approximating saddle points as equilibria of differential inclusions abstract motivated by nonlinear programming we study a process involving continuous adjustments of both primal and dual variables nonsmoothness in problem data or simple restrictions on variables make the velocity of the process possibly nonunique the process obtained is shown to converge locally under strict monotonicity conditions examples are given in terms of augmented lagrangians the results are believed to provide a basis for the construction of algorithms

on identification of active constraints ii the nonconvex case in this paper the results of burke and more 5 on the identification of active constraints are extended to the nonconvex constrained nonlinear programming problem the approach is motivated by the geometric structure of a certain polyhedral convex linearization of the constraint region at each iteration as in burke and more 5 questions of constraint identification are couched in terms of the faces of these polyhedra the main result employs a nondegeneracy condition due to dunn 7 and the linear independence condition to obtain a characterization of those algorithms that identify the optimal active constraints in a finite number of iterations the role of the linear independence condition is carefully examined and it is argued that it is required within the context of a first order theory of constraint identification in conclusion the characterization theorem is applied to the wilson han powell sequential quadratic programming algorithm j optim theory appl 22 1977 pp 297 309 proceedin

nondegenerate solutions and related concepts in affine variational inequalities the notion of a strictly complementary solution for complementarity problems is extended to that of a nondegenerate solution of variational inequalities several equivalent formulations of nondegeneracy are given in the affine case an existence theorem for a nondegenerate solution is given in terms of several related concepts which are shown to be equivalent in this context these include a weak sharp minimum the minimum principle sufficiency and error bounds the gap function associated with the variational inequality plays a central role in this existence theorem

projected newton methods and optimization of multicommodity flows a superlinearly convergent newton like method for linearly constrained optimization problems is adapted for solution of multicommodity network flow problems of the type arising in communication and transportation networks we show that the method can be implemented approximately by making use of conjugate gradient iterations without the need to compute explicitly the hessian matrix preliminary computational results suggest that this type of method is capable of yielding highly accurate solutions of nonlinear multicommodity flow problems far more efficiently than any of the methods available at present

stopping rules and backward error analysis for bound constrained optimization termination criteria for the iterative solution of bound constrained optimization problems are examined in the light of backward error analysis it is shown that the problem of determining a suitable perturbation on the problem s data corresponding to the definition of the backward error is analytically solvable under mild assumptions moreover a link between existing termination criteria and this solution is clarified indicating that some standard measures of criticality may be interpreted in the sense of backward error analysis the backward error problem is finally considered from the multicriteria optimization point of view and some numerical illustration is provided

convergence of the iterates of descent methods for analytic cost functions in the early eighties lojasiewicz in seminari di geometria 1982 1983 universita di bologna istituto di geometria dipartimento di matematica 1984 pp 115 117 proved that a bounded solution of a gradient flow for an analytic cost function converges to a well defined limit point in this paper we show that the iterates of numerical descent algorithms for an analytic cost function share this convergence property if they satisfy certain natural descent conditions the results obtained are applicable to a broad class of optimization schemes and strengthen classical weak convergence results for descent methods to strong limit point convergence for a large class of cost functions of practical interest the result does not require that the cost has isolated critical points and requires no assumptions on the convexity of the cost nor any nondegeneracy conditions on the hessian of the cost at critical points

pattern search algorithms for bound constrained minimization we present a convergence theory for pattern search methods for solving bound constrained nonlinear programs the analysis relies on the abstract structure of pattern search methods and an understanding of how the pattern interacts with the bound constraints this analysis makes it possible to develop pattern search methods for bound constrained problems while only slightly restricting the flexibility present in pattern search methods for unconstrained problems we prove global convergence despite the fact that pattern search methods do not have explicit information concerning the gradient and its projection onto the feasible region and consequently are unable to enforce explicitly a notion of sufficient feasible decrease

the henselization of a valued division algebra the main result of this paper is the following if f v is a valued field fh its henselization d a division algebra with center f and d f then v extends to d iff d ffh is a division algebra it is also shown that if p char f p 1if char f 0 then d f d f d f pb for some nonnegative integer b

new active set identification for general constrained optimization and minimax problems abstract the purpose of this paper is to discuss the problem of identifying the active constraints for general constrained nonlinear programming and constrained minimax problems at an isolated local solution facchinei fischer and kanzow 4 proposed an effective technique which can identify the active set in a neighborhood of an isolated local solution for nonlinear programming and han jian and li 5 improved and extended this to minimax problems in this work a new active constraint identification set is constructed not only is it tighter than the previous two identification sets but also it can be used effectively in penalty algorithms without strict complementarity and linear independence it is shown that the new identification technique can accurately identify the active constraints of nonlinear programming and constrained minimax problems some numerical examples illustrate the identification technique

one dimensional three state quantum walk we study a generalized hadamard walk in one dimension with three inner states the particle governed by the three state quantum walk moves in superposition both to the left and to the right according to the inner state in addition to these two degrees of freedom it is allowed to stay at the same position we calculate rigorously the wave function of the particle starting from the origin for any initial qubit state and show the spatial distribution of probability of finding the particle in contrast with the hadamard walk with two inner states on a line the probability of finding the particle at the origin does not converge to zero even after infinite time steps except special initial states this implies that the particle is trapped near the origin after a long time with high probability

