on existence of solutions to differential equations or inclusions remaining in a prescribed closed subset of a finite dimensional space abstract this paper is devoted to existence of trajectories to differential equations and inclusions remaining in a given closed or compact set topological properties of the boundary of this set enables us to conclude the existence of such trajectories in many cases

alternating projections on manifolds we prove that if two smooth manifolds intersect transversally then the method of alternating projections converges locally at a linear rate we bound the speed of convergence in terms of the angle between the manifolds which in turn we relate to the modulus of metric regularity for the intersection problem a natural measure of conditioning we discuss a variety of problem classes where the projections are computationally tractable and we illustrate the method numerically on a problem of finding a low rank solution of a matrix equation

semiconcave solutions of partial differential inclusions abstract we obtain new results on the propagation of singularities for semiconcave solutions of partial differential inclusions these results will be used to study the behavior of singularities of the value function for a reflected control problem

the clarke generalized gradient for functions whose epigraph has positive reach we consider the class of continuous functions that map an open set rn to r with an epigraph having locally positive reach with an additional property this class contains all finite convex and c1 1 functions but also ones that are not necessarily lipschitz continuous we provide a representation formula for the clarke generalized gradient of such functions using convex combinations and limits of gradients at differentiability points thus offering an alternative to the well known proximal normal formula by replacing a pointedness assumption by one of positive reach our proof consists of a detailed analysis of singularities using methods taken from both nonsmooth analysis and geometric measure theory and is based on an induction argument as an application we prove for a particular class of hamilton jacobi equations that an a e solution whose hypograph has positive reach and satisfies an additional property is indeed the unique viscosity solution

preservation of prox regularity of sets with applications to constrained optimization in this paper we first provide counterexamples showing that sublevels of prox regular functions and levels of differentiable mappings with lipschitz derivatives may fail to be prox regular then we prove the uniform prox regularity of such sets under usual verifiable qualification conditions the preservation of uniform prox regularity of intersection and inverse image under usual qualification conditions is also established applications to constrained optimization problems are given

new metric properties for prox regular sets in this paper we present diverse new metric properties that prox regular sets shared with convex ones at the heart of our work lie the legendre fenchel transform and complements of balls first we show that a connected prox regular set is completely determined by the legendre fenchel transform of a suitable perturbation of its indicator function then we prove that such a function is also the right tool to extend to the context of prox regular sets the famous connection between the distance function and the support function of a convex set on the other hand given a prox regular set we examine the intersection of complements of open balls containing the set we establish that the distance of a point to a prox regular set is the maximum of the distances of the point from boundaries of all such complements separating the set and the point this is in the line of the known result expressing the distance from a convex set in terms of separating hyperplanes to the best of our knowledge these results are new in the literature and show that the class of prox regular sets have good properties known in convex analysis mathematics subject classification 2010 49j52 49j53

on chebyshev functions and klee functions abstract associated to a lower semicontinuous function one can define its proximal mapping and farthest mapping the function is called chebyshev klee if its proximal mapping farthest mapping is single valued everywhere we show that the function f is 1 hypoconvex if its proximal mapping p f is single valued when the function f is bounded below and p f is single valued for every 0 the function must be convex similarly we show that the function f is 1 strongly convex if the farthest mapping q f is single valued when the function is the indicator function of a set this recovers the well known chebyshev problem and klee problem in r n we also give an example illustrating that a continuous proximal mapping farthest mapping needs not be locally lipschitz which answers one open question by hare and poliquin

invariant sets and lyapunov pairs for differential inclusions with maximal monotone operators we give different conditions for the invariance of closed sets with respect to differential inclusions governed by a maximal monotone operator defined on hilbert spaces which is subject to a lipschitz continuous perturbation depending on the state these sets are not necessarily weakly closed as in 3 4 while the invariance criteria are still written by using only the data of the system so no need to the explicit knowledge of neither the solution of this differential inclusion nor the semi group generated by the maximal monotone operator these invariant viability results are next applied to derive explicit criteria for a lyapunov pairs of lower semi continuous not necessarily weakly lsc functions associated to these differential inclusions the lack of differentiability of the candidate lyapunov functions and the consideration of general invariant sets possibly not convex or smooth are carried out by using techniques from nonsmooth analysis

an implementation of karmarkar s algorithm for linear programming this paper describes the implementation of power series dual affine scaling variants of karmarkar s algorithm for linear programming based on a continuous version of karmarkar s algorithm two variants resulting from first and second order approximations of the continuous trajectory are implemented and tested linear programs are expressed in an inequality form which allows for the inexact computation of the algorithm s direction of improvement resulting in a significant computational advantage implementation issues particular to this family of algorithms such as treatment of dense columns are discussed the code is tested on several standard linear programming problems and compares favorably with the simplex codeminos 4 0

estimating tangent and normal cones without calculus we study simple estimates of the tangent and normal cones to a set with a nonsmooth boundary we seek estimates that are robust relative to the base point and that depend only on collections of nearby points known to lie inside and outside of the set

super efficiency of vector optimization in banach spaces using variational analysis in terms of the clarke normal cone we consider super efficiency of vector optimization in banach spaces we establish some characterizations for super efficiency in particular dropping the assumption that the ordering cone has a bounded base we extend a result in borwein and zhuang j m borwein d zhuang super efficiency in vector optimization trans amer math soc 338 1993 105 122 to the nonconvex setting 2006 elsevier inc all rights reserved

