of the constant positive linear dependence constraint qualification cpld that we call rcpld this development is inspired by a recent generalization of the constant rank constraint qualification by minchenko and stakhovski that was called rcrcq we show that rcpld is enough to ensure the convergence of an augmented lagrangian algorithm and that it asserts the validity of an error bound we also provide proofs and counter examples that show the relations of rcrcq and rcpld with other known constraint qualifications in particular rcpld is strictly weaker than cpld and rcrcq while still stronger than abadie s constraint qualification we also verify that the second order necessary optimality condition holds under rcrcq

specifications for decidable hybrid games we introduce stormed hybrid games shg a generalization of stormed hybrid systems vladimerou et al 2008 33 which have natural specifications that allow rich continuous dynamics and various decidable properties we solve the control problem for shg using a reduction to bisimulation on finite game graphs this generalizes to a greater family of games which includes o minimal hybrid games bouyer et al 2006 6 we also solve the optimal cost reachability problem for weighted shg and prove decidability of wctl for weighted stormed hybrid systems

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 affine scaling algorithms for nonconvex quadratic programming we investigate the use of interior algorithms especially the affine scaling algorithm to solve nonconvex indefinite or negative definite quadratic programming qp problems although the nonconvex qp with a polytope constraint is a hard problem we show that the problem with an ellipsoidal constraint is easy when the hard qp is solved by successively solving the easy qp the sequence of points monotonically converge to a feasible point satisfying both the first and the second order optimality conditions

approximation metrics for discrete and continuous systems established system relationships for discrete systems such as language inclusion simulation and bisimulation require system observations to be identical when interacting with the physical world modeled by continuous or hybrid systems exact relationships are restrictive and not robust in this paper we develop the first framework of system approximation that applies to both discrete and continuous systems by developing notions of approximate language inclusion approximate simulation and approximate bisimulation relations we define a hierarchy of approximation pseudo metrics between two systems that quantify the quality of the approximation and capture the established exact relationships as zero sections our approximation framework is compositional for a synchronous composition operator algorithms are developed for computing the proposed pseudo metrics both exactly and approximately the exact algorithms require the generalization of the fixed point algorithms for computing simulation and bisimulation relations or dually the solution of a static game whose cost is the so called branching distance between the systems approximations for the pseudo metrics can be obtained by considering lyapunov like functions called simulation and bisimulation functions we illustrate our approximation framework in reducing the complexity of safety verification problems for both deterministic and nondeterministic continuous systems

a trust region algorithm with a worst case iteration complexity of mathcal o epsilon 3 2 o 3 2 for nonconvex optimization we propose a trust region algorithm for solving nonconvex smooth optimization problems for any overline epsilon in 0 infty 0 the algorithm requires at most mathcal o epsilon 3 2 o 3 2 iterations function evaluations and derivative evaluations to drive the norm of the gradient of the objective function below any epsilon in 0 overline epsilon 0 this improves upon the mathcal o epsilon 2 o 2 bound known to hold for some other trust region algorithms and matches the mathcal o epsilon 3 2 o 3 2 bound for the recently proposed adaptive regularisation framework using cubics also known as the arc algorithm our algorithm entitled trace follows a trust region framework but employs modified step acceptance criteria and a novel trust region update mechanism that allow the algorithm to achieve such a worst case global complexity bound importantly we prove that our algorithm also attains global and fast local convergence guarantees under similar assumptions as for other trust region algorithms we also prove a worst case upper bound on the number of iterations function evaluations and derivative evaluations that the algorithm requires to obtain an approximate second order stationary point

bisimilarity enforcement for discrete event systems using deterministic control for discrete event systems the control to enforce bisimilarity with respect to a given specification has been studied in in this note we consider the case when the control is required to be deterministic while a deterministic control is restrictive compared to a nondeterministic one the case of deterministic control has its own practical significance as it is easier to implement and computationally less expensive to verify we provide a necessary and sufficient condition for the existence of a bisimilarity enforcing deterministic control and discuss its computational complexity we also study properties related to the synthesis of subspecification and superspecification

