the expansion of functions under transformation and its application to optimization abstract this paper proposes the expansion of functions under transformation which represents infinite kinds of extensions of taylor s expansion based on this duffin s condensation formula is developed as a condensation formula of high order furthermore a theorem of efficiently solving mathematical programming by using the expansion of functions is proposed according to the condensation formula of high order two kinds of second order primal algorithms are proposed for solving generalized geometric programming with rapid and stable convergence which could be applied to a wide range of structural optimization problems finally an efficient fully stressed design method under the function transformation is proposed using these expansions of functions

domain contraction in nonlinear programming minimizing a quadratic concave objective over a polyhedron using linear underestimating approximations and their dual solutions we develop new results for the nonconvex problem of minimizing a quadratic concave function under linear constraints for solving this problem which has such important management applications as economies of scale fixed charge and quadratic assignment problems some results provide and tighten bounds on the global minimum value while others identify the domains of the variables which may be excluded from further consideration this leads to successive reduction of domains containing a global solution called domain contraction and ideally to a solution with the objective value within some acceptable error level for most results specific problem instances are provided to establish their use in theory in practice experience with solving the problems presented here indicates that a combination of the proposed domain contraction via linear underestimation and its dual information with the branch and bound approach offers a computa

a new accelerating method for global non convex quadratic optimization with non convex quadratic constraints abstract in this paper we combine the new global optimization method proposed by qu et al s j qu k c zhang y ji a global optimization algorithm using parametric linearization relaxation appl math comput 186 2007 763 771 with a suitable deleting technique to propose a new accelerating global optimization algorithm for solving the non convex quadratic optimization problems with non convex quadratic constraints nqp this technique offers a possibility to cut away a large part of the currently investigated region in which the global optimal solution of nqp does not exist and can be seen as an accelerating device for the global optimization algorithm of the nqp problems compared with the method in qu et al 2007 numerical results show that the computational efficiency is obviously improved by using this new technique in the number of iterations the required list length and the overall execution time of the algorithm

a quadratic programming approach to blind equalization and signal separation blind equalization and signal separation are two well established signal processing problems in this paper we present a quadratic programming algorithm for fast blind equalization and signal separation by introducing a special non mean square error mse objective function we reformulate fractionally spaced blind equalization into an equivalent quadratic programming problem based on a clear geometric interpretation and a formal proof we show that a perfect equalization solution is obtained at every local optimum of the quadratic program because blind source separation is by nature and mathematically a closely related problem we also generalize the algorithm for blind signal separation we show that by enforcing source orthogonalization through successive processing the quadratic programming approach can be applied effectively moreover the quadratic program is easily extendible to incorporate additional practical conditions such as jamming suppression constraints we also provide evidence of good performance through computer simulations

the anisotropic total variation flow we prove existence and uniqueness of solutions of the anisotropic total variation flow when the initial data is an l2 function and we give a characterization of such solutions that allows us to find explicit evolutions of sets in the presence of an anisotropy

reducing measurement error in student achievement estimation the achievement level is a variable measured with error that can be estimated by means of the rasch model teacher grades also measure the achievement level but they are expressed on a different scale this paper proposes a method for combining these two scores to obtain a synthetic measure of the achievement level based on the theory developed for regression with covariate measurement error in particular the focus is on ordinal scaled grades using the simex method for measurement error correction the result is a measure comparable across subjects with smaller measurement error variance an empirical application illustrates the method

heteroscedastic factor analysis summary two moment based model fitting procedures for the heteroscedastic factor analysis model are introduced and compared the procedures produce consistent parameter estimators and asymptotically valid inferences for heteroscedasticity without specifying the distributional forms for the factor and heteroscedastic errors also an individual specific inference procedure for the factor score is developed simulation studies show the practical usefulness of the procedures an example from a morphological measurement study is described

