a new general purpose algorithm for mixed integer bilevel linear programs bilevel optimization problems are very challenging optimization models arising in many important practical contexts including pricing mechanisms in the energy sector airline and telecommunication industry transportation networks critical infrastructure defense and machine learning in this paper we consider bilevel programs with continuous and discrete variables at both levels with linear objectives and constraints continuous upper level variables if any must not appear in the lower level problem we propose a general purpose branch and cut exact solution method based on several new classes of valid inequalities which also exploits a very effective bilevel specific preprocessing procedure an extensive computational study is presented to evaluate the performance of various solution methods on a common testbed of more than 800 instances from the literature and 60 randomly generated instances our new algorithm consistently outperforms often by a large margin alternative state of the art metho

error estimates in the numerical evaluation of some bem singular integrals in some applications of galerkin boundary element methods one has to compute integrals which after proper normalization are of the form b 2 x y dxdy a 1 x y y where a b 1 1 or a b a 1 or a b 1 b and f x y is a smooth function in this paper we derive error estimates for a numerical approach recently proposed to evaluate the above integral when a p or h p formulation of a galerkin method is used this approach suggests approximating the inner integral by a quadrature formula of interpolatory type that exactly integrates the cauchy kernel and the outer integral by a rule which takes into account the log endpoint singularities of its integrand some numerical examples are also given

error expansion of piecewise constant interpolation rule for certain two dimensional cauchy principal value integrals the classical composite rectangle midpoint rule for the computation of two dimensional singular integrals is discussed with the error functional of rectangle rule for computing two dimensional singular integrals the local coordinate of certain point and the convergence results o h2 are obtained when the local coordinate is coincided with certain priori known coordinates we get the convergence rate the same as the riemann integral at certain point at last numerical examples are presented to illustrate our theoretical analysis which agree with it very well

on the convergence of product formulas for the evaluation of certain two dimensional cauchy principal value integrals on etablit la convergence des regles de produits de type interpolatoire pour l evaluation numerique de certaines integrales bidimensionnelles a valeur principale de cauchy on obtient quelques estimations du reste pour plusieurs classes de fonctions

an exact approach for the bilevel knapsack problem with interdiction constraints and extensions we consider the bilevel knapsack problem with interdiction constraints an extension of the classic 0 1 knapsack problem formulated as a stackelberg game with two agents a leader and a follower that choose items from a common set and hold their own private knapsacks first the leader selects some items to be interdicted for the follower while satisfying a capacity constraint then the follower packs a set of the remaining items according to his knapsack constraint in order to maximize the profits the goal of the leader is to minimize the follower s total profit we derive effective lower bounds for the bilevel knapsack problem and present an exact method that exploits the structure of the induced follower s problem the approach strongly outperforms the current state of the art algorithms designed for the problem we extend the same algorithmic framework to the interval min max regret knapsack problem after providing a novel bilevel programming reformulation also for this problem the proposed approach outperforms the exact algorithms available in the literature

a smoothing heuristic for a bilevel pricing problem abstract in this paper we provide a heuristic procedure that performs well from a global optimality point of view for an important and difficult class of bilevel programs the algorithm relies on an interior point approach that can be interpreted as a combination of smoothing and implicit programming techniques although the algorithm cannot guarantee global optimality very good solutions can be obtained through the use of a suitable set of parameters the algorithm has been tested on large scale instances of a network pricing problem an application that fits our modeling framework preliminary results show that on hard instances our approach constitutes an alternative to solvers based on mixed 0 1 programming formulations

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

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

low roundoff noise and normal realizations of fixed point iir digital filters explicit design and performance equations are given in terms of the parameters of the desired transfer function h z for two classes of realizations these realizations offer the digital circuit designer an alternative to the usual designs based on direct forms in addition to low roundoff noise these new realizations offer other desirable properties such as low coefficient sensitivities and freedom from overflow oscillations numerical results are presented which detail the design process and verify the theory

a study on the computational complexity of the bilevel knapsack problem we analyze the computational complexity of three fundamental variants of the bilevel knapsack problem all three variants are shown to be complete for the second level of the polynomial hierarchy we also discuss the somewhat easier situation where the weight and profit coefficients in the knapsack problem are encoded in unary two of the considered bilevel variants become solvable in polynomial time whereas the third becomes np complete furthermore we design a polynomial time approximation scheme for this third variant whereas the other two variants cannot be approximated in polynomial time within any constant factor assuming p ne np

approximation of the hilbert transform on the real line using hermite zeros the authors study the hilbert transform on the real line they introduce some polynomial approximations and some algorithms for its numerical evaluation error estimates in uniform norm are given

