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

a closed formula for the product in simple integral extensions abstract let be an algebraic number and let q a closed formula for the coordinates of the product is given in terms of the coordinates of and and the companion matrix of the minimal polynomial of the formula as well as its proof extend to fairly general simple integral extensions

subalgebras of matrix algebras generated by companion matrices abstract let f g z x be monic polynomials of degree n and let c d m n z be the corresponding companion matrices we find necessary and sufficient conditions for the subalgebra z c d to be a sublattice of finite index in the full integral lattice m n z in which case we compute the exact value of this index in terms of the resultant of f and g if r is a commutative ring with identity we determine when r c d m n r in which case a presentation for m n r in terms of c and d is given

two stage stochastic optimization problems with stochastic ordering constraints on the recourse abstract we consider two stage risk averse stochastic optimization problems with a stochastic ordering constraint on the recourse function two new characterizations of the increasing convex order relation are provided they are based on conditional expectations and on integrated quantile functions a counterpart of the lorenz function we propose two decomposition methods to solve the problems and prove their convergence our methods exploit the decomposition structure of the risk neutral two stage problems and construct successive approximations of the stochastic ordering constraints numerical results confirm the efficiency of the methods

the solution of groundwater quality management problems with a nonconvex feasible region using a cutting plane optimization technique in groundwater quality management problems the concentration constraints have a nonlinear behavior which may be described either as a convex or a nonconvex function therefore the feasible region which is defined as the intersection of all of these constraints can be either a convex or a nonconvex set a review of existing optimization algorithms for the solution of the groundwater quality management problem indicates that the majority of them have an inability to determine a global optimum when nonconvexity occurs in an earlier paper that appeared in this journal karatzas and finder 1993 the outer approximation method a global optimization technique was presented for the solution of groundwater management problems with convex constraints the problem was formulated to minimize a concave objective function over a compact convex set of constraints in the present study the same concept is applied to problems with nonconvex constraints while the main concept of the current approach remains the same as that in our earlier study there is a significant difference in the determination of the cutting hyperplane the nonconvexity of the domain requires a special approach to insure that the introduction of the cutting hyperplane does not eliminate any part of the nonconvex feasible region in this work the theory of the developed algorithm is presented and subsequently applied to two groundwater quality problems in the first example a hypothetical aquifer is considered to illustrate the performance of the methodology in the second example a groundwater quality management problem in woburn massachusetts is solved results obtained are compared with those generated by minos 5 1

inverse stochastic dominance constraints and rank dependent expected utility theory we consider optimization problems with second order stochastic dominance constraints formulated as a relation of lorenz curves we characterize the relation in terms of rank dependent utility functions which generalize yaari s utility functions we develop optimality conditions and duality theory for problems with lorenz dominance constraints we prove that lagrange multipliers associated with these constraints can be identified with rank dependent utility functions the problem is numerically tractable in the case of discrete distributions with equally probable realizations

robust stochastic dominance and its application to risk averse optimization we introduce a new preference relation in the space of random variables which we call robust stochastic dominance we consider stochastic optimization problems where risk aversion is expressed by a robust stochastic dominance constraint these are composite semi infinite optimization problems with constraints on compositions of measures of risk and utility functions we develop necessary and sufficient conditions of optimality for such optimization problems in the convex case in the nonconvex case we derive necessary conditions of optimality under additional smoothness assumptions of some mappings involved in the problem

enhancement of the outer approximation method for the solution of concentration constrained optimal design groundwater remediation problems 1 nonlinearity and nonconvexity are two major characteristics of groundwater quality management models the classical solutions of such problems require enormous computational effort without ensuring a global optimum to circumvent this problem the outer approximation method a global optimization technique was introduced to solve groundwater quality management problems characterized by a nonconvex objective function with minima at the boundary of the feasible region and constraints that have convex or nonconvex behavior in this paper a more sophisticated and computationally efficient approach for the case of the nonconvex constraints is presented to illustrate the efficacy and efficiency of this new approach a hypothetical and a field scale problem are considered

the dual theory of choice under risk in this essay a new theory of choice under risk is being proposed it is a theory which in a sense that will become clear is dual to expected utility theory hence the title dual theory risky prospects are evaluated in this theory by a cardinal numerical scale which resembles an expected utility except that the roles of payments and probabilities are reversed this theme the reversal of the roles of probabilities and payments will recur throughout the paper i should emphasize that playing games with probabilities masquerading as payments and payments masquerading as probabilities is not my object rather i hope to convince the reader that the dual theory has intrinsic economic significance and that in some areas its predictions are superior to those of expected utility theory while in other areas the reverse will be the case two reasons have prompted me to look for an alternative to expected utility theory the first reason is methodological in expected utility theory the agent s attitude towards risk and the agent s attitude towards wealth are forever bonded together at the level of fundamental principles risk aversion and diminishing marginal utility of wealth which are synonymous under expected utility theory are horses of different colors the former expresses an attitute towards risk increased uncertainty hurts while the latter expresses an attitude towards wealth the loss of a sheep hurts more when the agent is poor than when the agent is rich a question arises therefore as to whether these two notions can be kept separate from each other in a full fledged theory of cardinal utility the dual theory will have this property the second reason that leads me to look for an alternative to expected utility theory is empirical behavior patterns which are systematic yet inconsistent with expected utility theory have often been observed two prominent references among many others are allais 1953 and kahneman tversky 1979 so deeply