universal coalgebra a theory of systems in the semantics of programming finite data types such as finite lists have traditionally been modelled by initial algebras later final coalgebras were used in order to deal with infinite data types coalgebras which are the dual of algebras turned out to be suited moreover as models for certain types of automata and more generally for transition and dynamical systems an important property of initial algebras is that they satisfy the familiar principle of induction such a principle was missing for coalgebras until the work of aczel 1988 on a theory of non wellfounded sets in which he introduced a proof principle nowadays called coinduction it was formulated in terms of bisimulation a notion originally stemming from the world of concurrent programming languages milner 1980 park 1981 using the notion of coalgebra homomorphism the definition of bisimulation on coalgebras can be shown to be formally dual to that of congruence on algebras aczel and mendler 1989 thus the three basic notions of universal algebra algebra homomorphism of algebras and congruence turn out to correspond to coalgebra homomorphism of coalgebras and bisimulation respectively in this paper the latter are taken as the basic ingredients of a theory called universal coalgebra some standard results from universal algebra are reformulated using the afore mentioned correspondence and proved for a large class of coalgebras leading to a series of results on e g the lattices of subcoalgebras and bisimulations simple coalgebras and coinduction and a covariety theorem for coalgebras similar to birkhoff s variety theorem

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

two new weak constraint qualifications and applications we present two new constraint qualifications cqs that are weaker than the recently introduced relaxed constant positive linear dependence rcpld cq rcpld is based on the assumption that many subsets of the gradients of the active constraints preserve positive linear dependence locally a major open question was to identify the exact set of gradients whose properties had to be preserved locally and that would still work as a cq this is done in the first new cq which we call the constant rank of the subspace component crsc cq this new cq also preserves many of the good properties of rcpld such as local stability and the validity of an error bound we also introduce an even weaker cq called the constant positive generator cpg which can replace rcpld in the analysis of the global convergence of algorithms we close this work by extending convergence results of algorithms belonging to all the main classes of nonlinear optimization methods sequential quadratic programming augmented lagrangians

on high order model regularization for constrained optimization in two recent papers regularization methods based on taylor polynomial models for minimization were proposed that only rely on holder conditions on the higher order employed derivatives grapiglia and nesterov considered cubic regularization with a sufficient descent condition that uses the current gradient and resembles the classical armijo s criterion cartis gould and toint used taylor models with arbitrary order regularization and defined methods that tackle convex constraints employing the descent criterion that compares actual reduction with predicted reduction the methods presented in this paper consider general not necessarily taylor models of arbitrary order employ a very mild descent criterion and handle general nonnecessarily convex constraints complexity results are compatible with the ones presented in the papers mentioned above

on the convergence and worst case complexity of trust region and regularization methods for unconstrained optimization a nonlinear stepsize control framework for unconstrained optimization was recently proposed by toint optim methods softw 28 82 95 2013 providing a unified setting in which the global convergence can be proved for trust region algorithms and regularization schemes the original analysis assumes that the hessians of the models are uniformly bounded in this paper the global convergence of the nonlinear stepsize control algorithm is proved under the assumption that the norm of the hessians can grow by a constant amount at each iteration the worst case complexity is also investigated the results obtained for unconstrained smooth optimization are extended to some algorithms for composite nonsmooth optimization and unconstrained multiobjective optimization as well

