exterior point algorithms for nearest points and convex quadratic programs we consider the problem of finding the nearest point by euclidean distance in a simplicial cone to a given point and develop an exterior penalty algorithm for it each iteration in the algorithm consists of a single newton step following a reduction in the value of the penalty parameter proofs of convergence of the algorithm are given various other versions of exterior penalty algorithms for nearest point problems in nonsimplicial polyhedral cones and for convex quadratic programs all based on a single descent step following a reduction in the value of the penalty parameter per iteration are discussed the performance of these algorithms in large scale computational experiments is very encouraging it shows that the number of iterations grows very slowly if at all with the dimension of the problem

an affine scaling interior trust region method for lc1 minimization subject to bounds on variables in this literature we extend the classical affine scaling interior trust region algorithm for smooth bounded constrained nonlinear programming to the nonsmooth case where the objective function is only locally lipschitzian we propose and analyze a new affine scaling trust region method in association with nonmonotonic interior backtracking line search technique for solving the linear equality constrained lc 1 optimization subject to bounds on variables at each iteration the trust region subproblem is defined by minimizing a quadratic function which is required both first and second order information of the objective function and a quadratic affine scaling matrix model subject only to an ellipsoidal constraint in a null space of the affine scaling linear equality constraints the second order derivative of the objective function is explicitly required to be lipschitzian each iterate switches to a very general setting of computing trial backtracking steps generated by the general trust region subproblem and then the line search technique projects the trial backtracking steps onto the strictly feasible set of the bounded constraints on variables the global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions where twice smoothness of the objective function is not required a nonmonotonic criterion should bring about speeding up the convergence progress in some ill conditioned cases applications of the algorithm to various nonsmooth optimization problems are discussed

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

a class of linear complementarity problems solvable in polynomial time we describe a condition number for the linear complementarity problem lcp which characterizes the degree of difficulty for its solution when a potential reduction algorithm is used consequently we develop a class of lcps solvable in polynomial time the result suggests that the convexity or positive semidefiniteness of the lcp may not be the basic issue that separates lcps solvable and not solvable in polynomial time

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

cofinality spectrum theorems in model theory set theory and general topology we connect and solve two longstanding open problems in quite different areas the modeltheoretic question of whether sop2 is maximal in keisler s order and the question from general topology set theory of whether p t the oldest problem on cardinal invariants of the continuum we do so by showing these problems can be translated into instances of a more fundamental problem which we state and solve completely using model theoretic methods by a cofinality spectrum problem s we essentially mean the data of a pair of models m m1 which code sufficient set theory possibly in an expanded language along with a distinguished set of formulas s which define linear orders in m1 let ts the treetops of s be the smallest regular cardinal such that one of a set of derived trees in m1 has a strictly increasing sequence with no upper bound let c s ts be the set of pairs of regular cardinals 1 2 such that 1 2 or what is equivalent such that d contains no cuts for cf is good we obtain several consequences notably existence of a minimum keisler class among the non simple theories

positive definite completions of partial hermitian matrices abstract the question of which partial hermitian matrices some entries specified some free may be completed to positive definite matrices is addressed it is shown that if the diagonal entries are specified and principal minors composed of specified entries are positive then if the undirected graph of the specified entries is chordal a positive definite completion necessarily exists furthermore if this graph is not chordal then examples exist without positive definite completions in case a positive definite completion exists there is a unique matrix in the class of all positive definite completions whose determinant is maximal and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial hermitian matrix additional observations regarding positive definite completions are made

approximation algorithms for indefinite quadratic programming we consider approximation schemes for indefinite quadratic programming we argue that such an approximation can be found in polynomial time for fixed andt wheret denotes the number of negative eigenvalues of the quadratic term our algorithm is polynomial in 1 for fixedt and exponential int for fixed we next look at the special case of knapsack problems showing that a more efficient polynomial int approximation algorithm exists

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

on some efficient interior point methods for nonlinear convex programming abstract we introduce two interior point algorithms for minimizing a convex function subject to linear constraints our algorithms require the solution of a nonlinear system of equations at each step we show that if sufficiently good approximations to the solutions of the nonlinear systems can be found then the primal dual gap becomes less that e in o n lne steps where n is the number of variables

non discrete euclidean buildings for the ree and suzuki groups we call a non discrete euclidean building a bruhat tits space if its automorphism group contains a subgroup that induces the subgroup generated by all the root groups of a root datum of the building at infinity this is the class of non discrete euclidean buildings introduced and studied by bruhat and tits we give the complete classification of bruhat tits spaces whose building at infinity is the fixed point set of a polarity of an ambient building of type b2 f4 or g2 associated with a ree or suzuki group endowed with the usual root datum in the b2 and g2 cases this fixed point set is a building of rank one in the f4 case it is a generalized octagon whose weyl group is not crystallographic we also show that each of these bruhat tits spaces has a natural embedding in the unique bruhat tits space whose building at infinity is the corresponding ambient building