convergence of rules based on nodal splines for the numerical evaluation of certain 2d cauchy principal value integrals abstract we consider interpolatory type integration rules for the numerical approximation of certain 2d cauchy principal value integrals based on tensor product of nodal splines we present convergence results which generalize those known in one dimensional case

numerical evaluation of two dimensional cauchy principal value integrals a twelve point cubature formula has been constructed for the numerical evaluation of two dimensional real cauchy principal value cpv integrals some particular cases of this rule have been discussed and the truncation error associated with the rules have been analysed

on the convergence of product formulas for the numerical evaluation of derivatives of cauchy principal value integrals product rules of interpolatory type on the zeros of generalized smooth jacobi polynomials for the numerical approximation of cauchy principal value integrals and their derivatives are considered convergence results for these rules are given and some estimates of the remainder are established in relation to the regularity of the density function

bilevel programming and the separation problem in recent years branch and cut algorithms have become firmly established as the most effective method for solving generic mixed integer linear programs milps methods for automatically generating inequalities valid for the convex hull of solutions to such milps are a critical element of branch and cut this paper examines the nature of the so called separation problem which is that of generating a valid inequality violated by a given real vector usually arising as the solution to a relaxation of the original problem we show that the problem of generating a maximally violated valid inequality often has a natural interpretation as a bilevel program in some cases this bilevel program can be easily reformulated as a simple single level mathematical program yielding a standard mathematical programming formulation for the separation problem in other cases no such polynomial size single level reformulation exists unless the polynomial hierarchy collapses to its first level an event considered extremely unlikely in computational complexity theory we illustrate our insights by considering the separation problem for two well known classes of valid inequalities

a constructive approach to nodal splines in the context of local spline interpolation methods nodal splines have been introduced as possible fundamental functions by de villiers and rohwer in 1988 the corresponding local spline interpolation operator possesses the desirable property of reproducing a large class of polynomials however it was remarked that their definition is rather intricate so that it seems desirable to reveal the actual origin of these splines the real source can be found in the martensenoperator which can be obtained by two point hermite spline interpolation problem posed and proved by martensen darstellung und entwicklung des restgliedes der gregoryschen quadraturformel mit hilfe von spline funktionen numer math 21 1973 70 80 on the one hand we will show how to represent the hermite martensen spline recursively and on the other hand explicitly in terms of the b spline by using the famous marsden identity having introduced the martensenoperator we will show that the nodal spline interpolation operator can be obtained by a special discretization of the occurring derivatives we will consider symmetric nodal splines of odd degree that can be obtained by our methods in a natural way

spline approximation for cauchy principal value integrals abstract in this paper we prove the uniform convergence of some quadrature formulas based on spline approximation for cauchy principal value integrals of the type f b a w x f k x x y d x k 0 1 and we present some numerical applications in particular we apply our rules to the well known prandtl s integral equation

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

design of state digital filters in this correspondence we develop a very simple algorithm for the design of a second order state space digital filter the design equations presented here were derived on the rendition that the sensitivity of the coefficients in the state space digital filter sdf are spread uniformly to all coefficients of the sdf the design algorithm has been shown to provide sdf realizations having a minimum output round off noise while preserving low coefficient sensitivity and freedom from limit cycles

optimal local spline interpolants abstract a constructive proof is given of the existence of a local spline interpolant which also approximates optimally in the sense that its associated operator reproduces polynomials of maximal order first it is shown that such an interpolant does not exist for orders higher than the linear case if the partition points of the appropriate spline space coincide with the given interpolation points next in the main result the desired existence of an optimal local spline interpolant for all orders is proved by increasing in a specified manner the set of partition points although our interpolant reproduces a more restricted function space than its quasi interpolant counterpart constructed by de boor and fix 1 it has the advantage of interpolating every real function at a given set of points finally we do some explicit calculations in the quadratic case

on the weights of certain quadratures for the numerical evaluation of cauchy principal value integrals and their derivatives we consider interpolatory quadrature rules of the type begin gathered frac d k d lambda k int limits 1 1 1 x alpha 1 x beta frac f x x lambda dx sum limits i 1 n w ni k lambda f x ni r n k f hfill k 0 1 alpha beta 0 hfill end gathered where the nodes x ni are the zeros of the jacobi polynomialp n x in particular we prove that sum limits i 1 n left w ni k lambda right o n k log n uniformly in 1 e 1 e furthermore we show that whenk 1 f x c p 1 1 p 1 andf p x h 1 1 0 1 we have r n 1 f o n p mu 1 delta 0 small as we like uniformly in 1 e 1 e