non monotone trust region algorithms for nonlinear optimization subject to convex constraints this paper presents two new trust region methods for solving nonlinear optimization problems over convex feasible domains these methods are distinguished by the fact that they do not enforce strict monotonicity of the objective function values at successive iterates the algorithms are proved to be convergent to critical points of the problem from any starting point extensive numerical experiments show that this approach is competitive with the lancelot package

preconditioned conjugate gradient algorithms for nonconvex problems with box constraints the paper describes new conjugate gradient algorithms for large scale nonconvex problems with box constraints in order to speed up convergence the algorithms employ scaling matrices which transform the space of original variables into the space in which hessian matrices of the problem s functionals have more clustered eigenvalues this is done by applying limited memory bfgs updating matrices once the scaling matrix is calculated the next few conjugate gradient iterations are performed in the transformed space the box constraints are treated efficiently by the projection we also present a limited memory quasi newton method which is a special version of our general algorithm the presented algorithms have strong global convergence properties in particular they identify constraints active at a solution in a finite number of iterations we believe that they are competitive to the l bfgs b method and present some numerical results which support our claim

modified subspace limited memory bfgs algorithm for large scale bound constrained optimization in this paper a subspace limited memory bfgs algorithm for solving large scale bound constrained optimization problems is developed it is modifications of the subspace limited memory quasi newton method proposed by ni and yuan q ni y x yuan a subspace limited memory quasi newton algorithm for large scale nonlinear bound constrained optimization math comput 66 1997 1509 1520 an important property of our proposed method is that more limited memory bfgs update is used under appropriate conditions the global convergence of the method is established the implementations of the method on cute test problems are presented which indicate the modifications are beneficial to the performance of the algorithm

error bound and reduced gradient projection algorithms for convex minimization over a polyhedral set consider the problem of minimizing over a polyhedral set the composition of an affine mapping with a strongly convex differentiable function the polyhedral set is expressed as the intersection of an affine set with a simpler polyhedral set and a new local error bound for this problem based on projecting the reduced gradient associated with the affine set onto the simpler polyhedral set is studied a class of reduced gradient projection algorithms for solving the case where the simpler polyhedral set is a box is proposed and this bound is used to show that algorithms in this class attain a linear rate of convergence included in this class are the gradient projection algorithm of goldstein and levitin and poljak and an algorithm of bertsekas a new algorithm in this class reminiscent of active set algorithms is also proposed some of the results presented here extend to problems where the objective function is extended real valued and to variational inequality problems

periodicity of lively quantum walks on cycles with generalized grover coin abstract in this paper we extend the study of three state lively quantum walks on cycles by considering the coin operator as linear sum of permutation matrices which is a generalization of the grover matrix first we provide a complete characterization of orthogonal matrices of order 3 3 which are linear sum of permutation matrices consequently we determine several groups of complex real and rational orthogonal matrices we establish that an orthogonal matrix of order 3 3 is a linear sum of permutation matrices if and only if it is permutative finally we determine period of lively quantum walk on cycles when the coin operator belongs to the orthogonal group of real linear sum of permutation matrices

on the identification property of a projected gradient method the authors study the convergence properties of a projected gradient algorithm for the general problem min f x x in omega where f r n to r is a mapping continuously differentiable on a closed convex set omega subseteq r n the algorithm which requires only one projection per iteration is a special version of the method of projection of the gradient by demyanov and rubinov approximate methods in optimization problems elsevier new york 1970 where the step choice is made according to a scheme similar to the one used by calamai and more math programming 39 1987 pp 93 116 the authors are mainly interested in analysing the identification property of the algorithm for the case where the set omega is a polyhedron that is the ability to identify in a finite number of steps the face in which the final solution lies the convergence results that are shown are very similar to those shown in 6 for the standard projected gradient method

on dubrovin valuation rings in crossed product algebras let f be a field and let v be a valuation ring in f if a is a central simple f algebra then v can be extended to a dubrovin valuation ring in a in this paper we consider the structure of dubrovin valuation rings with center v in crossed product algebras k f g f where k f is a finite galois extension with galois group g unramified over v and f is a normalized two cocycle in the case where v is indecomposed in k we introduce a family of orders naturally associated to f examine their basic properties and determine which of these orders is dubrovin in the case where v is decomposed we determine the structure in the case of certain special discrete finite rank valuations

pattern search methods for linearly constrained minimization we extend pattern search methods to linearly constrained minimization we develop a general class of feasible point pattern search algorithms and prove global convergence to a karush kuhn tucker point as in the case of unconstrained minimization pattern search methods for linearly constrained problems accomplish this without explicit recourse to the gradient or the directional derivative of the objective key to the analysis of the algorithms is the way in which the local search patterns conform to the geometry of the boundary of the feasible region

on the convergence of a new trust region algorithm in this paper we present a new trust region algorithm for general nonlinear constrained optimization problems the algorithm is based on the l infty exact penalty function under very mild conditions global convergence results for the algorithm are given local convergence properties are also studied it is shown that the penalty parameter generated by the algorithm will be eventually not less than the l 1 norm of the lagrange multipliers at the accumulation point it is proved that the method is equivalent to the sequential quadratic programming method for all large k hence superlinearly convergent results of the sqp method can be applied numerical results are also reported