a lack of fit test for quantile regression we propose an omnibus lack of fit test for linear or nonlinear quantile regression based on a cusum process of the gradient vector the test does not involve nonparametric smoothing but is consistent for all nonparametric alternatives without any moment conditions on the regression error in addition the test is suitable for detecting the local alternatives of any order arbitrarily close to n 1 2 from the null hypothesis the limiting distribution of the proposed test statistic is non gaussian but can be characterized by a gaussian process we propose a simple sequential resampling scheme to carry out the test whose nominal levels are well approximated in our empirical study for

subsmooth sets functional characterizations and related concepts prox regularity of a set poliquin rockafellar thibault 2000 or its global version proximal smoothness clarke stern wolenski 1995 plays an important role in variational analysis not only because it is associated with some fundamental properties as the local continuous differentiability of the function dist c or the local uniqueness of the projection mapping but also because in the case where c is the epigraph of a locally lipschitz function it is equivalent to the weak convexity lower c 2 property of the function in this paper we provide an adapted geometrical concept called sub smoothness which permits an epigraphic characterization of the approximate convex functions or lower c 1 property subsmooth sets turn out to be naturally situated between the classes of prox regular and of nearly radial sets this latter class has been recently introduced by lewis in 2002 we hereby relate it to the mifflin semismooth functions

mixed semicontinuous perturbations of nonconvex sweeping processes in this paper we prove a theorem on the existence of a global solution of a differential inclusion governed by a class of nonconvex sweeping processes with unbounded pertubations the perturbations are not required to be convex valued

the θ exterior sphere condition φ convexity and local semiconcavity abstract we consider sets s r n satisfying a locally uniform exterior sphere condition called the exterior sphere condition it is shown that under wedgedness of s this coincides with the convexity property studied by colombo and marigonda 2005 5 this generalizes and improves upon the main result of nour et al 2009 10 where the constant radius case was considered we also generalize certain known results concerning the regularity of the distance function associated with s as well as semiconcavity and other properties of functions with hypographs satisfying the exterior sphere condition

projection methods for a generalized system of nonconvex variational inequalities with different nonlinear operators abstract in this paper we introduce and consider a new generalized system of nonconvex variational inequalities with different nonlinear operators we establish the equivalence between the generalized system of nonconvex variational inequalities and the fixed point problems using the projection technique this equivalent alternative formulation is used to suggest and analyze a general explicit projection method for solving the generalized system of nonconvex variational inequalities our results can be viewed as a refinement and improvement of the previously known results for variational inequalities

global convergence of the affine scaling algorithm for convex quadratic programming in this paper we give a global convergence proof of the second order affine scaling algorithm for convex quadratic programming problems where the new iterate is the point which minimizes the objective function over the intersection of the feasible region with the ellipsoid centered at the current point and whose radius is a fixed fraction beta in 0 1 of the radius of the largest scaled ellipsoid inscribed in the nonnegative orthant the analysis is based on the local karmarkar potential function introduced by tsuchiya for any beta in 0 1 and without assuming any nondegeneracy assumption on the problem it is shown that the sequences of primal iterates and dual estimates converge to optimal solutions of the quadratic program and its dual respectively

on the asymptotics of constrained local m estimators we discuss in this paper asymptotics of locally optimal solutions of maximum likelihood and more generally m estimation procedures in cases where the true value of the parameter vector lies on the boundary of the parameter set s we give a counterexample showing that regularity of s in the sense of clarke is not sufficient for asymptotic equivalence of n consistent locally optimal m estimators we argue further that stronger properties such as so called near convexity or prox regularity of s are required in order to ensure that any two n consistent locally optimal m estimators have the same asymptotics

on reflecting boundary problem for optimal control this paper deals with mayer s problem for controlled systems with reflection on the boundary of a closed subset k the main result is the characterization of the possibly discontinuous value function in terms of a unique solution in a suitable sense to a partial differential equation of hamilton jacobi bellman type

a nonsmooth maximum principle for a controlled nonconvex sweeping process abstract for an optimal control problem governed by a controlled nonconvex sweeping process we provide using an exponential penalization technique existence of solution and nonsmooth necessary conditions in the form of the pontryagin maximum principle our results generalize known theorems in the literature including those in 3 and 23 in several directions indeed the main feature in our sweeping process inclusion is the presence of the subdifferential of a function that is c 1 1 in the interior of its domain instead of the usual normal cone the subdifferential of the indicator function moreover no convexity is assumed on the function and its domain or on the set f t x u and our control mapping f is merely assumed to be lipschitz in x

generalised prox regularity in reflexive smooth banach spaces with smooth dual norm abstract in this paper we continue the study of the properties of the generalised projection on nonconvex sets in reflexive smooth banach spaces with smooth dual norm started by the first author in 9 we define a new class of nonconvex sets in banach spaces containing the convex closed sets and having various important properties whenever the space is assumed to be uniformly convex with smooth norm

weak regularity of functions and sets in asplund spaces abstract in this paper we study a new concept of weak regularity of functions and sets in asplund spaces we show that this notion includes prox regular functions functions whose subdifferential is weakly submonotone and amenable functions in infinite dimension we establish also that weak regularity is equivalent to mordukhovich regularity in finite dimension finally we give characterizations of the weak regularity of epi lipschitzian sets in terms of their local representations

quadratic convergence in a primal dual method we show that the mizuno todd ye o nl iteration predictor corrector primal dual interior point algorithm for linear programming is quadratically convergent our proof does not assume that the problems be nondegenerate we do not assume that the iterate generated by the algorithm be convergent an assumption common to all previous asymptotic convergence analysis

