new algorithms for singly linearly constrained quadratic programs subject to lower and upper bounds there are many applications related to singly linearly constrained quadratic programs subjected to upper and lower bounds in this paper a new algorithm based on secant approximation is provided for the case in which the hessian matrix is diagonal and positive definite to deal with the general case where the hessian is not diagonal a new efficient projected gradient algorithm is proposed the basic features of the projected gradient algorithm are 1 a new formula is used for the stepsize 2 a recently established adaptive non monotone line search is incorporated and 3 the optimal stepsize is determined by quadratic interpolation if the non monotone line search criterion fails to be satisfied numerical experiments on large scale random test problems and some medium scale quadratic programs arising in the training of support vector machines demonstrate the usefulness of these algorithms

benchmarking optimization software with performance profiles we propose performance profiles distribution functions for a performance metric as a tool for benchmarking and comparing optimization software we show that performance profiles combine the best features of other tools for performance evaluation

on the moreau yosida regularization of the vector k norm related functions matrix optimization problems mops involving the ky fan k norm arise frequently from many applications in order to design algorithms to solve large scale mops involving the ky fan k norm we need to understand the first and second order properties of the moreau yosida regularization of the ky fan k norm function and the indicator function of the ky fan k norm ball according to the general theory on spectral functions in this paper we shall conduct a thorough study on the moreau yosida regularization of the vector k norm function and the indicator function of the vector k norm ball in particular we show that the proximal mappings associated with these two vector k norm related functions both admit fast and analytically computable solutions moreover we propose algorithms of low computational cost to compute the directional derivatives of these two proximal mappings and then completely characterize their frechet differentiability the work here thus builds the fundamental tools needed

g2 cubic transition between two circles with shape control this paper describes a method for joining two circles with an s shaped or with a broken back c shaped transition curve composed of at most two spiral segments in highway and railway route design or car like robot path planning it is often desirable to have such a transition it is shown that a single cubic curve can be used for blending or for a transition curve preserving g 2 continuity with local shape control parameter and more flexible constraints provision of the shape parameter and flexibility provide freedom to modify the shape in a stable manner which is an advantage over previous work by meek walton sakai and habib

symmetry maps of free form curve segments via wave propagation this paper presents an approach for computing the symmetries skeletons of an edge map consisting of a collection of curve segments this approach is a combination of analytic computations in the style of computational geometry and discrete propagations on a grid in the style of the numerical solutions of pde s specifically waves from each of the initial curve segments are initialized and propagated as a discrete wavefront along discrete directions in addition to avoid error built up due to the discrete nature of propagation shockwaves are detected and explicitly propagated along a secondary dynamic grid the propagation of shockwaves integrated with the propagation of the wavefront along discrete directions leads to an exact simulation of propagation by the eikonal equation the resulting symmetries are simply the collection of shockwaves formed in this process which can be manipulated locally exactly and efficiently under local changes in an edge map gap completion removal of spurious elements etc the ability to express grouping operations in the language of symmetry maps makes it an appropriate intermediate representation between low level edge maps and high level object hypotheses

an fptas for optimizing a class of low rank functions over a polytope we present a fully polynomial time approximation scheme fptas for optimizing a very general class of non linear functions of low rank over a polytope our approximation scheme relies on constructing an approximate pareto optimal front of the linear functions which constitute the given low rank function in contrast to existing results in the literature our approximation scheme does not require the assumption of quasi concavity on the objective function for the special case of quasi concave function minimization we give an alternative fptas which always returns a solution which is an extreme point of the polytope our technique can also be used to obtain an fptas for combinatorial optimization problems with non linear objective functions for example when the objective is a product of a fixed number of linear functions we also show that it is not possible to approximate the minimum of a general concave function over the unit hypercube to within any factor unless p np we prove this by showing a similar hardness of approximation result for supermodular function minimization a result that may be of independent interest

the use of cornu spirals in drawing planar curves of controlled curvature abstract cornu spirals or clothoids have been used in highway design for many years in the past the spirals have been found manually by draftsmen the purpose of this paper is to show that it is practical to find these spirals with a micro computer the design curve will be made up of arcs of circles and segments of cornu spirals joined in such a way that the curvature is continuous throughout and takes its largest values on the arcs of circles thus the radii of the circles used will limit and control the curvature of the whole design curve

an fptas for minimizing the product of two non negative linear cost functions we consider a quadratic programming qp problem of the form min x t c x subject to ax b x 0 where c in mathbb r n times n rm rank c 1 and a in mathbb r m times n b in mathbb r m we present an fully polynomial time approximation scheme fptas for this problem by reformulating the qp as a parameterized lp and rounding the optimal solution furthermore our algorithm returns an extreme point solution of the polytope therefore our results apply directly to 0 1 problems for which the convex hull of feasible integer solutions is known such as spanning tree matchings and sub modular flows they also apply to problems for which the convex hull of the dominant of the feasible integer solutions is known such as s t shortest paths and s t min cuts for the above discrete problems the quadratic program models the problem of obtaining an integer solution that minimizes the product of two linear non negative cost functions

