singular kahler einstein metrics we study degenerate complex monge ampere equations of the form omega dd c f n e t f mu where omega is a big semi kahler form on a compact kahler manifold x of dimension n t in r and mu f omega n is a positive measure with density f in l p x omega n p 1 we prove the existence and unicity of continuous o plurisubharmonic solutions in case x is projective and omega psi omega where psi x to v is a proper birational morphism to a normal projective variety omega in ns r v is an ample class and mu has only algebraic singularities we prove that the solution is smooth in the regular locus of the equation we use these results to construct singular kahler einstein metrics of non positive curvature on projective klt pairs in particular on canonical models of algebraic varieties of general type

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

solving stochastic mathematical programs with equilibrium constraints via approximation and smoothing implicit programming with penalization in this paper we consider the stochastic mathematical programs with linear complementarity constraints which include two kinds of models called here and now and lower level wait and see problems we present a combined smoothing implicit programming and penalty method for the problems with a finite sample space then we suggest a quasi monte carlo approximation method for solving a problem with continuous random variables a comprehensive convergence theory is included as well we further report numerical results with the so called picnic vender decision problem

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

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

an interior proximal method in vector optimization abstract this paper studies the vector optimization problem of finding weakly efficient points for maps from r n to r m with respect to the partial order induced by a closed convex and pointed cone c r m with nonempty interior we develop for this problem an extension of the proximal point method for scalar valued convex optimization problem with a modified convergence sensing condition that allows us to construct an interior proximal method for solving vop on nonpolyhedral set

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

a criterion for the properness of the k energy in a general kahler class in this paper we give a criterion for the properness of the k energy in a general kahler class of a compact kahler manifold by using song weinkove s result in commun pure appl math 61 2 210 229 2008 as applications we give some kahler classes on mathbb c mathbb p 2 3 overline mathbb c mathbb p 2 and mathbb c mathbb p 2 8 overline mathbb c mathbb p 2 in which the k energy is proper finally we prove song weinkove s result on the existence of critical points of hat j functional by the continuity method

nonmonotone gradient methods for vector optimization with a portfolio optimization application abstract this paper proposes two nonmonotone gradient algorithms for a class of vector optimization problems with a c convex objective function we establish both the global and local convergence results for the new algorithms we then apply the new algorithms to a portfolio optimization problem under multi criteria considerations

proximal methods in vector optimization we consider the vector optimization problem of finding weakly efficient points for maps from a hilbert space x to a banach space y with respect to the partial order induced by a closed convex and pointed cone c subset y with a nonempty interior we develop for this problem an extension of the proximal point method for scalar valued convex optimization in this extension the subproblems consist of finding weakly efficient points for suitable regularizations of the original map we present both an exact and an inexact version in which the subproblems are solved only approximately within a constant relative tolerance in both cases we prove weak convergence of the generated sequence to a weakly efficient point assuming convexity of the map with respect to c and c completeness of the initial section in cases where this last assumption fails we still establish that the generating sequence is in a suitable sense a minimizing one we also exhibit a particular instance of the algorithm for which under a mild hypothesis on c the weak limit of the generated sequence is an efficient rather than a weakly efficient point

stochastic mathematical programs with hybrid equilibrium constraints this paper considers a stochastic mathematical program with hybrid equilibrium constraints smphec which includes either here and now or wait and see type complementarity constraints an example is given to describe the necessity to study smphec in order to solve the problem the sampling average approximation techniques are employed to approximate the expectations and smoothing and penalty techniques are used to deal with the complementarity constraints limiting behaviors of the proposed approach are discussed preliminary numerical experiments show that the proposed approach is applicable

