on bilevel multi follower decision making general framework and solutions within the framework of any bilevel decision problem a leader s decision is influenced by the reaction of his or her follower when multiple followers who may have had a share in decision variables objectives and constraints are involved in a bilevel decision problem the leader s decision will be affected not only by the reactions of these followers but also by the relationships among these followers this paper firstly identifies nine different kinds of relationships s 1 to s 9 amongst followers by establishing a general framework for bilevel multi follower decision problems for each of the nine a corresponding bilevel multi follower decision model is then developed also this paper particularly proposes related theories focusing on an uncooperative decision problem i e s 1 model as this model is the most basic one for bilevel multi follower decision problems over the nine kinds of relationships moreover this paper extends the kuhn tucker approach for driving an optimal solution from the uncooperative decision model finally a real case study of a road network problem illustrates the application of the uncooperative bilevel decision model and the proposed extended kuhn tucker approach

a note on bilevel linear fractional programming problem abstract in this note the bilevel linear fractional programming problem is considered we examine a previous paper published by thirwani and arora cah cero 35 1993 135 we show that several proofs in that paper are incorrect and provide alternative proofs

two level linear programming decentralized planning has long been recognized as an important decision making problem many approaches based on the concepts of large scale system decomposition have generally lacked the ability to model the type of truly independent subsystems which often exist in practice multilevel programming models partition control over decision variables among ordered levels within a hierarchical planning structure a planner at one level of the hierarchy may have his objective function and set of feasible decisions determined in part by other levels however his control instruments may allow him to influence the policies at other levels and thereby improve his own objective function this paper examines the special case of the two level linear programming problem geometric characterizations and algorithms are presented with some examples the goal is to demonstrate the tractability of such problems and motivate a wider interest in their study

an extended kuhn tucker approach for linear bilevel programming kuhn tucker approach has been applied with remarkable success in linear bilevel programming blp however it still has some extent unsatisfactory and incomplete one principle challenges is that it could not well handle a linear blp problem when the constraint functions at the upper level are of arbitrary linear form this paper describes theoretical foundation of kuhn tucker approach and proposes an extended kuhn tucker approach to deal with the problem the results have demonstrated that the extended kuhn tucker approach can solve a wider class of linear blp problems can than current capabilities permit

estimation of distribution algorithm for a class of nonlinear bilevel programming problems in this paper a novel evolutionary algorithm called estimation of distribution algorithm eda is proposed for solving a special class of nonlinear bilevel programming problems blpps in which the lower level problem is a convex programming problem for each given upper level decision this special type of blpp is transformed into a equivalent single level constrained optimization problem using the karush kuhn er conditions of the lower level problem then we propose an eda based on the statistical information of the superior candidate solutions to solve the transformed problem we stress that the new population of individuals is sampled from the probabilistic distribution of those superior solutions thus one of the main advantages of eda over most other meta heuristics is its ability to adapt the operators to the structure of the problem although adaptation in eda is usually limited by the initial choice of the probabilistic model in addition two specific rules are established in the initialization procedure to make use of the hierarchical structure of blpps and to handle the constraints moreover without requiring the differentiability of the objective function or the convexity of the search space of the equivalent problem the proposed algorithm can address nonlinear blpps with non differentiable or non convex upper level objective function and upper level constraint functions finally the proposed algorithm has been applied to 16 benchmark problem in five of these problems all of the upper level variables and lower level variables are 10 dimensional the numerical results compared with those of other methods reveal the feasibility and effectiveness of the proposed algorithm

a collection of problems for which gaussian elimination with partial pivoting is unstable a significant collection of two point boundary value problems is shown to give rise to linear systems of algebraic equations on which gaussian elimination with row partial pivoting is unstable when standard solution techniques are used