search via quantum walk we propose a new method for designing quantum search algorithms for finding a marked element in the state space of a classical markov chain the algorithm is based on a quantum walk a la szegedy quantum speed up of markov chain based algorithms in proceedings of the 45th ieee symposium on foundations of computer science ieee computer society press 2004 pp 32 41 that is defined in terms of the markov chain the main new idea is to apply quantum phase estimation to the quantum walk in order to implement an approximate reflection operator this operator is then used in an amplitude amplification scheme as a result we considerably expand the scope of the previous approaches of ambainis quantum walk algorithm for element distinctness in proceedings of the 45th ieee symposium on foundations of computer science ieee computer society press 2004 pp 22 31 and szegedy 2004 our algorithm combines the benefits of these approaches in terms of being able to find marked elements incurring the smaller cost of the two and being applicable to a larger class of markov chains in addition it is conceptually simple and avoids some technical difficulties in the previous analyses of several algorithms based on quantum walk

a new active set algorithm for box constrained optimization an active set algorithm asa for box constrained optimization is developed the algorithm consists of a nonmonotone gradient projection step an unconstrained optimization step and a set of rules for branching between the two steps global convergence to a stationary point is established for a nondegenerate stationary point the algorithm eventually reduces to unconstrained optimization without restarts similarly for a degenerate stationary point where the strong second order sufficient optimality condition holds the algorithm eventually reduces to unconstrained optimization without restarts a specific implementation of the asa is given which exploits the recently developed cyclic barzilai borwein cbb algorithm for the gradient projection step and the recently developed conjugate gradient algorithm cg descent for unconstrained optimization numerical experiments are presented using box constrained problems in the cuter and minpack 2 test problem libraries

a robust sequential quadratic programming method the sequential quadratic programming method developed by wilson han and powell may fail if the quadratic programming subproblems become infeasible or if the associated sequence of search directions is unbounded this paper considers techniques which circumvent these difficulties by modifying the structure of the constraint region in the quadratic programming subproblems furthermore questions concerning the occurrence of an unbounded sequence of multipliers and problem feasibility are also addressed

on the linear convergence of descent methods for convex essentially smooth minimization consider the problem of minimizing over a polyhedral set the composition of an affine mapping with a strictly convex essentially smooth function a general result on the linear convergence of descent methods for solving this problem is presented by applying this result the linear convergence of both the gradient projection algorithm of goldstein and levitin and polyak and a matrix splitting algorithm using regular splitting is established the results do not require that the cost function be strongly convex or that the optimal solution set be bounded the key to the analysis lies in a new error bound for estimating the distance from a feasible point to the optimal solution set

a proximal method for composite minimization we consider minimization of functions that are compositions of convex or prox regular functions possibly extended valued with smooth vector functions a wide variety of important optimization problems fall into this framework we describe an algorithmic framework based on a subproblem constructed from a linearized approximation to the objective and a regularization term properties of local solutions of this subproblem underlie both a global convergence result and an identification property of the active manifold containing the solution of the original problem preliminary computational results on both convex and nonconvex examples are promising

a projected newton method for minimization problems with nonlinear inequality constraints recently developed projected newton methods for minimization problems in polyhedrons and cartesian products of euclidean balls are extended here to general convex feasible sets defined by finitely many smooth nonlinear inequalities iterate sequences generated by this scheme are shown to be locally superlinearly convergent to nonsingular extremals bar u and more specifically to local minimizers bar u satisfying the standard second order kuhn tucker sufficient conditions moreover all such convergent iterate sequences eventually enter and remain within the smooth manifold defined by the active constraints at bar u implementation issues are considered for large scale specially structured nonlinear programs and in particular for multistage discrete time optimal control problems in the latter case overall per iteration computational costs will typically increase only linearly with the number of stages sample calculations are presented for nonlinear programs in a right circular cylinder in 3

a globally convergent lagrangian barrier algorithm for optimization with general inequality constraints and simple bounds we consider the global and local convergence properties of a class of lagrangian barrier methods for solving nonlinear programming problems in such methods simple bound constraints may be treated separately from more general constraints the objective and general constraint functions are combined in a lagrangian barrier function a sequence of such functions are approximately minimized within the domain defined by the simple bounds global convergence of the sequence of generated iterates to a first order stationary point for the original problem is established furthermore possible numerical difficulties associated with barrier function methods are avoided as it is shown that a potentially troublesome penalty parameter is bounded away from zero this paper is a companion to previous work of ours on augmented lagrangian methods

a new proof of robinson s homeomorphism theorem for pl normal maps abstract we study a certain class of piecewise linear functions from r n to r n namely robinson s normal maps induced by linear mappings and polyhedral convex sets called pl normal maps robinson s homeomorphism theorem characterizes the pl normal maps which are homeomorphisms as those which have nonzero determinants of the same sign on all pieces of linearity this paper presents a new shorter proof of the result pl normal systems include many optimization and equilibrium problems they arise from variational inequalities or equivalently generalized equations specified by linear maps and polyhedral convex sets unique continuous solvability of these systems which is important in theory and computation is captured by the homeomorphism result

on the accurate identification of active set for constrained minimax problems abstract in this paper the problem of identifying the active constraints for constrained nonlinear programming and minimax problems at an isolated local solution is discussed the correct identification of active constraints can improve the local convergence behavior of algorithms and considerably simplify algorithms for inequality constrained problems so it is a useful adjunct to nonlinear optimization algorithms facchinei et al f facchinei a fischer c kanzow on the accurate identification of active constraints siam j optim 9 1998 14 32 introduced an effective technique which can identify the active set in a neighborhood of a solution for nonlinear programming in this paper we first improve this conclusion to be more suitable for infeasible algorithms such as the strongly sub feasible direction method and the penalty function method then we present the identification technique of active constraints for constrained minimax problems without strict complementarity and linear independence some numerical results illustrating the identification technique are reported

