a smoothing method for mathematical programs with equilibrium constraints the mathematical program with equilibrium constraints mpec is an optimization problem with variational inequality constraints mpec problems include bilevel programming problems as a particular case and have a wide range of applications mpec problems with strongly monotone variational inequalities are considered in this paper they are transformed into an equivalent one level nonsmooth optimization problem then a sequence of smooth regular problems that progressively approximate the nonsmooth problem and that can be solved by standard available software for constrained optimization is introduced it is shown that the solutions stationary points of the approximate problems converge to a solution stationary point of the original mpec problem numerical results showing viability of the approach are reported

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

outliers in the spectrum of iid matrices with bounded rank perturbations it is known that if one perturbs a large iid random matrix by a bounded rank error then the majority of the eigenvalues will remain distributed according to the circular law however the bounded rank perturbation may also create one or more outlier eigenvalues we show that if the perturbation is small then the outlier eigenvalues are created next to the outlier eigenvalues of the bounded rank perturbation but if the perturbation is large then many more outliers can be created and their law is governed by the zeroes of a random laurent series with gaussian coefficients on the other hand these outliers may be eliminated by enforcing a row sum condition on the final matrix

hierarchical optimization a satisfactory solution abstract hierarchical optimization or multi level programming techniques are extensions of stackelberg games for solving decentralized planning problems with multiple decision makers in a hierarchical organization they become more important for contemporary decentralized organizations where each unit or department seeks its own interests traditional approaches include vertex enumeration algorithms and approaches based on kuhn tucker conditions or penalty functions these are not only technically inefficient but also lead to a paradox that the follower s decision power dominates the leader s in this study concepts of memberships of optimalities and degrees of decision powers are proposed to solve the above problems efficiently in the proposed hierarchy decision process the leader first sets memberships of optimalities of his her possible objective values and decisions as well as his her decision power and then asks the follower for his her optima calculated in isolation under given constraints the follower s decision with the corresponding levels of optimalities and decision powers are submitted to and modified by the leader with considerations of overall benefit for the organization and distribution of decision power until a best preferred solution is reached

newton type method for a class of mathematical programs with complementarity constraints in this paper we present a newton type method for a class of mathematical programs with complementarity constraints under the mpec licq we use the definition of b stationary point to construct a constrained equations model and apply the newton method to solve the problem at the end of this paper numerical results are reported to show our method s validity

a novel neural network for nonlinear convex programming in this paper we present a neural network for solving the nonlinear convex programming problem in real time by means of the projection method the main idea is to convert the convex programming problem into a variational inequality problem then a dynamical system and a convex energy function are constructed for resulting variational inequality problem it is shown that the proposed neural network is stable in the sense of lyapunov and can converge to an exact optimal solution of the original problem compared with the existing neural networks for solving the nonlinear convex programming problem the proposed neural network has no lipschitz condition no adjustable parameter and its structure is simple the validity and transient behavior of the proposed neural network are demonstrated by some simulation results

nonlinear robust optimization via sequential convex bilevel programming in this paper we present a novel sequential convex bilevel programming algorithm for the numerical solution of structured nonlinear min max problems which arise in the context of semi infinite programming here our main motivation are nonlinear inequality constrained robust optimization problems in the first part of the paper we propose a conservative approximation strategy for such nonlinear and non convex robust optimization problems under the assumption that an upper bound for the curvature of the inequality constraints with respect to the uncertainty is given we show how to formulate a lower level concave min max problem which approximates the robust counterpart in a conservative way this approximation turns out to be exact in some relevant special cases and can be proven to be less conservative than existing approximation techniques that are based on linearization with respect to the uncertainties in the second part of the paper we review existing theory on optimality conditions for nonlinear lower level concave min max problems which arise in the context of semi infinite programming regarding the optimality conditions for the concave lower level maximization problems as a constraint of the upper level minimization problem we end up with a structured mathematical program with complementarity constraints mpcc the special hierarchical structure of this mpcc can be exploited in a novel sequential convex bilevel programming algorithm we discuss the surprisingly strong global and locally quadratic convergence properties of this method which can in this form neither be obtained with existing sqp methods nor with interior point relaxation techniques for general mpccs finally we discuss the application fields and implementation details of the new method and demonstrate the performance with a numerical example

