two stage optimization problems with multivariate stochastic order constraints we propose a two stage risk averse stochastic optimization problem with a stochastic order constraint on a vector valued function of the second stage decisions this model is motivated by a multiobjective second stage problem we formulate optimality conditions for the problem and analyse the lagrangian relaxation of the order constraint we propose two decomposition methods to solve the problems and prove their convergence the methods are based on lagrangian relaxation of the order constraints and on a construction of successive risk neutral two stage problems additionally we propose a new combinatorial method for verification of the multivariate order relation which is a key part of both methods we analyse a supply chain problem using our model and we apply our methods to solve the optimization problem numerical results confirm the efficiency of the proposed methods

on finitely terminating branch and bound algorithms for some global optimization problems global optimization algorithms are typically terminated with an epsilon approximate solution after a finite number of iterations this paper shows how existing infinitely convergent branch and bound algorithms can be augmented to guarantee finite termination with an exact global solution for problems having extreme point solutions

the number of maximal torsion cosets in subvarieties of tori we present sharp bounds on the number of maximal torsion cosets in a subvariety of the complex algebraic torus gm n our first main result gives a bound in terms of the degree of the defining polynomials a second result gives a bound in terms of the toric degree of the subvariety as a consequence we prove the conjectures of ruppert and of aliev and smyth on the number of isolated torsion points of a hypersurface these conjectures bound this number in terms of the multidegree and the volume of the newton polytope of a polynomial defining the hypersurface respectively

a cutting surface method for uncertain linear programs with polyhedral stochastic dominance constraints in this paper we study linear optimization problems with a newly introduced concept of multidimensional polyhedral linear second order stochastic dominance constraints by using the polyhedral properties of this dominance condition we present a cutting surface algorithm and show its finite convergence the cut generation problem is a difference of convex functions dc optimization problem we exploit the polyhedral structure of this problem to present a novel branch and cut algorithm that incorporates concepts from concave minimization and binary integer programming a linear programming problem is formulated for generating concavity cuts in our case where the polyhedra are unbounded we also present duality results for this problem relating the dual multipliers to utility functions without the need to impose constraint qualifications which again is possible because of the polyhedral nature of the problem numerical examples are presented showing the nature of solutions of our model a corrected pdf has been appended to the original

optimization with multivariate conditional value at risk constraints for many decision making problems under uncertainty it is crucial to develop risk averse models and specify the decision makers risk preferences based on multiple stochastic performance measures or criteria incorporating such multivariate preference rules into optimization models is a fairly recent research area existing studies focus on extending univariate stochastic dominance rules to the multivariate case however enforcing multivariate stochastic dominance constraints can often be overly conservative in practice as an alternative we focus on the widely applied risk measure conditional value at risk cvar introduce a multivariate cvar relation and develop a novel optimization model with multivariate cvar constraints based on polyhedral scalarization to solve such problems for finite probability spaces we develop a cut generation algorithm where each cut is obtained by solving a mixed integer problem we show that a multivariate cvar constraint reduces to finitely many univariate cvar co

bi objective multi mode project scheduling under risk aversion abstract the paper proposes a model for stochastic multi mode resource constrained project scheduling under risk aversion with the two objectives makespan and cost activity durations and costs are assumed as uncertain and modeled as random variables for the scheduling part of the decision problem the class of early start policies is considered in addition to the schedule the assignment of execution modes to activities has to be selected to take risk aversion into account the approach of optimization under multivariate stochastic dominance constraints recently developed in other fields is adopted for the resulting bi objective stochastic integer programming problem the pareto frontier is determined by means of an exact solution method incorporating a branch and bound technique based on the forbidden set branching scheme from stochastic project scheduling randomly generated test instances partially derived from a test case from the psplib are used to show the computational feasibility of the approach

a lift and project cutting plane algorithm for mixed 0 1 programs we propose a cutting plane algorithm for mixed 0 1 programs based on a family of polyhedra which strengthen the usual lp relaxation we show how to generate a facet of a polyhedron in this family which is most violated by the current fractional point this cut is found through the solution of a linear program that has about twice the size of the usual lp relaxation a lifting step is used to reduce the size of the lp s needed to generate the cuts an additional strengthening step suggested by balas and jeroslow is then applied we report our computational experience with a preliminary version of the algorithm this approach is related to the work of balas on disjunctive programming the matrix cone relaxations of lovasz and schrijver and the hierarchy of relaxations of sherali and adams