an ol n 3 primal interior point algorithm for convex quadratic programming we present a primal interior point method for convex quadratic programming which is based upon a logarithmic barrier function approach this approach generates a sequence of problems each of which is approximately solved by taking a single newton step it is shown that the method requires o sqrt n l iterations and o n3 5l arithmetic operations by using modified newton steps the number of arithmetic operations required by the algorithm can be reduced to o n3l

an interior newton method for quadratic programming quadratic programming represents an extremely important class of optimization problem in this paper we propose a new interior approach for the general quadratic programming problem we establish that our new method is globally and quadratically convergent published alternative interior approaches do not share such strong convergence properties for the nonconvex case we also report on the results of preliminary numerical experiments the results indicate that the proposed method has considerable prcatival potential

an extension of karmarkar projective algorithm for convex quadratic programming we present an extension of karmarkar s linear programming algorithm for solving a more general group of optimization problems convex quadratic programs this extension is based on the iterated application of the objective augmentation and the projective transformation followed by optimization over an inscribing ellipsoid centered at the current solution it creates a sequence of interior feasible points that converge to the optimal feasible solution in o ln iterations each iteration can be computed in o ln3 arithmetic operations wheren is the number of variables andl is the number of bits in the input in this paper we emphasize its convergence property practical efficiency and relation to the ellipsoid method

non discrete affine buildings and convexity abstract affine buildings are in a certain sense analogs of symmetric spaces it is therefore natural to ask for analogs of results for symmetric spaces we prove a version of kostant s convexity theorem for thick non discrete affine buildings kostant proves that the image of a certain orbit of a point x in the symmetric space under a projection onto a maximal flat is the convex hull of the weyl group orbit of x we obtain the same result for a projection onto an apartment in an affine building the methods are mostly borrowed from metric geometry our proof makes no appeal to the automorphism group of the building however the final result has an interesting application for groups acting nicely on non discrete buildings such as groups admitting a root datum with non discrete valuation along the proofs we obtain that segments are contained in apartments and that certain retractions are distance diminishing

a short cut potential reduction algorithm for linear programming as most interior point algorithms iterate they repeatedly perform costly matrix operations such as projections on the entire constraint matrix for large scale linear programming problems such operations consume the great majority of the computation time required however for problems where the number of variables far exceeds the number of constraints operations over the entire constraint matrix are unnecessary we will examine and extend decomposition techniques which greatly reduce the amount of work required by such interior point methods as the dual affine scaling and the dual potential reduction algorithms in an effort to judge the practical viability of the decompositioning we compare the performance of the dual potential reduction algorithm with and without decompositioning over a set of randomly generated transportation problems accompanying a theoretical justification of these techniques we focus on the implementation details and computational results of one such technique

a polynomial arc search interior point algorithm for convex quadratic programming abstract arc search is developed for linear programming in 24 25 the algorithms search for optimizers along an ellipse that is an approximation of the central path in this paper the arc search method is applied to primal dual path following interior point method for convex quadratic programming a simple algorithm with iteration complexity o n log 1 is devised several improvements on the simple algorithm which improve computational efficiency increase step length and further reduce duality gap in every iteration are then proposed and implemented it is intuitively clear that the iteration with these improvements will reduce the duality gap more than the iteration of the simple algorithm without the improvements though it is hard to show how much these improvements reduce the complexity bound the proposed algorithm is implemented in matlab and tested on quadratic programming problems originating from 13 the result is compared to the one obtained by loqo in 22 the proposed algorithm uses fewer iterations in all these problems and the number of total iterations is 27 fewer than the one obtained by loqo this preliminary result shows that the proposed algorithm is promising

coarse equivalences of euclidean buildings abstract we prove the following rigidity results coarse equivalences between metrically complete euclidean buildings preserve spherical buildings at infinity if all irreducible factors have dimension at least two then coarsely equivalent euclidean buildings are isometric up to scaling factors if in addition none of the irreducible factors is a euclidean cone then the isometry is unique and has finite distance from the coarse equivalence appendix a shows how these results can be extended to non complete euclidean buildings

asymptotic cones of finitely presented groups let g be a connected semisimple lie group with at least one absolutely simple factor s such that r rank s 2 and let be a uniform lattice in g a if ch holds then has a unique asymptotic cone up to homeomorphism b if ch fails then has 22 asymptotic cones up to homeomorphism