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

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

newton s method with a model trust region modification a modified newton method for unconstrained minimization is presented and analyzed the modification is based upon the model trust region approach this report contains a thorough analysis of the locally constrained quadratic minimizations that arise as subproblems in the modified newton iteration several promising alternatives are presented for solving these subproblems in ways that overcome certain theoretical difficulties exposed by this analysis very strong convergence results are presented concerning the minimization algorithm in particular the explicit use of second order information is justified by demonstrating that the iterates converge to a point which satisfies the second order necessary conditions for minimization with the exception of very pathological cases this occurs whenever the algorithm is applied to problems with continuous second partial derivatives

languages and models for hybrid automata a coalgebraic perspective abstract we study hybrid automata from a coalgebraic point of view we show that such a perspective supports a generic theory of hybrid automata with a rich palette of definitions and results this includes among other things notions of bisimulation and behaviour state minimisation techniques and regular expression languages

open maps behavioural equivalences and congruences abstract spans of open maps have been proposed by joyal nielsen and winskel as a way of adjoining an abstract equivalence bisimilarity to a category of models of computation ms where is an arbitrary subcategory of observations part of the motivation was to recast and generalise milner s well known strong bisimulation in this categorical setting an issue left open was the congruence properties of bisimilarity we address the following fundamental question given a category of models of computation ms and a category of observations are there any conditions under which algebraic constructs viewed as functors preserve bisimilarity we define the notion of functors being factorisable show how this ensures that bisimilarity is a congruence with respect to such functors guided by the definition of factorisability we show how it is possible to parametrise proofs of functors being factorisable with respect to the category of observations i e with respect to a behavioural equivalence

a smoothing sqp framework for a class of composite l_q lq minimization over polyhedron the composite l q 0 q 1 minimization problem over a general polyhedron has received various applications in machine learning wireless communications image restoration signal reconstruction etc this paper aims to provide a theoretical study on this problem first we derive the karush kuhn tucker kkt optimality conditions for local minimizers of the problem second we propose a smoothing sequential quadratic programming framework for solving this problem the framework requires a approximate solution of a convex quadratic program at each iteration finally we analyze the worst case iteration complexity of the framework for returning an epsilon kkt point i e a feasible point that satisfies a perturbed version of the derived kkt optimality conditions to the best of our knowledge the proposed framework is the first one with a worst case iteration complexity guarantee for solving composite l q minimization over a general polyhedron

a cone continuity constraint qualification and algorithmic consequences every local minimizer of a smooth constrained optimization problem satisfies the sequential approximate karush kuhn tucker akkt condition this optimality condition is used to define the stopping criteria of many practical nonlinear programming algorithms it is natural to ask for conditions on the constraints under which akkt implies kkt these conditions will be called strict constraint qualifications scqs in this paper we define a cone continuity property ccp that will be shown to be the weakest possible scq its relation to other constraint qualifications will also be clarified in particular it will be proved that ccp is strictly weaker than the constant positive generator constraint qualification

hierarchically consistent control systems large scale control systems typically possess a hierarchical architecture in order to manage complexity higher levels of the hierarchy utilize coarser models of the system resulting from aggregating the detailed lower level models in this layered control paradigm the notion of hierarchical consistency is important as it ensures the implementation of high level objectives by the lower level system in this paper we define a notion of modeling hierarchy for continuous control systems and obtain characterizations for hierarchically consistent linear systems with respect to controllability objectives as an interesting byproduct we obtain a hierarchical controllability criterion for linear systems from which we recover the best of the known controllability algorithms from numerical linear algebra

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

regularized newton methods for minimizing functions with holder continuous hessians in this paper we study the regularized second order methods for unconstrained minimization of a twice differentiable convex or nonconvex objective function for the current function these metho

the use of quadratic regularization with a cubic descent condition for unconstrained optimization cubic regularization and trust region methods with worst case first order complexity o varepsilon 3 2 and worst case second order complexity o varepsilon 3 have been developed in the last few years in this paper it is proved that the same complexities are achieved by means of a quadratic regularization method with a cubic sufficient descent condition instead of the more usual predicted reduction based descent asymptotic convergence and order of convergence results are also presented finally some numerical experiments comparing the new algorithm with a well established quadratic regularization method are shown