geometric and algorithmic developments for a hierarchical planning problem abstract this paper presents a new model for multiobjective planning in hierarchical systems that explicitly takes into consideration the order in which decisions are made interactions and conflicts that normally exist among the levels are introduced by specifying jointly controlled feasible regions and interdependent objective functions at each level in the system planners attempt to maximize net benefits in light of all higher level decisions and thus may influence but not control the behavior of others the resultant formulation leads to the multilevel programming problem the geometry of an all linear case is first examined wherein it is shown that the optimal solution must lie at a vertex of the original polyhedral constraint region next a set of first order optimality conditions is derived for the general case and used as the basis of an algorithm for the linear problem a number of examples are given to highlight the results

a new regularization method for mathematical programs with complementarity constraints with strong convergence properties mathematical programs with equilibrium or complementarity constraints mpecs form a difficult class of optimization problems the feasible set has a very special structure and violates most of the standard constraint qualifications therefore one typically applies specialized algorithms in order to solve mpecs one very prominent class of specialized algorithms are the regularization or relaxation methods the first regularization method for mpecs is due to scholtes siam j optim 11 2001 pp 918 936 but in the meantime there exist a number of different regularization schemes which try to relax the difficult constraints in different ways however almost all regularization methods converge to c stationary points only which is a very weak stationarity concept an exception is a recent method by kadrani dussault and benchakroun siam j optim 20 2009 pp 78 103 whose limit points are shown to be m stationary here we provide a new regularization method which also converges to m sta

a general methodology for designing globally convergent optimization neural networks we present a general methodology for designing optimization neural networks we prove that the neural networks constructed by using the proposed method are guaranteed to be globally convergent to solutions of problems with bounded or unbounded solution sets in contrast with the gradient methods whose convergence is not guaranteed we show that the proposed method contains both the gradient methods and nongradient methods employed in existing optimization neural networks as special cases based on the theoretical results of the proposed method we study the convergence and stability of general gradient models in the case of unisolated solutions using the proposed method we derive some new neural network models for a very large class of optimization problems in which the equilibrium points correspond to exact solutions and there is no variable parameter finally some numerical examples show the effectiveness of the method

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

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

an efficient point algorithm for a linear two stage optimization problem this paper presents an algorithm using sensitivity analysis to solve a linear two stage optimization problem the underlying theory rests on a set of first order optimality conditions that parallel the kuhn tucker conditions associated with a one dimensional parametric linear program the solution to the original problem is uncovered by systematically varying the parameter over the unit interval and solving the corresponding linear program finite convergence is established under nondegenerate assumptions the paper also discusses other solution techniques including branch and bound and vertex enumeration and gives an example highlighting their computational and storage requirements by these measures the algorithm presented here has an overall advantage finally a comparison is drawn between bicriteria and bilevel programming and underscored by way of an example

a continuation method for strongly monotone variational inequalities we consider the variational inequality problem denoted by vip x f wheref is a strongly monotone function and the convex setx is described by some inequality and possibly equality constraints this problem is solved by a continuation or interior point method which solves a sequence of certain perturbed variational inequality problems these perturbed problems depend on a parameter 0 it is shown that the perturbed problems have a unique solution for all values of 0 and that any sequence generated by the continuation method converges to the unique solution of vip x f under a well known linear independence constraint qualification licq we also discuss the extension of the continuation method to monotone variational inequalities and present some numerical results obtained with a suitable implementation of this method 1998 the mathematical programming society inc published by elsevier science b v

on a new class of bilevel programming problems and its use for reformulating mixed integer problems abstract we extend some known results about the bilevel linear problem blp a hierarchical two stage optimization problem showing how it can be used to reformulate any mixed integer linear problem then we introduce some new concepts which might be useful to fasten almost all the known algorithms devised for blp as this kind of reformulation appears to be somewhat artificial we define a natural generalization of blp the bilevel linear quadratic problem bl qp and show that most of the exact and or approximate algorithms originally devised for the blp such as gsa or k th best can be extended to this new class of bilevel programming problems for bl qp more natural reformulations of mips are available leading to the use of known nonexact algorithms for blp as heuristic approaches to mips we report some contrasting results obtained in the network design problem case showing that although the direct application of our dual gsa algorithm is not of any practical use we obtain as a by product a good theoretical characterization of the optimal solutions set of the ndp along with a powerful scheme for constructing fast algorithms for the minimum cost flow problem with piecewise convex linear cost functions