an algorithm for a singly constrained class of quadratic programs subject to upper and lower bounds this paper gives an o n algorithm for a singly constrained convex quadratic program using binary search to solve the kuhn tucker system computational results indicate that a randomized version of this algorithm runs in expected linear time and is suitable for practical applications for the nonconvex case ane approximate algorithm is proposed which is based on convex and piecewise linear approximations of the objective function

a framework of constraint preserving update schemes for optimization on stiefel manifold this paper considers optimization problems on the stiefel manifold x mathsf t x i p xtx ip where x in mathbb r n times p x rn p is the variable and i p ip is the p p by p p identity matrix a framework of constraint preserving update schemes is proposed by decomposing each feasible point into the range space of x x and the null space of x mathsf t xt while this general framework can unify many existing schemes a new update scheme with low complexity cost is also discovered then we study a feasible barzilai borwein like method under the new update scheme the global convergence of the method is established with an adaptive nonmonotone line search the numerical tests on the nearest low rank correlation matrix problem the kohn sham total energy minimization and a specific problem from statistics demonstrate the efficiency of the new method in particular the new method performs remarkably well for the nearest low rank correlation matrix problem in terms of speed and solution quality and is considerably competitive with the widely used scf iteration for the kohn sham total energy minimization

the gauss map for surfaces part 2 the euclidean case we study smooth maps t m ci of a riemann surface m into the grassmannian gi of oriented 2 planes in e2 and determine necessary and sufficient conditons on t in order that it be the gauss map of a conformai immersion x m e2 we sometimes view as an oriented riemannian vector bundle it is a subbundle of ej the trivial bundle over m with fibre e2 l the necessary and sufficient conditions obtained for simply connected m involve the curvatures of t and tx the orthogonal complement of t in e f as well as certain components of the tension of viewed as a map t m sc 1 where sc l is a unit sphere of dimension c that contains ci as a submanifold in a natural fashion if t satisfies a particular necessary condition then the results take two different forms depending on whether or not t is the gauss map of a conformai minimal immersion the case t m gl is also studied in some additional detail

tight bounds on indefinite separable singly constrained quadratic programs in linear time non convex knapsack separable quadratic optimization with compact box constraints is an np hard problem we present tight lower and upper bounding procedures that are computationally efficient as the problem size grows the lower bound is based on lagrangian relaxation and it is computed in linear time when the bound is not an exact global solution a worst case bound quality measure is developed moreover the lower bounding lb solution is improved to construct a feasible solution leading to an upper bound ub on the given problem the ub is based on fixing the convex variables and a subset of non convex and linear variables at the lb solution and considering the remaining problem which is always feasible the ub is computable with linear time complexity and it is the global optimum in certain verifiable cases when the duality gap is zero in our limited computational experiments the ub has very small relative gap with an exact global optimum solution when there is a nonzero duality gap performance of the bounds is demonstrated through a broad range of randomly generated problem instances and comparisons with existing global and non global solvers the proposed method on these indefinite problems of extremely large size is an order of magnitude faster than the alternative solvers including ibm s commercial software cplex12 6

decomposing trimmed surfaces using the voronoi diagram and a scan line algorithm many applications deal with the rendering of trimmed surfaces and the generation of grids for trimmed surfaces usually a structured or unstructured grid must be constructed in the parameter space of the trimmed surface trimmed surfaces not only cause problems in the context of grid generation but also when exchanging data between different cad systems this paper describes a new approach for decomposing the valid part of the parameter space of a trimmed surface into a set of four sided surfaces the boundaries of these four sided surfaces are line segments segments of the trimming curves themselves and segments of bisecting curves that are defined by a generalized voronoi diagram implied by the trimming curves in parameter space we use a triangular background mesh for the computation of the bisecting curves of the generalized voronoi diagram

medial axis transformation of a planar shape the medial axis transformation is a means first proposed by blum to describe a shape in this paper we present a 0 n log n algorithm for computing the medial axis of a planar shape represented by an n edge simple polygon the algorithm is an improvement over most previously known results interms of both efficiency and exactness and has been implemented in fortran some computer plotted output of the program are also shown in the paper

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

a unified formulation and fast accelerated proximal gradient method for classification binary classification is the problem of predicting the class a given sample belongs to to achieve a good prediction performance it is important to find a suitable model for a given dataset however it is often time consuming and impractical for practitioners to try various classification models because each model employs a different formulation and algorithm the difficulty can be mitigated if we have a unified formulation and an efficient universal algorithmic framework for various classification models to expedite the comparison of performance of different models for a given dataset in this paper we present a unified formulation of various classification models including c svm l2 svm svm mm fda mm mpm logistic regression distance weighted discrimination and develop a general optimization algorithm based on an accelerated proximal gradient apg method for the formulation we design various techniques such as backtracking line search and adaptive restarting strategy in order to speed up the practical convergence of our method we also give a theoretical convergence guarantee for the proposed fast apg algorithm numerical experiments show that our algorithm is stable and highly competitive to specialized algorithms designed for specific models e g sequential minimal optimization smo for svm

separable convex optimization problems with linear ascending constraints separable convex optimization problems with linear ascending inequality and equality constraints are addressed in this paper an algorithm that explicitly characterizes the optimum point in a finite number of steps is described the optimum value is shown to be monotone with respect to a partial order on the constraint parameters moreover the optimum value is convex with respect to these parameters this work generalizes the existing algorithms of morton von randow and ringwald math programming 32 1985 pp 238 241 and viswanath and anantharam ieee trans inform theory 48 2002 pp 1295 1318 to a wider class of separable convex objective functions computational experiments that compare the proposed algorithm with a standard convex optimization tool are also provided