solving quadratic convex bilevel programming problems using a smoothing method abstract in this paper we present a smoothing sequential quadratic programming to compute a solution of a quadratic convex bilevel programming problem we use the karush kuhn tucker optimality conditions of the lower level problem to obtain a nonsmooth optimization problem known to be a mathematical program with equilibrium constraints the complementary conditions of the lower level problem are then appended to the upper level objective function with a classical penalty these complementarity conditions are not relaxed from the constraints and they are reformulated as a system of smooth equations by mean of semismooth equations using fisher burmeister functional then using a quadratic sequential programming method we solve a series of smooth regular problems that progressively approximate the nonsmooth problem some preliminary computational results are reported showing that our approach is efficient

interactive balance space approach for solving multi level multi objective programming problems this paper studies a multi level multi objective decision making ml modm problems with linear or non linear constraints the objective functions at each level are non linear functions which are to be maximized or minimized this paper presents a three level multi objective decision making tl modm model and an interactive algorithm for solving such a model the algorithm simplifies three level multi objective decision making problems by transforming them into separate multi objective decision making problems at each level thereby avoiding the difficulty associated with non convex mathematical programming our algorithm is an extension of the work of shi and xia x shi h xia interactive bi level multi objective decision making journal of the operational research society 48 1997 943 949 which dealt with interactive bi level multi objective decision making problems with some modifications in assigning satisfactoriness to each objective function in all the levels of the tl modm problem also we solve each separate multi objective decision making problem of the tl modm problem by the balance space approach a new formula is introduced to interconnect the satisfactoriness and the proportions of deviation needed to reflect the relative importance of each objective function thus we have the proportions of deviation including satisfactoriness in addition we present new definitions for the satisfactoriness and the preferred solution in view of singular level multi objective decision making problems that corresponds to the h optimal solution of the balance space approach also new definitions for the feasible solution and the preferred solution h optimal point of the tl modm problem are presented an illustrative numerical example is given to demonstrate the algorithm

an extended kth best approach for linear bilevel programming kth best approach is one of the three popular and workable approaches for linear bilevel programming however it could not well deal with a linear bilevel programming problem when the constraint functions at the upper level are of arbitrary linear form based on our previous work this paper proposes theoretical properties of linear bilevel programming and presents an extended kth best approach for linear bilevel programming which can effectively solve above problem

the mixed integer linear bilevel programming problem a two person noncooperative game in which the players move in sequence can be modeled as a bilevel optimization problem in this paper we examine the case where each player tries to maximize the individual objective function over a jointly constrained polyhedron the decision variables are variously partitioned into continuous and discrete sets the leader goes first and through his choice may influence but not control the responses available to the follower for two reasons the resultant problem is extremely difficult to solve even by complete enumeration first it is not possible to obtain tight upper bounds from the natural relaxation and second two of the three standard fathoming rules common to branch and bound cannot be applied fully in light of these limitations we develop a basic implicit enumeration scheme that finds good feasible solutions within relatively few iterations a series of heuristics are then proposed in an effort to strike a balance between accuracy and speed the computational results suggest that some compromise is needed when the problem contains more than a modest number of integer variables

on the smallest poles of igusa s p adic zeta functions let k be a p adic field we explore igusa s p adic zeta function which is associated to a k analytic function on an open and compact subset of k n first we deduce a formula for an important coefficient in the laurent series of this meromorphic function at a candidate pole afterwards we use this formula to determine all values less than 1 2 for n 2 and less than 1 for n 3 which occur as the real part of a pole

fuzzy goal programming approach to multilevel programming problems the purpose of this paper is to propose a procedure for solving multilevel programming problems in a large hierarchical decentralized organization through linear fuzzy goal programming approach here the tolerance membership functions for the fuzzily described objectives of all levels as well as the control vectors of the higher level decision makers are defined by determining individual optimal solution of each of the level decision makers since the objectives are potentially conflicting in nature a possible relaxation of the higher level decision is considered for avoiding decision deadlock then fuzzy goal programming approach is used for achieving highest degree of each of the membership goals by minimizing negative deviational variables sensitivity analysis with variation of tolerance values on decision vectors is performed to present how the solution is sensitive to the change of tolerance values the efficiency of our concept is ascertained by comparing results with other fuzzy programming approaches