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

convexity of the k energy on the space of kahler metrics and uniqueness of extremal metrics we establish the convexity of mabuchi s k energy functional along weak geodesics in the space of kahler potentials on a compact kahler manifold thus confirming a conjecture of chen and give some applications in kahler geometry including a proof of the uniqueness of constant scalar curvature metrics or more generally extremal metrics modulo automorphisms the key ingredient is a new local positivity property of weak solutions to the homogeneous monge ampere equation on a product domain whose proof uses plurisubharmonic variation of bergman kernels

an approximation scheme for a class of risk averse stochastic equilibrium problems we consider two models for stochastic equilibrium one based on the variational equilibrium of a generalized nash game and the other on the mixed complementarity formulation each agent in the market solves a single stage risk averse optimization problem with both here and now investment variables and production wait and see variables a shared constraint couples almost surely the wait and see decisions of all the agents an important characteristic of our approach is that the agents hedge risk in the objective functions on costs or profits of their optimization problems which has a clear economic interpretation this feature is obviously desirable but in the risk averse case it leads to variational inequalities with set valued operators a class of problems for which no established software is currently available to overcome this difficulty we define a sequence of approximating differentiable variational inequalities based on smoothing the nonsmooth risk measure in the agents problems such as average or conditional value at risk the smoothed variational inequalities can be tackled by the path solver for example the approximation scheme is shown to converge including the case when smoothed problems are solved approximately an interesting by product of our proposal is that the smoothing approach allows us to show existence of an equilibrium for the original problem to assess the proposed technique numerical results are presented the first set of experiments is on randomly generated equilibrium problems for which we compare the proposed methodology with the standard smooth reformulation of average value at risk minimization using additional variables to rewrite the associated max function the second set of experiments deals with a part of the real life european gas network for which dantzig wolfe decomposition can be combined with the smoothing approach

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

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

the space of kahler metrics this paper the second of a series deals with the function space mathcal h of all smooth kahler metrics in any given n dimensional closed complex manifold v these metrics being restricted to a given fixed real cohomology class called a polarization of v this function space is equipped with a pre hilbert metric structure introduced by t mabuchi 10 who also showed that formally this metric has nonpositive curvature in the first paper of this series 4 the second author showed that the same space is a path length space he also proved that mathcal h is geodesically convex in the sense that for any two points of mathcal h there is a unique geodesic path joining them which is always length minimizing and of class c1 1 this partially verifies two conjectures of donaldson 8 on the subject in the present paper we show first of all that the space is as expected a path length space of nonpositive curvature in the sense of a d aleksandrov a second result is related to the theory of extremal kahler metrics namely that the gradient flow in mathcal h of the k energy of v has the property that it strictly decreases the length of all paths in mathcal h except those induced by one parameter families of holomorphic automorphisms of m

entropic approximation for mathematical programs with robust equilibrium constraints in this paper we consider a class of mathematical programs with robust equilibrium constraints represented by a system of semi infinite complementarity constraints sicc we propose a numerical scheme for tackling sicc specifically by relaxing the complementarity constraints and then randomizing the index set of sicc we employ the well known entropic risk measure to approximate the semi infinite constraints with a finite number of stochastic inequality constraints under some moderate conditions we quantify the approximation in terms of the feasible set and the optimal value the approximation scheme is then applied to a class of two stage stochastic mathematical programs with complementarity constraints in combination with the polynomial decision rules finally we extend the discussion to a mathematical program with distributionally robust equilibrium constraints which is essentially a one stage stochastic program with semi infinite stochastic constraints indexed by some probability measures from

approximate proximal methods in vector optimization abstract this paper studies the vector optimization problem of finding weakly efficient points for mappings in a banach space y with respect to the partial order induced by a closed convex and pointed cone c y with a nonempty interior the proximal method in vector optimization is extended to develop an approximate proximal method for this problem by virtue of the approximate proximal point method for finding a root of a maximal monotone operator in this approximate proximal method the subproblems consist of finding weakly efficient points for suitable regularizations of the original mapping we present both an absolute and a relative version in which the subproblems are solved only approximately weak convergence of the generated sequence to a weak efficient point is established in addition we also discuss an extension to bregman function based proximal algorithms for finding weakly efficient points for mappings

scalar curvature and stability of toric varieties we define a stability condition for a polarised algebraic variety and state a conjecture relating this to the existence of a kahler metric of constant scalar curvature the main result of the paper goes some way towards verifying this conjecture in the case of toric surfaces we prove that under the stability hypothesis the mabuchi functional is bounded below on invariant metrics and that minimising sequences have a certain convergence property in the reverse direction we give new examples of polarised surfaces which do not admit metrics of constant scalar curvature the proofs use a general framework developed by guillemin and abreu in which invariant kahler metrics correspond to convex functions on the moment polytope of a toric variety this paper is a step towards the solution of the general problem of finding conditions under which a complex projective variety admits a kahler metric of constant scalar curvature the pattern of the answer one expects is that this differential geometric condition should be equivalent to some notion of stability in the sense of geometric invariant theory this expectation is probably now an item of folklore going back to suggestions put forward by yau in the case of kahlereinstein metrics and the many results of tian and others in this case reinforced by a detailed formal picture which makes clear the analogy with the well established relation between the stability of vector bundles and yang mills connections 5 here we begin the investigation of