on the distribution of the largest eigenvalue in principal components analysis let x 1 denote the square of the largest singular value of an n p matrix x all of whose entries are independent standard gaussian variates equivalently x 1 is the largest principal component variance of the covariance matrix x x or the largest eigenvalue of a p variate wishart distribution on n degrees of freedom with identity covariance consider the limit of large p and n with n p gamma ge 1 when centered by mu p sqrt n 1 sqrt p 2 and scaled by sigma p sqrt n 1 sqrt p 1 sqrt n 1 1 sqrt p 1 3 the distribution of x 1 approaches the tracey widom law of order 1 which is defined in terms of the painleve ii differential equation and can be numerically evaluated and tabulated in software simulations show the approximation to be informative for n and p as small as 5 the limit is derived via a corresponding result for complex wishart matrices using methods from random matrix theory the result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts

concentration of norms and eigenvalues of random matrices we prove concentration results for lpn operator norms of rectangular random matrices and eigenvalues of self adjoint random matrices the random matrices we consider have bounded entries which are independent up to a possible self adjointness constraint our results are based on an isoperimetric inequality for product spaces due to talagrand

computational difficulties of bilevel linear programming we show using small examples that two algorithms previously published for the bilevel linear programming problem blp may fail to find the optimal solution and thus must be considered to be heuristics a proof is given that solving blp problems is np hard which makes it unlikely that there is a good exact algorithm

a generalized project metric algorithm for mathematical programs with equilibrium constraints this paper discusses a kind of mathematical programs with equilibrium constraints mpec for short by using a complementarity function and a kind of disturbed technique the original mpec problem is transformed into a nonlinear equality and inequality constrained optimization problem then we combine a generalized gradient projection matrix with penalty function technique to given a generalized project metric algorithm with arbitrary initial point for the mpec problems in order to avoid mataros effect a high order revised direction is obtained by an explicit formula under some relative weaker conditions the proposed method is proved to possess global convergence and superlinear convergence

convex two level optimization in this paper a model for a two level planning problem is presented in the form of a static stackelberg game by assumption play is sequential and noncooperative however the leader can influence the actions of the followers through a set of coordination variables while the followers responses may partly determine the leader s payoff under certain convexity assumptions it is shown that the feasible region induced by the leader is continuous in the original problem variables this observation coupled with two corollary results are used as a basis for a hybrid algorithm which clings to the inducible region until a local optimum is found a branching scheme is then employed to located other segments of the region eventually terminating with the global optimum a number of examples are given to highlight the results while the algorithm s performance is tested in a comparison with two other procedures

generalized nash games and quasi variational inequalities abstract a generalized nash game is an n person noncooperative game with nondisjoint strategy sets other names for this game form include social equilibria and pseudo nash games this paper explores both the qualitative and quantitative properties of such games through the use of quasi variational inequality theory several interesting relationships between the variational and quasi variational inequality forms of this class of games are described and the practical implementation of generalized nash games are explored at length

some feasibility issues in mathematical programs with equilibrium constraints this paper is concerned with some feasibility issues in mathematical programs with equilibrium constraints mpecs where additional joint constraints are present that must be satisfied by the state and design variables of the problems we introduce sufficient conditions that guarantee the feasibility of these mpecs it turns out that these conditions also guarantee the feasibility of the quadratic programming qp subproblems arising from the penalty interior point algorithm pipa and the sequential quadratic programming sqp algorithm for solving mpecs thus the same conditions ensure that these algorithms are applicable for solving this class of jointly constrained mpecs

mathematical programs with complementarity constraints convergence properties of a smoothing method in this paper optimization problems p with complementarity constraints are considered characterizations for local minimizers bar x of p of orders 1 and 2 are presented we analyze a parametric smoothing approach for solving these programs in which p is replaced by a perturbed problem p tau depending on a small parameter tau we are interested in the convergence behavior of the feasible set cal f tau and the convergence of the solutions bar x tau of p tau for tau to 0 in particular it is shown that under generic assumptions the solutions bar x tau are unique and converge to a solution bar x of p with a rate cal o sqrt tau moreover the convergence for the hausdorff distance d cal f tau cal f between the feasible sets of p tau and p is of order cal o sqrt tau

optimality tests for partitioning and sectional search algorithms this paper revives a familiar idea for use in solving optimization problems particularly unstructured nonlinear programs with many variables the idea involves partitioning variables into groups and successively performing a sectional search with respect to one group at a time the paper addresses various difficulties normally associated with this type of procedure and proposes ways to solve them these include testing optimality of the point at termination and developing ways to restart the process upon finding that the terminating point is not optimal the paper also briefly discusses some useful implications of the results on how to group variables in unstructured nonlinear programs so as to improve convergence

