bidual renormings of banach spaces abstract given a separable banach space x with no isomorphic copies of l 1 and a separable subspace y of its bidual we provide a sufficient condition on y to ensure that x admits an equivalent norm such that the restriction to y of the corresponding bidual norm is midpoint locally uniformly rotund this result applies to the separable subspaces of the bidual of a banach space with a shrinking unconditional schauder basis and to the bidual of the james space

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

a lasso for hierarchical interactions we add a set of convex constraints to the lasso to produce sparse interaction models that honor the hierarchy restriction that an interaction only be included in a model if one or both variables are marginally important we give a precise characterization of the effect of this hierarchy constraint prove that hierarchy holds with probability one and derive an unbiased estimate for the degrees of freedom of our estimator a bound on this estimate reveals the amount of fitting saved by the hierarchy constraint we distinguish between parameter sparsity the number of nonzero coefficients and practical sparsity the number of raw variables one must measure to make a new prediction hierarchy focuses on the latter which is more closely tied to important data collection concerns such as cost time and effort we develop an algorithm available in the r package hiernet and perform an empirical study of our 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

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 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

computing b stationary points of nonsmooth dc programs motivated by a class of applied problems arising from physical layer based security in a digital communication system in particular by a secrecy sum rate maximization problem this paper studies a nonsmooth difference of convex dc minimization problem the contributions of this paper are i clarify several kinds of stationary solutions and their relations ii develop and establish the convergence of a novel algorithm for computing a d stationary solution of a problem with a convex feasible set that is arguably the sharpest kind among the various stationary solutions iii extend the algorithm in several directions including a randomized choice of the subproblems that could help the practical convergence of the algorithm a distributed penalty approach for problems whose objective functions are sums of dc functions and problems with a specially structured nonconvex dc constraint for the latter class of problems a pointwise slater constraint qualification is introduced that facilitates the ver

a class of logic problems solvable by linear programming in propositional logic several problems such as satisfiability max sat and logical inference can be formulated as integer programs in this paper we consider sets of clauses for which the corresponding integer programs can be solved as linear programs we prove that balanced sets of clauses have this property

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

mathematical programs with complementarity constraints we study mathematical programs with complementarity constraints several stationarity concepts based on a piecewise smooth formulation are presented and compared the concepts are related to stat

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

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

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

mathematical programs with cardinality constraints reformulation by complementarity type conditions and a regularization method optimization problems with cardinality constraints are very difficult mathematical programs which are typically solved by global techniques from discrete optimization here we introduce a mixed integer formulation whose standard relaxation still has the same solutions in the sense of global minima as the underlying cardinality constrained problem the relation between the local minima is also discussed in detail since our reformulation is a minimization problem in continuous variables it allows us to apply ideas from that field to cardinality constrained problems here in particular we therefore also derive suitable stationarity conditions and suggest an appropriate regularization method for the solution of optimization problems with cardinality constraints this regularization method is shown to be globally convergent to a mordukhovich stationary point extensive numerical results are given to illustrate the behavior of this method

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

a study of the difference of convex approach for solving linear programs with complementarity constraints this paper studies the difference of convex dc penalty formulations and the associated difference of convex algorithm dca for computing stationary solutions of linear programs with complementarity constraints lpccs we focus on three such formulations and establish connections between their stationary solutions and those of the lpcc improvements of the dca are proposed to remedy some drawbacks in a straightforward adaptation of the dca to these formulations extensive numerical results including comparisons with an existing nonlinear programming solver and the mixed integer formulation are presented to elucidate the effectiveness of the overall dc approach

stability theory for absolutely continuous parts of commutative m tuples of self adjoint operators in complex hilbert spaces abstract by the method of direction wave operators we prove that absolutely continuous parts of commutative m tuples of self adjoint operators in complex hilbert spaces are invariant up to unitary equivalence under perturbations of class b p for p m m 2

a sequential algorithm for solving nonlinear optimization problems with chance constraints an algorithm is presented for solving nonlinear optimization problems with chance constraints i e those in which a constraint involving an uncertain parameter must be satisfied with at least a minimum probability in particular the algorithm is designed to solve cardinality constrained nonlinear optimization problems that arise in sample average approximations of chance constrained problems as well as in other applications in which it is only desired to enforce a minimum number of constraints the algorithm employs a novel exact penalty function which is minimized sequentially by solving quadratic optimization subproblems with linear cardinality constraints properties of minimizers of the penalty function in relation to minimizers of the corresponding nonlinear optimization problem are presented and convergence of the proposed algorithm to stationarity with respect to the penalty function is proved the effectiveness of the algorithm is demonstrated through numerical experiments with a nonlinear ca

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

computational study of a family of mixed integer quadratic programming problems we present computational experience with a branch and cut algorithm to solve quadratic programming problems where there is an upper bound on the number of positive variables such problems arise in financial applications the algorithm solves the largest real life problems in a few minutes of run time

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

invariance of essential spectra in the present paper we investigate the invariance of essential spectra of selfadjoint operators the following theorem is well known 5 xiii 14 if t and t are self adjoint operators and i t 1 i t 1 is compact then the essential spectra of t and t coincide theorem 13 in 3 states that the compactness of i t l i t 1 is equivalent to the following condition where e e denotes the spectral measure of t t t e j e j e j te j is compact for all bounded intervals j in ir and t t are mutually subordinate see 1 4 in sect 2 we show that a slightly weaker condition is still sufficient to ensure the coincidence of the essential spectra of t and t to illustrate the scope of applicability of our result we consider schr6dinger operators with strongly singular positive potentials in sect 3

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

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

some nonlinear maps and renormings of banach spaces we consider two classes of nonlinear maps between normed spaces which are relevant for the study of banach spaces that admit equivalent locally uniformly rotund and midpoint locally uniformly rotund norms it turns out to be a useful characterization of such maps in terms of optimization topology and probability using these characterizations we obtain some renorming results