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

a trust region strategy for minimization on arbitrary domains we present a trust region method for minimizing a general differentiable function restricted to an arbitrary closed set we prove a global convergence theorem the trust region method defines difficult subproblems that are solvable in some particular cases we analyze in detail the case where the domain is a euclidean ball for this case we present numerical experiments where we consider different hessian approximations

approximation of circular arcs and offset curves by bezier curves of high degree in this paper we present an exact error analysis for circle approximation by bezier curve the approximation method is a special case of floater s conic approximation method comput aided geom design 8 1991 135 using this we propose an approximation method of the offset curve of given plane bezier curve by bezier curve of the same degree

topics in surface modeling contains recent ideas and results in three areas of growing importance in curve and surface design algebraic methods variational surface design and some special applications leading researchers from throughout the world have contributed their latest work and provided several promising solutions to open issues in surface modeling

fast projection onto the simplex and the pmb l _ mathbf 1 l1 ball a new algorithm is proposed to project exactly and in finite time a vector of arbitrary size onto a simplex or an l 1 l1 norm ball it can be viewed as a gauss seidel like variant of michelot s variable fixing algorithm that is the threshold used to fix the variables is updated after each element is read instead of waiting for a full reading pass over the list of non fixed elements this algorithm is empirically demonstrated to be faster than existing methods

planar g 2 transition with a fair pythagorean hodograph quintic curve recently planar cubic and pythagorean hodograph quintic transition curves that are suitable for g2 blending were developed they are suitable for blending e g rounding corners or for smooth transition between two curves e g two circular arcs it was shown that a single cubic segment can be used as a transition curve with the guarantee that an s shaped transition curve will have no curvature extrema and a c shaped transition curve will have a single curvature extremum the results for the cubic curve are now extended to pythagorean hodograph quintic curves a pythagorean hodograph curve has the attractive properties that its arclength is a polynomial of its parameter and its offset is rational a quintic is the lowest degree pythagorean hodograph curve that may have an inflection point pythagorean hodograph curves with no curvature extrema for an s shaped transition and a single curvature extremum for a c shaped transition are suitable for the design of fair curves e g in highway design or for blending in cad applications

algorithms for the continuous nonlinear resource allocation problem new implementations and numerical studies patriksson 2008 provided a then up to date survey on the continuous separable differentiable and convex resource allocation problem with a single resource constraint since the publication of that paper the interest in the problem has grown several new applications have arisen where the problem at hand constitutes a subproblem and several new algorithms have been developed for its efficient solution this paper therefore serves three purposes first it provides an up to date extension of the survey of the literature of the field complementing the survey in patriksson 2008 with more then 20 books and articles second it contributes improvements of some of these algorithms in particular with an improvement of the pegging that is variable fixing process in the relaxation algorithm and an improved means to evaluate subsolutions third it numerically evaluates several relaxation primal and breakpoint dual algorithms incorporating a variety of pegging strategies as well as a quasi newton method our conclusion is that our modification of the relaxation algorithm performs the best at least for problem sizes up to 30 million variables the practical time complexity for the breakpoint and relaxation algorithms is linear

the barzilai and borwein gradient method for the large scale unconstrained minimization problem the barzilai and borwein gradient method for the solution of large scale unconstrained minimization problems is considered this method requires few storage locations and very inexpensive computations furthermore it does not guarantee descent in the objective function and no line search is required recently the global convergence for the convex quadratic case has been established however for the nonquadratic case the method needs to be incorporated in a globalization scheme in this work a nonmonotone line search strategy that guarantees global convergence is combined with the barzilai and borwein method this strategy is based on the nonmonotone line search technique proposed by grippo lampariello and lucidi siam j numer anal 23 1986 pp 707 716 numerical results to compare the behavior of this method with recent implementations of the conjugate gradient method are presented these results indicate that the global barzilai and borwein method may allow some significant reduction in the number of line searches and also in the number of gradient evaluations

approximation algorithms for the maximum hamiltonian path problem with specified endpoint s abstract this paper deals with the problem of constructing hamiltonian paths of optimal weight called hpps t if the two endpoints are specified hpps if only one endpoint is specified we show that hpps t is 1 2 differential approximable and hpps is 2 3 differential approximable moreover we observe that these problems cannot be differential approximable better than 741 742 based upon these results we obtain new bounds for standard ratio a 1 2 standard approximation for m ax hpps t and a 2 3 for m ax hpps which can be improved to 2 3 for m ax hpps t a 2a all the edge weights are within an interval a 2a to 5 6 for m ax hpps a 2a and to 2 3 for m in hpps t a 2a to 3 4 for m in hpps a 2a