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

an outer approximation algorithm for solving general convex programs a new outer approximation algorithm is proposed for solving general convex programs a remarkable advantage of the algorithm over existing outer approximation methods is that the approximation of the constraint set is not cumulative that is the algorithm solves at each iteration a quadratic program whose constraints depend only on the current estimate of an optimal solution convergence of the algorithm is proved and possible applications are discussed

uniform exponential convergence of sample average random functions under general sampling with applications in stochastic programming abstract sample average approximation saa is one of the most popular methods for solving stochastic optimization and equilibrium problems research on saa has been mostly focused on the case when sampling is independent and identically distributed iid with exceptions dai et al 2000 9 homem de mello 2008 16 in this paper we study saa with general sampling including iid sampling and non iid sampling for solving nonsmooth stochastic optimization problems stochastic nash equilibrium problems and stochastic generalized equations to this end we first derive the uniform exponential convergence of the sample average of a class of lower semicontinuous random functions and then apply it to a nonsmooth stochastic minimization problem exponential convergence of estimators of both optimal solutions and m stationary points characterized by mordukhovich limiting subgradients mordukhovich 2006 23 rockafellar and wets 1998 32 are established under mild conditions we also use the unform convergence result to establish the exponential rate of convergence of statistical estimators of a stochastic nash equilibrium problem and estimators of the solutions to a stochastic generalized equation problem

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

on the projected subgradient method for nonsmooth convex optimization in a hilbert space we consider the method for constrained convex optimization in a hilbert space consisting of a step in the direction opposite to anek subgradient of the objective at a current iterate followed by an orthogonal projection onto the feasible set the normalized stepsizesek are exogenously given satisfying k 0 k k 0 k2 0 we prove that the sequence generated in this way is weakly convergent to a minimizer if the problem has solutions and is unbounded otherwise among the features of our convergence analysis we mention that it covers the nonsmooth case in the sense that we make no assumption of differentiability off and much less of lipschitz continuity of its gradient also we prove weak convergence of the whole sequence rather than just boundedness of the sequence and optimality of its weak accumulation points thus improving over all previously known convergence results we present also convergence rate results 1998 the mathematical programming society inc published by elsevier science b v

a combined relaxation method for variational inequalities with nonlinear constraints a simple iterative method for solving variational inequalities with a set valued nonmonotone mapping and a convex feasible set is proposed this set can be defined by nonlinear functions the method is based on combining and extending ideas contained in various relaxation methods of nonsmooth optimization also a modification of the averaging method for the problem under consideration is proposed 1998 the mathematical programming society inc published by elsevier science b v

extremal metrics on toric surfaces a continuity method the paper develops an existence theory for solutions of the abreu equation which include extremal metrics on toric surfaces the technique employed is a continuity method combined with blow up arguments general existence results are obtained assuming a hypothesis the m condition on the solutions which is shown to be related to the injectivity radius

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 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

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

on the convergence of a class of outer approximation algorithms for convex programs abstract this paper presents a new class of outer approximation methods for solving general convex programs the methods solve at each iteration a subproblem whose constraints contain the feasible set of the original problem moreover the methods employ quadratic objective functions in the subproblems by adding a simple quadratic term to the objective function of the original problem while other outer approximation methods usually use the original objective function itself throughout the iterations by this modification convergence of the methods can be proved under mild conditions furthermore it is shown that generalized versions of the cut construction schemes in kelley cheney goldstein s cutting plane method and veinott s supporting hyperplane method can be incorporated with the present methods and a cut generated at each iteration need not be retained in the succeeding iterations

