solving mixed integer nonlinear programs by outer approximation a wide range of optimization problems arising from engineering applications can be formulated as mixed integer nonlinear programming problems minlps duran and grossmann 1986 suggest an outer approximation scheme for solving a class of minlps that are linear in the integer variables by a finite sequence of relaxed milp master programs and nlp subproblems

duality and statistical tests of optimality for two stage stochastic programs we present alternative methods for verifying the quality of a proposed solution to a two stage stochastic program with recourse our methods revolve around implications of a dual problem in which dual multipliers on the nonanticipativity constraints play a critical role using randomly sampled observations of the stochastic elements we introduce notions of statistical dual feasibility and sampled error bounds additionally we use the nonanticipativity multipliers to develop connections to reduced gradient methods finally we propose a statistical test based on directional derivatives we illustrate the applicability of these tests via some examples

an efficient trajectory method for probabilistic production inventory distribution problems we consider a supply chain operating in an uncertain environment the customers demand is characterized by a discrete probability distribution a probabilistic programming approach is adopted for constructing an inventory production distribution plan over a multiperiod planning horizon the plan does not allow the backlogging of the unsatisfied demand and minimizes the costs of the supply chain while enabling it to reach a prescribed nonstockout service level it is a strategic plan that hedges against undesirable outcomes and that can be adjusted to account for possible favorable realizations of uncertain quantities a modular integrated and computationally tractable method is proposed for the solution of the associated stochastic mixed integer optimization problems containing joint probabilistic constraints with dependent right hand side variables the concept of p efficiency is used to construct a finite number of demand trajectories which in turn are employed to solve problems with joint probabilistic constraints we complement this idea by designing a preordered set based preprocessing algorithm that selects a subset of promising p efficient demand trajectories finally to solve the resulting disjunctive mixed integer programming problem we implement a special column generation algorithm that limits the risk of congestion in the resources of the supply chain the methodology is validated on an industrial problem faced by a large chemical supply chain and turns out to be very efficient it finds a solution with a minimal integrality gap and provides substantial cost savings

the c 3 theorem and a d 2 algorithm for large scale stochastic mixed integer programming set convexification this paper considers the two stage stochastic integer programming problem with an emphasis on instances in which integer variables appear in the second stage drawing heavily on the theory of disjunctive programming we characterize convexifications of the second stage problem and develop a decomposition based algorithm for the solution of such problems in particular we verify that problems with fixed recourse are characterized by scenario dependent second stage convexifications that have a great deal in common we refer to this characterization as the c3 common cut coefficients theorem based on the c3 theorem we develop a decomposition algorithm which we refer to as disjunctive decomposition d2 in this new class of algorithms we work with master and subproblems that result from convexifications of two coupled disjunctive programs we show that when the second stage consists of 0 1 milp problems we can obtain accurate second stage objective function estimates after finitely many steps this result implies the convergence of the d2 algorithm

robust optimization for unconstrained simulation based problems in engineering design an optimized solution often turns out to be suboptimal when errors are encountered although the theory of robust convex optimization has taken significant strides over the past decade all approaches fail if the underlying cost function is not explicitly given it is even worse if the cost function is nonconvex in this work we present a robust optimization method that is suited for unconstrained problems with a nonconvex cost function as well as for problems based on simulations such as large partial differential equations pde solver response surface and kriging metamodels moreover this technique can be employed for most real world problems because it operates directly on the response surface and does not assume any specific structure of the problem we present this algorithm along with the application to an actual engineering problem in electromagnetic multiple scattering of aperiodically arranged dielectrics relevant to nanophotonic design the corresponding objective function is highly nonconvex and resides in a 100 dimensional design space starting from an optimized design we report a robust solution with a significantly lower worst case cost while maintaining optimality we further generalize this algorithm to address a nonconvex optimization problem under both implementation errors and parameter uncertainties