some new regularity properties for the minimal time function a minimal time problem with linear dynamics and convex target is considered it is shown essentially that the epigraph of the minimal time function t cdot is varphi convex i e it satisfies a kind of exterior sphere condition with locally uniform radius provided t cdot is continuous several regularity properties are derived from results in g colombo and a marigonda calc var partial differential equations 25 2005 pp 1 31 including twice a e differentiability of t cdot and local estimates on the total variation of dt

bv solutions of nonconvex sweeping process differential inclusion with perturbation abstract this paper is devoted to the study of differential inclusions particularly discontinuous perturbed sweeping processes in the infinite dimensional setting on the one hand the sets involved are assumed to be prox regular and to have a variation given by a function which is of bounded variation and right continuous on the other hand the perturbation satisfies a linear growth condition with respect to a fixed compact subset finally the case where the sets move in an absolutely continuous way is recovered as a consequence

existence and differentiability of metric projections in hilbert spaces this paper considers metric projections onto a closed subset s of a hilbert space if the set s is convex then it is well known that the corresponding metric projections always exist unique and directionally differentiable at boundary points of s these properties of metric projections are considered for possibly nonconvex sets s in particular existence and directional differentiability of metric projections for certain classes of sets are established and will be referred to as nearly convex sets

normality of the maximum principle for nonconvex constrained bolza problems we consider a bolza optimal control problem with state constraints it is well known that under some technical assumptions every strong local minimizer of this problem satisfies first order necessary optimality conditions in the form of a constrained maximum principle in general the maximum principle may be abnormal or even degenerate and so does not provide a sufficient information about optimal controls in the recent literature some sufficient conditions were proposed to guarantee that at least one maximum principle is nondegenerate cf a v arutyanov s m aseev investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints siam j control optim 35 1997 930 952 f rampazzo r b vinter a theorem on existence of neighbouring trajectories satisfying a state constraint with applications to optimal control ima 16 4 1999 335 351 f rampazzo r b vinter degenerate optimal control problems with state constraints siam j control optim 39 4 2000 989 1007 our aim is to show that actually conditions of a similar nature guarantee normality of every nondegenerate maximum principle in particular we allow the initial condition to be fixed and the state constraints to be nonsmooth to prove normality we use j yorke type linearization of control systems and show the existence of a solution to a linearized control system satisfying new state constraints defined in turn by linearization of the original set of constraints along an extremal trajectory

the problem of optimal control with reflection studied through a linear optimization problem stated on occupational measures abstract we obtain a linear programming characterization for the minimum cost associated with finite dimensional reflected optimal control problems in order to describe the value functions we employ an infinite dimensional dual formulation instead of using the characterization via hamilton jacobi partial differential equations in this paper we consider control problems with both infinite and finite horizons the reflection is given by the normal cone to a proximal retract set

uniform prox regularity of functions and epigraphs in hilbert spaces abstract we study the local uniform prox regularity of functions make the connection with the lower c 2 property and essentially with the prox regularity of epigraphs

the proximal normal formula in banach space approximation by proximal normals to the clarke generalized subdifferential for a distance function generated by a nonempty closed set and the normal cone to the set generated by the proximal normals are important tools in nonsmooth analysis we give simple general versions of such formulae in infinite dimensional banach spaces which satisfy different geometrical conditions our first class of spaces with uniformly g teaux differentiable norm includes the hilbert space case and the formulae is attained through dense subsets our second class of reflexive kadec smooth spaces is the most general for which such formulae can be obtained for all nonempty closed sets in the space our technique also allows us to establish the existence of solutions for a class of optimization problems substantially extending similar work of ekeland and lebourg

chaotic behavior of the affine scaling algorithm for linear programming the affine scaling algorithm for linear programming involves a step size parameter t that must be chosen in the interval 0 1 it is known that the algorithm converges to an optimal solution for values of t le 2 3 in this paper we examine the behavior of the algorithm for values of t 2 3 we show that for certain values of t in this range the algorithm can exhibit chaotic behavior

global convergence of a long step affine scaling algorithm for degenerate linear programming problems in this paper we present new global convergence results on a long step affine scaling algorithm obtained by means of the local karmarkar potential functions this development was triggered by dikin s interesting result on the convergence of the dual estimates associated with a long step afflne scaling algorithm for homogeneous lp problems with unique optimal solutions without requiring any assumption on degeneracy we show that moving a fixed proportion lambda up to two thirds of the way to the boundary at each iteration ensures convergence of the iterates to an interior point of the optimal face as well as the dual estimates to the analytic center of the dual optimal face where the asymptotic reduction rate of the value of the objective function is 1 lambda we also give an example showing that this result is tight to obtain convergence of the dual estimates to the analytic center of the dual optimal face

nonconvex policy search using variational inequalities policy search is a class of reinforcement learning algorithms for finding optimal policies in control problems with limited feedback these methods have been shown to be successful in high dimensional problems such as robotics control though successful current methods can lead to unsafe policy parameters that potentially could damage hardware units motivated by such constraints we propose projection based methods for safe policies these methods however can handle only convex policy constraints in this letter we propose the first safe policy search reinforcement learner capable of operating under nonconvex policy constraints this is achieved by observing for the first time a connection between nonconvex variational inequalities and policy search problems we provide two algorithms mann and two step iteration to solve the above problems and prove convergence in the nonconvex stochastic setting finally we demonstrate the performance of the algorithms on six benchmark dynamical systems and show that our new method is capable of outperforming previous methods under a variety of settings