optimization of limited network capacity with toll settings a non smooth optimization model is established for determining reserve capacity of a road network with toll settings optimization of limited network capacity with toll settings can be formulated as a mathematical program with equilibrium constraints mpec a quasi newton subgradient projection and contractive method qspc with global convergence is proposed numerical calculations are conducted using an example network where good results are obtained

a penalty function method for solving weak price control problem abstract the price control problem is an important linear bilevel programming problem in this paper we are concerned with a class of weak price control problems with non unique lower level solutions for such problems we study the existence of solution via a penalty method then we propose a simple algorithm for this problem and present the convergence of the algorithm the example illustrates that the method is feasible and efficient

exponential sums questions by denef sperber and igusa we prove the remaining part of the conjecture by denef and sperber denef j and sperber s exponential sums mod p n and newton poly hedra bull belg math soc suppl 2001 55 63 on nondegenerate local exponential sums modulo p m we generalize igusa s conjecture in the introduction of igusa j lectures on forms of higher degree lect math phys springer verlag 59 1978 from the homogeneous to the quasi homogeneous case and prove the nondegenerate case as well as the modulo p case we generalize some results by katz in katz n m estimates for singular exponential sums internat math res notices 1999 no 16 875 899 on finite field exponential sums to the quasi homogeneous case

interactive fuzzy programming for multi level 0 1 programming problems through genetic algorithms abstract in this paper we propose interactive fuzzy programming for multi level 0 1 programming problems through genetic algorithms our method is supposed to apply to hierarchical decision problems in which decision making at each level is sequential from upper to lower level and decision makers are essentially cooperative after determining the fuzzy goals of the decision makers at all levels a satisfactory solution is derived efficiently by updating the satisfactory degrees of the decision makers at the upper level with considerations of overall satisfactory balance among all levels an illustrative numerical example for three level 0 1 programming problems is provided to demonstrate the feasibility of the proposed method

computation of the optimal tolls on the traffic network abstract the present paper is devoted to the computation of optimal tolls on a traffic network that is described as fuzzy bilevel optimization problem as a fuzzy bilevel optimization problem we consider bilinear optimization problem with crisp upper level and fuzzy lower level an effective algorithm for computation optimal tolls for the upper level decision maker is developed under assumption that the lower level decision maker chooses the optimal solution as well the algorithm is based on the membership function approach this algorithm provides us with a global optimal solution of the fuzzy bilevel optimization problem

the motivic zeta function and its smallest poles abstract let f be a regular function on a nonsingular complex algebraic variety of dimension d we prove a formula for the motivic zeta function of f in terms of an embedded resolution this formula is over the grothendieck ring itself and specializes to the formula of denef and loeser over a certain localization we also show that the space of n jets satisfying f 0 can be partitioned into locally closed subsets which are isomorphic to a cartesian product of some variety with an affine space of dimension d n 2 finally we look at the consequences for the poles of the motivic zeta function

approximate inertial manifolds for the kuramoto sivashinsky equation analysis and computations abstract we evaluate several alternative methods for the approximation of inertial manifolds for the one dimensional kuramoto sivashinsky equation kse a method motivated by the dynamics originally developed for the navier stokes equation is adapted for the kse rigorous error estimates are obtained and compared to those of other methods introduced in the literature formal relationships between these other methods and the one introduced here are established numerical bifurcation diagrams of the various approximate inertial forms for the kse are presented we discuss the correspondence between the rigorous error estimates and the accuracy of the computational results these methods can be adapted to other dissipative partial differential equations

the close relationships between methods for solving two point boundary value problems the well conditioning of a boundary value problem is shown to be related to bounding two quantities one involving the boundary conditions and the other involving the green s function in the case of a solution dichotomy they are related to known stability results these results easily explain why difficulties arise using superposition and reduced superposition it is shown how multiple shooting overcomes these difficulties by relating its matrix conditioning to the underlying boundary value problem some factorization methods for the multiple shooting method are considered these show the close relationships between multiple shooting the stabilized march and invariant imbedding and one factorization leads to an efficient new way to implement each of these three methods many of the results apply for collocation and finite difference methods too finally it is shown both theroetically and computationally when matrix compactification can lead to difficulty

