exact penalty results for mathematical programs with vanishing constraints a mathematical program with vanishing constraints mpvc is a constrained optimization problem arising in certain engineering applications the feasible set has a complicated structure so that the most familiar constraint qualifications are usually violated this in turn implies that standard penalty functions are typically non exact for mpvcs we therefore develop a new mpvc tailored penalty function which is shown to be exact under reasonable assumptions this new penalty function can then be used to derive or recover suitable optimality conditions for mpvcs

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

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

elastic mode algorithms for mathematical programs with equilibrium constraints global convergence and stationarity properties the elastic mode formulation of the problem of minimizing a nonlinear function subject to equilibrium constraints has appealing local properties in that for a finite value of the penalty parameter local solutions satisfying first and second order necessary optimality conditions for the original problem are also first and second order points of the elastic mode formulation here we study global convergence properties of methods based on this formulation which involve generating an exact or inexact first or second order point of the formulation for nondecreasing values of the penalty parameter under certain regularity conditions on the active constraints we establish finite or asymptotic convergence to points having a certain stationarity property such as strong stationarity m stationarity or c stationarity numerical experience with these approaches is discussed in particular our analysis and the numerical evidence show that exact complementarity can be achieved finitely even when the elastic mode formulation is solved inexactly

strong duality and kkt conditions in nonconvex optimization with a single equality constraint and geometric constraint some topological and geometric characterizations of strong duality for a non convex optimization problem under a single equality and geometric constraints are established in particular a hidden convexity of the conic hull of joint range of the pair of functions associated to the original problem is obtained applications to derive a characterization of the validity of kkt conditions without standard constraints qualification are also discussed it goes beyond the exact penalization technique several examples showing our results provide much more information than those appearing elsewhere are given finally the standard quadratic problem involving a non necessarily polyhedral cone is analyzed in detail

stationary conditions for mathematical programs with vanishing constraints using weak constraint qualifications abstract we consider a class of optimization problems that is called a mathematical program with vanishing constraints mpvc for short this class has some similarities to mathematical programs with equilibrium constraints mpecs for short and typically violates standard constraint qualifications hence the well known karush kuhn tucker conditions do not provide necessary optimality criteria in order to obtain reasonable first order conditions under very weak assumptions we introduce several mpvc tailored constraint qualifications discuss their relation and prove an optimality condition which may be viewed as the counterpart of what is called m stationarity in the mpec field

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

exact penalization of mathematical programs with equilibrium constraints we study theoretical and computational aspects of an exact penalization approach to mathematical programs with equilibrium constraints mpecs in the first part we prove that a mangasarian fromovitz type condition ensures the existence of a stable local error bound at the root of a real valued nonnegative piecewise smooth function a specification to nonsmooth formulations of equilibrium constraints e g complementarity conditions or normal equations provides conditions which guarantee the existence of a nonsmooth exact penalty function for mpecs in the second part we study a trust region minimization method for a class of composite nonsmooth functions which comprises exact penalty functions arising from mpecs we prove a global convergence result for the general method and incorporate a penalty update rule a further specification results in an sqp trust region method for mpecs based on an ell 1 penalty function

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

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

stochastic programming approach to optimization under uncertainty in this paper we discuss computational complexity and risk averse approaches to two and multistage stochastic programming problems we argue that two stage say linear stochastic programming problems can be solved with a reasonable accuracy by monte carlo sampling techniques while there are indications that complexity of multistage programs grows fast with increase of the number of stages we discuss an extension of coherent risk measures to a multistage setting and in particular dynamic programming equations for such problems

some nonlinear brascamp lieb inequalities and applications to harmonic analysis abstract we use the method of induction on scales to prove certain diffeomorphism invariant nonlinear brascamp lieb inequalities we provide applications to multilinear convolution inequalities and the restriction theory for the fourier transform extending to higher dimensions recent work of bejenaru herr tataru and bennett carbery wright

the coc algorithm part ii topological geometrical and generalized shape optimization abstract after outlining analytical methods for layout optimization and illustrating them with examples the coc algorithm is applied to the simultaneous optimization of the topology and geometry of trusses with many thousand potential members the numerical results obtained are shown to be in close agreement up to twelve significant digits with analytical results finally the problem of generalized shape optimization finding the best boundary topology and shape is discussed