censored regression quantiles abstract the object of this paper is to demonstrate how the lad estimation method for the censored regression model can be extended to more general quantiles in this paper the form of the conditional quantiles for the censored regression models is heuristically derived and discussed the resulting estimators of the regression coefficients which include the censored lad estimator as a special case are shown to be consistent and asymptotically normally distributed under appropriately translated versions of the corresponding assumptions for the former approach consistent estimation of the asymptotic covariance matrix when the error terms are i i d is also treated the paper discussed how several quantile estimators might be combined to improve efficiency when the error terms are identically distributed and how tests of homoskedasticity and symmetry of the error distribution can be constructed using particular differences of the estimated coefficients

prox regular sets and epigraphs in uniformly convex banach spaces various regularities and other properties we continue the study of prox regular sets that we began in a previous work in the setting of uniformly convex banach spaces endowed with a norm both uniformly smooth and uniformly convex e g l p w m p spaces we prove normal and tangential regularity properties for these sets and in particular the equality between mordukhovich and proximal normal cones we also compare in this setting the proximal normal cone with different holderian normal cones depending on the power types s q of moduli of smoothness and convexity of the norm in the case of sets that are epigraphs of functions we show that j primal lower regular functions have prox regular epigraphs and we compare these functions with poliquin s primal lower nice functions depending on the power types s q of the moduli the preservation of prox regularity of the intersection of finitely many sets and of the inverse image is obtained under a calmness assumption a conical derivative formula for the metric projection mapping of prox regular sets is also established among other results of the paper it is proved that the attouch wets convergence preserves the uniform r prox regularity property and that the metric projection mapping is in some sense continuous with respect to this convergence for such sets

prox regular functions in hilbert spaces this paper studies the prox regularity concept for functions in the general context of hilbert space in particular a subdifferential characterization is established as well as several other properties it is also shown that the moreau envelopes of such functions are continuously differentiable

sweeping by a continuous prox regular set abstract the evolution problem known as sweeping process is considered for a class of nonconvex sets called prox regular or convex assuming essentially that such sets contain in the interior a suitable subset and move continuously w r t the hausdorff distance we prove local and global existence as well as uniqueness of solutions which are continuous functions with bounded variation some examples are presented

least absolute deviations estimation for the censored regression model this paper proposes an alternative to maximum likelihood estimation of the parameters of the censored regression or censored tobit model the proposed estimator is a generalization of least absolute deviations estimation for the standard linear model and unlike estimation methods based on the assumption of normally distributed error terms the estimator is consistent and asymptotically normal for a wide class of error distributions and is also robust to heteroscedasticity the paper gives the regularity conditions and proofs of these large sample results and proposes classes of consistent estimators of the asymptotic covariance matrix for both homoscedastic and heteroscedastic disturbances

global convergence of the affine scaling methods for degenerate linear programming problems in this paper we show the global convergence of the affine scaling methods without assuming any condition on degeneracy the behavior of the method near degenerate faces is analyzed in detail on the basis of the equivalence between the affine scaling methods for homogeneous lp problems and karmarkar s method it is shown that the step size 1 8 where the displacement vector is normalized with respect to the distance in the scaled space is sufficient to guarantee the global convergence of the affine scaling methods

convergence of subdifferentials and normal cones in locally uniformly convex banach space abstract in this paper we study the behaviour of normal cones and subdifferentials with respect to two types of convergence of sets and functions mosco and attouch wets convergences our analysis is devoted to proximal frechet and mordukhovich limiting normal cones and subdifferentials the results obtained can be seen as extensions of the attouch theorem to the context of non convex functions on locally uniformly convex banach space they also generalize to sequences of subsmooth sets or functions various results in the literature

a variation on karmarkar s algorithm for solving linear programming problems the algorithm described here is a variation on karmarkar s algorithm for linear programming it has several advantages over karmarkar s original algorithm in the first place it applies to the standard form of a linear programming problem and produces a monotone decreasing sequence of values of the objective function the minimum value of the objective function does not have to be known in advance secondly in the absence of degeneracy the algorithm converges to an optimal basic feasible solution with the nonbasic variables converging monotonically to zero this makes it possible to identify an optimal basis before the algorithm converges

a numerical scheme for a class of sweeping processes the aim of this paper is to study a whole class of first order differential inclusions which fit into the framework of perturbed sweeping process by uniformly prox regular sets after obtaining well posedness results we propose a numerical scheme based on a prediction correction algorithm and we prove its convergence finally we apply these results to a problem coming from the modelling of crowd motion

existence of solutions to the nonconvex sweeping process abstract in this note we prove the existence of solutions for the sweeping process problem x t nc t x t a e x t c t x 0 x0 c 0 where c is an arbitrary hausdorff lipschitzean multifunction from i 0 t onto the set of nonempty closed subsets of r d this generalizes a well known result of j j moreau 1971 in sem d analyse convexe montpelier exp 15 in the convex case

an extension of attouch s theorem and its application to second order epi differentiation of convexly composite functions in 1977 hedy attouch established that a sequence of closed proper convex functions epi converges to a convex function if and only if the graphs of the subdierentials converge in the mosco sense to the subdifferential of the limiting function and roughly speaking there is a condition that fixes the constant of integration we show that the theorem is valid if instead one considers functions that are the composition of a closed proper convex function with a twice continuously differentiable mapping in addition a constraint qualification is imposed