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

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

a new algorithm for linearly constrained c convex vector optimization with a supply chain network risk application abstract we study a class of vector optimization problems with a c convex objective function under linear constraints we extend the proximal point algorithm used in scalar optimization to vector optimization we analyze both the global and local convergence results for the new algorithm we then apply the proximal point algorithm to a supply chain network risk management problem under bi criteria considerations

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 weighted monge ampere energy of quasiplurisubharmonic functions abstract we study degenerate complex monge ampere equations on a compact kahler manifold x we show that the complex monge ampere operator d d c n is well defined on the class e x of plurisubharmonic functions with finite weighted monge ampere energy the class e x is the largest class of psh functions on which the monge ampere operator is well defined and the comparison principle is valid it contains several functions whose gradient is not square integrable we give a complete description of the range of the operator d d c n on e x as well as on some of its subclasses we also study uniqueness properties extending calabi s result to this unbounded and degenerate situation and we give applications to complex dynamics and to the existence of singular kahler einstein metrics

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

a class of smoothing saa methods for a stochastic mathematical program with complementarity constraints abstract a class of smoothing sample average approximation saa methods is proposed for solving the stochastic mathematical program with complementarity constraints smpcc considered by birbil et al s i birbil g gurkan o listes solving stochastic mathematical programs with complementarity constraints using simulation math oper res 31 2006 739 760 the almost sure convergence of optimal solutions of the smoothed saa problem to that of the true problem is established by the notion of epi convergence in variational analysis it is demonstrated that under suitable conditions any accumulation point of karash kuhn tucker points of the smoothed saa problem is almost surely a kind of stationary point of smpcc as the sample size tends to infinity moreover under a strong second order sufficient condition for smpcc the exponential convergence rate of the sequence of karash kuhn tucker points of the smoothed saa problem is investigated through an application of robinson s stability theory some preliminary numerical results are reported to show the efficiency of proposed method

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

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

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

the mabuchi geometry of finite energy classes abstract we introduce different finsler metrics on the space of smooth kahler potentials that will induce a natural geometry on various finite energy classes e x motivated by questions raised by r berman v guedj and y rubinstein we characterize the underlying topology of these spaces in terms of convergence in energy and give applications of our results to existence of kahler einstein metrics on fano manifolds

a steepest descent method for vector optimization in this work we propose a cauchy like method for solving smooth unconstrained vector optimization problems when the partial order under consideration is the one induced by the nonnegative orthant we regain the steepest descent method for multicriteria optimization recently proposed by fliege and svaiter we prove that every accumulation point of the generated sequence satisfies a certain first order necessary condition for optimality which extends to the vector case the well known gradient equal zero condition for real valued minimization finally under some reasonable additional hypotheses we prove global convergence to a weak unconstrained minimizer as a by product we show that the problem of finding a weak constrained minimizer can be viewed as a particular case of the so called abstract equilibrium problem

relative k stability and modified k energy on toric manifolds in this paper we give a sufficient condition for both the relative k stability and the properness of modified k energy associated to calabi s extremal metrics on toric manifolds in addition several examples of toric manifolds which satisfy the sufficient condition are presented

the self regulation problem as an inexact steepest descent method for multicriteria optimization abstract in this paper we study an inexact steepest descent method for multicriteria optimization whose step size comes with armijo s rule we show that this method is well defined moreover by assuming the quasi convexity of the multicriteria function we prove full convergence of any generated sequence to a pareto critical point as an application we offer a model for the psychology s self regulation problem using a recent variational rationality approach

a relaxed projection method for solving multiobjective optimization problems abstract in this paper we propose an algorithm for solving multiobjective minimization problems on nonempty closed convex subsets of the euclidean space the proposed method combines a reflection technique for obtaining a feasible point with a projected subgradient method under suitable assumptions we show that the sequence generated using this method converges to a pareto optimal point of the problem we also present some numerical results