on the definition of linear bilevel programming solution linear bilevel programming theory has been studied for many years by a number of researchers from different aspects yet it still remains to some extent unsatisfactory and incomplete the main challenge is how to solve a linear bilevel programming problem when the upper level s constraint functions are of arbitrary linear form this paper proposes a definition for linear bilevel programming solution the performance comparisons have demonstrated that the new model can solve a wider class of problems than current capabilities permit

fuzzy multiple level programming the use of fuzzy set concept to overcome the computational difficulties of multi level optimization for large systems appears to be a powerful approach and can solve practical large problems in a very simplified manner furthermore unlike the duo ploy or multi ploy approaches the proposed interactive concept provides flexibility which is essential due to the different styles of management in different organizations the approach is formulated based on exploring the fuzziness or indefiniteness which is almost always present in a decentralized large hierarchy organization the advantage of this approach is that not only the problem is much more simplified for solution purposes the original problem is also much more realistically represented

stackelberg solutions for fuzzy random two level linear programming through probability maximization with possibility this paper considers stackelberg solutions for decision making problems in hierarchical organizations under fuzzy random environments taking into account vagueness of judgments of decision makers fuzzy goals are introduced into the formulated fuzzy random two level linear programming problems on the basis of the possibility and necessity measures that each objective function fulfills the corresponding fuzzy goal together with the introduction of probability maximization criterion in stochastic programming we propose new two level fuzzy random decision making models which maximize the probabilities that the degrees of possibility and necessity are greater than or equal to certain values through the proposed models it is shown that the original two level linear programming problems with fuzzy random variables can be transformed into deterministic two level linear fractional programming problems for the transformed problems extended concepts of stackelberg solutions are defined and computational methods are also presented a numerical example is provided to illustrate the proposed methods

an extended branch and bound algorithm for linear bilevel programming abstract for linear bilevel programming the branch and bound algorithm is the most successful algorithm to deal with the complementary constraints arising from kuhn tucker conditions however one principle challenge is that it could not well handle a linear bilevel programming problem when the constraint functions at the upper level are of arbitrary linear form this paper proposes an extended branch and bound algorithm to solve this problem the results have demonstrated that the extended branch and bound algorithm can solve a wider class of linear bilevel problems can than current capabilities permit

a fuzzy approach for bi level integer non linear programming problem abstract bi level programming a tool for modeling decentralized decisions consists of the objective of the leader at its first level and that is of the follower at the second level integer programming deals with the mathematical programming problems in which some or all the variables are to be integer this paper studies a bi level integer non linear programming problem with linear or non linear constraints and in which the non linear objective function at each level are to maximized the bi level integer non linear programming bli nlp problem can be thought as a static version of the stackelberg strategy which is used leader follower game in which a stackelberg strategy is used by the leader or the higher level decision maker hldm given the rational reaction of the follower or the lower level decision maker lldm this paper proposes a two planner integer model and a solution method for solving this problem this method uses the concept of tolerance membership function and the branch and bound technique to develop a fuzzy max min decision model for generating pareto optimal solution for this problem an illustrative numerical example is given to demonstrate the obtained results

application of particle swarm optimization algorithm for solving bi level linear programming problem bi level linear programming is a technique for modeling decentralized decision it consists of the upper level and lower level objectives this paper attempts to develop an efficient method based on particle swarm optimization pso algorithm with swarm intelligence the performance of the proposed method is ascertained by comparing the results with genetic algorithm ga using four problems in the literature and an example of supply chain model the results illustrate that the pso algorithm outperforms ga in accuracy