global and asymptotic convergence rate estimates for a class of projected gradient processes projected gradient processes of the goldstein levitin polyak type are considered for constrained minimization problems min omega f with omega a convex set in a hilbert space x and f x to mathbb r 1 a differentiable functional global and local convergence theorems are established for a large class of these processes including those generated with implicit step length rules proposed by bertsekas and goldstein in this analysis traditional uniform strong positivity conditions on the hessian nabla 2 f are replaced by weaker pseudoconvexity conditions and growth conditions on f when f has a unique minimizes in omega convergence rates are shown to depend on how rapidly the function gamma sigma inf r f x f xi mid x in omega x xi geqq sigma grows with increasing sigma 0 if gamma sigma geqq b sigma nu for some b 0 the processes f n in question converge to inf f like o n nu nu 2 linear

optimization by direct search new perspectives on some classical and modern methods direct search methods are best known as unconstrained optimization techniques that do not explicitly use derivatives direct search methods were formally proposed and widely applied in the 1960s but fell out of favor with the mathematical optimization community by the early 1970s because they lacked coherent mathematical analysis nonetheless users remained loyal to these methods most of which were easy to program some of which were reliable in the past fifteen years these methods have seen a revival due in part to the appearance of mathematical analysis as well as to interest in parallel and distributed com puting this review begins by briefly summarizing the history of direct search methods and considering the special properties of problems for which they are well suited our focus then turns to a broad class of methods for which we provide a unifying framework that lends itself to a variety of convergence results the underlying principles allow general ization to handle bound constraints and linear constraints we also discuss extensions to problems with nonlinear constraints

division algebras over henselian fields in this chapter we focus on the tame division algebras d with center a field f with henselian valuation v as usual we approach this by first obtaining results for graded division algebras then lifting back from operatorname mathsf gr d to d this is facilitated by results in 8 1 on existence and uniqueness of lifts of tame subalgebras from operatorname mathsf gr d to d in 8 2 we describe four fundamental canonical up to conjugacy subalgebras of d that reflect its valuative structure the rest of the chapter is devoted to brauer group factorizations of d corresponding to the noncanonical direct product decomposition of operatorname mathit br t f given in cor 7 85 the factor operatorname mathit hom c operatorname mathcal g overline f mathbb t gamma f is represented by a type of division algebra n called decomposably semiramified defined in 8 3 and characterized by the property that n contains a maximal subfield inertial over f and another totally ramified over f we show in 8 4 that every tame division algebra d is brauer equivalent to some s f t where s is inertially split and t is tame and totally ramified over f we show further that every inertially split division algebra s is brauer equivalent to some i f n where i is inertial over f and n is decomposably semiramified the classes overline i for the i appearing in the i f n decompositions of s are shown to range over a single coset of mathit dec z overline s overline f in operatorname mathit br overline f called the specialization coset of s in the final subsection 8 4 6 we summarize what happens in the special case that v is discrete of rank 1 where substantial simplifications occur

barrier operators and associated gradient like dynamical systems for constrained minimization problems we study some continuous dynamical systems associated with constrained optimization problems for that purpose we introduce the concept of elliptic barrier operators and develop a unified framework to derive and analyze the associated class of gradient like dynamical systems called a driven descent method a dm prominent methods belonging to this class include several continuous descent methods studied earlier in the literature such as steepest descent method continuous gradient projection methods and newton type methods as well as continuous interior descent methods such as lotka volterra type differential equations and riemannian gradient methods related discrete iterative methods such as proximal interior point algorithms based on bregman functions and second order homogeneous kernels can also be recovered within our framework and allow for deriving some new and interesting dynamics we prove global existence and strong viability results of the corresponding trajectories of a dm for a smooth objective function when the objective function is convex we analyze the asymptotic behavior at infinity of the trajectory produced by the proposed class of dynamical systems a dm in particular we derive a general criterion ensuring the global convergence of the trajectory of a dm to a minimizer of a convex function over a closed convex set this result is then applied to several dynamics built upon specific elliptic barrier operators throughout the paper our results are illustrated with many examples

value functions on central simple algebras in this paper we study noncommutative valuation rings as defined by dubrovin while there is in general no valuation associated to a dubrovin valuation ring we show that there is a value function associated to any dubrovin valuation ring integral over its center by using value functions we obtain information on dubrovin valuation rings in a tensor product both generalizing and giving a much simpler proof of a result about valued division algebras by being able to work directly with central simple algebras we gain new information about division algebras over henselian fields

optimal control of the temperature in a catalytic converter abstract this paper is concerned with the optimization of the dynamics of the gas flow in an exhaust pipe as a prototypical question the start up heating of the catalytic converter is considered reaching the optimal temperature in the converter rapidly competes against the costs i e releasing too much unburnt fuel in the exhaust gases the underlying model is a one dimensional small mach number asymptotic gas dynamic model for a mixture of burnt and unburnt gases formulated on a network consisting of the various pieces of the exhaust tube for the optimization adjoint calculus is applied on the continuous level the resulting system is then discretized finally numerical examples show the validity of this approach