local differentiability of distance functions recently clarke stern and wolenski characterized in a hilbert space the closed subsets c for which the distance function dc is continuously differentiable everywhere on an open tube of uniform thickness around c here a corresponding local theory is developed for the property of dc being continuously differentiable outside of c on some neighborhood of a point x c this is shown to be equivalent to the prox regularity of c at x which is a condition on normal vectors that is commonly fulfilled in variational analysis and has the advantage of being verifiable by calculation additional characterizations are provided in terms of dc being locally of class c 1 or such that dc 2 is convex around x for some 0 prox regularity of c at x corresponds further to the normal cone mapping nc having a hypomonotone truncation around x and leads to a formula for pc by way of nc the local theory also yields new insights on the global level of the clarke stern wolenski results and on a property of sets introduced by shapiro as well as on the concept of sets with positive reach considered by federer in the finite dimensional setting

hypographs satisfying an external sphere condition and the regularity of the minimum time function abstract we prove that if the hypograph of a continuous function f admits at every boundary point a supporting ball then it has essentially positive reach i e the hypograph of the restriction of f outside a closed set of zero measure has locally positive reach hence such a function enjoys some properties of a concave function in particular a e twice differentiability we apply this result to a minimum time problem in the case of a nonlinear smooth dynamics and a target satisfying internal sphere condition

superlinear convergence of the affine scaling algorithm in this paper we show that a variant of the long step affine scaling algorithm with variable stepsizes is two step superlinearly convergent when applied to general linear programming lp problems superlinear convergence of the sequence of dual estimates is also established for homogeneous lp problems having the origin as the unique optimal solution we also show that 2 3 is a sharp upper bound on the fixed stepsize that provably guarantees that the sequence of primal iterates converge to the optimal solution along a unique direction of approach since the point to which the sequence of dual estimates converge depend on the direction of approach of the sequence of primal iterates this result gives a plausible but not accurate theoretical explanation for why 2 3 is a sharp upper bound on the fixed stepsize that guarantees the convergence of the dual estimates

a non convex setup for multivalued differential equations driven by oblique subgradients abstract the paper proves the existence and uniqueness of the solution for the following multivalued deterministic variational inequality with oblique subgradients considered in a non convex domain x t h t x t x t g t x t t 0 x 0 x 0 where stands for the frechet subdifferential of the semiconvex function and the matrix application x h x is a lipschitz mapping the presence of the oblique reflection brought by the term h leads to the use of different techniques comparing to the cases of standard reflection in non convex domains or oblique reflection in convex domains the last section of the article is focused on the qualitative analysis of a non convex skorohod problem with generalized reflection and as applications stochastic variational inequalities driven by oblique frechet subgradients are addressed

existence of solutions for second order perturbed nonconvex sweeping process abstract we prove a theorem on the existence solutions for a second order differential inclusion governed by a class of nonconvex sweeping process our result improves and generalizes many results from the literature of sweeping process

iterative algorithms for a new class of extended general nonconvex set valued variational inequalities abstract some new classes of extended general nonconvex set valued variational inequalities and the extended general wiener hopf inclusions are introduced by the projection technique equivalence between the extended general nonconvex set valued variational inequalities and the fixed point problems as well as the extended general nonconvex wiener hopf inclusions is proved then by using this equivalent formulation we discuss the existence of solutions of the extended general nonconvex set valued variational inequalities and construct some new perturbed finite step projection iterative algorithms with mixed errors for approximating the solutions of the extended general nonconvex set valued variational inequalities we also verify that the approximate solutions obtained by our algorithms converge to the solutions of the extended general nonconvex set valued variational inequalities the results presented in this paper extend and improve some known results from the literature

convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities in this paper we analyze and discuss the well posedness of two new variants of the so called sweeping process introduced by moreau in the early 70s moreau in sem anal convexe montpellier 1971 with motivation in plasticity theory the first new variant is concerned with the perturbation of the normal cone to the moving convex subset c t c t supposed to have a bounded variation by a lipschitz mapping under some assumptions on the data we show that the perturbed differential measure inclusion has one and only one right continuous solution with bounded variation the second variant for which a large analysis is made concerns a first order sweeping process with velocity in the moving set c t c t this class of problems subsumes as a particular case the evolution variational inequalities widely used in applied mathematics and unilateral mechanics duvaut and lions in inequalities in mechanics and physics springer berlin 1976 assuming that the moving subset c t c t has a continuous variation for every t in 0 t t 0 t with c 0 c 0 bounded we show that the problem has at least a lipschitz continuous solution the well posedness of this class of sweeping process is obtained under the coercivity assumption of the involved operator we also discuss some applications of the sweeping process to the study of vector hysteresis operators in the elastoplastic model krej i in eur j appl math 2 281 292 1991 to the planning procedure in mathematical economy henry in j math anal appl 41 179 186 1973 and cornet in j math anal appl 96 130 147 1983 and to nonregular electrical circuits containing nonsmooth electronic devices like diodes acary et al nonsmooth modeling and simulation for switched circuits lecture notes in electrical engineering springer new york 2011 the theoretical results are supported by some numerical simulations to prove the efficiency of the algorithm used in the existence proof our methodology is based only on tools from convex analysis like other papers in this collection we show in this presentation how elegant modern convex analysis was influenced by moreau s seminal work

existence of generalized equilibria we consider the problem of the existence of equilibria generalized equilibria and 8xed points of a correspondence f de8ned on a compact subset m of rn with values in rn when the set m is neither assumed to be convex nor smooth 1 the general framework of the article is the following let f be a correspondence from m to rn that is a map from m to the set of all the subsets of rn the correspondence f is said to be upper semicontinuous u s c if the set x m f x v is open in m for every open set v rn let n be a normal cone correspondence i e at this stage n is a correspondence from m to rn such that for every x m n x