joint optimization of product family configuration and scaling design by stackelberg game abstract product family design is generally characterized by two types of approaches module based and scale based while the former aims to enable product variety based on module configuration the latter is to variegate product design by scaling up or down certain design parameters the prevailing practice is to treat module configuration and scaling design as separate decisions or aggregate two design problems as a single level all in one optimization problem in practice optimization of scaling variables is always enacted within a specific modular platform and meanwhile an optimal module configuration depends on how design parameters are to be scaled the key challenge is how to deal with explicitly the coupling of these two design optimization problems this paper formulates a stackelberg game theoretic model for joint optimization of product family configuration and scaling design in which a bilevel decision structure reveals coupled decision making between module configuration and parameter scaling a bilevel mixed 0 1 non linear programming model is developed and solved comprising an upper level optimization problem and a lower level optimization problem the upper level seeks for an optimal configuration of modules and module attributes by maximizing the shared surplus of an entire product family the lower level entails parametric optimization of attribute values for optimal technical performance of each individual module a case study of electric motors demonstrates that the bilevel joint optimization model excels in leveraging optimal scaling in conjunction with optimal module configuration which is advantageous over the existing paradigm of product family scaling design that cannot change the product family configuration

applications of optimally local interpolation to interpolatory approximants and compactly supported wavelets the objective of this paper is to introduce a general scheme for the construction of interpolatory approximation formulas and compactly supported wavelets by using spline functions with arbitrary nonuniform knots both construction procedures are based on certain optimally local interpolatory fundamental spline functions which are not required to possess any approximation property

error expansion of trapezoidal rule for certain two dimensional cauchy principal value integrals abstract in this article we discuss the classical composite trapezoidal rule for the computation of two dimensional singular integrals the purpose is to obtain the convergence results o h 2 which is the same as the riemann integral convergence rate at certain points of the classical composite trapezoidal rule with the error functional of trapezoidal rule for computing two dimensional singular integrals we get the superconvergence phenomenon when the special function in error functional is equal to zero at last some numerical examples are reported to illustrate our theoretical analysis which agree with it very well

on the evaluation of certain two dimensional singular integrals with cauchy kernels in this paper we consider product formulas of interpolatory type for the two dimensional cauchy principal value integral i f eta 1 eta 2 int limits 1 1 int limits 1 1 omega x y frac f x y x eta 1 y eta 2 dxdy where 1 2 1 1 x y 1 x 2 y and 1 x and 2 x are two absolutely integrable weight functions the integral is approximated by f n m f eta 1 eta 2 sum limits i 0 n sum limits j 0 m a i 1 eta 1 a j 2 eta 2 f x i y i where the nodes x i and y j are the zeros of the chebyshev polynomials of the first kind commonly named classical abscissas or the clenshaw points often called practical abscissas we present convergence results for these rules

a nodal spline interpolant for the gregory rule of even order the gregory rule is a well known example in numerical quadrature of a trapezoidal rule with endpoint corrections of a given order in the literature the methods of constructing the gregory rule have in contrast to newton cotes quadrature not been based on the integration of an interpolant in this paper after first characterizing an even order gregory interpolant by means of a generalized lagrange interpolation operator we proceed to explicitly construct such an interpolant by employing results from nodal spline interpolation as established in recent work by the author and c h rohwer nonoptimal order error estimates for the gregory rule of even order are then easily obtained

new second order state space structures for realizing low roundoff noise digital filters several new second order state space digital filter structures are introduced each of the new structures requires one or two less multiplies than the minimum noise structure while achieving a round off noise performance which is nearly as good thus they provide an excellent means of realizing second order subfilters when such subfilters are used to construct nth order filters the reduction in multiplies as compared to using minimum noise subfilters is from about 4n to about 3n an approach to developing simple algebraic real arithmetic design equations for l 2 norm scaled second order structures is presented and then applied to obtaining design equations for the new structures and for the minimum noise structure