worst case complexity of smoothing quadratic regularization methods for non lipschitzian optimization in this paper we propose a smoothing quadratic regularization sqr algorithm for solving a class of nonsmooth nonconvex perhaps even non lipschitzian minimization problems which has wide applications in statistics and sparse reconstruction the proposed sqr algorithm is a first order method at each iteration the sqr algorithm solves a strongly convex quadratic minimization problem with a diagonal hessian matrix which has a simple closed form solution that is inexpensive to calculate we show that the worst case complexity of reaching an epsilon scaled stationary point is o epsilon 2 moreover if the objective function is locally lipschitz continuous the sqr algorithm with a slightly modified updating scheme for the smoothing parameter and iterate can obtain an epsilon clarke stationary point in at most o epsilon 3 iterations

bisimulation through probabilistic testing abstract we propose a language for testing concurrent processes and examine its strength in terms of the processes that are distinguished by a test by using probabilistic transition systems as the underlying semantic model we show how a testing algorithm can distinguish with a probability arbitrarily close to one between processes that are not bisimulation equivalent we also show a similar result in a slightly stronger form for a new process relation called 2 3 bisimulation which lies strictly between that of simulation and bisimulation finally the ultimately strength of the testing language is shown to identify a new process relation called probabilistic bisimulation which is strictly stronger than bisimulation

hierarchical hybrid control systems a lattice theoretic formulation the notion of dynamical consistency is extended to hybrid systems so as to define the set of dynamically consistent hybrid partition machines associated with a continuous system s following the formulation of the notions of between block and in block controllable hybrid partition machines the lattice hibc s of hybrid in block controllable partition machines is defined and investigated based on these constructions the definition and properties of hierarchical hybrid control systems are presented together with an example

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

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

bisimilar control affine systems the notion of bisimulation plays a very important role in theoretical computer science where it provides several notions of equivalence between models of computation these equivalences are in turn used to simplify verification and synthesis for these models as well as to enable compositional reasoning in systems theory a similar notion is also of interest in order to develop modular verification and design tools for purely continuous or hybrid control systems in this paper we introduce two notions of bisimulation for nonlinear systems we present differential geometric characterizations of these notions and show that bisimilar systems of different dimensions are obtained by factoring out certain invariant distributions furthermore we also show that all bisimilar systems of different dimension are of this form

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

a new sequential optimality condition for constrained optimization and algorithmic consequences necessary first order sequential optimality conditions provide adequate theoretical tools to justify stopping criteria for nonlinear programming solvers sequential optimality conditions are satisfied by local minimizers of optimization problems independently of the fulfillment of constraint qualifications a new condition of this type is introduced in the present paper it is proved that a well established augmented lagrangian algorithm produces sequences whose limits satisfy the new condition practical consequences are discussed

local factorization of trajectory lifting morphisms for single input affine control systems trajectory preserving and lifting maps have been implicitly used in many recursive or hierarchical control design techniques well known systems theoretic concepts such as differential flatness or more recent ones such as simulation and bisimulation can be also understood through the trajectory lifting maps they define in this paper we initiate a study of trajectory preserving and lifting maps between affine control systems our main result shows that any trajectory lifting map between two single input control affine systems can be locally factored as the composition of two special trajectory lifting maps a projection onto a quotient system followed by a differentially flat output with respect to another control system we use this decomposition result to show that under mild regularity conditions trajectory preserving maps between single input affine control systems also lift trajectories as an additional application of the main result we also show how the hierarchical stabilization method known as back stepping can be used based on the existence of a trajectory preserving and lifting map having a feedback stabilizable control system as codomain

low dimensional hybrid systems decidable undecidable don t know even though many attempts have been made to define the boundary between decidable and undecidable hybrid systems the affair is far from being resolved more and more low dimensional systems are being shown to be undecidable with respect to reachability and many open problems in between are being discovered in this paper we present various two dimensional hybrid systems for which the reachability problem is undecidable we show their undecidability by simulating minsky machines their proximity to the decidability frontier is understood by inspecting the most parsimonious constraints necessary to make reachability over these automata decidable we also show that for other two dimensional systems the reachability question remains unanswered by proving that it is as hard as the reachability problem for piecewise affine maps on the real line which is a well known open problem

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