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

the adaptive convexification algorithm for semi infinite programming with arbitrary index sets a numerical solution method for semi infinite optimization problems with arbitrary not necessarily box shaped index sets is presented following the ideas of floudas and stein siam j optim 18 1187 1208 2007 convex relaxations of the lower level problem are adaptively constructed and then reformulated as mathematical programs with complementarity constraints and solved although the index set is arbitrary this approximation produces feasible iterates for the original problem the convex relaxations and needed parameters are constructed with ideas of the bb method of global optimization and interval methods it is shown that after finitely many steps an epsilon stationary point of the original semi infinite problem is reached a numerical example illustrates the performance of the proposed method

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

convergent semidefinite programming relaxations for global bilevel polynomial optimization problems in this paper we consider a bilevel polynomial optimization problem where the objective and the constraint functions of both the upper and the lower level problems are polynomials we present methods for finding its global minimizers and global minimum using a sequence of semidefinite programming sdp relaxations and provide convergence results for the methods our scheme for problems with a convex lower level problem involves solving a transformed equivalent single level problem by a sequence of sdp relaxations whereas our approach for general problems involving a nonconvex polynomial lower level problem solves a sequence of approximation problems via another sequence of sdp relaxations

maximum likelihood estimation of rician distribution parameters the problem of parameter estimation from rician distributed data e g magnitude magnetic resonance images is addressed the properties of conventional estimation methods are discussed and compared to maximum likelihood ml estimation which is known to yield optimal results asymptotically in contrast to previously proposed methods ml estimation is demonstrated to be unbiased for high signal to noise ratio snr and to yield physical relevant results for low snr

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

semi infinite programming theory methods and applications starting from a number of motivating and abundant applications in 2 including control of robots eigenvalue computations mechanical stress of materials and statistical design the authors describe a class of optimization problems which are referred to as semi infinite because their constraints bound functions of a finite number of variables on a whole region in 3 5 first and second order optimality conditions are derived for general nonlinear problems as well as a procedure for reducing the problem locally to one with only finitely many constraints another main effort for achieving simplification is through duality in 6 there algebraic properties of finite linear programming are brought to bear on duality theory in semi infinite programming section 7 treats numerical methods based on either discretization or local reduction with the emphasis on the design of superlinearly convergent sqp type methods taking this differentiable point of view this paper can be considered to be complementar

monotonic optimization problems and solution approaches problems of maximizing or minimizing monotonic functions of n variables under monotonic constraints are discussed a general framework for monotonic optimization is presented in which a key role is given to a property analogous to the separation property of convex sets the approach is applicable to a wide class of optimization problems including optimization problems dealing with functions representable as differences of increasing functions d i functions

on generalized semi infinite optimization and bilevel optimization the paper studies the connections and differences between bilevel problems bl and generalized semi infinite problems gsip under natural assumptions gsip can be seen as a special case of a bl we consider the so called reduction approach for bl and gsip leading to optimality conditions and newton type methods for solving the problems we show by a structural analysis that for gsip problems the regularity assumptions for the reduction approach can be expected to hold generically at a solution but for general bl problems not the genericity behavior of bl and gsip is in particular studied for linear problems

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

on polya random walks lattice green functions and the bond percolation threshold based on numerical evidence the authors conjecture a connection between the bond percolation threshold of bravais lattices in three or more dimensions and the value at the origin of a lattice green function related to the probability of return to the origin for a polya random walk

interval analysis theory and applications we give an overview on applications of interval arithmetic among others we discuss verification methods for linear systems of equations nonlinear systems the algebraic eigenvalue problem initial value problems for odes and boundary value problems for elliptic pdes of second order we also consider the item software in this field and give some historical remarks