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

a recurrent neural network for solving bilevel linear programming problem in this brief based on the method of penalty functions a recurrent neural network nn modeled by means of a differential inclusion is proposed for solving the bilevel linear programming problem blpp compared with the existing nns for blpp the model has the least number of state variables and simple structure using nonsmooth analysis the theory of differential inclusions and lyapunov like method the equilibrium point sequence of the proposed nns can approximately converge to an optimal solution of blpp under certain conditions finally the numerical simulations of a supply chain distribution model have shown excellent performance of the proposed recurrent nns

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

a multi level non linear multi objective decision making under fuzziness this paper studies a three level non linear multi objective decision making tln modm problem with linear or non linear constraints and in which the objective function at every level are non linear functions which are to be maximized this paper makes an extension work of abo sinna j operat res soc india opsearch 38 5 2001 484 495 which deal with a bi level non linear multi objective decision making problem under fuzziness the three level programming tlp problem whether from the stand point of the three planner stackelberg behavior or from the interactive organizational behavior is a very practical problem and encountered frequently in actual practice this paper proposes a three planner multi objective decision making model and solution method for solving this problem this method uses the concepts of tolerance membership function and multi objective optimization at each level to develop a fuzzy max min decision model for generating pareto optimal satisfactory solution for tln modm problem the first level decision maker fldm specifies his her objective functions and decisions with possible tolerances which are described by membership functions of fuzzy set theory then the second level decision maker sldm specifies his her objective functions and decisions in the view of the fldm with possible tolerances which are described by membership functions of fuzzy set theory finally the third level decision maker tldm uses the preference information for the fldm and sldm to solves his her problem subject to the two upper level decision makers restrictions an illustrative numerical example is given to demonstrate the obtained results

linear bilevel programming with interval coefficients in this paper we address linear bilevel programs when the coefficients of both objective functions are interval numbers the focus is on the optimal value range problem which consists of computing the best and worst optimal objective function values and determining the settings of the interval coefficients which provide these values we prove by examples that in general there is no precise way of systematizing the specific values of the interval coefficients that can be used to compute the best and worst possible optimal solutions taking into account the properties of linear bilevel problems we prove that these two optimal solutions occur at extreme points of the polyhedron defined by the common constraints moreover we develop two algorithms based on ranking extreme points that allow us to compute them as well as determining settings of the interval coefficients which provide the optimal value range

trust region algorithms for solving nonsmooth equations two globally convergent trust region algorithms are presented for solving nonsmooth equations where the functions are only locally lipschitzian the first algorithm is an extension of the classic levenberg marquardt method by approximating the locally lipschitzian function with a smooth function and using the derivative of the smooth function in the algorithm wherever a derivative is needed global convergence for this algorithm is established under a regular condition in the second algorithm successive smooth approximation functions and their derivatives are used global convergence for the second algorithm is established under mild assumptions both objective functions of subproblems of these two algorithms are quadratic functions

a neural network approach for solving nonlinear bilevel programming problem a neural network model is presented for solving nonlinear bilevel programming problem which is a np hard problem the proposed neural network is proved to be lyapunov stable and capable of generating approximal optimal solution to the nonlinear bilevel programming problem the asymptotic properties of the neural network are analyzed and the condition for asymptotic stability solution feasibility and solution optimality are derived the transient behavior of the neural network is simulated and the validity of the network is verified with numerical examples