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

duality and sensitivity in nonconvex quadratic optimization over an ellipsoid abstract in this paper a duality framework is discussed for the problem of optimizing a nonconvex quadratic function over an ellipsoid additional insight is obtained from the observation that this nonconvex problem is in a sense equivalent to a convex problem of the same type from which known necessary and sufficient conditions for optimality readily follow based on the duality results some existing solution procedures are interpreted as in fact solving the dual the duality relations are also shown to provide a natural framework for sensitivity analysis

on some properties of quadratic programs with a convex quadratic constraint in this paper we consider the problem of minimizing a possibly nonconvex quadratic function with a quadratic constraint we point out some new properties of the problem in particular in the first part of the paper we show that i given a kkt point that is not a global minimizer it is easy to find a better feasible point ii strict complementarity holds at the local nonglobal minimizer in the second part of this paper we show that the original constrained problem is equivalent to the unconstrained minimization of a piecewise quartic merit function using the unconstrained formulation we give in the nonconvex case a new second order necessary condition for global minimizers in the third part of this paper algorithmic applications of the preceding results are briefly outlined and some preliminary numerical experiments are reported

an interior point algorithm of o m ln e iterations for c 1 convex programming we present a theoretical result on a path following algorithm for convex programs the algorithm employs a nonsmooth newton subroutine it starts from a near center of a restricted constraint set performs a partial nonsmooth newton step in each iteration and converges to a point whose cost is within accuracy of the optimal cost in o sqrt m left ln varepsilon right iterations wherem is the number of constraints in the problem unlike other interior point methods the analyzed algorithm only requires a first order lipschitzian condition and a generalized hessian similarity condition on the objective and constraint functions therefore our result indicates the theoretical feasibility of applying interior point methods to certainc1 optimization problems instead ofc2 problems since the complexity bound is unchanged compared with similar algorithms forc2 convex programming the result shows that the smoothness of functions may not be a factor affecting the complexity of interior point methods

a convergence proof for an affine scaling algorithm for convex quadratic programming without nondegeneracy assumptions this paper presents a theoretical result on convergence of a primal affine scaling method for convex quadratic programs it is shown that as long as the stepsize is less than a threshold value which depends on the input data only ye and tse s interior ellipsoid algorithm for convex quadratic programming is globally convergent without nondegeneracy assumptions in addition its local convergence rate is at least linear and the dual iterates have an ergodically convergent property

an affine scaling projective reduced hessian algorithm for minimum optimization with nonlinear equality and linear inequality constraints abstract in this paper we propose a nonmonotonic interior point backtracking strategy to modify the reduced projective affine scaling trust region algorithm for solving minimum optimization subject to both nonlinear equality and linear inequality constraints the general full trust region subproblem for solving the minimum optimization is decomposed to a pair of trust region subproblems in horizontal and vertical subspaces of linearize equality constraints and extended affine scaling equality constraints by qr decomposition of an affine scaling matrix and an orthonormal basis on the null subspace the horizontal subproblem in the proposed algorithm is defined by minimizing a quadratic projective reduced hessian function subject only to an ellipsoidal trust region constraint while the vertical subproblem is also defined by the least squares subproblem subject only to an ellipsoidal trust region constraint combining trust region strategy with line search technique will switch to strictly feasible interior point step generated by a component direction of the two trust region subproblems by adopting the l1 penalty function as the merit function the global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions the second order correction step and a nonmonotonic criterion are used to overcome maratos effect and speed up the convergence progress in some ill conditioned cases respectively

partial affine scaling for linearly constrained minimization we propose an interior point method for finding a stationary point of a nonlinear program with linear equality and nonnegativity constraints this method maintains a basis at each iteration and updates the iterate by taking an affine scaling step in the space of nonbasic variables in the general case we propose to choose the basis according to a rule that maximizes the basic components of the iterate in the case where the feasible region is the product of simplices we propose an alternative rule that minimizes the basic components of the cost gradient we analyze the convergence of the method under each rule a key feature of this method is that it can be implemented much like the simplex method

an active set strategy in an interior point method for linear programming we will present a potential reduction method for linear programming where only the constraints with relatively small dual slacks termed active constraints will be taken into account to form the ellipsoid constraint at each iteration of the process the algorithm converges to the optimal feasible solution in o sqrt n l iterations with the same polynomial bound as in the full constraints case wheren is the number of variables andl is the data length if a small portion of the constraints is active near the optimal solution the computational cost to find the next direction of movement in one iteration may be considerably reduced by the proposed strategy

partial matrix contractions and intersections of matrix balls abstract the balls of maximal volume in the intersection of two concentric matrix balls are computed the results are applied to the 2 2 contractive completion problem where the diagonal elements are given

divergence in lattices in semisimple lie groups and graphs of groups divergence functions of a metric space estimate the length of a path connecting two points a b at distance 3 s is a finite set of valuations of a number field k including all infinite valuations and o s is the corresponding ring of s integers