clinical dt mri estimation smoothing and fiber tracking with log euclidean metrics diffusion tensor magnetic resonance imaging dt mri or dti is an imaging modality that is gaining importance in clinical applications however in a clinical environment data have to be acquired rapidly often at the expense of the image quality this often results in dti datasets that are not suitable for complex postprocessing like fiber tracking we propose a new variational framework to improve the estimation of dt mri in this clinical context most of the existing estimation methods rely on a log gaussian noise gaussian noise on the image logarithms or a gaussian noise that do not reflect the rician nature of the noise in mr images with a low signal to noise ratio snr with these methods the rician noise induces a shrinking effect the tensor volume is underestimated when other noise models are used for the estimation in this paper we propose a maximum likelihood strategy that fully exploits the assumption of a rician noise to further reduce the influence of the noise we optimally exploit the spatial correlation by coupling the estimation with an anisotropic prior previously proposed on the spatial regularity of the tensor field itself which results in a maximum a posteriori estimation optimizing such a nonlinear criterion requires adapted tools for tensor computing we show that riemannian metrics for tensors and more specifically the log euclidean metrics are a good candidate and that this criterion can be efficiently optimized experiments on synthetic data show that our method correctly handles the shrinking effect even with very low snr and that the positive definiteness of tensors is always ensured results on real clinical data demonstrate the truthfulness of the proposed approach and show promising improvements of fiber tracking in the brain and the spinal cord

impact of an improved combination of signals from array coils in diffusion tensor imaging an improved method for the combination of signals from array coils is presented as a way to reduce the influence of the noise floor on the estimation of diffusion tensor imaging dti parameters by an optimized combination of signals from the array channels and complex averaging of measurements this method leads to a significant reduction of the noise bias this combination algorithm allows computation of accurate tensors by using the simple two point method and is shown to provide results similar to the ones obtained using the standard signal combination and a nonlinear regression method with noise parameter estimation in many applications the use of this combination method would result in a scan time reduction in comparison to the current standard the effects of the improved combination on diffusion decay curves fractional anisotropy maps and apparent diffusion coefficient adc profiles are demonstrated

a constrained variational principle for direct estimation and smoothing of the diffusion tensor field from complex dwi in this paper we present a novel constrained variational principle for simultaneous smoothing and estimation of the diffusion tensor field from complex valued diffusion weighted images dwi the constrained variational principle involves the minimization of a regularization term of l sup p norms subject to a nonlinear inequality constraint on the data the data term we employ is the original stejskal tanner equation instead of the linearized version usually employed in literature the complex valued nonlinear form leads to a more accurate when compared to the linearized version estimate of the tensor field the inequality constraint requires that the nonlinear least squares data term be bounded from above by a known tolerance factor finally in order to accommodate the positive definite constraint on the diffusion tensor it is expressed in terms of cholesky factors and estimated the constrained variational principle is solved using the augmented lagrangian technique in conjunction with the limited memory quasi newton method experiments with complex valued synthetic and real data are shown to depict the performance of our tensor field estimation and smoothing algorithm

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

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

on the percolation threshold for a d dimensional simple hypercubic lattice the authors use 1 sigma expressions sigma 2d 1 to investigate systematically a number of conjectures and approximations for the bond and site percolation thresholds of a d dimensional simple hypercubic lattice the technique provides convincing evidence that none of the conjectures are correct and gives a quantitative understanding of the quality of the approximations

a projected lagrangian algorithm for semi infinite programming a globally convergent algorithm is presented for the solution of a wide class of semi infinite programming problems the method is based on the solution of a sequence of equality constrained quadratic programming problems and usually has a second order convergence rate numerical results illustrating the method are given