linear system solvers for boundary value odes abstract we investigate the stability properties of several linear system solvers for solving boundary value odes we consider the compactification algorithm gaussian elimination with row partial pivoting and a qr algorithm applied to linear systems arising from solving bvps for which the matrix is block bidiagonal except for bordering along the last n rows and columns we will particularly compare auto s original linear solver an lu decomposition with partial pivoting and our implementation of the analogous qr algorithm to auto two other factors the underlying continuation strategy and mesh selection strategy may affect the stability of the linear system solver for ode continuation codes as well and are also discussed in our numerical investigations

a value function based exact approach for the bilevel mixed integer programming problem we examine bilevel mixed integer programs whose constraints and objective functions depend on both upper and lower level variables the class of problems we consider allows for nonlinear terms to appear in both the constraints and the objective functions requires all upper level variables to be integer and allows a subset of the lower level variables to be integer this class of bilevel problems is difficult to solve because the upper level feasible region is defined in part by optimality conditions governing the lower level variables which are difficult to characterize because of the nonconvexity of the follower problem we propose an exact finite algorithm for these problems based on an optimal value function reformulation we demonstrate how this algorithm can be tailored to accommodate either optimistic or pessimistic assumptions on the follower behavior computational experiments demonstrate that our approach outperforms a state of the art algorithm for solving bilevel mixed integer linear programs

a neural network for solving a convex quadratic bilevel programming problem a neural network is proposed for solving a convex quadratic bilevel programming problem based on lyapunov and lasalle theories we prove strictly an important theoretical result that for an arbitrary initial point the trajectory of the proposed network does converge to the equilibrium which corresponds to the optimal solution of a convex quadratic bilevel programming problem numerical simulation results show that the proposed neural network is feasible and efficient for a convex quadratic bilevel programming problem

comparative tests of solution methods for signal controlled road networks in this paper we consider a signal controlled road network with link capacity expansions this network design problem can be formulated as a constrained optimization subject to equilibrium flows a projected quasi newton method is proposed to find good local optimal solutions numerical calculations are conducted using a real data road network and large scale grid networks

a control theoretic view on incentives the idea of declaring a reward punishment for a decision maker according to his particular choice of action in order to induce certain desired behavior on the part of the decision maker is known as incentive threat this practice is age old however only in recent years have the notions been formalized in the development of a control theoretic view on incentives we first investigate the deterministic version of the incentive problem this reveals the basic simple idea behind the problem it also illustrates the different possibilities introduced by the presence of dynamics and multi follower nature of the problem this is followed by two variants of the stochastic version of the problem where we concentrate on the role of uncertainties relationship to economic literature is also discussed

efficient solutions for the linear bilevel programming problem abstract one of the interesting features of the bilevel programming problem is that its optimal solution may not be pareto optimal this behavior may be economically inadmissible this paper will present a post optimality analysis it is intended to provide some rational efficient solutions for decision makers after a non pareto optimal solution is obtained for a linear bilevel programming problem an efficient solution which obeys some properties will be adopted by decision makers according to their decision preferences

some numerical aspects of computing inertial manifolds various aspects of the problem of computing inertial manifolds for dissipative partial differential equations are investigated in particular methods are proposed which generalize previous ones and improve upon some of their limitations the basic ideas and techniques are exemplified mainly for the kuramoto sivashinsky equation this equation is chosen because it is fairly simple but the dynamics are sufficiently complicated and it is a classical case for which a considerable amount of computational experience has been well documented