optimization with multivariate stochastic dominance constraints we consider stochastic optimization problems where risk aversion is expressed by a stochastic ordering constraint the constraint requires that a random vector depending on our decisions stochastically dominates a given benchmark random vector we identify a suitable multivariate stochastic order and describe its generator in terms of von neumann morgenstern utility functions we develop necessary and sufficient conditions of optimality and duality relations for optimization problems with this constraint assuming convexity we show that the lagrange multipliers corresponding to dominance constraints are elements of the generator of this order thus refining and generalizing earlier results for optimization under univariate stochastic dominance constraints furthermore we obtain necessary conditions of optimality for non convex problems under additional smoothness assumptions

numerical methods for stochastic programs with second order dominance constraints with applications to portfolio optimization inspired by the successful applications of the stochastic optimization with second order stochastic dominance ssd model in portfolio optimization we study new numerical methods for a general ssd model where the underlying functions are not necessarily linear specifically we penalize the ssd constraints to the objective under slater s constraint qualification and then apply the well known stochastic approximation sa method and the level function method to solve the penalized problem both methods are iterative the former requires to calculate an approximate subgradient of the objective function of the penalized problem at each iterate while the latter requires to calculate a subgradient under some moderate conditions we show that w p 1 the sequence of approximated solutions generated by the sa method converges to an optimal solution of the true problem as for the level function method the convergence is deterministic and in some cases we are able to estimate the number of iterations for a given precision both methods are applied to portfolio optimization problem where the return functions are not necessarily linear and some numerical test results are reported

sample average approximation of stochastic dominance constrained programs in this paper we study optimization problems with second order stochastic dominance constraints this class of problems allows for the modeling of optimization problems where a risk averse decision maker wants to ensure that the solution produced by the model dominates certain benchmarks here we deal with the case of multi variate stochastic dominance under general distributions and nonlinear functions we introduce the concept of mathcal c dominance which generalizes some notions of multi variate dominance found in the literature we apply the sample average approximation saa method to this problem which results in a semi infinite program and study asymptotic convergence of optimal values and optimal solutions as well as the rate of convergence of the feasibility set of the resulting semi infinite program as the sample size goes to infinity we develop a finitely convergent method to find an epsilon optimal solution of the saa problem an important aspect of our contribution is the construction of practical statistical lower and upper bounds for the true optimal objective value we also show that the bounds are asymptotically tight as the sample size goes to infinity

augmented lagrangian methods for solving optimization problems with stochastic order constraints we investigate risk averse stochastic optimization problems with a risk shaping constraint in the form of a stochastic order relation both univariate and multivariate orders are considered we extend the notion of a linear multivariate order adding flexibility with respect to the controlled portion of the distributions we propose several methods for the numerical solution of these problems based on augmented lagrangian framework and analyze their convergence the methods construct finite dimensional risk neutral approximations of the optimization problem whose solutions converge to the solution of the original problem in the case of univariate stochastic dominance we explore augmented lagrangian functionals based on inverse formulations of the stochastic order constraint the performance of the methods is compared to other extant numerical methods and shows the numerical advantage of the augmented lagrangian framework the proposed numerical approach is particularly successful when applied to problem

stable polynomials and crystalline measures explicit examples of positive crystalline measures and fourier quasicrystals are constructed using pairs of stable polynomials answering several open questions in the area

optimization with stochastic dominance constraints we introduce stochastic optimization problems involving stochastic dominance constraints we develop necessary and sufficient conditions of optimality and duality theory for these models and show that the lagrange multipliers corresponding to dominance constraints are concave nondecreasing utility functions the models and results are illustrated on a portfolio optimization problem