weighted linear least squares estimation of diffusion mri parameters strengths limitations and pitfalls abstract purpose linear least squares estimators are widely used in diffusion mri for the estimation of diffusion parameters although adding proper weights is necessary to increase the precision of these linear estimators there is no consensus on how to practically define them in this study the impact of the commonly used weighting strategies on the accuracy and precision of linear diffusion parameter estimators is evaluated and compared with the nonlinear least squares estimation approach methods simulation and real data experiments were done to study the performance of the weighted linear least squares estimators with weights defined by a the squares of the respective noisy diffusion weighted signals and b the squares of the predicted signals which are reconstructed from a previous estimate of the diffusion model parameters results the negative effect of weighting strategy a on the accuracy of the estimator was surprisingly high multi step weighting strategies yield better performance and in some cases even outperformed the nonlinear least squares estimator conclusion if proper weighting strategies are applied the weighted linear least squares approach shows high performance characteristics in terms of accuracy precision and may even be preferred over nonlinear estimation methods

on the mathematical foundations of nondifferentiable optimization in engineering design it is shown by example that a large class of engineering design problems can be transcribed into the form of a canonical optimization problem with inequality constraints involving mar functions such problems are commonly referred to as semi infinite optimization problems the bulk of this paper is devoted to the development of a mathematical theory for the construction of first order nondifferentiable optimization algorithms related to phase i phase ii methods of feasible directions which solve these semi infinite optimization problems the applicability of the theory is illustrated with examples that are relevant to engineering design

the adaptive convexification algorithm a feasible point method for semi infinite programming we present a new numerical solution method for semi infinite optimization problems its main idea is to adaptively construct convex relaxations of the lower level problem replace the relaxed lower level problems equivalently by their karush kuhn tucker conditions and solve the resulting mathematical programs with complementarity constraints this approximation produces feasible iterates for the original problem the convex relaxations are constructed with ideas from the alpha bb method of global optimization the necessary upper bounds for second derivatives of functions on box domains can be determined using the techniques of interval arithmetic where our algorithm already works if only one such bound is available for the problem we show convergence of stationary points of the approximating problems to a stationary point of the original semi infinite problem within arbitrarily given tolerances numerical examples from chebyshev approximation and design centering illustrate the performance of the method

resolution of complex tissue microarchitecture using the diffusion orientation transform dot this article describes an accurate and fast method for fiber orientation mapping using multidirectional diffusion weighted magnetic resonance mr data this novel approach utilizes the fourier transform relationship between the water displacement probabilities and diffusion attenuated mr signal expressed in spherical coordinates the radial part of the fourier integral is evaluated analytically under the assumption that mr signal attenuates exponentially the values of the resulting functions are evaluated at a fixed distance away from the origin the spherical harmonic transform of these functions yields the laplace series coefficients of the probabilities on a sphere of fixed radius alternatively probability values can be computed nonparametrically using legendre polynomials orientation maps calculated from excised rat nervous tissue data demonstrate this technique s ability to accurately resolve crossing fibers in anatomical regions such as the optic chiasm this proposed methodology has a trivial extension to multiexponential diffusion weighted signal decay the developed methods will improve the reliability of tractography schemes and may make it possible to correctly identify the neural connections between functionally connected regions of the nervous system

least squares for diffusion tensor estimation revisited propagation of uncertainty with rician and non rician signals abstract least squares ls and its minimum variance counterpart weighted least squares wls have become very popular when estimating the diffusion tensor dt to the point that they are the standard in most of the existing software for diffusion mri they are based on the linearization of the stejskal tanner equation by means of the logarithmic compression of the diffusion signal due to the rician nature of noise in traditional systems a certain bias in the estimation is known to exist this artifact has been made patent through some experimental set ups but it is not clear how the distortion translates in the reconstructed dt and how important it is when compared to the other source of error contributing to the mean squared error mse in the estimate i e the variance in this paper we propose the analytical characterization of log rician noise and its propagation to the components of the dt through power series expansions we conclude that even in highly noisy scenarios the bias for log rician signals remains moderate when compared to the corresponding variance yet with the advent of parallel imaging pmri the rician model is not always valid we make our analysis extensive to a number of modern acquisition techniques through the study of a more general non central chi nc model since wls techniques were initially designed bearing in mind rician noise it is not clear whether or not they still apply to pmri an important finding in our work is that the common implementation of wls is nearly optimal when nc noise is considered unfortunately the bias in the estimation becomes far more important in this case to the point that it may nearly overwhelm the variance in given situations furthermore we evidence that such bias cannot be removed by increasing the number of acquired gradient directions a number of experiments have been conducted that corroborate our analytical findings while in vivo data have been used to test the actual relevance of the bias in the estimation