duality gaps in nonconvex stochastic optimization we consider multistage stochastic optimization models containing nonconvex constraints e g due to logical or integrality requirements we study three variants of lagrangian relaxations and of the corresponding decomposition schemes namely scenario nodal and geographical decomposition based on convex equivalents for the lagrangian duals we compare the duality gaps for these decomposition schemes the first main result states that scenario decomposition provides a smaller or equal duality gap than nodal decomposition the second group of results concerns large stochastic optimization models with loosely coupled components the results provide conditions implying relations between the duality gaps of geographical decomposition and the duality gaps for scenario and nodal decomposition respectively

a finite branch and bound algorithm for two stage stochastic integer programs this paper addresses a general class of two stage stochastic programs with integer recourse and discrete distributions we exploit the structure of the value function of the second stage integer problem to develop a novel global optimization algorithm the proposed scheme departs from those in the current literature in that it avoids explicit enumeration of the search space while guaranteeing finite termination computational experiments on standard test problems indicate superior performance of the proposed algorithm in comparison to those in the existing literature

regularization methods for optimization problems with probabilistic constraints we analyze nonlinear stochastic optimization problems with probabilistic constraints on nonlinear inequalities with random right hand sides we develop two numerical methods with regularization for their numerical solution the methods are based on first order optimality conditions and successive inner approximations of the feasible set by progressive generation of p efficient points the algorithms yield an optimal solution for problems involving concave probability distributions for arbitrary distributions the algorithms solve the convex hull problem and provide upper and lower bounds for the optimal value and nearly optimal solutions the methods are compared numerically to two cutting plane methods

lagrangian decomposition and mixed integer quadratic programming reformulations for probabilistically constrained quadratic programs abstract probabilistically constrained quadratic programming pcqp problems arise naturally from many real world applications and have posed a great challenge in front of the optimization society for years due to the nonconvex and discrete nature of its feasible set we consider in this paper a special case of pcqp where the random vector has a finite discrete distribution we first derive second order cone programming socp relaxation and semidefinite programming sdp relaxation for the problem via a new lagrangian decomposition scheme we then give a mixed integer quadratic programming miqp reformulation of the pcqp and show that the continuous relaxation of the miqp is exactly the socp relaxation this new miqp reformulation is more efficient than the standard miqp reformulation in the sense that its continuous relaxation is tighter than or at least as tight as that of the standard miqp we report preliminary computational results to demonstrate the tightness of the new convex relaxations and the effectiveness of the new miqp reformulation

a gaussian upper bound for gaussian multi stage stochastic linear programs this paper deals with two stage and multi stage stochastic programs in which the right hand sides of the constraints are gaussian random variables such problems are of interest since the use of gaussian estimators of random variables is widespread we introduce algorithms to find upper bounds on the optimal value of two stage and multi stage stochastic minimization programs with gaussian right hand sides the upper bounds are obtained by solving deterministic mathematical programming problems with dimensions that do not depend on the sample space size the algorithm for the two stage problem involves the solution of a deterministic linear program and a simple semidefinite program the algorithm for the multi stage problem invovles the solution of a quadratically constrained convex programming problem

an approximation scheme for stochastic integer programs arising in capacity expansion planning for capacity expansion forms a crucial part of the strategic level decision making in many applications consequently quantitative models for economic capacity expansion planning have been the subject of intense research however much of the work in this area has been restricted to linear cost models and or limited degree of uncertainty to make the problems analytically tractable this paper addresses a stochastic capacity expansion problem where the economies of scale in expansion costs are handled via fixed charge cost functions and forecast uncertainties in the problem parameters are explicitly considered by specifying a set of scenarios the resulting formulation is a multistage stochastic integer program we develop a fast linear programming based approximation scheme that exploits the decomposable structure and is guaranteed to produce feasible solutions for this problem through a probabilistic analysis we prove that the optimality gap of the heuristic solution almost surely vanishes asymptotically as the problem size increases