a new hybrid method for solving global optimization problem abstract in this paper we present a new hybrid method called the sasp method the purpose of this method is the hybridization of the simulated annealing sa with the descent method where we estimate the gradient using simultaneous perturbation firstly the new hybrid method finds a local minimum using the descent method then sa is executed in order to escape from the currently discovered local minimum to a better one from which the descent method restarts a new local search and so on until convergence the new hybrid method can be widely applied to a class of global optimization problems for continuous functions with constraints experiments on 30 benchmark functions including high dimensional functions show that the new method is able to find near optimal solutions efficiently in addition its performance as a viable optimization method is demonstrated by comparing it with other existing algorithms numerical results improve the robustness and efficiency of the method presented

an effective linear approximation method for separable programming problems among the numerous applications of piecewise linearization methods include data fitting network analysis logistics and statistics in the early 1950s a concave function was found to be able to be linearized by introducing 0 1 variables most textbooks in operations research offer such methods for expressing linear approximations various methods of linearization have also been developed in recent literature nevertheless the transformed linear scheme has a severe shortcoming most standard procedures for linearizing typically involve a large increase in the number of binary variables consequently the gains to be derived from dealing with linear functions are quite likely to be nullified by the increase in the size of the problem conventional methods for linearizing a concave function with m break points require m 1 binary variables however when m becomes large the computation will be very time consuming and may cause a heavy computational burden this study proposes an effective approach in which only log 2 m 1 binary variables are used the proposed method has the following features i it offers more convenient and efficient means of expressing a piecewise linear function ii fewer 0 1 variables are used iii the computational results show that the proposed method is much more efficient and faster than the conventional one especially when the number of break points becomes large

the total variation flow in rn the purpose of this chapter is to prove existence and uniqueness of the minimizing total variation flow in n frac partial u partial t div left frac du left du right right in left 0 infty right times mathbb r n 3 1 coupled with the initial condition u 0 x u 0 x text x in mathbb r n 3 2 when u0 l loc 1 l n

d c sets d c functions and nonlinear equations ad c set is a set which is the difference of two convex sets we show that any set can be viewed as the image of a d c set under an appropriate linear mapping using this universality we can convert any problem of finding an element of a given compact set in n into one of finding an element of a d c set on the basis of this approach a method is developed for solving a system of nonlinear equations inequations unlike newton type methods our method does not require either convexity differentiability assumptions or an initial approximate solution

a nonisolated optimal solution for special reverse convex programming problems in this paper an efficient algorithm is proposed for globally solving special reverse convex programming problems with more than one reverse convex constraints the proposed algorithm provides a nonisolated global optimal solution which is also stable under small perturbations of the constraints and it turns out that such an optimal solution is adequately guaranteed to be feasible and to be close to the actual optimal solution convergence of the algorithm is shown and the numerical experiment is given to illustrate the feasibility of the presented algorithm

comparison principles for infinity laplace equations in finsler metrics abstract in this paper we study comparison principles for normalized finsler infinity laplace operators with nonhomogeneous terms that depend on solutions and their gradients this is achieved by using a combination of sup convolution methods to approximate viscosity solutions by semiconvex functions and finite difference approximation schemes characterizations of viscosity subsolutions and supersolutions of equations with constant nonhomogeneous terms through comparison with quadratic finsler cone are also discussed

a characterization of convex calibrable sets in the main purpose of this paper is to characterize the calibrability of bounded convex sets in open image in new window by the mean curvature of its boundary extending the known analogous result in dimension 2 as a by product of our analysis we prove that any bounded convex set c of class c1 1 has a convex calibrable set k in its interior and and for any volume v k c the solution of the perimeter minimizing problem with fixed volume v in the class of sets contained in c is a convex set as a consequence we describe the evolution of convex sets in open image in new window by the minimizing total variation flow