active set newton methods and partial smoothness diverse optimization algorithms correctly identify in finite time intrinsic constraints that must be active at optimality analogous behavior extends beyond optimization to systems involving part

algorithms for bound constrained quadratic programming problems we present an algorithm which combines standard active set strategies with the gradient projection method for the solution of quadratic programming problems subject to bounds we show in particular that if the quadratic is bounded below on the feasible set then termination occurs at a stationary point in a finite number of iterations moreover if all stationary points are nondegenerate termination occurs at a local minimizer a numerical comparison of the algorithm based on the gradient projection algorithm with a standard active set strategy shows that on mildly degenerate problems the gradient projection algorithm requires considerable less iterations and time than the active set strategy on nondegenerate problems the number of iterations typically decreases by at least a factor of 10 for strongly degenerate problems the performance of the gradient projection algorithm deteriorates but it still performs better than the active set method

an active set truncated newton method for large scale bound constrained optimization an active set truncated newton method for large scale bound constrained optimization is proposed the active sets are guessed by an identification technique the search direction consists of two parts some of the components are simply defined the other components are determined by the truncated newton method the method based on a nonmonotone line search technique is shown to be globally convergent numerical experiments are presented using bound constrained problems in the cuter test problem library the numerical performance reveals that our method is effective and competitive with the famous algorithm tron

modified fixed point equations and related iterative methods for variational inequalities abstract in this paper we study the equivalence characterizations of several modified fixed point equations to variational inequalities vi based on these equations we give some applications in constructing iterative methods for the solution of the vi especially we show global convergence the sublinear convergence and the finite termination of a new iterative algorithm under certain conditions

weak sharp minima on riemannian manifolds this is the first paper dealing with the study of weak sharp minima for constrained optimization problems on riemannian manifolds which are important in many applications we consider the notions of local weak sharp minima boundedly weak sharp minima and global weak sharp minima for such problems and establish their complete characterizations in the case of convex problems on finite dimensional riemannian manifolds and hadamard manifolds a number of the results obtained in this paper are also new for the case of conventional problems in finite dimensional euclidean spaces our methods involve appropriate tools of variational analysis and generalized differentiation on riemannian and hadamard manifolds developed and efficiently implemented in this paper

convergence properties of trust region methods for linear and convex constraints we develop a convergence theory for convex and linearly constrained trust region methods which only requires that the step between iterates produce a sufficient reduction in the trust region subproblem global convergence is established for general convex constraints while the local analysis is for linearly constrained problems the main local result establishes that if the sequence converges to a nondegenerate stationary point then the active constraints at the solution are identified in a finite number of iterations as a consequence of the identification properties we develop rate of convergence results by assuming that the step is a truncated newton method our development is mainly geometrical this approach allows the development of a convergence theory without any linear independence assumptions

newton s method for large bound constrained optimization problems we analyze a trust region version of newton s method for bound constrained problems our approach relies on the geometry of the feasible set not on the particular representation in terms of constraints the convergence theory holds for linearly constrained problems and yields global and superlinear convergence without assuming either strict complementarity or linear independence of the active constraints we also show that the convergence theory leads to an efficient implementation for large bound constrained problems

on branching numbers of normal manifolds kuhn and lowen l and robinson 2 have provided deep and elegant results that characterize the homeomorphic members of certain classes of piecewise linear pl mappings the purpose of this paper is to show that the class of mappings examined by robinson is contained in the class of mappings examined by kuhn and lowen let 312 be a finite piecewise linear pl manifold of r and p r r be pl with respect to 3n i e p is affine on each set in 9x the branching number of 9x l is the maximum number of sets in 317 that can contain any common face of codimension 2 in l theorem 5 31 it is shown that if 312 has a branching number less than or equal to 4 then p is a homeomorphism if and only if it is coherently oriented i e the determinants of p on each set in 312 have the same nonzero sign let c be a nonempty polyhedral convex set in real n dimensional space ir and a belong to the space of real n x n matrices f robinson 2 defines a finite pl manifold 32 of k 2 proposition 2 41 called the normal manifold of c and the normal map a i t r induced by a c which is pl with respect to e 2 proposition 2 51 the main result 2 theorem 4 31 shows that ac is homeomorphic if and only if it is coherently oriented see ralph 3 for an alternative proof and robinson 4 for a much simpler proof in the special case that a is symmetric it is pointed out in 2 section l that if xc could be shown to have a branching number less than or equal to 4 then 2 theorem 4 31 would follow from l theorem 5 31 indeed showing that normal manifolds have this branching number property is precisely our goal throughout the paper c and a will be as above let q be the euclidean projection onto c that is rcc maps each point of ir to its nearest point in c with respect to the euclidean norm the identity in ir x is i

tangent projection equations and general variational inequalities in this paper we establish the equivalence between the general variational inequalities and tangent projection equations this equivalence is used to discuss the local convergence analysis of a wide class of iterative methods for solving the general variational inequalities we show that some existing methods can identify the optimal face after finitely many iterations under the degenerate assumption