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

a neurodynamic optimization approach to bilevel quadratic programming this paper presents a neurodynamic optimization approach to bilevel quadratic programming bqp based on the karush kuhn tucker kkt theorem the bqp problem is reduced to a one level mathematical program subject to complementarity constraints mpcc it is proved that the global solution of the mpcc is the minimal one of the optimal solutions to multiple convex optimization subproblems a recurrent neural network is developed for solving these convex optimization subproblems from any initial state the state of the proposed neural network is convergent to an equilibrium point of the neural network which is just the optimal solution of the convex optimization subproblem compared with existing recurrent neural networks for bqp the proposed neural network is guaranteed for delivering the exact optimal solutions to any convex bqp problems moreover it is proved that the proposed neural network for bilevel linear programming is convergent to an equilibrium point in finite time finally three numerical examples are elaborated to substantiate the efficacy of the proposed approach

random matrices nonbacktracking walks and orthogonal polynomials several well known results from the random matrix theory such as wigner s law and the marchenko pastur law can be interpreted and proved in terms of nonbacktracking walks on a certain graph orthogonal polynomials with respect to the limiting spectral measure play a role in this approach

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

quantum field theory on spacetimes with a compactly generated cauchy horizon we prove two theorems which concern difficulties in the formulation of the quantum theory of a linear scalar field on a spacetime with a compactly generated cauchy horizon these theorems demonstrate the breakdown of the theory at certain base points of the cauchy horizon which are defined as past terminal accumulation points of the horizon generators thus the theorems may be interpreted as giving support to hawking s chronology protection conjecture according to which the laws of physics prevent one from manufacturing a time machine specifically we prove

the polynomial hierarchy and a simple model for competitive analysis the multi level linear programs of candler norton and townsley are a simple class of sequenced move games in which players are restricted in their moves only by common linear constraints and each seeks to optimize a fixed linear criterion function in his her own continuous variables and those of other players all data of the game and earlier moves are known to a player when he she is to move the one player case is just linear programming

convergence properties of a regularization scheme for mathematical programs with complementarity constraints we study the convergence behavior of a sequence of stationary points of a parametric nlp which regularizes a mathematical program with equilibrium constraints mpec in the form of complementarity conditions accumulation points are feasible points of the mpec they are c stationary if the mpec linear independence constraint qualification holds they are m stationary if in addition an approaching subsequence satisfies second order necessary conditions and they are b stationary if in addition an upper level strict complementarity condition holds these results complement recent results of fukushima and pang convergence of a smoothing continuation method for mathematical programs with equilibrium constraints in ill posed variational problems and regularization techniques springer verlag new york 1999 we further show that every local minimizer of the mpec which satisfies the linear independence upper level strict complementarity and a second order optimality condition can be embedded into a locally unique piecewise smooth curve of local minimizers of the parametric nlp

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

completeness of cauchy horizon generators it is proved that every compactly generated future cauchy horizon has past complete generators and dually no condition on the differentiability of the horizon is imposed

a new approach for solving linear bilevel problems using genetic algorithms bilevel programming involves two optimization problems where the constraint region of the first level problem is implicitly determined by another optimization problem this paper develops a genetic algorithm for the linear bilevel problem in which both objective functions are linear and the common constraint region is a polyhedron taking into account the existence of an extreme point of the polyhedron which solves the problem the algorithm aims to combine classical extreme point enumeration techniques with genetic search methods by associating chromosomes with extreme points of the polyhedron the numerical results show the efficiency of the proposed algorithm in addition this genetic algorithm can also be used for solving quasiconcave bilevel problems provided that the second level objective function is linear

universality at the edge of the spectrum in wigner random matrices we prove universality at the edge for rescaled correlation functions of wigner random matrices in the limit n as a corollary we show that after proper rescaling the 1th 2nd 3rd etc eigenvalues of wigner random hermitian resp real symmetric matrix weakly converge to the distributions established by tracy and widom in g u e g o e cases

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

on the differentiability of cauchy horizons chru ciel and galloway constructed a cauchy horizon that is nondifferentiable on a dense set we prove that in a certain class of cauchy horizons densely nondifferentiable cauchy horizons form a dense set we show that our class of densely nondifferentiable cauchy horizons implies the existence of densely nondifferentiable cauchy horizons arising from partial cauchy surfaces and also the existence of densely nondifferentiable black hole event horizons

a fuzzy rule based approach for screening international distribution centres in practical situations a decision maker often has to deal with numerous alternatives in evaluating and selecting the most suitable international distribution centre idc location for enhancing the organizational competitiveness to improve the effectiveness of the decision making process it is critical to reduce the number of alternatives through screening so that the decision maker can focus on the most attractive alternatives for further consideration this paper presents a fuzzy rule based approach to effectively screen decision alternatives in adequately solving the problem of evaluating and selecting idc locations case based reasoning with the use of fuzzy production rules is adopted to screen out unattractive alternatives based on past experience in a specific decision making situation a decision support system based on the proposed approach is presented to improve the effectiveness of the screening process an example is given to demonstrate the applicability of the proposed approach in improving the effectiveness of the decision making process in multicriteria decision making