utility deviation risk portfolio selection we here provide a comprehensive study of the utility deviation risk portfolio selection problem by considering the first order condition for the corresponding objective function we first derive the necessary condition that the optimal terminal wealth satisfying two mild regularity conditions solves for a primitive static problem called the nonlinear moment problem we then illustrate the application of this general necessity result by revisiting the nonexistence of the optimal solution for the mean semivariance problem second we establish an alternative version of the verification theorem serving as the sufficient condition that the solution which satisfies another mild condition different from that for necessity of the nonlinear moment problem is the optimal terminal wealth of the original utility deviation risk portfolio selection problem we then apply this general sufficiency result to revisit the various well posed mean risk problems already known in the literature and to also establish the ex

on the nonlinear brascamp lieb inequality we prove a nonlinear variant of the general brascamp lieb inequality our proof consists of running an efficient or tight induction on scales argument which uses the existence of gaussian near extremizers to the underlying linear brascamp lieb inequality lieb s theorem in a fundamental way a key ingredient is an effective version of lieb s theorem which we establish via a careful analysis of near minimizers of weighted sums of exponential functions instances of this inequality are quite prevalent in mathematics and we illustrate this with some applications in harmonic analysis

on l p resolvent estimates and the density of eigenvalues for compact riemannian manifolds we address an interesting question raised by dos santos ferreira kenig and salo forum math 2014 about regions mathcal r g subset mathbb c for which there can be uniform l frac 2n n 2 to l frac 2n n 2 resolvent estimates for delta g zeta zeta in mathcal r g where delta g is the laplace beltrami operator with metric g on a given compact boundaryless riemannian manifold of dimension n geq 3 this is related to earlier work of kenig ruiz and the third author duke math j 55 329 347 1987 for the euclidean laplacian in which case the region is the entire complex plane minus any disc centered at the origin presently we show that for the round metric on the sphere s n the resolvent estimates in dos santos ferreira et al in forum math 2014 involving a much smaller region are essentially optimal we do this by establishing sharp bounds based on the distance from zeta to the spectrum of delta s n in the other direction we also show that the bounds in dos santos ferreira et al in forum math 2014 can be sharpened logarithmically for manifolds with nonpositive curvature and by powers in the case of the torus mathbb t n mathbb r n mathbb z n with the flat metric the latter improves earlier bounds of shen int math res not 1 1 31 2001 the work of dos santos ferreira et al in forum math 2014 and shen in int math res not 1 1 31 2001 was based on hadamard parametrices for delta g zeta 1 ours is based on the related hadamard parametrices for cos t sqrt delta g and it follows ideas in sogge in ann math 126 439 447 1987 of proving l p multiplier estimates using small time wave equation parametrices and the spectral projection estimates from sogge in j funct anal 77 123 138 1988 this approach allows us to adapt arguments in berard math z 155 249 276 1977 and hlawka monatsh math 54 1 36 1950 to obtain the aforementioned improvements over dos santos ferreira et al in forum math 2014 and shen in int math res not 1 1 31 2001 further improvements for the torus are obtained using recent techniques of the first author bourgain in israel j math 193 1 441 458 2013 and his work with guth bourgain and guth in geom funct anal 21 1239 1295 2011 based on the multilinear estimates of bennett carbery and tao math z 2 261 302 2006 our approach also allows us to give a natural necessary condition for favorable resolvent estimates that is based on a measurement of the density of the spectrum of sqrt delta g and moreover a necessary and sufficient condition based on natural improved spectral projection estimates for shrinking intervals as opposed to those in sogge in j funct anal 77 123 138 1988 for unit length intervals we show that the resolvent estimates are sensitive to clustering within the spectrum which is not surprising given sommerfeld s original conjecture sommerfeld in physikal zeitschr 11 1057 1066 1910 about these operators

optimal dynamic trading strategies with risk limits value at risk var has emerged in recent years as a standard tool to measure and control the risk of trading portfolios yet existing theoretical analysis of the optimal behavior of a trader subject to var limits has produced a negative view of var as a risk control tool in particular var limits have been found to induce increased risk exposure in some states and an increased probability of extreme losses however these conclusions are based on models that are either static or dynamically inconsistent in this paper we formulate a dynamically consistent model of optimal portfolio choice subject to var limits and show that the concerns expressed in earlier papers do not apply if consistently with common practice the var limit is reevaluated dynamically in particular we find that the optimal risk exposure of a trader subject to a var limit is always lower than that of an unconstrained trader and that the probability of extreme losses is also lower we also consider risk limits formulated in terms of tail conditional expectation tce a coherent risk measure often advocated as an alternative to var and show that in our dynamic setting it is always possible to transform a tce limit into an equivalent var limit and conversely

on the cone multiplier in r3 abstract we prove the sharp l 3 bounds for the cone multiplier in r 3 and the associated square function which is known as mockenhaupt s square function