correspondences between valued division algebras and graded division algebras abstract if d is a tame central division algebra over a henselian valued field f then the valuation on d yields an associated graded ring gd which is a graded division ring and is also central and graded simple over gf after proving some properties of graded central simple algebras over a graded field including a cohomological characterization of its graded brauer group it is proved that the map d gd g yields an index preserving isomorphism from the tame part of the brauer group of f to the graded brauer group of gf this isomorphism is shown to be functorial with respect to field extensions and corestrictions and using this it is shown that there is a correspondence between f subalgebras of d with center tame over f and graded gf subalgebras of gd

an inequality constrained nonlinear kalman bucy smoother by interior point likelihood maximization kalman bucy smoothers are often used to estimate the state variables as a function of time in a system with stochastic dynamics and measurement noise this is accomplished using an algorithm for which the number of numerical operations grows linearly with the number of time points all of the randomness in the model is assumed to be gaussian including other available information for example a bound on one of the state variables is non trivial because it does not fit into the standard kalman bucy smoother algorithm in this paper we present an interior point method that maximizes the likelihood with respect to the sequence of state vectors satisfying inequality constraints the method obtains the same decomposition that is normally obtained for the unconstrained kalman bucy smoother hence the resulting number of operations grows linearly with the number of time points we present two algorithms the first is for the affine case and the second is for the nonlinear case neither algorithm requires the optimization to start at a feasible sequence of state vector values both the unconstrained affine and unconstrained nonlinear kalman bucy smoother are special cases of the class of problems that can be handled by these algorithms

a fast algorithm for sparse reconstruction based on shrinkage subspace optimization and continuation we propose a fast algorithm for solving the ell 1 regularized minimization problem min x in mathbb r n mu x 1 ax b 2 2 for recovering sparse solutions to an undetermined system of linear equations ax b the algorithm is divided into two stages that are performed repeatedly in the first stage a first order iterative shrinkage method yields an estimate of the subset of components of x likely to be nonzero in an optimal solution restricting the decision variables x to this subset and fixing their signs at their current values reduces the ell 1 norm x 1 to a linear function of x the resulting subspace problem which involves the minimization of a smaller and smooth quadratic function is solved in the second phase our code fpc as embeds this basic two stage algorithm in a continuation homotopy approach by assigning a decreasing sequence of values to mu this code exhibits state of the art performance in terms of both its speed and its ability to recover sparse signals

asymptotic convergence analysis of the forward backward splitting algorithm the asymptotic convergence of the forward backward splitting algorithm for solving equations of type 0 t z is analyzed where t is a multivalued maximal monotone operator in the n dimensional euclidean space when the problem has a nonempty solution set and t is split in the form t h with being maximal monotone and h being co coercive with modulus greater than convergence rates are shown under mild conditions to be linear superlinear or sublinear depending on how rapidly 1 and h 1 grow in the neighborhoods of certain specific points as a special case when both and h are polyhedral functions we get r linear convergence and 2 step q linear convergence without any further assumptions on the strict monotonicity on t or on the uniqueness of the solution as another special case when h 0 the splitting algorithm reduces to the proximal point algorithm and we get new results which complement r t rockafellar s and f j luque s earlier results on the proximal point algorithm

optimality identifiability and sensitivity around a solution of an optimization problem an identifiable subset of the feasible region is one containing all nearby solutions after small perturbations to the problem a quest for only the most essential ingredients of sensitivity analysis leads us to consider identifiable sets that are minimal this new notion lays a broad and intuitive variational analytic foundation for optimality conditions sensitivity and active set methods

conditional subgradient optimization theory and applications abstract we generalize the subgradient optimization method for nondifferentiable convex programming to utilize conditional subgradients firstly we derive the new method and establish its convergence by generalizing convergence results for traditional subgradient optimization secondly we consider a particular choice of conditional subgradients obtained by projections which leads to an easily implementable modification of traditional subgradient optimization schemes to evaluate the subgradient projection method we consider its use in three applications uncapacitated facility location two person zero sum matrix games and multicommodity network flows computational experiments show that the subgradient projection method performs better than traditional subgradient optimization in some cases the difference is considerable these results suggest that our simply modification may improve subgradient optimization schemes significantly this finding is important as such schemes are very popular especially in the context of lagrangean relaxation

tackling box constrained optimization via a new projected quasi newton approach numerous scientific applications across a variety of fields depend on box constrained convex optimization box constrained problems therefore continue to attract research interest we address box constrained strictly convex problems by deriving two new quasi newton algorithms our algorithms are positioned between the projected gradient j b rosen j siam 8 1960 pp 181 217 and projected newton d p bertsekas siam j control optim 20 1982 pp 221 246 methods we also prove their convergence under a simple armijo step size rule we provide experimental results for two particular box constrained problems nonnegative least squares nnls and nonnegative kullback leibler nnkl minimization for both nnls and nnkl our algorithms perform competitively as compared to well established methods on medium sized problems for larger problems our approach frequently outperforms the competition