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

curvature extrema of planar parametric polynomial cubic curves parametric polynomial cubic curve segments are widely used in computer aided design and computer aided geometric design applications because their flexibility makes them suitable for use in the interactive design of curves and surfaces shape properties of these segments have been studied and results on the occurrence of cusps and whether the segment is s or c shaped are available in the literature their critical points however are not as well known this paper presents results on the number and location of curvature extrema of these segments all possible numbers of curvature extrema are found and it is demonstrated that all possible curvature extrema can be obtained numerically for any planar parametric cubic curve these results are useful in the study of the fairness of curves designed with parametric polynomial cubic curve segments

on the asymptotic behaviour of some new gradient methods the barzilai borwein bb gradient method and some other new gradient methods have shown themselves to be competitive with conjugate gradient methods for solving large dimension nonlinear unconstrained optimization problems little is known about the asymptotic behaviour even when applied to n dimensional quadratic functions except in the case that n 2 we show in the quadratic case how it is possible to compute this asymptotic behaviour and observe that as n increases there is a transition from superlinear to linear convergence at some value of n 4 depending on the method by neglecting certain terms in the recurrence relations we define simplified versions of the methods which are able to predict this transition the simplified methods also predict that for larger values of n the eigencomponents of the gradient vectors converge in modulus to a common value which is a similar to a property observed to hold in the real methods some unusual and interesting recurrence relations are analysed in the course of the study

a planar cubic be zier spiral a planar cubic bezier curve segment that is a spiral i e its curvature varies monotonically with arc length is discussed since this curve segment does not have cusps loops and inflection points except for a single inflection point at its beginning it is suitable for applications such as highway design in which the clothoid has been traditionally used since it is polynomial it can be conveniently incorporated in cad systems that are based on b splines bezier curves or nurbs nonuniform rational b splines and is thus suitable for general curve design applications in which fair curves are important

an efficient global algorithm for a class of indefinite separable quadratic programs we present a global algorithm for indefinite knapsack separable quadratic programs with bound constraints the upper bounds on variables with nonconvex terms are assumed to be infinite in the algorithmic development by characterizing optimal solutions of the problem we enumerate a subset of kkt points to determine a global optimum the enumeration is made efficient by developing a theory for shrinking and partitioning the search domain of kkt multipliers the global algorithmic procedure is developed based on interval and point testing techniques that have roots in solving strictly convex problems our algorithm has quadratic worst case complexity and its performance is demonstrated on a broad range of randomly generated problem instances and comparisons with existing global and local solvers it turns out that our method can solve these nonconvex problems of extremely large size in a fraction of the time relative to commercial software such as cplex 12 6 on a desktop computer furthermore a feasible upper bounding scheme is developed for the case when the variables with nonconvex terms are allowed to have finite upper limits using an iterative application of the above global algorithm and the computational experience is presented

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

on the solution of concave knapsack problems we consider a version of the knapsack problem which gives rise to a separable concave minimization problem subject to bounds on the variables and one equality constraint we characterize strict local miniimizers of concave minimization problems subject to linear constraints and use this characterization to show that although the problem of determining a global minimizer of the concave knapsack problem is np hard it is possible to determine a local minimizer of this problem with at most o n logn operations and 1 logn evaluations of the function if the function is quadratic this algorithm requires at most o n logn operations

differential approximation results for the traveling salesman problem with distances 1 and 2 abstract we prove that both minimum and maximum traveling salesman problems on complete graphs with edge distances 1 and 2 denoted by min tsp12 and max tsp12 respectively are approximable within 3 4 based upon this result we improve the standard approximation ratio known for maximum traveling salesman with distances 1 and 2 from 3 4 to 7 8 finally we prove that for any 0 it is np hard to approximate both problems better than within 741 742 the same results hold when dealing with a generalization of min and max tsp12 where instead of 1 and 2 edges are valued by a and b

robust geometrically based automatic two dimensional mesh generation a technical description of the algorithms employed in the modified quadtree mesh generator is given although the basis of the mesh generator is the same as the original version developed by yerry and shephard 1 2 the actual algorithms on which it is built have been entirely changed for the purpose of ensuring the robustness of the technique as demonstrated in the paper the algorithmic changes made do ensure the robustness of the approach but introduce additional algorithmic difficulties the solutions of which are also presented in addition to examples showing the capability of the mesh generator the linear computational growth rate of the mesh generator is demonstrated

a class of nonlinear nonseparable continuous knapsack and multiple choice knapsack problems this paper considers a general class of continuous nonlinear and nonseparable knapsack problems special cases of which arise in numerous operations and financial contexts we develop important properties of optimal solutions for this problem class based on the properties of a closely related class of linear programs using these properties we provide a solution method that runs in polynomial time in the number of decision variables while also depending on the time required to solve a particular one dimensional optimization problem thus for the many applications in which this one dimensional function is reasonably well behaved e g unimodal the resulting algorithm runs in polynomial time we next develop a related solution approach to a class of continuous nonlinear and nonseparable multiple choice knapsack problems this algorithm runs in polynomial time in both the number of variables and the number of variants per item while again dependent on the complexity of the same one dimensional optimization problem as for the knapsack problem computational testing demonstrates the power of the proposed algorithms over a commercial global optimization software package

quadratic programming and combinatorial minimum weight product problems we present a fully polynomial time approximation scheme fptas for minimizing an objective a t x b t x under linear constraints a x d examples of such problems are combinatorial minimum weight product problems such as the following given a graph g v e and two edge weights a b e rightarrow mathbb r find an s t path p that minimizes a p b p the product of its edge weights relative to a and b