interactive compensatory fuzzy programming for decentralized multi level linear programming dmllp problems abstract this paper presents interactive compensatory fuzzy programming for decentralized multi level linear programming dmllp problems by adjusting the cooperative decision making process between the different levels and also between the decision makers of the same level our aim is to obtain a preferred compensatory compromise pareto optimal solution for dmllp for this the weights of objectives at each level are assigned by the next upper level decision maker dm by using analytic hierarchy process ahp or any other weighting methods the weight of any objective for whole system is equal to the product of the weights on the path tying it to the top decision maker dm 0 using these weights equivalence is established such that the satisfactory levels of all objectives are proportional to their own weights werners compensatory fuzzy and operator is offered to solve dmllp problem the most important idea to be emphasized is that equivalence is established such that the satisfactory levels of all objectives are proportional to their own weights thanks to this equivalence dmllp problem has been transformed to the multi objective linear programming molp problem at level 0 the equivalence is reflected to the compensatory model within the constraints and the equivalence also enables all dms to obtain proportional satisfactions with their weights as much as possible so in our compensatory model a reduction on equivalent satisfactory level of one dm can be compensated for by an increase in the equivalent satisfactory level of another dm furthermore being developed a finite interactive iterative procedure with maximum interaction step a set of compensatory solutions which are also pareto optimal is obtained depending on compensation parameter giving a theorem we will show that the solutions generated by werners compensatory fuzzy and operator do guarantee pareto optimality for our dmllp problem and comparing it with some other computational efficient compensatory fuzzy aggregation operators we will conclude that this operator is more appropriate for dmllp illustrative numerical example is provided to demonstrate the feasibility and efficiency of the proposed interactive fuzzy compensatory method for dmllp

bilevel multiplicative problems a penalty approach to optimality and a cutting plane based algorithm bilevel programming has been proposed for dealing with decision processes involving two decision makers with a hierarchical structure they are characterised by the existence of two optimisation problems in which the constraint region of the upper level problem is implicitly determined by the lower level optimisation problem in this paper we focus on the class of bilevel problems in which the upper level objective function is linear multiplicative the lower level one is linear and the common constraint region is a bounded polyhedron after replacing the lower level problem by its karush kuhn tucker conditions the existence of an extreme point which solves the problem is proved by using a penalty function approach besides an algorithm based on the successive introduction of valid cutting planes is developed obtaining a global optimal solution finally we generalise the problem by including upper level constraints which involve both level variables

a penalty function method based on kuhn tucker condition for solving linear bilevel programming abstract using the kuhn tucker optimality condition of the lower level problem we transform the linear bilevel programming problem into a corresponding single level programming the complementary and slackness condition of the lower level problem is appended to the upper level objective with a penalty then we decompose the linear bilevel programming into a series of linear programming problems and get the optimal solution of the linear bilevel programming using linear programming method

lower bound for the poles of igusa s p adic zeta functions let k be a p adic field r the valuation ring of k p the maximal ideal of r and q the cardinality of the residue field r p let f be a polynomial over r in n 1 variables and let be a character of r times let mi u be the number of solutions of f u in r pi n for i in mathbb z geq 0 and u in r p i these numbers are related with igusa s p adic zeta function zf s of f we explain the connection between the mi u and the smallest real part of a pole of zf s we also prove that mi u is divisible by q ulcorner n 2 i 1 urcorner where the corners indicate that we have to round up this will imply our main result zf s has no poles with real part less than n 2 we will also consider arbitrary k analytic functions f

numerical solution of constrained optimal control problems with parameters this article presents a numerical solution technique for constrained optimal control problems that contain parameters here the state control and parameter inequality constraints are accommodated via an extended penalty function this penalty function takes on large values when the constraints are violated and small values when the constraints are satisfied using the calculus of variation it is shown that the first order necessary conditions for optimality are in the form of a two point boundary value problem involving differential and algebraic equations bvp dae a multiple shooting continuation method is developed for solving this bvp dae two examples are presented to demonstrate the effectiveness of the solution approach developed in the paper

solving bi level linear programming problem through hybrid of immune genetic algorithm and particle swarm optimization algorithm bi level linear programming consisting of upper level and lower level objectives is a technique for modeling decentralized decision this study presents a hybrid of immune genetic algorithm and vector controlled particle swarm optimization igvpso to solve the bi level linear programming problem blpp it is applied to a supply chain model that is a blpp using four problems from the literature and the supply chain distribution models the computational results indicate that the proposed method is superior to some algorithms