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

constraint qualifications and kkt conditions for bilevel programming problems in this paper we consider the bilevel programming problem blpp which is a sequence of two optimization problems where the constraint region of the upper level problem is determined implicitly by the solution set to the lower level problem we extend well known constraint qualifications for nonlinear programming problems such as the abadie constraint qualification the kuhn tucker constraint qualification the zangwill constraint qualification the arrow hurwicz uzawa constraint qualification and the weak reverse convex constraint qualification to blpps and derive a karash kuhn tucker kkt type necessary optimality condition under these constraint qualifications without assuming the lower level problem satisfying the mangasarian fromovitz constraint qualification relationships among various constraint qualifications are also given

characterizing fj and kkt conditions in nonconvex mathematical programming with applications in this paper we analyze the fritz john and karush kuhn tucker kkt conditions for a g teaux differentiable nonconvex optimization problem with inequality constraints and a geometric constraint set the fritz john condition is characterized in terms of an alternative theorem which covers beyond standard situations while characterizations of kkt conditions without assuming constraints qualifications are related to strong duality of a suitable linear approximation of the given problem and the properties of its associated image mapping such characterizations are suitable for dealing with some problems in structural optimization where most of the known constraint qualifications fail in particular several examples are given showing the usefulness and optimality in a certain sense of our results which provide much more information than those including the mordukhovich normal cone or clarke s appearing elsewhere the case with a single inequality constraint is discussed in detail by establishing

lipschitz and holder stability of optimization problems and generalized equations this paper studies stability aspects of solutions of parametric mathematical programs and generalized equations respectively with disjunctive constraints we present sufficient conditions that under some constraint qualifications ensuring metric subregularity of the constraint mapping continuity results of upper lipschitz and upper holder type respectively hold furthermore we apply the above results to parametric mathematical programs with equilibrium constraints and demonstrate how some classical results for the nonlinear programming problem can be recovered and even improved by our theory

on the multilinear restriction and kakeya conjectures we prove d linear analogues of the classical restriction and kakeya conjectures in rd our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians closely related to heat flow we conclude by giving some applications to the corresponding variable coefficient problems and the so called joints problem as well as presenting some n linear analogues for n d

a bundle free implicit programming approach for a class of elliptic mpecs in function space using a standard first order optimality condition for nonsmooth optimization problems a general framework for a descent method is developed this setting is applied to a class of mathematical programs with equilibrium constraints in function space from which a new algorithm is derived global convergence of the algorithm is demonstrated in function space and the results are then illustrated by numerical experiments

mathematical programs with vanishing constraints optimality conditions and constraint qualifications we consider a difficult class of optimization problems that we call a mathematical program with vanishing constraints problems of this kind arise in various applications including optimal topology design problems of mechanical structures we show that some standard constraint qualifications like licq and mfcq usually do not hold at a local minimum of our program whereas the abadie constraint qualification is sometimes satisfied we also introduce a suitable modification of the standard abadie constraint qualification as well as a corresponding optimality condition and show that this modified constraint qualification holds under fairly mild assumptions we also discuss the relation between our class of optimization problems with vanishing constraints and a mathematical program with equilibrium constraints

dynamic portfolio choice when risk is measured by weighted var we seek to characterize the trading behavior of an agent in the context of a continuous time portfolio choice model if she measures the risk by a so called weighted value at risk var which is a generalization of both var and conditional var we show that when bankruptcy is allowed the agent displays extreme risk taking behaviors unless the downside risk is significantly penalized in which case an asymptotically optimal strategy is to invest a very small amount of money in an extremely risky but highly rewarding lottery and save the rest in the risk free asset when bankruptcy is prohibited extreme risk taking behaviors are prevented in most cases in which the asymptotically optimal strategy is to spend a very small amount of money in an extremely risky but highly rewarding lottery and put the rest in an asset with moderate risk finally we show that the trading behaviors remain qualitatively the same if the weighted var is replaced by a law invariant coherent risk measure