variational gram functions convex analysis and optimization we introduce a new class of convex penalty functions called variational gram functions vgfs that can promote pairwise relations such as orthogonality among a set of vectors in a vector space these functions can serve as regularizers in convex optimization problems arising from hierarchical classification multitask learning and estimating vectors with disjoint supports among other applications we study convexity of vgfs and give characterizations for their convex conjugates subdifferentials proximal operators and related quantities we discuss efficient optimization algorithms for regularized loss minimization problems where the loss admits a common yet simple variational representation and the regularizer is a vgf these algorithms enjoy a simple kernel trick an efficient line search as well as computational advantages over first order methods based on the subdifferential or proximal maps we also establish a general representer theorem for such learning problems last numerical experi

voronoi diagram and medial axis algorithm for planar domains with curved boundaries i theoretical foundations in this first installment of a two part paper the underlying theory for an algorithm that computes the voronoi diagram and medial axis of a planar domain bounded by free form polynomial or rational curve segments is presented an incremental approach to computing the voronoi diagram is used wherein a single boundary segment is added to an existing boundary segment set at each step the introduction of each new segment entails modifying the voronoi regions of the existing boundary segments and constructing the voronoi region of the new segment we accomplish this by i computing the bisector of the new segment with each of the current boundary segments ii updating the voronoi regions of the current boundary segments by partitioning them with these bisectors and iii constructing the voronoi region of the new segment as a union of regions obtained from the partitioning in ii when all boundary segments are included and their voronoi regions have been constructed the voronoi diagram of the boundary is obtained as the union of the voronoi polygons for each boundary segment to construct the medial axis of a planar domain we first compute the voronoi diagram of its boundary the medial axis is then obtained from the voronoi diagram by i removing certain edges of the voronoi diagram that do not belong to the medial axis and ii adding certain edges that do belong to the medial axis but are absent from the voronoi diagram unambiguous characterizations for edges in both these categories are given details of algorithms based on this theory are deferred to the second installment of this two part paper

on approximation algorithms for concave mixed integer quadratic programming concave mixed integer quadratic programming is the problem of minimizing a concave quadratic polynomial over the mixed integer points in a polyhedral region in this work we describe an algorithm that finds an epsilon approximate solution to a concave mixed integer quadratic programming problem the running time of the proposed algorithm is polynomial in the size of the problem and in 1 epsilon provided that the number of integer variables and the number of negative eigenvalues of the objective function are fixed the running time of the proposed algorithm is expected unless mathcal p mathcal np

some np complete problems in quadratic and nonlinear programming in continuous variable smooth nonconvex nonlinear programming we analyze the complexity of checking whether a a given feasible solution is not a local minimum and b the objective function is not bounded below on the set of feasible solutions we construct a special class of indefinite quadratic programs with simple constraints and integer data and show that checking a or b on this class is np complete as a corollary we show that checking whether a given integer square matrix is not copositive is np complete

on the steplength selection in gradient methods for unconstrained optimization abstract the seminal paper by barzilai and borwein 1988 has given rise to an extensive investigation leading to the development of effective gradient methods several steplength rules have been first designed for unconstrained quadratic problems and then extended to general nonlinear optimization problems these rules share the common idea of attempting to capture in an inexpensive way some second order information however the convergence theory of the gradient methods using the previous rules does not explain their effectiveness and a full understanding of their practical behaviour is still missing in this work we investigate the relationships between the steplengths of a variety of gradient methods and the spectrum of the hessian of the objective function providing insight into the computational effectiveness of the methods for both quadratic and general unconstrained optimization problems our study also identifies basic principles for designing effective gradient methods

on floyd and rivest s select algorithm we show that several versions of floyd and rivest s algorithm select for finding the kth smallest of n elements require at most n min k n k o n comparisons on average and with high probability this rectifies the analysis of floyd and rivest and extends it to the case of nondistinct elements our computational results confirm that select may be the best algorithm in practice

geometry processing for design and manufacturing this text includes papers covering topics in geometry processing applications such as surface surface intersections and offset surfaces present methods fundamental to geometric modelling are highlighted

clothoid spline transition spirals highway and railway designers use clothoid splines planar g2 curves consisting of straight line segments circular arcs and clothoid segments as center lines in route location this paper considers the problem of finding a clothoid spline transition spiral which joins two given points and matches given curvatures and unit tangents at the two points conditions are given for the existence and uniqueness of the clothoid spline transition spirals and algorithms for finding them are outlined

geometric constraints on quadratic bezier curves using minimal length and energy this paper derives expressions for the arc length and the bending energy of quadratic bezier curves the formulas are in terms of the control point coordinates for fixed start and end points of the bezier curve the locus of the middle control point is analyzed for curves of fixed arc length or bending energy in the case of arc length this locus is convex for bending energy it is not given a line or a circle and fixed end points the locus of the middle control point is determined for those curves that are tangent to a given line or circle for line tangency this locus is a parallel line in the case of the circle the locus can be classified into one of six major types in some of these cases the locus contains circular arcs these results are then used to implement fast algorithms that construct quadratic bezier curves tangent to a given line or circle with given end points that minimize bending energy or arc length