asymptotic chi square tests for a large class of factor analysis models asymptotic behavior of the test statistics for three types of 2 goodness of fit tests is investigated for the factor analysis model with linearly or nonlinearly restricted factor loadings under weak assumptions on the factor vector and the error vector in particular the limiting 2 result for the three tests is shown to hold for the factor vector and for the error vector with any distribution having finite second order moments provided that the components of the error vector are independent as special cases the result holds for exploratory and confirmatory factor analysis models and for certain non normal structural equation models

a parametric linear relaxation algorithm for globally solving nonconvex quadratic programming abstract in this article we present a parametric linear relaxation algorithm for globally solving the nonconvex quadratic programming nqp in this algorithm a new parametric linearized technique is proposed for generating parametric linear relaxation programming plrp of the nqp which can be used to determine the lower bound of global minimum value of the nqp to improve the convergent speed of the proposed algorithm a pruning operation is employed to compress the investigated region by subdividing subsequently the initial domain and solving subsequently a series of parametric linear relaxation programming problems over the subdivided domain the proposed algorithm is convergent to the global minimum of the nqp finally an engineering problem for the design of heat exchanger network and some test examples are used to verify the effectiveness of the proposed algorithm

a new technique for generating quadratic programming test problems this paper describes a new technique for generating convex strictly concave and indefinite bilinear or not quadratic programming problems these problems have a number of properties that make them useful for test purposes for example strictly concave quadratic problems with their global maximum in the interior of the feasible domain and with an exponential number of local minima with distinct function values and indefinite and jointly constrained bilinear problems with nonextreme global minima can be generated unlike most existing methods our construction technique does not require the solution of any subproblems or systems of equations in addition the authors know of no other technique for generating jointly constrained bilinear programming problems

a new rectangle branch and pruning approach for generalized geometric programming generalized geometric programming ggp problem occurs frequently in engineering design and management in this paper a branch and pruning global optimization algorithm is proposed for ggp by utilizing some transformations a linear relaxation of the problem ggp is obtained based on the linear lower bound functions of objective and constraint functions inside some hyperrectangle region then a new pruning technique is given to accelerate the convergence of the given algorithm and this pruning technique offers the possibility to cut away a large part of the current investigated region in which there no exist global optimum solution the proposed algorithm which connects branch and bound method with the pruning technique successfully is convergent to the global minimum according to the successive refinement of the linear relaxation of feasible region of the objective function and the solutions of a series of linear relaxation problems and finally numerical experiment is given to illustrate the feasibility and efficiency of the proposed algorithm

a parallel algorithm for constrained concave quadratic global minimization the global minimization of large scale concave quadratic problems over a bounded polyhedral set using a parallel branch and bound approach is considered the objective function consists of both a concave part nonlinear variables and a strictly linear part which are coupled by the linear constraints these large scale problems are characterized by having the number of linear variables much greater than the number of nonlinear variables a linear underestimating function to the concave part of the objective is easily constructed and minimized over the feasible domain to get both upper and lower bounds on the global minimum function value at each minor iteration of the algorithm the feasible domain is divided into subregions and linear underestimating problems over each subregion are solved in parallel branch and bound techniques can then be used to eliminate parts of the feasible domain from consideration and improve the upper and lower bounds it is shown that the algorithm guarantees that a solution is obtained to within any specified tolerance in a finite number of steps computational results are presented for problems with 25 and 50 nonlinear variables and up to 400 linear variables these results were obtained on a four processor cray2 using both sequential and parallel implementations of the algorithm the average parallel solution time was approximately 15 seconds for problems with 400 linear variables and a relative tolerance of 0 001 for a relative tolerance of 0 1 the average computation time appears to increase only linearly with the number of linear variables

facet bending driven by the planar crystalline curvature with a generic nonuniform forcing term abstract we study crystalline driven curvature flow with spatially nonuniform driving force term we assume special monotonicity properties of the driving term which are motivated by our previous work on berg s effect we consider special initial data which we call bent rectangles we prove existence of solutions for a generic forcing term as well as generic subclass of bent rectangles we show the initially flat facets may begin to bend provided loosely speaking they are too large moreover depending on the initial configuration we notice instantaneous loss of regularity of the moving curve