approximation and regularization of lipschitz functions convergence of the gradients we examine the possible extensions to the lipschitzian setting of the classical result on c 1 convergence first approximation if a sequence f n of functions of class c 1 from r n to r converges uniformly to a function f of class c 1 then the gradient of f is a limit of gradients of f n in the sense that graph f c lim inf n graph f n second regularization the functions f n can be chosen to be of class c and c 1 converging to f in the sense that lim n f n f f n f 0 in other words the space of c functions is dense in the space of c 1 functions endowed with the c 1 pseudo norm we first deepen the properties of warga s counterexample 1981 for the extension of the approximation part to the lipschitzian setting this part cannot be extended even if one restricts the approximation schemes to the classical convolution and the lasry lions regularization we thus make more precise various results in the literature on the convergence of sub differentials we then show that the regularization part can be extended to the lipschitzian setting namely if f r n r is a locally lipschitz function we build a sequence of smooth functions f n n n such that 0 in other words the space of c functions is dense in the space of locally lipschitz functions endowed with an appropriate lipschitz pseudo distance up to now rockafellar and wets 1998 have shown that the convolution procedure permits us to have the equality lim sup n graph f n graph f which cannot provide the exactness of our result as a consequence we obtain a similar result on the regularization of epi lipschitz sets with both functional and set parts we improve previous results in the literature on the regularization of functions and sets

limiting behavior of the affine scaling continuous trajectories for linear programming problems we consider the continuous trajectories of the vector field induced by the primal affine scaling algorithm as applied to linear programming problems in standard form by characterizing these trajectories as solutions of certain parametrized logarithmic barrier families of problems we show that these trajectories tend to an optimal solution which in general depends on the starting point by considering the trajectories that arise from the lagrangian multipliers of the above mentioned logarithmic barrier families of problems we show that the trajectories of the dual estimates associated with the affine scaling trajectories converge to the so called centered optimal solution of the dual problem we also present results related to asymptotic direction of the affine scaling trajectories we briefly discuss how to apply our results to linear programs formulated in formats different from the standard form finally we extend the results to the primal dual amne scaling algorithm

quantile regression methods for recursive structural equation models abstract two classes of quantile regression estimation methods for the recursive structural equation models of chesher 2003 identification in nonseparable models econometrica 71 1405 1441 are investigated a class of weighted average derivative estimators based directly on the identification strategy of chesher is contrasted with a new control variate estimation method the latter imposes stronger restrictions achieving an asymptotic efficiency bound with respect to the former class an application of the methods to the study of the effect of class size on the performance of dutch primary school students shows that i reductions in class size are beneficial for good students in language and for weaker students in mathematics ii larger classes appear beneficial for weaker language students and iii the impact of class size on both mean and median performance is negligible

boundary regularity of reachable sets of control systems abstract the reachable set of a differential inclusion has nonsmooth boundary in general now we give sufficient conditions on the compact initial set and the differential inclusion x t f t x t u t a e so that this topological boundary has positive erosion i e it satisfies a uniform interior sphere condition the proof is based on the close relation between the semiconcave value function of a suitable mayer problem and its epigraph with positive erosion

finding best approximation pairs relative to a convex and prox regular set in a hilbert space we study the convergence of an iterative projection reflection algorithm originally proposed for solving what are known as phase retrieval problems in optics there are two features that frustrate any analysis of iterative methods for solving the phase retrieval problem nonconvexity and infeasibility the algorithm that we developed called relaxed averaged alternating reflections raar was designed primarily to address infeasibility though our strategy has advantages for nonconvex problems as well in the present work we investigate the asymptotic behavior of the raar algorithm for the general problem of finding points that achieve the minimum distance between two closed convex sets in a hilbert space with empty intersection and for the problem of finding points that achieve a local minimum distance between one closed convex set and a closed prox regular set also possibly nonintersecting the nonconvex theory includes and expands prior results limited to convex sets with nonempty intersection to place the raar algorithm in context we develop parallel statements about the standard alternating projections algorithm and gradient descent all of the various algorithms are unified as instances of iterated averaged alternating proximal reflectors applied to a sum of regularized maximal monotone mappings

convergence analysis of the projective scaling algorithm based on a long step homogeneous affine scaling algorithm in this paper we give a new convergence analysis of a projective scaling algorithm we consider a long step affine scaling algorithm applied to a homogeneous linear programming problem obtained from the original linear programming problem this algorithm takes a fixed fraction 2 3 of the way towards the boundary of the nonnegative orthant at each iteration the iteration sequence for the original problem is obtained by pulling back the homogeneous iterates onto the original feasible region with a conical projection which generates the same search direction as the original projective scaling algorithm at each iterate the recent convergence results for the long step affine scaling algorithm by the authors are applied to this algorithm to obtain some convergence results on the projective scaling algorithm specifically we will show i polynomiality of the algorithm with complexities of o nl and o n2l iterations for 2 3 and 2 3 respectively ii global covnergence of the algorithm when the optimal face is unbounded iii convergence of the primal iterates to a relative interior point of the optimal face iv convergence of the dual estimates to the analytic center of the dual optimal face and v convergence of the reduction rate of the objective function value to 1

second order sweeping process with a lipschitz perturbation abstract in this paper we prove a theorem concerning the existence of solutions for a second order sweeping process with a lipschitz single valued perturbation in hilbert space