active set algorithm for mathematical programs with linear complementarity constraints in the paper an incomplete active set algorithm is given for mathematical programs with linear complementarity constraints mplcc at each iteration a finite number of inner iterations are contained for approximately solving the relaxed nonlinear optimization problem if the feasible region of the mplcc is bounded under the uniform linear independence constraint qualification licq any cluster point of the sequence generated from the algorithm is a b stationary point of the mplcc preliminary numerical tests show that the algorithm is promising

on stability of the feasible set of a mathematical problem with complementarity problems the feasible set of mathematical programs with complementarity constraints mpcc is considered we discuss local stability of the feasible set with respect to perturbations up to first order of the defining functions here stability refers to homeomorphy invariance under small perturbations for stability we propose a kind of mangasarian fromovitz condition mfc and its stronger version smfc mfc is a natural constraint qualification for c stationarity and smfc is a generalization of the well known clarke s maximal rank condition it turns out that smfc implies local stability mfc and smfc coincide in the case where the number of complementarity constraints k equals the dimension of the state space n moreover the equivalence of mfc and smfc is also proven for the cases k 2 as well as under linear independence constraint qualification licq for mpcc

fuzzy multi level minimum cost flow problems both multiple objectives and multiple hierarchies minimum cost flow problems with fuzzy costs and fuzzy capacities in the arcs are investigated to reduce the complexity a possibility programming is used to handle the vagueness in the parameters fuzzy approach can considerably simplify the problem and thus a fairly general multi level problem can be solved reasonably easily in spite of the fact that the multi level problem is np hard and very difficult to solve several numerical examples are considered to illustrate the approach

a new look at independence the concentration of measure phenomenon in product spaces is a farreaching abstract generalization of the classical exponential inequalities for sums of independent random variables we attempt to explain in the simplest possible terms the basic concepts underlying this phenomenon the basic method to prove concentration inequalities and the meaning of several of the most useful inequalities

identification of cohesive crack fracture parameters by evolutionary search this paper is concerned with the application and performance of two evolutionary search techniques to identify the parameters characterizing mode i cohesive crack models in particular genetic based heuristic schemes implemented via a classical genetic algorithm and also as a differential evolution process are considered actual experimental data are used to assess the schemes for both piecewise linear and nonlinear softening models the results of these two adaptive algorithms are also compared with the performance of a previously proposed local optimization approach involving formulation of the inverse problem as a mathematical program with equilibrium constraints

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

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

unconstrained duals to partially separable constrained programs in this note partially separable convex programs are dualized in such a way that under certain assumptions unconstrained concave duals arise a return formula is given by which the solution of the primal is directly computed if a solution of the dual is known further the solvability of both the primal and the dual is shown to depend essentially on the behaviour of the lower dimensional programs for determining the fenchel conjugates

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

weak linear bilevel programming problems existence of solutions via a penalty method we are concerned with a class of weak linear bilevel programs with nonunique lower level solutions for such problems we give via an exact penalty method an existence theorem of solutions then we propose an algorithm

on stochastic gradient and subgradient methods with adaptive steplength sequences traditionally stochastic approximation sa schemes have been popular choices for solving stochastic optimization problems however the performance of standard sa implementations can vary significantly based on the choice of the steplength sequence and in general little guidance is provided about good choices motivated by this gap we present two adaptive steplength schemes for strongly convex differentiable stochastic optimization problems equipped with convergence theory that aim to overcome some of the reliance on user specific parameters the first scheme referred to as a recursive steplength stochastic approximation rsa scheme optimizes the error bounds to derive a rule that expresses the steplength at a given iteration as a simple function of the steplength at the previous iteration and certain problem parameters the second scheme termed as a cascading steplength stochastic approximation csa scheme maintains the steplength sequence as a piecewise constant decreasing function with the reduction in the steplength occurring when a suitable error threshold is met then we allow for nondifferentiable objectives but with bounded subgradients over a certain domain in such a regime we propose a local smoothing technique based on random local perturbations of the objective function that leads to a differentiable approximation of the function assuming a uniform distribution on the local randomness we establish a lipschitzian property for the gradient of the approximation and prove that the obtained lipschitz bound grows at a modest rate with problem size this facilitates the development of an adaptive steplength stochastic approximation framework which now requires sampling in the product space of the original measure and the artificially introduced distribution