global minimization of a linearly constrained concave function by partition of feasible domain the problem of minimizing a concave function subject to linear inequality constraints may have many local solutions therefore finding the global constrained minimum is a computationally difficult problem a computational method is described which finds the global minimum of a smooth concave function over a polyhedron in rn the feasible domain is partitioned into a rectangular domain which can be excluded from further consideration and r 2n subdomains at least one of which contains the global minimum a known algorithm can be applied sequentially or in parallel to each of these r subdomains to compute the global minimum a method is also presented appendix b for the construction of nontrivial test problems for which the global minimum point is known given an arbitrary polyhedron and a selected vertex it is shown how to determine a concave quadratic function generally with many local minima with its global minimum at the selected vertex

on generalized geometric programming problems with non positive variables generalized geometric programming ggp problems occur frequently in engineering design and management some exponential based decomposition methods have been developed for solving global optimization of ggp problems however the use of logarithmic exponential transformations restricts these methods to handle the problems with strictly positive variables this paper proposes a technique for treating non positive variables with integer powers in ggp problems by means of variable transformation the ggp problem with non positive variables can be equivalently solved with another one having positive variables in addition we present some computationally efficient convexification rules for signomial terms to enhance the efficiency of the optimization approach numerical examples are presented to demonstrate the usefulness of the proposed method in ggp problems with non positive variables

a global optimization algorithm using linear relaxation various local optimization approaches have been developed for solving the non convex quadratically constrained quadratic programs qp but up to now less work has been devoted to solving global optimization of qp problem due to the inherent difficulty this paper considers the global minimum of qp that arise in various practice problems by utilizing the inherent property of the quadratic function an approximations a linear relaxation of qp is then obtained thus initial qp is reduced to a sequence of linear programming problems through the successive refinement of a linear relaxation of feasible region of the objective function the proposed algorithm is convergent to the global minimum of qp by means of the subsequent solutions of a series of linear programming problems test results indicate that the proposed algorithm is extremely robust and can be used successfully to solve the global minimum of qp on a microcomputer

quasiconjugates of functions duality relationship between quasiconvex minimization under a reverse convex constraint and quasiconvex maximization under a convex constraint and applications abstract in this paper we introduce a concept of quasiconjugate for functions defined on rn whose values are in r the conjugacy correspondence between functions and their quasiconjugates is one to one and symmetric in a class of quasiconvex functions whose minimizer on rn is located at the origin by using the concept of quasiconjugate we obtain a duality relationship between quasiconvex minimization under a reverse convex constraint and quasiconvex maximization under a convex constraint this duality relationship allows us to establish a primal dual pair in a class of nonconvex optimization problems without the duality gap several applications are given

computational method for the profit bounds of inventory model with interval demand and unit cost changing economic conditions make the selling price and demand quantity more and more uncertain in the market the conventional inventory models determine the selling price and order quantity for a retailer s maximal profit with exactly known parameters this paper develops a solution method to derive the imprecise profit of the inventory model when the demand quantity and unit cost are imprecise since imprecise parameters contained in the inventory model lie in intervals the profit value calculated from the model should be in a range as well we formulate a pair of two level mathematical programming model to derive the upper bound and lower bound of the profit based on the duality theorem of geometric programming the pair of two level mathematical programs is transformed into a pair of conventional geometric programs solving the corresponding pair of geometric programs produces the interval of profit an example illustrates the idea in this paper that example also shows that the interval profit contains more information for determining inventory policy

accelerating method of global optimization for signomial geometric programming abstract signomial geometric programming sgp has been an interesting problem for many authors recently many methods have been provided for finding locally optimal solutions of sgp but little progress has been made for global optimization of sgp in this paper we propose a new accelerating method for global optimization algorithm of sgp using a suitable deleting technique this technique offers a possibility to cut away a large part of the currently investigated region in which the globally optimal solution of sgp does not exist and can be seen as an accelerating device for global optimization algorithm of sgp problem compared with the method of shen and zhang global optimization of signomial geometric programming using linear relaxation appl math comput 150 2004 99 114 numerical results show that the computational efficiency is improved obviously by using this new technique in the number of iterations the required saving list length and the execution time of the algorithm