an approximate programming method based on the simplex method for bilevel programming problem many real problems can be modeled to the problems with a hierarchical structure and bilevel programming is a useful tool to solve the hierarchical optimization problems so the bilevel programming is widely applied and numerous methods have been proposed to solve this programming in this paper we propose an approximate programming algorithm to solve bilevel nonlinear programming problem finally the example illustrates the feasibility of the proposed algorithm

on two level optimization decentralized planning has long been recognized as an important decision making problem many approaches based on the concepts of large scale system decomposition have generally lacked the ability to model the type of truly independent subsystems which have often existed in practice similarly tractable game theoretic approaches have not allowed for a continuous solution space in which one player s decision restricts the feasible choices of another this correspondence offers a formulation to help model this type of decision making hierarchy characterizations and algorithmic approaches for a class of two level problems are discussed

interactive fuzzy programming for two level linear fractional programming problems with fuzzy parameters in this paper we present interactive fuzzy programming for two level linear fractional programming problems with fuzzy parameters using the level sets of fuzzy parameters the corresponding nonfuzzy two level linear fractional programming problem is introduced in our interactive method after determining fuzzy goals of decision makers at both levels a satisfactory solution is derived efficiently by updating a minimal satisfactory level of the decision maker at the upper level with considerations of overall satisfactory balance between both levels the satisfactory solution well balanced between both levels is easily computed by combined use of the bisection method the phase one of the simplex method and the variable transformation method by charnes and cooper an illustrative numerical example for two level linear fractional programming problems with fuzzy parameters is provided to demonstrate the feasibility of the proposed method

interactive random fuzzy two level programming through possibility based probability model this paper considers interactive decision making methods for random fuzzy two level linear programming problems assuming that the decision makers concern about the probabilities that their own objective function values are smaller than or equal to certain target values fuzzy goals of the decision makers for the probabilities are introduced then the possibility based probability model to maximize the degrees of possibility with respect to the attained probability is considered interactive fuzzy nonlinear programming to obtain a satisfactory solution for the decision maker at the upper level in consideration of the cooperative relation between decision makers is presented an illustrative numerical example demonstrates the feasibility and efficiency of the proposed method

a modified simplex approach for solving bilevel linear programming problems abstract a method for solving bilevel linear programming problems is proposed the inner problem is represented by its kuhn tucker conditions and the complementary slackness equations are moved to the outer objective as a penalty function the global optimum of the resulting quadratic programme is reached by iterative application of a modified simplex algorithm

linear bilevel programming with upper level constraints depending on the lower level solution abstract focus in the paper is on the definition of linear bilevel programming problems the existence of optimal solutions and necessary as well as sufficient optimality conditions in the papers c shi g zhang j lu an extended kuhn tucker approach for linear bilevel programming appl math comput 162 2005 51 63 and c shi g zhang j lu on the definition of linear bilevel programming solution appl math comput 160 2005 169 176 the authors claim to suggest a refined definition of linear bilevel programming problems and related optimality conditions mainly their attempt reduces to shifting upper level constraints involving both the upper and the lower level variables into the lower level we investigate such a shift in more details and show that it is not allowed in general we show that an optimal solution of the bilevel program exists under the conditions in c shi g zhang j lu on the definition of linear bilevel programming solution appl math comput 160 2005 169 176 if we add the assumption that the inducible region is not empty the necessary optimality condition reduces to check optimality in one linear programming problem optimality of one feasible point for a certain number of linear programs implies optimality for the bilevel problem

a bm lambda cut and goal programming based algorithm for fuzzy linear multiple objective bilevel optimization bilevel programming techniques are developed to handle decentralized problems with two level decision makers which are leaders and followers who may have more than one objective to achieve this paper proposes a cut and goal programming based algorithm to solve fuzzy linear multiple objective bilevel flmob decision problems first based on the definition of a distance measure between two fuzzy vectors using cut a fuzzy linear bilevel goal flbg model is formatted and related theorems are proved then using a cut for fuzzy coefficients and a goal programming strategy for multiple objectives a cut and goal programming based algorithm to solve flmob decision problems is presented a case study for a newsboy problem is adopted to illustrate the application and executing procedure of this algorithm finally experiments are carried out to discuss and analyze the performance of this algorithm