a better differential approximation ratio for symmetric tsp in this paper we study the approximability properties of symmetric tsp under an approximation measure called the differential ratio more precisely we improve up to 3 email protected for any e 0 the best differential ratio of 2 3 known so far given in hassin and khuller r hassin s khuller z approximations j algorithms 41 2 2001 429 442

differential approximation algorithms for some combinatorial optimization problems we use a new approximation measure the differential approximation ratio to derive polynomial time approximation algorithms for minimum set covering for both weighted and unweighted cases minimum graph coloring and bin packing we also propose differential approximation ratio preserving reductions linking minimum coloring minimum vertex covering by cliques minimum edge covering by cliques and minimum edge covering of a bipartite graph by complete bipartite graphs

formulation analysis and numerical study of an optimization based conservative interpolation remap of scalar fields for arbitrary lagrangian eulerian methods we develop and study the high order conservative and monotone optimization based remap obr of a scalar conserved quantity mass between two close meshes with the same connectivity the key idea is to phrase remap as a global inequality constrained optimization problem for mass fluxes between neighboring cells the objective is to minimize the discrepancy between these fluxes and the given high order target mass fluxes subject to constraints that enforce physically motivated bounds on the associated primitive variable density in so doing we separate accuracy considerations handled by the objective functional from the enforcement of physical bounds handled by the constraints the resulting obr formulation is applicable to general unstructured heterogeneous grids under some weak requirements on grid proximity but not on the cell types we prove that the obr algorithm is linearity preserving in one two and three dimensions the paper also examines connections between the obr and the recently proposed flux corrected remap fcr liska et al 1 we show that the fcr solution coincides with the solution of a modified version of obr m obr which has the same objective but a simpler set of box constraints derived by using a worst case scenario because m obr fcr has a smaller feasible set preservation of linearity may be lost and accuracy may suffer for some grid configurations our numerical studies confirm this and show that obr delivers significant increases in robustness and accuracy preliminary efficiency studies of obr reveal that it is only a factor of 2 1 slower than fcr but admits 1 5 times larger time steps

about strongly polynomial time algorithms for quadratic optimization over submodular constraints we present new strongly polynomial algorithms for special cases of convex separable quadratic minimization over submodular constraints the main results are an o nm log n 2 m algorithm for the problemnetwork defined on a network onm arcs andn nodes an o n logn algorithm for thetree problem onn variables an o n logn algorithm for thenested problem and a linear time algorithm for thegeneralized upper bound problem these algorithms are the best known so far for these problems the status of the general problem and open questions are presented as well

fast optimization based conservative remap of scalar fields through aggregate mass transfer abstract we develop a fast efficient and accurate optimization based algorithm for the high order conservative and local bound preserving remap constrained interpolation of a scalar conserved quantity between two close meshes with the same connectivity the new formulation is as robust and accurate as the flux variable flux target optimization based remap fvft obr 1 2 yet has the computational efficiency of an explicit remapper the coupled system of linear inequality constraints resulting from the flux form of remap is the main efficiency bottleneck in fvft obr while advection based remappers use the flux form to directly enforce mass conservation the optimization setting allows us to treat mass conservation as one of the constraints to take advantage of this fact we consider an alternative mass variable mass target mvmt obr formulation in which the optimization variables are the net mass updates per cell and a single linear constraint enforces the conservation of mass in so doing we change the structure of the obr problem from a global linear inequality constrained qp to a singly linearly constrained qp with simple bounds using the structure of the mvmt obr problem and the fact that in remap the old and new grids are close we are able to develop a simple efficient and easily parallelizable optimization algorithm for the primal mvmt obr qp numerical studies on a variety of affine and non affine grids confirm that mvmt obr is as accurate and robust as fvft obr but has the same computational cost as the explicit state of the art fcr

voronoi diagram and medial axis algorithm for planar domains with curved boundaries ii detailed algorithm description abstract details of algorithms to construct the voronoi diagrams and medial axes of planars domain bounded by free form polynomial or rational curve segments are presented based on theoretical foundations given in the first installment ramamurthy and farouki j comput appl math 1999 102 119 141 of this two part paper in particular we focus on key topological and computational issues that arise in these constructions the topological issues include i the data structures needed to represent various geometrical entities bisectors voronoi regions etc and ii the boolean operations i e union intersection and difference on planar sets required by the algorithm specifically representations for the voronoi polygons of boundary segments and for individual voronoi diagram or medial axis edges are proposed since these edges may be segments of a nonrational algebraic curves curve curve bisectors b rational curves point curve bisectors or c straight lines point point bisectors data structures tailored to each of these geometrical entities are introduced the computational issues addressed include the curve intersection algorithms required in the boolean operations and iterative schemes used to precisely locate bifurcation or n prong points n 3 of the voronoi diagram and medial axis a selection of computed voronoi diagram and medial axis examples is included to illustrate the capabilities of the algorithm

on the continuous quadratic knapsack problem we introduce a new algorithm for the continuous bounded quadratic knapsack problem this algorithm is motivated by the geometry of the problem is based on the iterative solution of a series of simple projection problems and is easy to understand and implement in practice the method compares favorably to other well known algorithms some of which have superior worst case complexity on problem sizes up ton 4000