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

estimation for the multivariate errors in variables model with estimated error covariance matrix the errors in variables model in which the unobserved true values satisfy multiple linear restrictions is considered under the assumptions that the unobservable true values are normally distributed and that an estimator of the covariance matrix of the measurement error is available the maximum likelihood estimators are derived the limiting properties of the estimators are obtained for a wide range of assumptions including the assumption of fixed true values

global optimization for sum of generalized fractional functions this paper considers the solution of generalized fractional programming gfp problem which contains various variants such as a sum or product of a finite number of ratios of linear functions polynomial fractional programming generalized geometric programming etc over a polytope for such problems we present an efficient unified method in this method by utilizing a transformation and a two part linearization method a sequence of linear programming relaxations of the initial nonconvex programming problem are derived which are embedded in a branch and bound algorithm numerical results are given to show the feasibility and effectiveness of the proposed algorithm

methods for global concave minimization a bibliographic survey the global concave minimization problem is to find the constrained global minimum of a concave function since such a function may have many local minima finding the global minimum is a computationally difficult problem in this bibliographic survey which includes most of the recent papers on constrained global concave minimization we have attempted to briefly summarize the main ideas in each paper these recent papers include those concerned with large scale global concave minimization and bilinear programming

total variation and cheeger sets in gauss space abstract the aim of this paper is to study the isoperimetric problem with fixed volume inside convex sets and other related geometric variational problems in the gauss space in both the finite and infinite dimensional case we first study the finite dimensional case proving the existence of a maximal cheeger set which is convex inside any bounded convex set we also prove the uniqueness and convexity of solutions of the isoperimetric problem with fixed volume inside any convex set then we extend these results in the context of the abstract wiener space and for that we study the total variation denoising problem in this context

a global optimization using linear relaxation for generalized geometric programming many local optimal solution methods have been developed for solving generalized geometric programming ggp but up to now less work has been devoted to solving global optimization of ggp problem due to the inherent difficulty this paper considers the global minimum of ggp problems by utilizing an exponential variable transformation and the inherent property of the exponential function and some other techniques the initial nonlinear and nonconvex ggp problem is reduced to a sequence of linear programming problems the proposed algorithm is proven that it is convergent to the global minimum through the solutions of a series of linear programming problems test results indicate that the proposed algorithm is extremely robust and can be used successfully to solve the global minimum of ggp on a microcomputer

an algorithm for indefinite integer quadratic programming abstract we present an algorithm for finding the global minimum of an indefinite quadratic function over the integers contained in a compact convex set to find this minimum the algorithm first transforms the problem into an equivalent problem with a separable objective function it then uses a branch and bound search on the values of the constraints rather than the variables of the transformed problem

relative entropy relaxations for signomial optimization signomial programs sps are optimization problems specified in terms of signomials which are weighted sums of exponentials composed with linear functionals of a decision variable sps are nonconvex optimization problems in general and families of np hard problems can be reduced to sps in this paper we describe a hierarchy of convex relaxations to obtain successively tighter lower bounds of the optimal value of sps this sequence of lower bounds is computed by solving increasingly larger sized relative entropy optimization problems which are convex programs specified in terms of linear and relative entropy functions our approach relies crucially on the observation that the relative entropy function by virtue of its joint convexity with respect to both arguments provides a convex parametrization of certain sets of globally nonnegative signomials with efficiently computable nonnegativity certificates via the arithmetic geometric mean inequality by appealing to representation theorems from real algeb