on integer solutions of parsell vinogradov systems we prove a sharp upper bound on the number of integer solutions of the parsell vinogradov system in every dimension d ge 2

stationarity conditions and constraint qualifications for mathematical programs with switching constraints in optimal control switching structures demanding at most one control to be active at any time instance appear frequently discretizing such problems a so called mathematical program with switching constraints is obtained although these problems are related to other types of disjunctive programs like optimization problems with complementarity or vanishing constraints their inherent structure makes a separate consideration necessary since standard constraint qualifications are likely to fail at the feasible points of switching constrained optimization problems stationarity notions which are weaker than the associated karush kuhn tucker conditions need to be investigated in order to find applicable necessary optimality conditions furthermore appropriately tailored constraint qualifications need to be formulated in this paper we introduce suitable notions of weak mordukhovich and strong stationarity for mathematical programs with switching constraints and present some associated constraint qualifications our findings are exploited to state necessary optimality conditions for discretized optimal control problems with switching constraints furthermore we apply our results to optimization problems with either or constraints first a novel reformulation of such problems using switching constraints is presented second the derived surrogate problem is exploited to obtain necessary optimality conditions for the original program

the proof of the l 2 decoupling conjecture we prove the l 2 decoupling conjecture for compact hypersurfaces with positive denite second fundamental form and also for the cone this has a wide range of important consequences one of them is the validity of the discrete restriction conjecture which implies the full range of expected l p strichartz estimates for both the rational and up to n losses the irrational torus another one is an improvement in the range for the discrete restriction theorem for lattice points on the sphere various applications to additive combinatorics incidence geometry and number theory are also discussed our argument relies on the interplay between linear and multilinear restriction theory

how stringent is the linear independence assumption for mathematical programs with complementarity constraints the linear independence constraint qualifications licq plays an important role in the analysis of mathematical programs with complementarity constraints mpccs and is a vital ingredient to convergence analyses of sqp type or smoothing methods cf e g fukushima and pang 1999 luo et al 1996 scholtes and sta hr 1999 scholtes 2001 sta hr 2000 we will argue in this paper that licq is not a particularly stringent assumption for mpccs our arguments are based on an extension of jongen s 1977 genericity analysis to mpccs his definitions of nondegenerate critical points and regular programs extend naturally to mpccs and his genericity results generalize straightforwardly to mpccs in standard form an extension is not as straightforward for mpccs with the particular structure induced by lower level stationarity conditions for variational inequalities or optimization problems we show that licq remains a generic property for this class of mpccs

a mixed integer programming for robust truss topology optimization with stress constraints this paper presents a mixed integer programming mip formulation for robust topology optimization of trusses subjected to the stress constraints under the uncertain load a design dependent uncertainty model of the external load is proposed for dealing with the variation of truss topology in the course of optimization for a truss with the discrete member cross sectional areas it is shown that the robust topology optimization problem can be reduced to an mip problem which is solved globally numerical examples illustrate that the robust optimal topology of a truss depends on the magnitude of uncertainty

theoretical and numerical comparison of relaxation methods for mathematical programs with complementarity constraints mathematical programs with equilibrium constraints mpecs are difficult optimization problems whose feasible sets do not satisfy most of the standard constraint qualifications hence mpecs cause difficulties both from a theoretical and a numerical point of view as a consequence a number of mpec tailored solution methods have been suggested during the last decade which are known to converge under suitable assumptions among these mpec tailored solution schemes the relaxation methods are certainly one of the most prominent class of solution methods several different relaxation schemes are available in the meantime and the aim of this paper is to provide a theoretical and numerical comparison of these schemes more precisely in the theoretical part we improve the convergence theorems of several existing relaxation methods there we also take a closer look at the properties of the feasible sets of the relaxed problems and show which standard constraint qualifications are satisfied for these relaxed problems finally the numerical comparison is based on the macmpec test problem collection

local stress constrained and slope constrained sand topology optimisation summary we study the alternative simultaneous analysis and design sand formulation of the local stress and slope constrained topology design problem it is demonstrated that a standard trust region lagrange newton sequential quadratic programming sqp type algorithm based in this case on strictly convex and separable approximate subproblems may converge to singular optima of the local stress constrained problem without having to resort to relaxation or perturbation techniques moreover due to the negation of the sensitivity analyses in sand the density and displacement variables are independent and the immense sparsity of the sand problem solutions to large scale problem instances may be obtained in a reasonable amount of computation time this article is protected by copyright all rights reserved

on a reverse form of the brascamp lieb inequality we prove a reverse form of the multidimensional brascamp lieb inequality our method also gives a new way to derive the brascamp lieb inequality and is rather convenient for the study of equality cases