pessimistic bilevel optimization we study a variant of the pessimistic bilevel optimization problem which comprises constraints that must be satisfied for any optimal solution of a subordinate lower level optimization problem we present conditions that guarantee the existence of optimal solutions in such a problem and we characterize the computational complexity of various subclasses of the problem we then focus on problem instances that may lack convexity but that satisfy a certain independence property we develop convergent approximations for these instances and we derive an iterative solution scheme that is reminiscent of the discretization techniques used in semi infinite programming we also present a computational study that illustrates the numerical behavior of our algorithm on standard benchmark instances

a new generating function for calculating the igusa local zeta function abstract a new method is devised for calculating the igusa local zeta function z f of a polynomial f x 1 x n over a p adic field this involves a new kind of generating function g f that is the projective limit of a family of generating functions and contains more data than z f this g f resides in an algebra whose structure is naturally compatible with operations on the underlying polynomials facilitating calculation of local zeta functions this new technique is used to expand significantly the set of quadratic polynomials whose local zeta functions have been calculated explicitly local zeta functions for arbitrary quadratic polynomials over p adic fields with p odd are presented as well as for polynomials over unramified 2 adic fields of the form q l where q is a quadratic form and l is a linear form such that q and l have disjoint variables for a quadratic form over an arbitrary p adic field with odd p this new technique makes clear precisely which of the three candidate poles are actual poles

a note on the definition of a linear bilevel programming solution an alternative definition of the linear bilevel programming problem blp has recently been proposed by lu shi and zhang this note shows that the proposed definition is a restriction of blp indeed the new definition is equivalent to transferring the first level constraints involving second level variables into the second level resulting in a special case of blp in which there are no first level constraint involving second level variables thus contrary to what is stated by the authors who suggested the new definition this does not allow to solve a wider class of problems but rather relaxes the feasible region allowing for infeasible points to be considered as feasible

solving a bilevel linear program when the inner decision maker controls few variables abstract this paper presents an algorithm that always finds an optimum of a bilevel linear program blp problem if one exists moreover if the inner decision maker does not control too many variables the algorithm is efficient in particular if the number of variables controlled by the inner decision maker can be considered to be constant then the algorithm is a polynomial time algorithm the algorithm is also well suited to parallel implementation

decoupling and stability of algorithms for boundary value problems the ordinary differential equations occurring in linear boundary value problems characteristically have both stable and unstable solution modes therefore a stable numerical algorithm should avoid both forward and backward integration of solutions on large intervals it is shown that most methods like multiple shooting collocation invariant imbedding and difference methods derive their stability from the fact that they all decouple the continuous or the discrete problem sooner or later for instance when solving a linear system this decoupling is related to the dichotomy of the ordinary differential equations in fact it turns out that the inherent initial value instability is an important prerequisite for a stable utilization of the decoupled representations from which the solutions are computed how this stability is related to the use of the boundary conditions is also investigated

integrating goal programming kuhn tucker conditions and penalty function approaches to solve linear bi level programming problems abstract bi level programming a tool for modeling decentralized decisions consists of the objective s of the leader at its first level and the follower s at the second level three level programming results when the second level itself is a bi level programming which helps us develop the idea of bi level programming to multi level programs with any number of levels there are many approaches for encountering with bi multi level problems the method introduced by chenggen shi et al chenggen shi jie lu guangquan zhang an extended kuhn tucker approach for linear bi level programming applied mathematics and computation 162 2005 51 63 uses extended kuhn tucker conditions to solve bi level programming problems in this paper we integrate goal programming gp kuhn tucker conditions kkt and penalty function approaches to solve linear bi level programming problems by using a numerical example we illustrate efficiency of our methodology