a milp formulation for generalized geometric programming using piecewise linear approximations abstract generalized geometric programming ggp problems are converted to mixed integer linear programming milp problems using piecewise linear approximations our approach is to approximate a multiple term log sum function of the form log x 1 x 2 x n in terms of a set of linear equalities or inequalities of log x 1 log x 2 and log x n where x 1 x n are strictly positive the advantage of this approach is its simplicity and readiness to implement and solve using commercial milp solvers while milp problems in general are no easier than ggp problems this approach is justified by the phenomenal progress of computing power of both personal computers and commercial milp solvers the limitation of this approach is discussed along with numerical tests

global optimization of signomial geometric programming problems abstract this paper presents a global optimization approach for solving signomial geometric programming problems in most cases nonconvex optimization problems with signomial parts are difficult np hard problems to solve for global optimality but some transformation and convexification strategies can be used to convert the original signomial geometric programming problem into a series of standard geometric programming problems that can be solved to reach a global solution the tractability and effectiveness of the proposed successive convexification framework is demonstrated by seven numerical experiments some considerations are also presented to investigate the convergence properties of the algorithm and to give a performance comparison of our proposed approach and the current methods in terms of both computational efficiency and solution quality

range division and contraction algorithm for a class of global optimization problems abstract this article presents a range division and contraction algorithm for a class of global optimization problems in the algorithm the original problem is first converted into an equivalent monotonic optimization problem whose objective function is just a simple univariate by exploiting the particularity of this monotonicity the variable bound in a particular node is then tightened and the bounds on the constraints are calculated to remove the region which doesn t contain optimal solutions the proposed method can reach an approximate solution within an acceptable error in which such solution is adequately guaranteed to be feasible and to be close to the actual global optimal solution several numerical examples are given to illustrate the feasibility and efficiency of the present algorithm

global optimization of mixed integer nonlinear programs a theoretical and computational study this work addresses the development of an efficient solution strategy for obtaining global optima of continuous integer and mixed integer nonlinear programs towards this end we develop novel relaxation schemes range reduction tests and branching strategies which we incorporate into the prototypical branch and bound algorithm in the theoretical algorithmic part of the paper we begin by developing novel strategies for constructing linear relaxations of mixed integer nonlinear programs and prove that these relaxations enjoy quadratic convergence properties we then use lagrangian linear programming duality to develop a unifying theory of domain reduction strategies as a consequence of which we derive many range reduction strategies currently used in nonlinear programming and integer linear programming this theory leads to new range reduction schemes including a learning heuristic that improves initial branching decisions by relaying data across siblings in a branch and bound tree finally we incorporate these relaxation and reduction strategies in a branch and bound algorithm that incorporates branching strategies that guarantee finiteness for certain classes of continuous global optimization problems in the computational part of the paper we describe our implementation discussing wherever appropriate the use of suitable data structures and associated algorithms we present computational experience with benchmark separable concave quadratic programs fractional 0 1 programs and mixed integer nonlinear programs from applications in synthesis of chemical processes engineering design just in time manufacturing and molecular design

a comparison principle for singular diffusion equations with spatially inhomogeneous driving force for graphs we introduce the notions of viscosity super and subsolutions suitable for singular diffusion equations of non divergence type with a general spatially inhomogeneous driving term in particular the viscosity super and subsolutions support facets and allow a possible facet bending we prove a comparison principle by a modified doubling variables technique finally we present examples of viscosity solutions our results apply to a general crystalline curvature flow with a spatially inhomogeneous driving term for a graph like curve

global minimization of large scale constrained concave quadratic problems by separable programming the global minimization of a large scale linearly constrained concave quadratic problem is considered the concave quadratic part of the objective function is given in terms of the nonlinear variablesx r n while the linear part is in terms ofy r k for large scale problems we may havek much larger thann the original problem is reduced to an equivalent separable problem by solving a multiple cost row linear program with 2n cost rows the solution of one additional linear program gives an incumbent vertex which is a candidate for the global minimum and also gives a bound on the relative error in the function value of this incumbent ana priori bound on this relative error is obtained which is shown to be 0 25 in important cases if the incumbent is not a satisfactory approximation to the global minimum a guaranteede approximate solution is obtained by solving a single liner zero one mixed integer programming problem this integer problem is formulated by a simple piecewise linear underestimation of the separable problem