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

on using the elastic mode in nonlinear programming approaches to mathematical programs with complementarity constraints we investigate the possibility of solving mathematical programs with complementarity constraints mpccs using algorithms and procedures of smooth nonlinear programming although mpccs do not satisfy a constraint qualification we establish sufficient conditions for their lagrange multiplier set to be nonempty mpccs that have nonempty lagrange multiplier sets and satisfy the quadratic growth condition can be approached by the elastic mode with a bounded penalty parameter in this context the elastic mode transforms mpcc into a nonlinear program with additional variables that has an isolated stationary point and local minimum at the solution of the original problem which in turn makes it approachable by sequential quadratic programming sqp algorithms one such algorithm is shown to achieve local linear convergence once the problem is relaxed under stronger conditions we also prove superlinear convergence to the solution of an mpcc using an adaptive elastic mode approach for an sqp algorithm recently analyzed in an mpcc context in r fletcher s leyffer s sholtes and d ralph local convergence of sqp methods for mathematical programs with equilibrium constraints tech report na 210 university of dundee dundee uk 2002 our assumptions are more general since we do not use a critical assumption from that reference in addition we show that the elastic parameter update rule will not interfere locally with the superlinear convergence once the penalty parameter is appropriately chosen

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

the c index a new stability concept for quadratic programs with complementarity constraints we introduce nondegeneracy and the c index for c stationary points of a qpcc that is for a mathematical program with a quadratic objective function and linear complementarity constraints the c index characterizes the qualitative local behavior of a qpcc around a nondegenerate c stationary point the article focuses on the structure of the c stationary set of qpccs depending on a real parameter we show that for generic qpcc data the c index changes exactly at turning points of the c stationary set and that it changes by exactly one to illustrate this concept we introduce and analyze two homotopy methods for finding c stationary points numerical results illustrate that for randomly generated test problems the two homotopy methods very often identify b stationary points

convergence analysis of an augmented lagrangian method for mathematical programs with complementarity constraints abstract in this paper a mathematical program with complementarity constraints mpcc is reformulated as a nonsmooth constrained optimization problem by using the fischer burmeister function an augmented proximal lagrangian method is applied to tackle the resulting constrained optimization problem the augmented lagrangian problems are in general nonsmooth we derive first and second order optimality conditions for the augmented lagrangian problems using an approximate smooth variational principle and establish that the limit point of a sequence of points that satisfy the second order necessary optimality conditions of the augmented lagrangian problems is a strongly stationary point of the original mpcc if the limit point is feasible to mpcc and the linear independence constraint qualification for mpcc and the upper level strict complementarity condition hold at the limit point

optimal design and relaxation of variational problems iii abstract this paper has attempted a synthesis of three subjects that have developed separately optimal design homogenization and relaxation in the calculus of variations the underlying theme is that of oscillations they occur with certain definite patterns in minimizing sequences for the variational problem the same patterns recur in nearly optimal designs and in the coefficients of the equation being homogenized each problem is resolved by determining the structure of those oscillations and above all their macroscopic averages it is those averages that solve the relaxed problem they also determine the optimal design and the solution of the homogenized equation we emphasize also the role of optimization the homogenized equations and composite materials of interest here come not from arbitrary mixtures but from those that are extremal since they correspond to minimizing sequences for the variational problem the effective properties of these composites attain the optimal bounds for given materials in fixed proportions

sensitivity analysis for cone constrained optimization problems under the relaxed constraint qualifications we present the local sensitivity analysis for cone constrained optimization problems under the cq type conditions significantly weaker than those traditionally used in this context our basic sensitivity results are established under the first or second order sufficient optimality conditions combined with the estimate of the distance to the feasible set of the perturbed problem we demonstrate how such an estimate can be obtained under the assumptions weaker than robinson s cq and establish the corresponding sensitivity results finally we apply our results to sensitivity analysis and relaxation schemes for mathematical programs with complementarity constraints