how to solve a semi infinite optimization problem abstract after an introduction to main ideas of semi infinite optimization this article surveys recent developments in theory and numerical methods for standard and generalized semi infinite optimization problems particular attention is paid to connections with mathematical programs with complementarity constraints lower level wolfe duality semi smooth approaches as well as branch and bound techniques in adaptive convexification procedures a section on recent genericity results includes a discussion of the symmetry effect in generalized semi infinite optimization

smooth sqp methods for mathematical programs with nonlinear complementarity constraints mathematical programs with nonlinear complementarity constraints are reformulated using better posed but nonsmooth constraints we introduce a class of functions parameterized by a real scalar to approximate these nonsmooth problems by smooth nonlinear programs this smoothing procedure has the extra benefits that it often improves the prospect of feasibility and stability of the constraints of the associated nonlinear programs and their quadratic approximations we present two globally convergent algorithms based on sequential quadratic programming sqp as applied in exact penalty methods for nonlinear programs global convergence of the implicit smooth sqp method depends on existence of a lower level nondegenerate strictly complementary limit point of the iteration sequence global convergence of the explicit smooth sqp method depends on a weaker property i e existence of a limit point at which a generalized constraint qualification holds we also discuss some practical matters relating to computer implementations

nonlinear integer bilevel programming abstract in many decentralized organizations resource planning with sequential decision making can be formulated as a multi level programming problem in such cases the decision variables are partitioned among the decision makers each of the decision makers optimizes his her own objective function this paper presents an algorithm using parametric analysis to solve a typical kind of nonlinear integer multilevel programming problems called separable integer monotone bilevel programming simbp and then extends the algorithm for solving a parametric simbp problem a numerical example with application of reliability optimization is given to illustrate the solution method

area theorem and smoothness of compact cauchy horizons we obtain an improved version of the area theorem for not necessarily differentiable horizons which in conjunction with a recent result on the completeness of generators allows us to prove that under the null energy condition every compactly generated cauchy horizon is smooth and compact we explore the consequences of this result for time machines topology change black holes and cosmic censorship for instance it is shown that compact cauchy horizons cannot form in a non empty spacetime which satisfies the stable dominant energy condition wherever there is some source content

a numerical approach to optimization problems with variational inequality constraints optimization problems with variational inequality constraints are converted to constrained minimization of a local lipschitz function to this minimization a non differentiable optimization method is used the required subgradients of the objective are computed by means of a special adjoint equation besides tests with some academic examples the approach is applied to the computation of the stackelberg cournot nash equilibria and to the numerical solution of a class of quasi variational inequalities

matrix infinitely divisible series tail inequalities and their applications in this paper we study tail inequalities of the largest eigenvalue of a matrix infinitely divisible i d series which is a finite sum of fixed matrices weighted by i d random variables we obtain several types of tail inequalities including bennett type and bernstein type inequalities this allows us to further bound the expectation of the spectral norm of a matrix i d series moreover by developing a new lower bound function for q s s 1 log s 1 s that appears in the bennett type inequality we derive a tighter tail inequality of the largest eigenvalue of the matrix i d series than the bernstein type inequality when the matrix dimension is high the resulting lower bound function is of independent interest and can improve any bennett type concentration inequality that involves the function q s the class of i d probability distributions is large and includes gaussian and poisson distributions among many others therefore our results encompass the existing work on matrix gaussian series as a special case lastly we show that the tail inequalities of a matrix i d series have applications in several optimization problems including the chance constrained optimization problem and the quadratic optimization problem with orthogonality constraints in addition we also use the resulting tail bounds to show that random matrices constructed from i d random variables satisfy the restricted isometry property rip when it acts as a measurement matrix in compressed sensing

a solution method for the linear static stackelberg problem using penalty functions a new solution technique is presented for the linear constrained static stackelberg problem for a given value of x the leader s decision vector the follower is at its rational reaction set when the duality gap of the second level problem becomes zero the outer problem is solved by appending to the leader s objective a function that minimizes the duality gap of the follower s problem this structure leads to the decomposition of the composite problem into a series of linear programs leading to an efficient algorithm it is proved that optimality is reached for an exact penalty function and the method is illustrated with some examples