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

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

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

generalized semi infinite programming theory and methods generalized semi infinite optimization problems gsip are considered the difference between gsip and standard semi infinite problems sip is illustrated by examples by applying the reduction ansatz optimality conditions for gsip are derived numerical methods for solving gsip are considered in comparison with methods for sip from a theoretical and a practical point of view it is investigated under which assumptions a gsip can be transformed into an sip

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

estimation of fiber orientation probability density functions in high angular resolution diffusion imaging abstract an estimator of the orientation probability density function opdf of fiber tracts in the white matter of the brain from high angular resolution diffusion data is presented unlike q balls which use the funk radon transform to estimate the radial projection of the 3d probability density function the jacobian of the spherical coordinates is included in the funk radon approximation to the radial integral thus true angular marginalizations are computed which allows a strict probabilistic interpretation extensive experiments with both synthetic and real data show the better capability of our method to characterize complex micro architectures compared to other related approaches q balls and diffusion orientation transform especially for low values of the diffusion weighting parameter

maximum a posteriori estimation of diffusion tensor parameters using a rician noise model why how and but the diffusion tensor is a commonly used model for diffusion weighted mr image data the parameters are typically estimated by ordinary or weighted least squares on log transformed data assuming normal or log normal distribution of measurement errors respectively this may not be adequate when using high b values and or performing high resolution scans resulting in poor snr in which case the difference between the assumed and the true rician noise model becomes important as a consequence the estimated diffusion parameters will be biased underestimating the true diffusion in this paper a computational framework is presented where parameters pertaining to a spectral decomposition of the diffusion tensor are estimated using a rician noise model the parameters are estimated using a fisher scoring scheme which gives robust and rapid convergence it is demonstrated how the fisher information matrix can be used as a generic tool for designing optimal experiments it is shown that the rician noise model leads to significantly less biased estimates for a large range of b values and snr but that the rician estimates have a poorer precision compared to the gaussian model for very low snr by pooling the rician estimates of uncertainty over neighbouring voxel estimates with higher precision but still not as high as with a gaussian model can be obtained we suggest the use of a rician estimator when it is important with truly quantitative values and when comparing different predictive models the higher precision of the gaussian estimates may be more important when the objective is to compare diffusion related parameters over time or across groups

global solution of semi infinite programs optimization problems involving a finite number of decision variables and an infinite number of constraints are referred to as semi infinite programs sips existing numerical methods for solving nonlinear sips make strong assumptions on the properties of the feasible set e g convexity and or regularity or solve a discretized approximation which only guarantees a lower bound to the true solution value of the sip here a general deterministic algorithm for solving semi infinite programs to guaranteed global optimality is presented a branch and bound bb the lower bounding problem is formulated as a convex relaxation of a discretized approximation to the sip the sip b b algorithm is shown to converge finitely to optimality when the subdivision and discretization procedures used to formulate the node subproblems are known to retain certain convergence characteristics other than the properties assumed by globally convergent b b methods for finitely constrained nlps this sip algorithm makes only one additional assumption for every minimizer x of the sip there exists a sequence of slater points xn for which cf section 5 4 numerical results for test problems in the sip literature are presented the exclusion test and a modified upper bounding problem are also investigated as heuristic approaches for alleviating the computational cost of solving a nonlinear sip to guaranteed global optimality

generalized semi infinite programming a tutorial this tutorial presents an introduction to generalized semi infinite programming gsip which in recent years became a vivid field of active research in mathematical programming a gsip problem is characterized by an infinite number of inequality constraints and the corresponding index set depends additionally on the decision variables there exist a wide range of applications which give rise to gsip models some of them are discussed in the present paper furthermore geometric and topological properties of the feasible set and in particular the difference to the standard semi infinite case are analyzed by using first order approximations of the feasible set corresponding constraint qualifications are developed then necessary and sufficient first and second order optimality conditions are presented where directional differentiability properties of the optimal value function of the so called lower level problem are used finally an overview of numerical methods is given