a bilinear approach to the restriction and kakeya conjectures the purpose of this paper is to investigate bilinear variants of the restriction and kakeya conjectures to relate them to the standard formulations of these conjectures and to give applications of this bilinear approach to existing conjectures the methods used are based on several observations and results of bourgain see 2 6 together with some refinements by moyua vargas and vega 17 18 this paper is organized as follows in the first section we discuss bilinear restriction estimates and show how one can pass back and forth between these estimates and the standard restriction estimates we also generalize the 12 7 bilinear restriction estimate of 18 to higher dimensions in the second section we give analogues of the above results for the kakeya operator in particular we give a bilinear improvement to wolff s kakeya theorem in arbitrary dimension in the third section we give applications of these bilinear estimates in three dimensions for example we are able to improve the 42 11 exponent in wolff s restriction theorem to 34 9 we are also able to prove a sharp lp lq restriction theorem which improves on the classical l2 l4 tomas stein theorem and also give some concrete progress on a bilinear restriction conjecture of klainerman and machedon we also give a non bilinear approach to these estimates which gives weaker results but is more direct and probably has a wider range of application

the endpoint case of the bennett carbery tao multilinear kakeya conjecture we prove the endpoint case of the multilinear kakeya conjecture of bennett carbery and tao the proof uses the polynomial method introduced by dvir

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

gaussian kernels have only gaussian maximizers a gaussian integral kernel g x y on rn x rn is the exponential of a quadratic form in x and y the fourier transform kernel is an example the problem addressed here is to find the sharp bound of g as an operator from l p rp to l q rn and to prove that the l p rn functions that saturate the bound are necessarily gaussians this is accomplished generally for 1 q in some special cases besides greatly extending previous results in this area the proof technique is also essentially different from earlier ones a corollary of these results is a fully multidimensional multilinear generalization of young s inequality

decoupling for moment manifolds associated to arkhipov chubarikov karatsuba systems abstract we prove l p l p decoupling inequalities for a class of moment manifolds these inequalities imply optimal mean value estimates for multidimensional weyl sums of the kind considered by arkhipov chubarikov and karatsuba and by parsell in our proofs we take a new point of view on the bourgain demeter guth induction on scales argument this point of view substantially simplifies even the proof of l 2 l p decoupling for the moment curve

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

a note on the sample average approximation method for stochastic mathematical programs with complementarity constraints abstract meng and xu 2006 3 proposed a sample average approximation saa method for solving a class of stochastic mathematical programs with complementarity constraints smpccs after showing that under some moderate conditions a sequence of weak stationary points of saa problems converge to a weak stationary point of the original smpcc with probability approaching one at exponential rate as the sample size tends to infinity the authors proposed an open question that is whether similar results can be obtained under some relatively weaker conditions in this paper we try to answer the open question based on the reformulation of stationary condition of mpccs and new stability results on generalized equations we present a similar convergence theory without any information of second order derivative and strict complementarity conditions moreover we carry out convergence analysis of the regularized saa method proposed by meng and xu 2006 3 where the convergence results have not been considered

an implementable active set algorithm for computing a b stationary point of a mathematical program with linear complementarity constraints erratum in m fukushima and p tseng siam j optim 12 2002 pp 724 739 an active set algorithm was proposed for solving a mathematical program with a smooth objective function and linear inequality complementarity constraints it is asserted therein that under a uniform licq on the feasible set this algorithm generates iterates whose cluster points are b stationary points of the problem however the proof has a gap and shows only that each cluster point is an m stationary point we discuss this gap and show that b stationarity can be achieved if the algorithm is modified and an additional error bound condition holds

convolutions of singular measures and applications to the zakharov system abstract uniform l 2 estimates for the convolution of singular measures with respect to transversal submanifolds are proved in arbitrary space dimension the results of bennett bez are used to extend previous work of bejenaru herr tataru as an application it is shown that the 3d zakharov system is locally well posed in the full subcritical regime