generalized hessian properties of regularized nonsmooth functions we take up the question of second order expansions for a class of functions of importance in optimization namely moreau envelope regularizations of nonsmooth functions f it is shown that when f is prox regular which includes convex functions and the extended real valued functions representing problems of nonlinear programming the many second order properties that can be formulated around the existence and stability of expansions of the envelopes of f or of their gradient mappings are linked by surprisingly extensive lists of equivalences with each other and with generalized differentiation properties of f itself this clarifies the circumstances conducive to developing computational methods based on envelope functions such as second order approximations in nonsmooth optimization and variants of the proximal point algorithm the results establish that generalized second order expansions of moreau envelopes at least can be counted on in most situations of interest in finite dimensional optimization

a low complexity interior point algorithm for linear programming this paper describes an interior point algorithm for linear programming that is almost as simple as the affine scaling method and yet achieves the currently best complexity of o sqrt n t iterations to attain precision t the basic algorithm needs neither dual estimates nor lower bounds although its analysis is based on ye s results for the primal dual potential function some computationally preferable variants are also presented

boundary behavior of interior point algorithms in linear programming this paper studies the boundary behavior of some interior point algorithms for linear programming the algorithms considered are karmarkar s projective rescaling algorithm the linear rescaling algorithm which was proposed as a variation on karmarkar s algorithm and the logarithmic barrier technique the study includes both the continuous trajectories of the vector fields induced by these algorithms and also the discrete orbits it is shown that although the algorithms are defined on the interior of the feasible polyhedron they actually determine differentiable vector fields on the closed polyhedron conditions are given under which a vector field gives rise to trajectories that each visit the neighborhoods of all the vertices of the klee minty cube the linear rescaling algorithm satisfies these conditions thus limits of such trajectories obtained when a starting point is pushed to the boundary may have an exponential number of breakpoints it is shown that limits of projective rescaling trajectories may have a linear number of such breakpoints however projective rescaling trajectories may visit the neighborhoods of linearly many vertices the behavior of the linear rescaling algorithm near vertices is analyzed it is proved that all the trajectories have a unique asymptotic direction of convergence to the optimum

approximate convexity and submonotonicity abstract it is shown that a locally lipschitz function is approximately convex if and only if its clarke subdifferential is a submonotone operator consequently in finite dimensions the class of locally lipschitz approximately convex functions coincides with the class of lower c 1 functions directional approximate convexity is introduced and shown to be a natural extension of the class of lower c 1 functions in infinite dimensions the following characterization is established a multivalued operator is maximal cyclically submonotone if and only if it coincides with the clarke subdifferential of a locally lipschitz directionally approximately convex function which is unique up to a constant furthermore it is shown that in asplund spaces every regular function is generically approximately convex

using a signed distance function for the simulation of metal forming processes formulation of the contact condition and mesh adaptation from a lagrangian approach to an eulerian approach this paper proposes to use the metric properties of the distance function between two bodies in contact or gap function in simulations involving contact problems first the normal vectors which are involved in the formulation of the contact condition are defined through the gradient of this distance function this definition avoids to deal with the numerical penetration parameter which is generally introduced otherwise furthermore it allows the contact problem to be extended in a simple way to an eulerian formulation second this paper investigates two mesh adaptation strategies based on the properties of the distance function the first strategy consists in building a size map according to the values of this function in order to refine locally the mesh and consequently to improve the description of the contact surface the second strategy consists in adapting locally the mesh to the geometry of the contact surface this anisotropic adaptation is performed by constructing a metric map that allows the mesh size to be imposed in the direction of the distance function gradient a lot of elements are saved when compared with the isotropic case throughout this paper many numerical simulations are presented in the context of the forging process the deformable material is pressed between two rigid tools furthermore the algorithm used to calculate the signed distance to a surface mesh is detailed in appendix of this paper

integration of subdifferentials of lower semicontinuous functions on banach spaces abstract this paper studies the lower semicontinuous functions on banach spaces whose subdifferentials are not disassociated from several conditions are given to ensure that two lower semicontinuous functions with the same subdifferential are equal up to an additive constant

integration of subdifferentials of nonconvex functions in nonsmooth analysis and optimization subgradients come in many different flavors e g approximate dini proximal clarke generalized see 2 5 6 12 14 191 these subgradients are important and valuable tools however many questions remain unsolved concerning the exact link between the function and its subgradients for instance can two functions not differing by an additive constant have the same subgradients in this paper we study the fundamental problem of determining functions that can be recovered up to an additive constant from the knowledge of their subgradients this integration problem is not very well understood and very few functions or classes of functions are known to be recoverable from their subgradients in section 4 we show that if the basic constraint qualification holds at r then the composition of a closed i e lowersemicontinuous proper convex function with a twice continuously differentiable mapping is determined up to an additive constant by its generalized subgradients actually in this case all above mentioned subgradients are the same beside the obvious theoretical interest of this integration problem it is our hope or perhaps our long term goal that once this problem is better understood we can then tackle the question of uniqueness of solutions to generalized differential equations involving subgradients in place of partial derivatives an example of such an equation that well deserves study is the extended hamilton jacobi equation used in optimal control see clarke 2 let us also mention that a problem similar to the integration problem is the one of determining the set valued mappings that are in fact subgradient set valued mappings uniqueness is not mandatory for a contribution to this problem see janin 8 before we look at some of the known cases where the function can be recovered from its subgradients let us look at some negative examples it is clear that not every function can be recovered up to an additive constant from its subgradients we only need to look at the following two functions 0 x50 0 x10 f x 1 x o g x 2 x o