a primal method for the solution of the groundwater quality management problem a new algorithm for the solution of the groundwater quality management problem is presented the method assumes that most of the computational effort in such problems involves evaluation of the concentrations and their derivatives with respect to the pumping rates at the control points the methodology proposed herein is a combination of the cutting plane method and the primal method in this approach each line search moves from a feasible point towards the solution of a subproblem with linear constraints and the same objective function as the original problem sensitivity analysis is used to compute the derivative of a single concentration with respect to all the pumping rates by solving only one linear system at each time step the method has been tested with analytical convex problems and a model example a comparison of the algorithm s performance was conducted using minos 5 1 for the case of a linear cost function the method shows a computational growth rate almost linearly proportional to the number of decision variables

optimization problems with second order stochastic dominance constraints duality compact formulations and cut generation methods for stochastic optimization problems with second order stochastic dominance constraints we develop a new form of the duality theory featuring measures on the product of the probability space and the real line we present two formulations involving small numbers of variables and exponentially many constraints primal and dual the dual formulation reveals connections between dominance constraints generalized transportation problems and the theory of measures with given marginals both formulations lead to two classes of cutting plane methods finite convergence of both methods is proved in the case of finitely many events numerical results for a portfolio problem are provided

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

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

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

tractable almost stochastic dominance abstract ll almost stochastic dominance ll asd is a relaxation of the stochastic dominance sd concept proposed by leshno and levy that explains more of realistic preferences observed in practice than sd alone does unfortunately numerical applications of this concept such as identifying if a given portfolio is efficient or determining a marketed portfolio that dominates a given benchmark are computationally prohibitive due to the structure of ll asd we propose a new almost stochastic dominance asd concept that is computationally tractable for instance a marketed dominating portfolio can be identified by solving a simple linear programming problem moreover the new concept performs well on all the intuitive examples from the literature and in some cases leads to more realistic predictions than the earlier concept we develop some properties of asd formulate efficient optimization models and apply the concept to analyzing investors preferences between bonds and stocks for the long run

groundwater management using numerical simulation and the outer approximation method for global optimization groundwater quantity management problems with fixed charges have been formulated in the past as mixed integer and linear programming problems in this paper a new methodology is presented where the fixed charges are incorporated into the objective function in an exponential form and the problem is solved as a concave minimization problem the principal difficulty in the minimization of a concave function over a linear or nonlinear set of constraints is that the local minima which are determined by the classical minimization algorithms may not be global in an effort to circumvent this problem the outer approximation method is introduced this method is applicable to the global minimization of a concave function over a compact set of constraints in the present work the outer approximation is applied to concave minimization problems over a convex compact set of constraints two applications of the method to groundwater management problems are presented herein and the results are compared with an existing solution obtained using a different optimization approach

development of objective oriented groundwater models 2 robust experimental design 1 this paper continues the discussion in part 1 by considering the data collection strategy problem when the existing data are judged to be insufficient for constructing a reliable model designing an experiment for identifying a distributed parameter is very difficult because the identification of a more complex parameter structure requires more data moreover without knowing the sufficiency of a design finding an optimal design becomes meaningless these difficulties can be avoided if we turn to the construction of objective oriented models the identifiability of a distributed parameter as defined in this paper contains the reducibility of parameter structure sufficient conditions for this kind of identifiability are given when the structure error associated with a structure reduction is too large these conditions may not be satisfied no matter how much data are collected in this paper we formulate a new experimental design problem that consists of two objectives minimizing the cost and maximizing the information content with robustness and feasibility as constraints we develop an algorithm that can find a cost effective robust design for objective oriented parameter identification we also present a heuristic algorithm that can find a suboptimal design with less computational effort for real case studies the proposed methodology is used to design a pumping test for identifying a distributed hydraulic conductivity we verify the robustness of the obtained design by assuming that the true parameter may have continuous discrete random and fractured structures finally the presented procedure of constructing objective oriented models is described step by step

dual stochastic dominance and related mean risk models we consider the problem of constructing mean risk models which are consistent with the second degree stochastic dominance relation by exploiting duality relations of convex analysis we develop the quantile model of stochastic dominance for general distributions this allows us to show that several models using quantiles and tail characteristics of the distribution are in harmony with the stochastic dominance relation we also provide stochastic linear programming formulations of these models

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

subjective probability and expected utility without additivity comonotic acts are defined and an axiom of comonotomic independence 19 introduced that weakens the von newmann morgenstern axiom of independence if a nondegenerate continuous and monotonic state independent weak order over acts satisfies comonotonic independence then it induces a unique non necessarily additive probability and a von neumann morgenstern utility an extension of theexpected utility theory covers situations as the ellsberg paradox which are inconsistent with additive expected utility the concept of uncertainty aversion and interpretation of comonotonic independence in the context of social welfare functions are included