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

kahler einstein metrics with positive scalar curvature in this paper we prove that the existence of kahler einstein metrics implies the stability of the underlying kahler manifold in a suitable sense in particular this disproves a long standing conjecture that a compact kahler manifold admits kahler einstein metrics if it has positive first chern class and no nontrivial holomorphic vector fields we will also establish an analytic criterion for the existence of kahler einstein metrics our arguments also yield that the analytic criterion is satisfied on stable kahler manifolds provided that the partial c 0 estimate posed in t6 is true

combined monte carlo sampling and penalty method for stochastic nonlinear complementarity problems in this paper we consider a new formulation with recourse for a class of stochastic nonlinear complementarity problems we show that the new formulation is equivalent to a smooth semi infinite program that no longer contains recourse variables we then propose a combined monte carlo sampling and penalty method for solving the problem in which the underlying sample space is assumed to be compact furthermore we suggest a compact approximation approach for the case where the sample space is unbounded two preliminary numerical examples are included as well

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

regularized mathematical programs with stochastic equilibrium constraints estimating structural demand models the article considers a particular class of optimization problems involving set valued stochastic equilibrium constraints we develop a solution procedure that relies on an approxima tion scheme for the equilibrium constraints based on regularization we replace the approximated equilibrium constraints by those involving only single valued lipschitz continuous functions in ad dition sampling has the further effect of replacing the simplified equilibrium constraints by more manageable ones obtained by implicitly discretizing the given probability measure so as to render the problem computationally tractable convergence is obtained by relying in particular on the graphical convergence of the approximated equilibrium constraints the problem of estimating the characteristics of a demand model a widely studied problem in microeconometrics serves both as motivation and illustration of the regularization and sampling procedure

nonsmooth multiobjective programming with quasi newton methods this paper proposes a new algorithm to solve nonsmooth multiobjective programming the algorithm is a descent direction method to obtain the critical point a necessary condition for pareto optimality we analyze both global and local convergence results under some assumptions numerical tests are also given

a relaxed projection method for variational inequalities this paper presents a modification of the projection methods for solving variational inequality problems each iteration of the proposed algorithm consists of projection onto a halfspace containing the given closed convex set rather than the latter set itself the algorithm can thus be implemented very easily and its global convergence to the solution can be established under suitable conditions

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

the complex monge ampere equation on compact kahler manifolds we consider the complex monge ampere equation on a compact kahler manifold m g when the right hand side f has rather weak regularity in particular we prove that estimate of triangle phi and the gradient estimate hold when f is in w 1 p 0 for any p 0 2n as an application we show that there exists a classical solution in w 3 p 0 for the complex monge ampere equation when f is in w 1 p 0

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

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

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

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

smoothing methods for nonsmooth nonconvex minimization we consider a class of smoothing methods for minimization problems where the feasible set is convex but the objective function is not convex not differentiable and perhaps not even locally lipschitz at the solutions such optimization problems arise from wide applications including image restoration signal reconstruction variable selection optimal control stochastic equilibrium and spherical approximations in this paper we focus on smoothing methods for solving such optimization problems which use the structure of the minimization problems and composition of smoothing functions for the plus function x many existing optimization algorithms and codes can be used in the inner iteration of the smoothing methods we present properties of the smoothing functions and the gradient consistency of subdifferential associated with a smoothing function moreover we describe how to update the smoothing parameter in the outer iteration of the smoothing methods to guarantee convergence of the smoothing methods to a stationary point of the original minimization problem

a subgradient method for vector optimization problems vector optimization problems are a significant extension of scalar optimization and have many real life applications we consider an extension of the projected subgradient method to convex vector optimization which works directly with vector valued functions without using scalar valued objectives we eliminate the scalarization approach a popular strategy for solving vector optimization problems exploring strongly the structure of these kinds of problems under suitable assumptions we show that the sequence generated by the algorithm converges to a weakly efficient optimum point

newton s method for multiobjective optimization we propose an extension of newton s method for unconstrained multiobjective optimization multicriteria optimization this method does not use a priori chosen weighting factors or any other form of a priori ranking or ordering information for the different objective functions newton s direction at each iterate is obtained by minimizing the max ordering scalarization of the variations on the quadratic approximations of the objective functions the objective functions are assumed to be twice continuously differentiable and locally strongly convex under these hypotheses the method as in the classical case is locally superlinear convergent to optimal points again as in the scalar case if the second derivatives are lipschitz continuous the rate of convergence is quadratic our convergence analysis uses a kantorovich like technique as a byproduct existence of optima is obtained under semilocal assumptions