partial smoothness tilt stability and generalized hessians we compare two recent variational analytic approaches to second order conditions and sensitivity analysis for nonsmooth optimization we describe a broad setting where computing the generalized hessian of mordukhovich is easy in this setting the idea of tilt stability introduced by poliquin and rockafellar is equivalent to a classical smooth second order condition

global convergence property of the affine scaling methods for primal degenerate linear programming problems in this paper we investigate the global convergence property of the affine scaling method under the assumption of dual nondegeneracy the behavior of the method near degenerate vertices is analyzed in detail on the basis of the equivalence between the affine scaling methods for homogeneous lp problems and karmarkar s method it is shown that the step size 1 8 where the displacement vector is normalized with respect to the distance in the scaled space is sufficient to guarantee the global convergence for dual nondegenerate lp problems the result can be regarded as a counterpart to dikin s global convergence result on the affine scaling method assuming primal nondegeneracy

linear regularity for a collection of subsmooth sets in banach spaces using variational analysis we study the linear regularity for a collection of finitely many closed sets in particular we extend duality characterizations of the linear regularity for a collection of finitely many closed convex sets to the possibly nonconvex setting moreover the sharpest linear regularity constant can also be dually represented under the subsmoothness assumption

the consistency of nonlinear regression minimizing the l_1 norm we consider conditions in a nonlinear regression model for the consistency of the estimator obtained by minimizing the l 1 norm i e the sum of absolute deviations

partial extensions of attouch s theorem with applications to proto derivatives of subgradient mappings attouch s theorem which gives on a reflexive banach space the equivalence between the mosco epi convergence of a sequence of convex functions and the graph convergence of the associated sequence of subgradients has many important applications in convex optimization in particular generalized derivatives have been defined in terms of the epi convergence or graph convergence of certain difference quotient mappings and attouch s theorem has been used to relate these various generalized derivatives these relations can then be used to study the stability of the solution mapping associated with a parameterized family of optimization problems we prove in a hilbert space several partial extensions of attouch s theorem to functions more general than convex these functions are called primal lower nice furthermore we use our extensions to derive a relationship between the second order epi derivatives of primal lower nice functions and the proto derivative of their associated subgradient mappings

on the convergence of the affine scaling algorithm the affine scaling algorithm first proposed by dikin is presently enjoying great popularity as a potentially effective means of solving linear programs an outstanding question about this algorithm concerns its convergence in the presence of degeneracy in this paper we give new convergence results for this algorithm that do not require any non degeneracy assumption on the problem in particular we show that if the stepsize choice of either dikin or barnes or vanderbei et al is used then the algorithm generates iterates that converge at least linearly with a convergence ratio of 1 beta sqrt n wheren is the number of variables and 0 1 is a certain stepsize ratio for one particular stepsize choice which is an extension of that of barnes we show that the cost of the limit point is within o 1 of the optimal cost and for sufficiently small roughly proportional to how close the cost of the nonoptimal vertices are to the optimal cost is exactly optimal we prove the latter result by using an unusual proof technique that of analyzing the ergodic convergence of the corresponding dual vectors for the special case of network flow problems with integer data we show that it suffices to take 1 6mc wherem is the number of constraints andc is the sum of the cost coefficients to attain exact optimality

letter to the editor extragradient methods for solving nonconvex variational inequalities in this paper we introduce and consider a new class of variational inequalities which are called the nonconvex variational inequalities using the projection technique we suggest and analyze an extragradient method for solving the nonconvex variational inequalities we show that the extragradient method is equivalent to an implicit iterative method the convergence of which requires only pseudo monotonicity a weaker condition than monotonicity this clearly improves on the previously known result our method of proof is very simple as compared with other techniques

regularization of differential variational inequalities with locally prox regular sets this paper studies for a differential variational inequality involving a locally prox regular set a regularization process with a family of classical differential equations whose solutions converge to the solution of the differential variational inequality the concept of local prox regularity will be termed in a quantified way as r alpha prox regularity

strong and weak convexity of sets and functions in this paper we study two classes of sets strongly and weakly convex sets for each class we derive a series of properties which involve either the concept of supporting ball an obvious extension of the concept of supporting hyperplane or the normal cone to the set we also study a class of functions denoted convex which satisfy for arbitrary points x1 and x2 and any value 0 1 the classical inequality of convex functions up to a term 1 x1 x2 2 depending on the sign of the constant the function is said to be strongly or weakly convex we provide characteristic properties of this class of sets and we relate it to strongly and weakly convex sets via the epigraph and the level sets finally we give three applications a separation theorem a sufficient condition for global optimum of a nonconvex programming problem and a sufficient geometrical condition for a set to be a manifold

reflected differential games this paper concerns the existence of a possible discontinuous value for a zero sum two player reflected differential game under isaacs condition we characterize the value function as the unique solution in a suitable sense to a variational inequality namely the hamilton jacobi isaacs differential inclusion

an interior point algorithm for nonlinear quantile regression abstract a new algorithm for computing quantile regression estimates for problems in which the response function is nonlinear in parameters is described the nonlinear l 1 estimation problem is a special median case the algorithm is closely related to recent developments on interior point methods for solving linear programs performance of the algorithm on a variety of test problems including the censored linear quantile regression problem of powell 1986 is reported

sweeping process with regular and nonregular sets abstract differential inclusions involving the normal cone to a moving set are investigated a special attention is paid to the sweeping process associated with sets for which no regularity assumption is required