high quality point sampling for b spline fitting of parametric curves with feature recognition abstract determining a sequence of points from parametric curves is essential for the construction of least squares b spline fitting curves in geometric modeling and related applications uniform sampling methods always determine points uniformly in arc length curvature or a weighted form but feature points intuitively indicating the curve profile are generally ignored in this paper we focus on high quality point sampling so as to better capture the original curve shape firstly all the feature points inflection points and extreme curvature points on the curve are recognized by a parabolic interpolation method then more auxiliary points are adaptively added according to the characteristic function defined as a weighted sum of arc length and total curvature characteristics by adjusting the weight in the characteristic function it provides more flexibility and controllability to obtain auxiliary points finally we convert the original curve into a b spline curve by interpolating these feature points and auxiliary points using the progressive iterative approximation method numerical examples demonstrate the effectiveness and high quality of our proposed method

multimaterial topology optimization by volume constrained allen cahn system and regularized projected steepest descent method abstract a new computational algorithm is introduced in the present study to solve multimaterial topology optimization problems it is based on the penalization of the objective functional by the multiphase volume constrained ginzburg landau energy functional the update procedure is based on the gradient flow of the objective functional by a fractional step projected steepest descent method in the first step the new design is found based on the projected steepest descent method to ensure the reduction in the objective functional simultaneously satisfying the control constraints in the second step regularization step an h 1 regularity of the solution is ensured while keeping the feasibility of solution with respect to the set of control constraints the presented algorithm could be accounted as a constrained h 1 optimization algorithm which according to our knowledge has not been reported to solve such kind of problems yet the success and efficiency of the presented method are shown through several test problems numerical results show that the presented algorithm ends with a near 0 1 topology and its computational cost scales sub linearly by the number of phases for the sake of reader convenience and the ease of further extension the matlab implementation of the presented algorithm is included in the appendix

strongly polynomial algorithms for the quadratic transportation problem with a fixed number of sources the transportation problem with a linear objective function is known to be solvable in strongly polynomial time whereas instances with a convex nonquadratic objective function are not even for cases with the integrality constraints relaxed we present a linear time algorithm for the continuous quadratic transportation problem qtp with two source nodes further we show how problem instances with a fixed number of source or destination nodes k could be solved in strongly polynomial time onk 1 using an algebraic tree computation model the algorithms exploit a relationship between the transportation problem and the resource allocation problem the strong polynomiality of the algorithms presented here imply the existence of strongly polynomial algorithms for integer quadratic transportation problems with a fixed number of source nodes as well

the nonlinear knapsack problem algorithms and applications abstract we present a survey of algorithms and applications for the nonlinear knapsack problem or the nonlinear resource allocation problem in its most general form the nonlinear knapsack problem will be defined as a nonlinear optimization problem with just one constraint bounds on the variables and in some cases a set of specially structured constraints such as generalized upper bounds gubs this problem is encountered either directly or as a subproblem in a variety of applications including production planning financial modeling stratified sampling and capacity planning in manufacturing health care and computer networks by taking advantage of the special structure of the problem efficient solution methods can be developed problem classes addressed here include continuous and integer problems convex and nonconvex problems separable and nonseparable problems and problems with additional specially structured constraints

breakpoint searching algorithms for the continuous quadratic knapsack problem we give several linear time algorithms for the continuous quadratic knapsack problem in addition we report cycling and wrong convergence examples in a number of existing algorithms and give encouraging computational results for large scale problems

projected barzilai borwein methods for large scale box constrained quadratic programming this paper studies projected barzilai borwein pbb methods for large scale box constrained quadratic programming recent work on this method has modified the pbb method by incorporating the grippo lampariello lucidi gll nonmonotone line search so as to enable global convergence to be proved we show by many numerical experiments that the performance of the pbb method deteriorates if the gll line search is used we have therefore considered the question of whether the unmodified method is globally convergent which we show not to be the case by exhibiting a counter example in which the method cycles a new projected gradient method pabb is then considered that alternately uses the two barzilai borwein steplengths we also give an example in which this method may cycle although its practical performance is seen to be superior to the pbb method with the aim of both ensuring global convergence and preserving the good numerical performance of the unmodified methods we examine other recent work on nonmonotone line searches and propose a new adaptive variant with some attractive features further numerical experiments show that the pabb method with the adaptive line search is the best bb like method in the positive definite case and it compares reasonably well against the gpcg algorithm of more and toraldo in the indefinite case the pbb method with the adaptive line search is shown on some examples to find local minima with better solution values and hence may be preferred for this reason

disaggregation and resource allocation using convex knapsack problems with bounded variables the allocation of a specific amount of a given resource among competitive alternatives can often be modelled as a knapsack problem this model formulation is extremely efficient because it allows convex cost representation with bounded variables to be solved without great computational efforts practical applications of this problem abound in the fields of operations management finance manpower planning marketing etc in particular knapsack problems emerge in hierarchical planning systems when a first level of decisions need to be further allocated among specific activities which have been previously treated in an aggregate way in this paper we provide a recursive procedure to solve such problems the method differs from classical optimization algorithms of convex programming in that it determines at each iteration the optimal value of at least one variable applications and computational results are presented