curvature driven flows a variational approach this paper introduces a new mathematical approach to the study of time evolutions of solids k t in n space whose boundaries move with velocity equal to the weighted mean curvature derived from the boundary surface energy phi partial k t these flat phi curvature flows are limits of sequences of solutions to variational problems in which a sum of surface and bulk energy is minimized the construction works equally well for smooth elliptic phi s for nondifferentiable crystalline phi s and for anything in between the flows agree with classical smooth flows when the data is smooth and elliptic in any dimension and coincide with motion by crystalline curvature for polyhedral curves in the plane

global optimization for sum of geometric fractional functions this paper presents an efficient branch and bound algorithm for globally solving sum of geometric fractional functions under geometric constraints which arise in various practical problems by using an equivalent transformation and a new linear relaxation technique a linear relaxation programming problem of the equivalent problem is obtained the proposed algorithm is convergent to the global optimal solution by means of the subsequent solutions of a series of linear programming problems numerical results are reported to show the feasibility of our algorithm

global optimization of generalized linear fractional programming with nonlinear constraints this paper presents a branch and bound algorithm for globally solving a wide class of generalized linear fractional programming problems glfp this class includes such problems as minimizing a sum or error for product of a finite number of ratios of linear functions linear multiplicative programming polynomial programming etc over nonconvex feasible region first a problem q is derived which is equivalent to problem glfp in the algorithm lower bounds are derived by solving a sequence of linear relaxation programming problems which is based on the construction of the linear lower bounding functions for the objective function and constraint functions of problem q over the feasible region convergent property of the presented algorithm is proved and numerical results are given to show the feasibility of the proposed algorithm

range reduction techniques for improving computational efficiency in global optimization of signomial geometric programming problems many global optimization approaches for solving signomial geometric programming problems are based on transformation techniques and piecewise linear approximations of the inverse transformations since using numerous break points in the linearization process leads to a significant increase in the computational burden for solving the reformulated problem this study integrates the range reduction techniques in a global optimization algorithm for signomial geometric programming to improve computational efficiency in the proposed algorithm the non convex geometric programming problem is first converted into a convex mixed integer nonlinear programming problem by convexification and piecewise linearization techniques then an optimization based approach is used to reduce the range of each variable tightening variable bounds iteratively allows the proposed method to reach an approximate solution within an acceptable error by using fewer break points in the linearization process therefore decreasing the required cpu time several numerical experiments are presented to demonstrate the advantages of the proposed method in terms of both computational efficiency and solution quality

clusters minimizing area plus length of singular curves soap bubble clusters which seek the least area way to enclose and separate given volumes of air often have been used as models for metallic structures which likewise seek to minimize interfacial areas between regions for metals however there may well be additional energy associated with the singular curves affecting both the global and local structure cf mt bo see fig 1 we give the first existence and regularity results for clusters of prescribed volumes in r 3 minimizing area plus length of singular curves almgren s general theory of soap bubble clusters a2 does not apply because it provides no control over the singular curves although some of his ideas and lemmas prove useful

a new bound and reduce approach of nonconvex quadratic programming problems abstract for the nonconvex quadratic programming problem a new linear programming relaxation bound and reduce algorithm is proposed and its convergence is proved in this algorithm a new hyper rectangle partition technique and a new linear programming relaxation tactics are used at the same time the hyper rectangular reduction method is used to raise its convergent speed the numerical results demonstrate the effectiveness and feasibility of the proposed algorithm