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

some extremal problems for positive definite matrices and operators let g be a real valued function defined on the set pn f of all positive definite complex hermitian or real symmetric matrices according as f c the complex field or f r the real field suppose a b pn f we study the optimization problems of 1 finding max g x subject to a x b x positive semidefinite 2 finding min g x subject to x a x b positive semidefinite for a general class of functions g we construct the optimal solutions and give conditions under which the solutions obtained are unique the particular case of the determinant function which motivated this work is studied in detail we then extend the results to the infinite dimensional case using the theory of symmetrically normed ideas similar optimization problems with more constraints are also briefly discussed

projected newton methods for optimization problems with simple constraints we consider the problem min f x x geqq 0 and propose algorithms of the form x k 1 x k alpha k d k nabla f x k where cdot denotes projection on the positive orthant alpha k is a stepsize chosen by an armijo like rule and d k is a positive definite symmetric matrix which is partly diagonal we show that d k can be calculated simply on the basis of second derivatives of f so that the resulting newton like algorithm has a typically superlinear rate of convergence with other choices of d k convergence at a typically linear rate is obtained the algorithms are almost as simple as their unconstrained counterparts they are well suited for problems of large dimension such as those arising in optimal control while being competitive with existing methods for low dimensional problems the effectiveness of the newton like algorithm is demonstrated via computational examples involving as many as 10 000 variables extensions to general linearly constrained pr

recovering optimal dual solutions on karmarkar s polynomial algorithm for linear programming this paper presents a standard form variant of karmarkar s algorithm for linear programming the tecniques of using duality and cutting objective are combined in this variant to maintain polynomial time complexity and to bypass the difficulties found in karmarkar s original algorithm the variant works with problems in standard form and simultaneously generates sequences of primal and dual feasible solutions whose objective function values converge to the unknown optimal value some computational results are also reported

an o n 3 l primal dual potential reduction algorithm for solving convex quadratic programs in this paper we combine partial updating and an adaptation of anstreicher s safeguarded linesearch of the primal dual potential function with kojima mizuno and yoshise s potential reduction algorithm for the linear complementarity problem to obtain an o n3l algorithm for convex quadratic programming our modified algorithm is a long step method that requires at most o sqrt n l steps

ls galleries the path model and mv cycles we give an interpretation of the path model of a representation 18 of a complex semisimple algebraic group g in terms of the geometry of its affine grassmannian in this setting the paths are replaced by ls galleries in the affine coxeter complex associated to the weyl group of g to explain the connection with geometry consider a demazure hansen bott samelson desingularization of the closure of an orbit g c t in the affine grassmannian the homology of has a basis given by bia lynicki birula cell s which are indexed by the t fixed points in now the points of can be identified with galleries of a fixed type in the affine tits building associated to g and the t fixed points correspond in this language to combinatorial galleries of a fixed type in the affine coxeter complex we determine those galleries such that the associated cell has a non empty intersection with g c t identified with an open subset of and we show that the closures of the strata associated to ls galleries are exactly the mv cycles 24 which form a basis of the representation v for the langland s dual group g

algorithms for the solution of quadratic knapsack problems we consider the problem of minimizing a quadratic function with a knapsack constraint quadratic knapsack problems have numerous applications including the least distance problem quadratic programming defined on the convex hull of a set of points and the maximum clique problem we propose and analyze three algorithms for solving quadratic knapsack problems two algorithms are based on recently developed interior point methods the first solves convex quadratic problems and the second computes a stationary point for the indefinite case for both computational results on a variety of test problems are presented the third algorithm based on simplicial partitioning and convex underestimating functions can be used to compute the global optimum of indefinite quadratic knapsack problems

nonmonotonic back tracking trust region interior point algorithm for linear constrained optimization in this paper we modify the trust region interior point algorithm proposed by bonnans and pola in siam j optim 7 3 1997 717 for linear constrained optimization a mixed strategy using both trust region and line search techniques is adopted which switches to back tracking steps when a trial step produced by the trust region subproblem may be unacceptable the global convergence and local convergence rate of the improved algorithm are established under some reasonable conditions a nonmonotonic criterion is used to speed up the convergence progress in some ill conditioned cases the results of numerical experiments are reported to show the effectiveness of the proposed algorithm