a bilevel bottleneck programming problem this paper studies a bilevel programming problem with linear constraints and in which the objective functions at both levels are concave bottleneck functions which are to be minimized the problem is a non convex programming problem it is shown that an optimal solution to the problem is attainable at an extreme point of the underlying region the outer level objective function values are ranked in increasing order until a value is reached one of the solutions corresponding to which is feasible for the problem this solution is then the required global optimal solution

multiplier convergence in trust region methods with application to convergence of decomposition methods for mpecs we study piecewise decomposition methods for mathematical programs with equilibrium constraints mpecs for which all constraint functions are linear at each iteration of a decomposition method one step of a nonlinear programming scheme is applied to one piece of the mpec to obtain the next iterate our goal is to understand global convergence to b stationary points of these methods when the embedded nonlinear programming solver is a trust region scheme and the selection of pieces is determined using multipliers generated by solving the trust region subproblem to this end we study global convergence of a linear trust region scheme for linearly constrained nlps that we call a trust search method the trust search has two features that are critical to global convergence of decomposition methods for mpecs a robustness property with respect to switching pieces and a multiplier convergence result that appears to be quite new for trust region methods these combine to clarify and strengthen global convergence of decomposition methods without resorting either to additional conditions such as eventual inactivity of the trust region constraint or more complex methods that require a separate subproblem for multiplier estimation

modeling and computing two settlement oligopolistic equilibrium in a congested electricity network a model of two settlement electricity markets is introduced which accounts for flow congestion demand uncertainty system contingencies and market power we formulate the subgame perfect nash equilibrium for this model as an equilibrium problem with equilibrium constraints epec in which each firm solves a mathematical program with equilibrium constraints mpec the model assumes linear demand functions quadratic generation cost functions and a lossless dc network resulting in equilibrium constraints as a parametric linear complementarity problem lcp we introduce an iterative procedure for solving this epec through repeated application of an mpec algorithm this mpec algorithm is based on solving quadratic programming subproblems and on parametric lcp pivoting numerical examples demonstrate the effectiveness of the mpec and epec algorithms and the tractability of the model for realistic size power systems

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

an algorithm for solving the general bilevel programming problem the conflict that naturally arises in a hierarchical system can often be modeled as a multistage optimization problem this paper considers the sequential uncooperative problem in which two decision makers wish to maximize their own objective functions over a feasible region defined by interactive strategy sets the resultant problem is known as the bilevel program such programs are inherently nonconvex and resistant to standard nonlinear programming solution techniques such as piecewise linearization and convex underestimating envelopes alternatively a grid search algorithm is offered which exhibits the desirable property of monotonicity the algorithm is based on two sets of necessary conditions previously developed and combined here to provide an operational check for stationarity and local optimality potential solutions are obtained from a parameterized master program whose feasible region approximates that of the original problem as the one dimensional parameter is varied over the unit interval

mpcc critical point theory we study mathematical programs with complementarity constraints mpcc from a topological point of view special focus will be on c stationary points under the linear independence constraint qualification licq we derive an equivariant morse lemma at nondegenerate c stationary points then two basic theorems from morse theory deformation theorem and cell attachment theorem are proved outside the c stationary point set continuous deformation of lower level sets can be performed as a consequence the topological data such as the number of connected components then remain invariant however when passing a c stationary level the topology of the lower level set changes via the attachment of a q dimensional cell the dimension q equals the stationary c index of the nondegenerate c stationary point the stationary c index depends on both the restricted hessian of the lagrangian and the lagrange multipliers related to biactive complementarity constraints finally some relations with other stationarity concepts such as w a m s and b stationarity are discussed

on bilevel programming part i general nonlinear cases this paper is concerned with general nonlinear nonconvex bilevel programming problems blpp we derive necessary and sufficient conditions at a local solution and investigate the stability and sensitivity analysis at a local solution in the blpp we then explore an approach in which a bundle method is used in the upper level problem with subgradient information from the lower level problem two algorithms are proposed to solve the general nonlinear blpp and are shown to converge to regular points of the blpp under appropriate conditions the theoretical analysis conducted in this paper seems to indicate that a sensitivity based approach is rather promising for solving general nonlinear blpp