on a crystalline variational problem part ii bv regularity and structure of minimizers on facets for a nonsmooth positively one homogeneous convex function n 0 it is possible to introduce the class n of smooth boundaries with respect to to define their mean curvature and to prove that for e n l e 9 based on these results we continue the analysis on the structure of e and on the regularity properties of we prove that a facet f of e is lipschitz up to negligible sets and that has bounded variation on f further properties of the jump set of are inspected in particular in three space dimensions we relate the sublevel sets of on f to the geometry of the wulff shape 1

convergent lagrangian and contour cut method for nonlinear integer programming with a quadratic objective function in this paper we present an efficient exact solution method for solving nonlinear separable integer programming problems with a quadratic objective function the proposed method combines the lagrangian dual method with a duality reduction scheme using contour cut at each iteration of the algorithm lower and upper bounds of the problem are determined by the lagrangian dual search to eliminate the duality gap a novel cut and partition scheme is derived by exploring the special structure of the quadratic contour the method finds an exact solution of the problem in a finite number of iterations computational results are reported for problems with up to 2000 integer variables comparison results with other methods are also presented

global optimization of signomial geometric programming using linear relaxation various local optimization approaches have been developed for solving signomial geometric programming sgp problems but up to now less work has been devoted to solving global optimization of sgp due to the inherent difficulty this paper considers the global minimum of sgp that arise in various practice problems by utilizing an exponential variable transformation and tangential hypersurfaces and convex envelop approximations a linear relaxation of sgp is then obtained thus initial nonconvex nonlinear problem sgp is reduced to a sequence of linear programming problems through the successive refinement of a linear relaxation of feasible region of the objective function the proposed algorithm is convergent to the global minimum of sgp by means of the subsequent solutions of a series of linear programming problems test results indicate that the proposed algorithm is extremely robust and can be used successfully to solve global minimum of sgp on a microcomputer

a new rectangle branch and reduce approach for solving nonconvex quadratic programming problems the paper presents a new rectangle branch and reduce approach for solving nonconvex quadratic programming problems in which a new lower approximate linear functions of the quadratic function over an n rectangle is given to determine a lower bound of the global optimal value of the original problem over each rectangle and a simple two partition technique on rectangle is used as well as the tactics on reducing and deleting subrectangles is used to accelerate the convergence of the proposed algorithm the proposed algorithm is proved to be convergent and shown to be effective with numerical results

linearization method of global optimization for generalized geometric programming many methods for solving generalized geometric programming ggp problem can only find locally optimal solutions but up to now less work has been devoted to solving global optimization of ggp due to the inherent difficulty this paper gives a method for finding the globally optimal solutions of ggp utilizing an exponentially variable transformation and some other techniques the initial nonlinear and nonconvex ggp problem is reduced to a sequence of linear programming problems the proposed algorithm is proven that it is convergent to the global minimum through the solutions of a series of linear programming problems several ggp examples in the literatures are tested to demonstrate that the proposed method can systematically solve these examples to find the global optimum within a prespecified error

on a crystalline variational problem part i first variation and global l regularity let n 0 be a given positively one homogeneous convex function and let 1 pursuing our interest in motion by crystalline mean curvature in three dimensions we introduce and study the class n of smooth boundaries in the relative geometry induced by the ambient banach space n it can be seen that even when is a polytope n cannot be reduced to the class of polyhedral boundaries locally resembling curved portions must be necessarily included and this fact as well as the nonsmoothness of is the source of several technical difficulties related to the geometry of lipschitz manifolds given a boundary e in the class n we rigorously compute the first variation of the corresponding anisotropic perimeter which leads to a variational problem on vector fields defined on e it turns out that the minimizers have a uniquely determined intrinsic tangential divergence on e we define such a divergence to be the mean curvature of e the function is expected to be the initial velocity of e whenever e is considered as the initial datum for the corresponding anisotropic mean curvature flow we prove that is bounded on e and that its sublevel sets are characterized through a variational inequality