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

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 of product formulas for the numerical evaluation of certain two dimensional cauchy principal value integrals we consider product rules of interpolatory type for the numerical approximation of certain two dimensional cauchy principal value integrals we present convergence results which generalize those known in the one dimensional case

interlacing properties of the zeros of the orthogonal polynomials and approximation of the hilbert transform abstract in this paper starting from interlacing properties of the zeros of the orthogonal polynomials the authors propose a new method to approximate the finite hilbert transform for this method they give error estimates in uniform norm

on the complexity of quantified linear systems in this paper we explore the computational complexity of the conjunctive fragment of the first order theory of linear arithmetic quantified propositional formulas of linear inequalities with k 1 quantifier alternations are log space complete in s k p or p k p depending on the initial quantifier we show that when we restrict ourselves to quantified conjunctions of linear inequalities i e quantified linear systems the complexity classes collapse to polynomial time in other words the presence of universal quantifiers does not alter the complexity of the linear programming problem which is known to be in p our result reinforces the importance of sentence formats from the perspective of computational complexity

gregory type quadrature based on quadratic nodal spline interpolation using a method based on quadratic nodal spline interpolation we define a quadrature rule with respect to arbitrary nodes and which in the case of uniformly spaced nodes corresponds to the gregory rule of order two i e the lacroix rule which is an important example of a trapezoidal rule with endpoint corrections the resulting weights are explicitly calculated and peano kernel techniques are then employed to establish error bounds in which the associated error constants are shown to grow at most linearly with respect to the mesh ratio parameter specializing these error estimates to the case of uniform nodes we deduce non optimal order error constants for the lacroix rule which are significantly smaller than those calculated by cruder methods in previous work and which are shown here to compare favourably with the corresponding error constants for the simpson rule

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

the numerical evaluation of a 2 d cauchy principal value integral arising in boundary integral equation methods in this paper we consider the problem of computing 2 d cauchy principal value integrals of the form

capacity planning with competitive decision makers trilevel milp formulation degeneracy and solution approaches abstract capacity planning addresses the decision problem of an industrial producer investing on infrastructure to satisfy future demand with the highest profit traditional models neglect the rational behavior of some external decision makers by assuming either static competition or captive markets we propose a mathematical programing formulation with three levels of decision makers to capture the dynamics of duopolistic markets the trilevel model is transformed into a bilevel optimization problem with mixed integer variables in both levels by replacing the third level linear program with its optimality conditions we introduce new definitions required for the analysis of degeneracy in multilevel models and develop two novel algorithms to solve these challenging problems each algorithm is shown to converge to a different type of degenerate solution the computational experiments for capacity expansion in industrial gas markets show that no algorithm is strictly superior in terms of performance

on the convergence of interpolatory type quadrature rules for evaluating cauchy integrals the aim of this work is to analyse the stability and the convergence for the quadrature rule of interpolatory type based on the trigonometric approximation for the discretization of the cauchy principal value integrals 11 f t d we prove that the quadrature rule has almost optimal stability property behaving in the form o logn 1 sin2x x cos t using this result we show that the rule has an exponential convergence rate when the function f is differentiable enough when f possesses continuous derivatives up to order p 0 and the derivative f p t satisfies holder continuity of order we can also prove that the rule has the convergence rate of the form o a b log n n2 np p where is as small as we like a and b are constants depending only on x

error bounds for spline based quadrature methods for strongly singular integrals abstract for the numerical evaluation of finite part integrals with singularities of order p 1 we give error bounds for quadrature methods based on spline approximation these bounds behave in the same way as the optimal ones the ideas of the proof are also useful for methods based on other approximation processes

simultaneous convexification of bilinear functions over polytopes with application to network interdiction we study the simultaneous convexification of graphs of bilinear functions g k x y y intercal a k x over x in ensuremath xi x in 0 1 n e x geq f and y in delta m y in r m 1 intercal y leq 1 we propose a constructive procedure to obtain a linear description of the convex hull of the resulting set this procedure can be applied to derive convex and concave envelopes of certain bilinear functions to study unary expansions of integer variables in mixed integer bilinear sets and to obtain convex hulls of sets with complementarity constraints exploiting the structure of xi the procedure naturally yields stronger linearizations for bilinear terms in a variety of practical settings in particular we demonstrate the effectiveness of the approach by strengthening the traditional dual formulation of network interdiction problems and report encouraging preliminary numerical results

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

smoothness and error bounds of martensen splines martensen splines m f of degree n interpolate f and its derivatives up to the order n 1 at a subset of the knots of the spline space have local support and exactly reproduce both polynomials and splines of degree n an approximation error estimate has been provided for f c n 1 this paper aims to clarify how well the martensen splines m f approximate smooth functions on compact intervals assuming that f c n 1 approximation error estimates are provided for d j f j 0 1 n 1 where d j is the j th derivative operator moreover a set of sufficient conditions on the sequence of meshes are derived for the uniform convergence of d j m f to d j f for j 0 1 n 1

approximation of the hilbert transform on the real semiaxis using laguerre zeros the authors propose two new algorithms for the computation of cauchy principal value integrals on the real semiaxis the proposed quadrature rules use zeros of laguerre polynomials theoretical error estimates are proved and some numerical examples are showed