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

value functions and associated graded rings for semisimple algebras we introduce a type of value function y called a gauge on a finite dimensional semisimple algebra a over a field f with valuation v the filtration on a induced by y yields an associated graded ring gr y a which is a graded algebra over the graded field gr v f key requirements for y to be a gauge are that gr y a be graded semisimple and that dim gr v f gr y a dim f a it is shown that gauges behave well with respect to scalar extensions and tensor products when v is henselian and a is central simple over f it is shown that gr y a is simple and graded brauer equivalent to gr w d where d is the division algebra brauer equivalent to a and w is the valuation on d extending v on f the utility of having a good notion of value function for central simple algebras not just division algebras and with good functorial properties is demonstrated by giving new and greatly simplified proofs of some difficult earlier results on valued division algebras

generic optimality conditions for semialgebraic convex programs we consider linear optimization over a nonempty convex semialgebraic feasible region f semidefinite programming is an example if f is compact then for almost every linear objective there is a unique optimal solution lying on a unique active manifold around which f is partly smooth and the second order sufficient conditions hold perturbing the objective results in smooth variation of the optimal solution the active manifold consists locally of these perturbed optimal solutions it is independent of the representation of f and is eventually identified by a variety of iterative algorithms such as proximal and projected gradient schemes these results extend to unbounded sets f

extensions of valuation rings in central simple algebras certain subrings r of simple algebras q finite dimensional over their center k are studied these rings are called q valuation rings since they share many properties with commutative valuation rings let v be a valuation ring of k the center of q and let m be the set of q valuation rings r in q with r n k v then 1s 1 this extension theorem which does not hold if one considers only total valuation rings was proved by n i dubrovin here first a somewhat different proof of this result is given and then information about the set m is obtained theorem the elements in j are conjugate if v has finite rank theorem the elements in m are total valuation rings if m contains one total valuation ring in this case q is a division ring theorem i i 1 if 5 contains an invariant total valuation ring

a new subspace limited memory bfgs algorithm for large scale bound constrained optimization an active set limited memory bfgs algorithm for large scale bound constrained optimization is introduced the active sets are based on guessing technique to be identified at each iteration the search direction in free subspace is determined by limited memory bfgs l bfgs algorithm which provides an efficient means for attacking large scale optimization problems the implementations of the method on cute test problems are described

some developments in general variational inequalities general variational inequalities provide us with a unified natural novel and simple framework to study a wide class of equilibrium problems arising in pure and applied sciences in this paper we present a number of new and known numerical techniques for solving general variational inequalities using various techniques including projection wiener hopf equations updating the solution auxiliary principle inertial proximal penalty function dynamical system and well posedness we also consider the local and global uniqueness of the solution and sensitivity analysis of the general variational inequalities as well as the finite convergence of the projection type algorithms our proofs of convergence are very simple as compared with other methods our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems since the general variational inequalities include quasi variational inequalities and quasi implicit complementarity problems as special cases results presented here continue to hold for these problems several open problems have been suggested for further research in these areas

convergence properties of minimization algorithms for convex constraints using a structured trust region in this paper we present a class of trust region algorithms for minimization problems within convex feasible regions in which the structure of the problem is explicitly used in the definition of the trust region this development is intended to reflect the possibility that some parts of the problem may be more accurately modelled than others a common occurrence in large scale nonlinear applications after describing the structured trust region mechanism we prove global convergence for all algorithms in our class

on the identification of active constraints nondegeneracy conditions that guarantee that the optimal active constraints are identified in a finite number of iterations are studied results of this type have only been established for a few algorithms and then under restrictive hypothesis the main result is a characterization of those algorithms that identify the optimal constraints in a finite number of iterations this result is obtained with a nondegeneracy assumption which is equivalent in the standard nonlinear programming problem to the assumption that there is a set of strictly complementary lagrange multipliers as an important consequence of the authors results the way that this characterization applies to gradient projection and sequential quadratic programming algorithms is shown

a linearly convergent dual based gradient projection algorithm for quadratically constrained convex minimization this paper presents a new dual formulation for quadratically constrained convex programs the special structure of the derived dual problem allows us to apply the gradient projection algorithm to produce a simple explicit method involving only elementary vector matrix operations proven to converge at a linear rate

on the accurate identification of active constraints we consider nonlinear programs with inequality constraints and we focus on the problem of identifying those constraints which will be active at an isolated local solution the correct identification of active constraints is important from both a theoretical and a practical point of view such an identification removes the combinatorial aspect of the problem and locally reduces the inequality constrained minimization problem to an equality constrained problem which can be more easily dealt with we present a new technique which identifies active constraints in a neighborhood of a solution and which requires neither complementary slackness nor uniqueness of the multipliers we also present extensions to variational inequalities and numerical examples illustrating the identification technique

an efficient algorithm for large scale nonlinear programming problems with simple bounds on the variables in this paper we present a new conjugate gradient algorithm for nonlinear programming problems with simple bounds on the variables the method is an extension of our conjugate gradient algorithm described in r pytlak ima j numer anal 14 1994 pp 443 460 the simple constraints on the variables are treated by a projection under mild assumptions the method is globally convergent the algorithm has the property whereby many constraints can leave or enter an active set of constraints at one iteration if the strict complementarity condition is satisfied the active set at a solution is identified in a finite number of iterations the algorithm has been tested on problems with more than 10 000 variables

solving variational inequality and fixed point problems by line searches and potential optimization we introduce a general adaptive line search framework for solving fixed point and variational inequality problems our goals are to develop iterative schemes that i compute solutions when the underlying map satisfies properties weaker than contractiveness for example weaker forms of nonexpansiveness ii are more efficient than the classical methods even when the underlying map is contractive and iii unify and extend several convergence results from the fixed point and variational inequality literatures to achieve these goals we introduce and study joint compatibility conditions imposed upon the underlying map and the iterative step sizes at each iteration and consider line searches that optimize certain potential functions as a special case we introduce a modified steepest descent method for solving systems of equations that does not require a previous condition from the literature the square of the jacobian matrix is positive definite since the line searches we propose might be difficult to perform exactly we also consider inexact line searches