exact penalization level function method and modified cutting plane method for stochastic programs with second order stochastic dominance constraints level function methods and cutting plane methods have been recently proposed to solve stochastic programs with stochastic second order dominance ssd constraints a level function method requires an exact penalization setup because it can only be applied to the objective function not the constraints slater constraint qualification scq is often needed for deriving exact penalization it is well known that ssd usually does not satisfy scq and various relaxation schemes have been proposed so that the relaxed problem satisfies the scq in this paper we show that under some moderate conditions the desired constraint qualification can be guaranteed through some appropriate reformulation of the constraints rather than relaxation exact penalization schemes based on l 1 norm and l infty norm are subsequently derived through robinson s error bound on convex systems and clarke s exact penalty function theorem moreover we propose a modified cutting plane method which constructs a cutting plane through t

convergence analysis of stationary points in sample average approximation of stochastic programs with second order stochastic dominance constraints sample average approximation saa method has recently been applied to solve stochastic programs with second order stochastic dominance ssd constraints in particular hu et al math program 133 171 201 2012 presented a detailed convergence analysis of epsilon optimal values and epsilon optimal solutions of sample average approximated stochastic programs with polyhedral ssd constraints in this paper we complement the existing research by presenting convergence analysis of stationary points when saa is applied to a class of stochastic minimization problems with ssd constraints specifically under some moderate conditions we prove that optimal solutions and stationary points obtained from solving sample average approximated problems converge with probability one to their true counterparts moreover by exploiting some recent results on large deviation of random functions and sensitivity analysis of generalized equations we derive exponential rate of convergence of stationary points

optimization with stochastic preferences based on a general class of scalarization functions it is of crucial importance to develop risk averse models for multicriteria decision making under uncertainty a major stream of the related literature studies optimization problems that feature multivariate stochastic benchmarking constraints these problems typically involve a univariate stochastic preference relation often based on stochastic dominance or a coherent risk measure such as conditional value at risk which is then extended to allow the comparison of random vectors by the use of a family of scalarization functions all scalarized versions of the vector of the uncertain outcomes of a decision are required to be preferable to the corresponding scalarizations of the benchmark outcomes while this line of research has been dedicated almost entirely to linear scalarizations the corresponding deterministic literature uses a wide variety of scalarization functions that among other advantages offer a high degree of modeling flexibility in this paper we aim to incorporate these scalarizations i

the deligne mumford compactification of the real multiplication locus and teichmuller curves in genus 3 in the moduli space mathcal m g of genus g riemann surfaces consider the locus mathcal r mathcal m mathcal o of riemann surfaces whose jacobians have real multiplication by the order mathcal o in a totally real number field f of degree g if g 3 we compute the closure of mathcal r mathcal m mathcal o in the deligne mumford compactification of mathcal m g and the closure of the locus of eigenforms over mathcal r mathcal m mathcal o in the deligne mumford compactification of the moduli space of holomorphic 1 forms for higher genera we give strong necessary conditions for a stable curve to be in the boundary of mathcal r mathcal m mathcal o boundary strata of mathcal r mathcal m mathcal o are parameterized by configurations of elements of the field f satisfying a strong geometry of numbers type restriction

cut generation for optimization problems with multivariate risk constraints we consider a class of stochastic optimization problems that features benchmarking preference relations among random vectors representing multiple random performance measures criteria of interest given a benchmark random performance vector preference relations are incorporated into the model as constraints which require the decision based random vector to be preferred to the benchmark according to a relation based on multivariate conditional value at risk cvar or second order stochastic dominance ssd we develop alternative mixed integer programming formulations and solution methods for cut generation problems arising in optimization under such multivariate risk constraints the cut generation problems for cvar and ssd based models involve the epigraphs of two distinct piecewise linear concave functions which we refer to as reverse concave sets we give the complete linear description of the linearization polytopes of these two non convex substructures we present computational results that show the effectiveness of our proposed models and methods

small points on subvarieties of a torus let v be a subvariety of a torus de ned over the algebraic numbers we give a qualitative and quantitative description of the set of points of v of height bounded by invariants associated to any variety containing v especially we determine whether such a set is or is not dense in v we then prove that these sets can always be written as the intersection of v with a nite union of translates of tori of which we control the sum of the degrees as a consequence we prove a conjecture by the rst author and david up to a logarithmic factor