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

three modeling paradigms in mathematical programming celebrating the sixtieth anniversary since the zeroth international symposium on mathematical programming was held in 1949 this paper discusses several promising paradigms in mathematical programming that have gained momentum in recent years but have yet to reach the main stream of the field these are competition dynamics and hierarchy the discussion emphasizes the interplay between these paradigms and their connections with existing subfields including disjunctive equilibrium and nonlinear programming and variational inequalities we will describe the modeling approaches mathematical formulations and recent results of these paradigms and sketch some open mathematical and computational challenges arising from the resulting optimization and equilibrium problems our goal is to elucidate the need for a systematic study of these problems and to inspire new research in the field

stability analysis of two stage stochastic mathematical programs with complementarity constraints via nlp regularization this paper presents numerical approximation schemes for a two stage stochastic programming problem where the second stage problem has a general nonlinear complementarity constraint first the complementarity constraint is approximated by a parameterized system of inequalities with a well known regularization approach by scholtes siam j optim 11 2001 pp 918 936 in deterministic mathematical programs with equilibrium constraints the distribution of the random variables of the regularized two stage stochastic program is then approximated by a sequence of probability measures by treating the approximation problems as a perturbation of the original true problem we carry out a detailed stability analysis of the approximated problems including continuity and local lipschitz continuity of optimal value functions and outer semicontinuity and continuity of the set of optimal solutions and stationary points a particular focus is given to the case where the probability distribution is approximated by

structural properties of affine sparsity constraints we introduce a new constraint system for sparse variable selection in statistical learning such a system arises when there are logical conditions on the sparsity of certain unknown model parameters that need to be incorporated into their selection process formally extending a cardinality constraint an affine sparsity constraint asc is defined by a linear inequality with two sets of variables one set of continuous variables and the other set represented by their nonzero patterns this paper aims to study an asc system consisting of finitely many affine sparsity constraints we investigate a number of fundamental structural properties of the solution set of such a non standard system of inequalities including its closedness and the description of its closure continuous approximations and their set convergence and characterizations of its tangent cones for use in optimization based on the obtained structural properties of an asc system we investigate the convergence of b ouligand stationary solutions when the asc is approximated by surrogates of the step ell 0 function commonly employed in sparsity representation our study lays a solid mathematical foundation for solving optimization problems involving these affine sparsity constraints through their continuous approximations

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

on the time transformation of mixed integer optimal control problems using a consistent fixed integer control function nonlinear control systems with instantly changing dynamical behavior can be modeled by introducing an additional control function that is integer valued in contrast to a control function that is allowed to have continuous values the discretization of a mixed integer optimal control problem miocp leads to a non differentiable optimization problem and the non differentiability is caused by the integer values the paper is about a time transformation method that is used to transform a miocp with integer dependent constraints into an ordinary optimal control problem differentiability is achieved by replacing a variable integer control function with a fixed integer control function and a variable time allows to change the sequence of active integer values in contrast to other contributions so called control consistent fixed integer control functions are taken into account here it is shown that these control consistent fixed integer control functions allow a better accuracy in the resulting trajectories in particular in the computed switching times the method is verified on analytical and numerical examples

solving bilevel programs with the kkt approach bilevel programs bl form a special class of optimization problems they appear in many models in economics game theory and mathematical physics bl programs show a more complicated structure than standard finite problems we study the so called kkt approach for solving bilevel problems where the lower level minimality condition is replaced by the kkt or the fj condition this leads to a special structured mathematical program with complementarity constraints we analyze the kkt approach from a generic viewpoint and reveal the advantages and possible drawbacks of this approach for solving bl problems numerically

combining the regularization strategy and the sqp to solve mpcc a matlab implementation mathematical program with complementarity constraints mpcc plays a very important role in many fields such as engineering design economic equilibrium multilevel games and mathematical programming theory itself in theory its constraints fail to satisfy a standard constraint qualification such as the linear independence constraint qualification licq or the mangasarian fromovitz constraint qualification mfcq at any feasible point as a result the developed nonlinear programming theory may not be applied to mpcc class directly nowadays a natural and popular approach is trying to find some suitable approximations of an mpcc so that it can be solved by solving a sequence of nonlinear programs this work aims to solve the mpcc using nonlinear programming techniques namely the sqp and the regularization scheme some algorithms with two iterative processes the inner and the external were developed a set of ampl problems from macmpec database leyffer 2000 8 were tested the comparative analysis regarding performance of algorithms was carried out