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

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

the falsity of a conjecture concerning the percolation threshold a recent conjecture relating the critical concentration for bond percolation and a lattice green function is shown to be false

a new methodology for the estimation of fiber populations in the white matter of the brain with the funk radon transform the funk radon transform frt is a powerful tool for the estimation of fiber populations with high angular resolution diffusion imaging hardi it is used in q ball imaging qbi and other hardi techniques such as the recent orientation probability density transform opdt to estimate fiber populations with very few restrictions on the diffusion model the frt consists in the integration of the attenuation signal sampled by the mri scanner on the unit sphere along equators orthogonal to the directions of interest it is easily proved that this calculation is equivalent to the integration of the diffusion propagator along such directions although a characteristic blurring with a bessel kernel is introduced under a different point of view the frt can be seen as an efficient way to compute the angular part of the integral of the attenuation signal in the plane orthogonal to each direction of the diffusion propagator in this paper stoke s theorem is used to prove that the frt can in fact be used to compute accurate estimates of the true integrals defining the functions of interest in hardi keeping the diffusion model as little restrictive as possible varying the assumptions on the attenuation signal we derive new estimators of fiber orientations generalizing both q balls and the opdt extensive experiments with both synthetic and real data have been intended to show that the new techniques improve existing ones in many situations

decision rule bounds for two stage stochastic bilevel programs we study two stage stochastic bilevel programs where the leader chooses a binary here and now decision and the follower responds with a continuous wait and see decision using modern decision rule approximations we construct lower bounds on an optimistic version and upper bounds on a pessimistic version of the leader s problem both bounding problems are equivalent to explicit mixed integer linear programs that are amenable to efficient numerical solution the method is illustrated through a facility location problem involving sellers and customers with conflicting preferences

feasible method for semi infinite programs a new numerical method is presented for semi infinite optimization problems which guarantees that each iterate is feasible for the original problem the basic idea is to construct concave relaxations of the lower level problem to compute the optimal values of the relaxation problems explicitly and to solve the resulting approximate problems with finitely many constraints the concave relaxations are constructed by replacing the objective function of the lower level problem by its concave upper bound functions under mild conditions we prove that every accumulation point of the solutions of the approximate problems is an optimal solution of the original problem an adaptive subdivision algorithm is proposed to solve semi infinite optimization problems it is shown that the karush kuhn tucker points of the approximate problems converge to a karush kuhn tucker point of the original problem within arbitrarily given tolerances numerical experiments show that our algorithm is much faster than the existi

relaxation based bounds for semi infinite programs finite formulations are presented for the calculation of lower and upper bounds on the optimal solution value of semi infinite programs sips involving smooth potentially nonconvex objective function and constraints the lower bounding problem is obtained by a formulation that combines the first and second order kkt necessary conditions of the lower level problem with a discretization of the parameter set the resulting mathematical program with equilibrium constraints mpec is a relaxation of the original sip and furnishes valid lower bounds if the parameter set is subdivided the optimal solution value of the lower bounding problem converges to the optimal solution value of the sip the upper bounding problem is based on convex and linear relaxations of the lower level problem resulting in a restriction of the sip if the parameter set is subdivided the constructed relaxations converge to the original lower level program the existence of sip slater points ensures convergence of the upper bounding problems to the optimal solution value of the sip several alternatives for the upper bounding problem are presented and discussed numerical results are presented for a number of test problems from the literature