local minima for indefinite quadratic knapsack problems we consider the complexity of finding a local minimum for the nonconvex quadratic knapsack problem global minimization for this example of quadratic programming is np hard more and vavasis have investigated the complexity of local minimization for the strictly concave case of qkp here we extend their algorithm to the general indefinite case our main result is an algorithm that computes a local minimum in o n logn 2 steps our approach involves eliminating all but one of the convex variables through parametrization yielding a nondifferentiable problem we use a technique from computational geometry to address the nondifferentiable problem

a filtered bucket clustering method for projection onto the simplex and the ell _1 l1 ball we propose in this paper a new method processing the projection of an arbitrary size vector onto the probabilistic simplex or the ell 1 ball our method merges two principles the first one is an original search of the projection using a bucket algorithm the second one is a filtering on the fly of the values that cannot be part of the projection the combination of these two principles offers a simple and efficient algorithm whose worst case complexity is linear with respect to the vector size furthermore the proposed algorithm exploits the representation of numeric values in digital computers to define the number of buckets and to accelerate the filtering

power diagrams properties algorithms and applications the power pow x s of a point x with respect to a sphere s in euclidean d space e d is given by d 2 x z r 2 where d denotes the euclidean distance function and z and r are the center and the radius of s the power diagram of a finite set s of spheres in e d is a cell complex that associates each s in s with the convex domain x in e d operatorname pow x s operatorname pow x t text for all t in s s the close relationship to convex hulls and arrangements of hyperplanes is investigated and exploited efficient algorithms that compute the power diagram and its order k modifications are obtained among the applications of these results are algorithms for detecting k sets for union and intersection problems for cones and paraboloids and for constructing weighted voronoi diagrams and voronoi diagrams for spheres upper space bounds for these geometric problems are derived

on mutual concavity and strategically zero sum bimatrix games we study the fundamental problem 2nash of computing a nash equilibrium ne point in bimatrix games we start by proposing a novel characterization of the ne set via a bijective map to the solution set of a parameterized quadratic program neqp whose feasible space is the highly structured set of correlated equilibria ce this is to our knowledge the first characterization of the subset of ce points that are in 1 1 correspondence with the ne set of the game and contributes to the quite lively discussion on the relation between the spaces of ce and ne points in a bimatrix game e g 15 26 33 we proceed with studying a property of bimatrix games which we call mutually concavity mc that assures polynomial time tractability of 2nash due to the convexity of a proper parameterized quadratic program either neqp or a parameterized variant of the mangasarian stone formulation 23 for a particular value of the parameter we prove various characterizations of the mc games which eventually lead us to the conclusion that this class is equivalent to the class of strategically zero sum szs games of moulin vial 25 this gives an alternative explanation of the polynomial time tractability of 2nash for these games not depending on the solvability of zero sum games moreover the recognition of the mc property for an arbitrary game is much faster than the recognition szs property this along with the comparable time complexity of linear programs and convex quadratic programs leads us to a much faster algorithm for 2nash in mc games we conclude our discussion with a comparison of mc games or szs games to k rank games which are known to admit for 2nash a fptas when k is fixed 18 and a polynomial time algorithm for k 1 2 we finally explore some closeness properties under well known ne set preserving transformations of bimatrix games

quasi newton updates with bounds we develop a quasi newton method which preserves known bounds on the jacobian matrix we show that this update can be computed with the same amount of work as competitive methods in particular we prove that the number of operations required to obtain this update is proportional to the number of nonzeros in the sparsity pattern of the jacobian matrix the method is also shown to share the local convergence properties of broyden s and schubert s method

generalization of voronoi diagrams in the plane in this paper we study the voronoi diagram for a set of n line segments and circles in the euclidean plane the diagram is a generalization of the voronoi diagram for a set of points in the plane and has applications in wire layout facility location clustering and contouring problems we present an o n log n 2 algorithm for constructing the diagram it is an improvement of a previous known result which takes o nc sqrt log n time the algorithm described in this paper is also shown to be applicable under a more general metric if certain conditions are satisfied

evaluating the boundary and covering degree of planar minkowski sums and other geometrical convolutions algorithms are developed based on topological principles to evaluate the boundary and internal structure of the minkowski sum of two planar curves a graph isotopic to the envelope curve is constructed by computing its characteristic points the edges of this graph are in one to one correspondence with a set of monotone envelope segments a simple formula allows a degree to be assigned to each face defined by the graph indicating the number of times its points are covered by the minkowski sum the boundary can then be identified with the set of edges that separate faces of zero and non zero degree and the boundary segments corresponding to these edges can be approximated to any desired geometrical accuracy for applications that require only the minkowski sum boundary the algorithm minimizes geometrical computations on the internal envelope edges that do not contribute to the final boundary in other applications this internal structure is of interest and the algorithm provides comprehensive information on the covering degree for different regions within the minkowski sum extensions of the algorithm to the computation of minkowski sums in r 3 and other forms of geometrical convolution are briefly discussed

the complexity of approximating a nonlinear program we consider the problem of finding the maximum of a multivariate polynomial inside a convex polytope we show that there is no polynomial time approximation algorithm for this problem even one with a very poor guarantee unless p np we show that even when the polynomial is quadratic i e quadratic programming there is no polynomial time approximation unless np is contained in quasi polynomial time