multicell coordinated beamforming with rate outage constraint part ii efficient approximation algorithms this paper studies the coordinated beamforming cobf design for the multiple input single output interference channel provided that only channel distribution information is known to the transmitters the problem under consideration is a probabilistically constrained optimization problem which maximizes a predefined system utility subject to constraints on rate outage probability and power budget of each transmitter our recent analysis has shown that the outage constrained cobf problem is intricately difficult e g np hard therefore the focus of this paper is on suboptimal but computationally efficient algorithms specifically by leveraging on the block successive upper bound minimization bsum method in optimization we propose a gauss seidel type algorithm called distributed bsum algorithm which can handle differentiable monotone and concave system utilities by exploiting a weighted minimum mean square error wmmse reformulation we further propose a jocobi type algorithm called distributed wmmse algorithm which can optimize the weighted sum rate utility in a fully parallel manner both algorithms are shown to converge to the stationary points of the original np hard problems to further provide a performance benchmark a relaxed approximation method based on polyblock outer approximation is also proposed simulation results show that the proposed algorithms are significantly superior to the existing successive convex approximation method in both performance and computational efficiency and can yield promising approximation performance by comparing with the performance benchmark

weak sharp minima revisited part iii error bounds for differentiable convex inclusions the notion of weak sharp minima unifies a number of important ideas in optimization part i of this work provides the foundation for the theory of weak sharp minima in the infinite dimensional setting part ii discusses applications of these results to linear regularity and error bounds for nondifferentiable convex inequalities this work applies the results of part i to error bounds for differentiable convex inclusions a number of standard constraint qualifications for such inclusions are also examined

complete characterizations of local weak sharp minima with applications to semi infinite optimization and complementarity abstract in this paper we identify a favorable class of nonsmooth functions for which local weak sharp minima can be completely characterized in terms of normal cones and subdifferentials or tangent cones and subderivatives or their mixture in finite dimensional spaces the results obtained not only extend previous ones in the literature but also allow us to provide new types of criteria for local weak sharpness applications of the developed theory are given to semi infinite programming and to a new class of semi infinite complementarity problems

an extension of the projected gradient method to a banach space setting with application in structural topology optimization for the minimization of a nonlinear cost functional under convex constraints the relaxed projected gradient process is a well known method the analysis is classically performed in a hilbert space we generalize this method to functionals which are differentiable in a banach space the search direction is calculated by a quadratic approximation of the cost functional using the idea of the projected gradient thus it is possible to perform e g an l 2 gradient method if the cost functional is only differentiable in l infty we show global convergence using armijo backtracking for the step length selection and allow the underlying inner product and the scaling of the derivative to change in every iteration as an application we present a structural topology optimization problem based on a phase field model where the reduced cost functional is differentiable in h 1 cap l infty the presented numerical results using the h 1 inner product and a pointwise chosen metric including second order inform

solving convex programs by means of ordinary differential equations within a hilbert space we consider nonsmooth convex programs with sharp constraints examples include all problem instances that are bounded and strictly feasible to solve such programs we pursue an absolutely continuous trajectory generated by a differential inclusion of subgradient type whenever this inclusion offers some freedom of choice we select a steepest descent direction it is shown that the proposed algorithm converges to an optimal solution in finite time

on finite convergence and constraint identification of subgradient projection methods this paper deals with a continuous time subgradient projection algorithm shown to generate trajectories that accumulate to the solution set under a strong convexity assumption we show that convergence is exponential in norm a sharpness condition yields convergence in finite time and the necessary lapse is estimated invoking a constraint qualification and a non degeneracy assumption we demonstrate that optimally active constraints are identified in finite time

active set identification for linearly constrained minimization without explicit derivatives we consider active set identification for linearly constrained optimization problems in the absence of explicit information about the derivative of the objective function we begin by presenting some general results on active set identification that are not tied to any particular algorithm these general results are sufficiently strong that given a sequence of iterates converging to a karush kuhn tucker point it is possible to identify binding constraints for which there are nonzero multipliers we then focus on generating set search methods a class of derivative free direct search methods we discuss why these general results which are posed in terms of the direction of steepest descent apply to generating set search even though these methods do not have explicit recourse to derivatives nevertheless there is a clearly identifiable subsequence of iterations at which we can reliably estimate the set of constraints that are binding at a solution we discuss how active set estimation can be used to accelerate generating set search methods and illustrate the appreciable improvement that can result using several examples from the cuter test suite we also introduce two algorithmic refinements for generating set search methods the first expands the subsequence of iterations at which we can make inferences about stationarity the second is a more flexible step acceptance criterion

a limited memory quasi newton trust region method for box constrained optimization by means of wolfe conditions strategy we propose a quasi newton trust region method to solve box constrained optimization problems this method is an adequate combination of the compact limited memory bfgs and the trust region direction while the generated point satisfies the wolfe conditions and therefore maintains a positive definite approximation to the hessian of the objective function the global convergence and the quadratic convergence of this method are established under suitable conditions finally we compare our algorithms iwtral and ibwtral with an active set trust region algorithm astral xu and burke 2007 on the cutest box constrained test problems presented by gould et al 2015 numerical results show that the presented method is competitive and totally interesting for solving box constrained optimization