stochastically weighted stochastic dominance concepts with an application in capital budgeting abstract the problem of comparing random vectors arises in many applications we propose three new concepts of stochastically weighted dominance for comparing random vectors x and y the main idea is to use a random vector v to scalarize x and y as v t x and v t y and subsequently use available concepts from stochastic dominance and stochastic optimization for comparison for the case where the distributions of x y and v have finite support we give mixed integer linear inequalities that can be used for random vector comparison as well as for modeling of optimization problems where one of the random vectors depends on decisions to be optimized some advantages of the proposed new concepts are illustrated with the help of a capital budgeting example

the number of solutions of decomposable form equations has only finitely many solutions the diophantine approximation techniques of thue and mahler and improvements by siegel dyson roth and bombieri made it possible to derive good explicit upper bounds for the number of solutions of 1 1 the best such upper bound to date due to bombieri 1 is 2 12r bombieri assumed that f is irreducible and r 6 which was not essential in his proof for t 0 i e f x y 1 in x y z bombieri and schmidt 2 derived the upper bound constant r which is best possible in terms of r

robust decision making using a general utility set abstract we address the problem of ambiguity and inconsistency in a decision maker s dm assessed utility function by using a maxmin framework in this framework dm s utility function belongs to a set of functions the member functions of the set are nondecreasing and satisfy additional boundary and auxiliary conditions the maxmin framework provides robustness in decision making alternatively it allows us to perform sensitivity or parametric analysis for the optimal decision when a reference utility function is known for this purpose we use a cost of ambiguity concept and show that this cost is increasing and concave when extent of ambiguity is parametrically increased next we develop a lagrangian based solution approach for the decision problem we show that under suitable conditions a sample average approximation saa of the lagrangian model can be solved using a mixed integer linear program milp we also show that the set of optimal solutions of the saa converges to that of its true counterpart and the optimum objective value of the saa converges to the true objective value at an exponential rate we use this convergence property to develop a heuristic for identifying a solution of the saa milp with increasing sample size we illustrate the properties of the maxmin model using two examples of portfolio investment and streaming bandwidth we provide a discussion on the performance of a commercial solver used to solve saa milps and the quality of solutions generated by the proposed heuristic

arithmetic group symmetry and finiteness properties of torelli groups we examine groups whose resonance varieties characteristic varieties and sigma invariants have a natural arithmetic group symmetry and we explore implications on various niteness properties of subgroups we compute resonance varieties characteristic varieties and alexander polynomials of torelli groups and we show that all subgroups containing the johnson kernel have nite rst betti number when the genus is at least 4 we also prove that in this range the i adic completion of the alexander invariant is nite dimensional and the kahler property for the torelli group implies the nite generation of the johnson kernel

two stage stochastic programming under multivariate risk constraints with an application to humanitarian relief network design in this study we consider two classes of multicriteria two stage stochastic programs in finite probability spaces with multivariate risk constraints the first stage problem features multivariate stochastic benchmarking constraints based on a vector valued random variable representing multiple and possibly conflicting stochastic performance measures associated with the second stage decisions in particular the aim is to ensure that the decision based random outcome vector of interest is preferable to a specified benchmark with respect to the multivariate polyhedral conditional value at risk or a multivariate stochastic order relation in this case the classical decomposition methods cannot be used directly due to the complicating multivariate stochastic benchmarking constraints we propose an exact unified decomposition framework for solving these two classes of optimization problems and show its finite convergence we apply the proposed approach to a stochastic network design problem in the context of pre disaster humanitarian logistics and conduct a computational study concerning the threat of hurricanes in the southeastern part of the united states the numerical results provide practical insights about our modeling approach and show that the proposed algorithm is computationally scalable

dynamical anomalous subvarieties structure and bounded height theorems abstract according to medvedev and scanlon 14 a polynomial f x q x of degree d 2 is called disintegrated if it is not conjugate to x d or to c d x where c d is the chebyshev polynomial of degree d let n n let f 1 f n q x be disintegrated polynomials of degrees at least 2 and let f 1 f n be the corresponding coordinate wise self map of p 1 n let x be an irreducible subvariety of p 1 n of dimension r defined over q we define the anomalous locus of x which is related to the periodic subvarieties of p 1 n we prove that the anomalous locus of x is zariski closed this is a dynamical analogue of a theorem of bombieri masser and zannier 4 we also prove that the points in the intersection of x with the union of all irreducible periodic subvarieties of p 1 n of codimension r have bounded height outside the anomalous locus of x this is a dynamical analogue of habegger s theorem 8 which was previously conjectured in 4 the slightly more general self maps f 1 f n where each f i q x is a disintegrated rational function are also treated at the end of the paper

local systems on analytic germ complements abstract we prove that the cohomology jump loci of rank one local systems on the complement in a small ball of a germ of a complex analytic set are finite unions of torsion translates of subtori this is a generalization of the classical monodromy theorem stating that the eigenvalues of the monodromy on the cohomology of the milnor fiber of a germ of a holomorphic function are roots of unity

estimating distribution functions from survey data summary a simple method for estimating population distribution functions and associated quantiles from sample survey data is described and some asymptotic theory for it presented the method assumes a model based approach to survey estimation and allows auxiliary population information to be directly incorporated into the estimation process monte carlo results comparing the proposed method with conventional design based methods are given these suggest that the model based approach offers significant gains when the auxiliary population information is linearly related to the survey variables of interest

ambiguity in risk preferences in robust stochastic optimization abstract we consider robust stochastic optimization problems for risk averse decision makers where there is ambiguity about both the decision maker s risk preferences and the underlying probability distribution we propose and analyze a robust optimization problem that accounts for both types of ambiguity first we derive a duality theory for this problem class and identify random utility functions as the lagrange multipliers second we turn to the computational aspects of this problem we show how to evaluate our robust optimization problem exactly in some special cases and then we consider some tractable relaxations for the general case finally we apply our model to both the newsvendor and portfolio optimization problems and discuss its implications

stability analysis of stochastic programs with second order dominance constraints in this paper we present a stability analysis of a stochastic optimization problem with stochastic second order dominance constraints we consider a perturbation of the underlying probability measure in the space of regular measures equipped with pseudometric discrepancy distance romisch in stochastic programming elsevier amsterdam pp 483 554 2003 by exploiting a result on error bounds in semi infinite programming due to gugat math program ser b 88 255 275 2000 we show under the slater constraint qualification that the optimal value function is lipschitz continuous and the optimal solution set mapping is upper semicontinuous with respect to the perturbation of the probability measure in particular we consider the case when the probability measure is approximated by an empirical probability measure and show an exponential rate of convergence of the sequence of optimal solutions obtained from solving the approximation problem the analysis is extended to the stationary points

optimality and duality theory for stochastic optimization problems with nonlinear dominance constraints we consider a new class of optimization problems involving stochastic dominance constraints of second order we develop a new splitting approach to these models optimality conditions and duality theory these results are used to construct special decomposition methods

a criterion for weak convergence on berkovich projective space we give a criterion for the weak convergence of unit borel measures on the n dimensional berkovich projective space bf p n k over a complete non archimedean field k as an application we give a sufficient condition for a certain type of equidistribution on bf p n k in terms of a weak zariski density property on the scheme theoretic projective space mathbb p n tilde k vphantom 0 over the residue field tilde k as a second application in the case of residue characteristic zero we give an ergodic theoretic equidistribution result for the powers of a point a in the n dimensional unit torus mathbb t n k over k this is a non archimedean analogue of a well known result of weyl over mathbb c and its proof makes essential use of a theorem of mordell lang type for mathbb g m n due to laurent

intersections of polynomial orbits and a dynamical mordell lang conjecture we prove that if nonlinear complex polynomials of the same degree have orbits with infinite intersection then the polynomials have a common iterate we also prove a special case of a conjectured dynamical analogue of the mordell lang conjecture

o minimality and the andre oort conjecture for mathbb c n we give an unconditional proof of the andre oort conjecture for arbitrary products of modular curves we establish two generalizations the first includes the manin mumford conjecture for arbitrary products of elliptic curves defined over q as well as lang s conjecture for torsion points in powers of the multiplicative group the second includes the manin mumford conjecture for abelian varieties defined over q our approach uses the theory of o minimal structures a part of model theory and follows a strategy proposed by zannier and implemented in three recent papers a new proof of the manin mumford conjecture by pila zannier a proof of a special but new case of pink s relative manin mumford conjecture by masser zannier and new proofs of certain known results of andre oort manin mumford type by pila 2010 mathematics subject classification 11g18 03c64