solving monotone stochastic variational inequalities and complementarity problems by progressive hedging the concept of a stochastic variational inequality has recently been articulated in a new way that is able to cover in particular the optimality conditions for a multistage stochastic programming problem one of the long standing methods for solving such an optimization problem under convexity is the progressive hedging algorithm that approach is demonstrated here to be applicable also to solving multistage stochastic variational inequality problems under monotonicity thus increasing the range of applications for progressive hedging stochastic complementarity problems as a special case are explored numerically in a linear two stage formulation

solving lagrangian variational inequalities with applications to stochastic programming lagrangian variational inequalities feature both primal and dual elements in expressing first order conditions for optimality in a wide variety of settings where multipliers in a very general sense need to be brought in their stochastic version relates to problems of stochastic programming and covers not only classical formats with inequality constraints but also composite models with nonsmooth objectives the progressive hedging algorithm as a means of solving stochastic programming problems has however focused so far only on optimality conditions that correspond to variational inequalities in primal variables alone here that limitation is removed by appealing to a recent extension of progressive hedging to multistage stochastic variational inequalities in general

an augmented lagrangian method for distributed optimization we propose a novel distributed method for convex optimization problems with a certain separability structure the method is based on the augmented lagrangian framework we analyze its convergence and provide an application to two network models as well as to a two stage stochastic optimization problem the proposed method compares favorably to two augmented lagrangian decomposition methods known in the literature as well as to decomposition methods based on the ordinary lagrangian function

pattern based modeling and solution of probabilistically constrained optimization problems we propose a new modeling and solution method for probabilistically constrained optimization problems the methodology is based on the integration of the stochastic programming and combinatorial pattern recognition fields it permits the fast solution of stochastic optimization problems in which the random variables are represented by an extremely large number of scenarios the method involves the binarization of the probability distribution and the generation of a consistent partially defined boolean function pdbf representing the combination f p of the binarized probability distribution f and the enforced probability level p we show that the pdbf representing f p can be compactly extended as a disjunctive normal form dnf the dnf is a collection of combinatorial p patterns each defining sufficient conditions for a probabilistic constraint to hold we propose two linear programming formulations for the generation of p patterns that can be subsequently used to derive a linear programming inner approximation of the original stochastic problem a formulation allowing for the concurrent generation of a p pattern and the solution of the deterministic equivalent of the stochastic problem is also proposed the number of binary variables included in the deterministic equivalent formulation is not an increasing function of the number of scenarios used to represent uncertainty results show that large scale stochastic problems in which up to 50 000 scenarios are used to describe the stochastic variables can be consistently solved to optimality within a few seconds

an outer approximation algorithm for a class of mixed integer nonlinear programs an outer approximation algorithm is presented for solving mixed integer nonlinear programming problems of a particular class linearity of the integer or discrete variables and convexity of the nonlinear functions involving continuous variables are the main features in the underlying mathematical structure based on principles of decomposition outer approximation and relaxation the proposed algorithm effectively exploits the structure of the problems and consists of solving an alternating finite sequence of nonlinear programming subproblems and relaxed versions of a mixed integer linear master program convergence and optimality properties of the algorithm are presented as well as a general discussion on its implementation numerical results are reported for several example problems to illustrate the potential of the proposed algorithm for programs in the class addressed in this paper finally a theoretical comparison with generalized benders decomposition is presented on the lower bounds predicted by the relaxed master programs

single commodity stochastic network design under probabilistic constraint with discrete random variables single commodity networks are considered where demands at the nodes are random the problem is to find minimum cost optimal built in capacities at the nodes and arcs subject to the constraint that all demands should be met on a prescribed probability level reliability constraint and some deterministic constraints should be satisfied the reliability constraint is formulated in terms of the gale hoffman feasibility inequalities but their number is reduced by elimination technique the concept of a p efficient point is used in a smart way to convert and then relax the problem into an lp the p efficient points are simultaneously generated with the solution of the lp the joint distribution of the demands is used to obtain the p efficient points for all stochastic inequalities that were not eliminated and the solution of a multiple choice knapsack problem is used to generate p efficient points the model can be applied to planning in interconnected power systems flood control networks design of shelter

uncertain convex programs randomized solutions and confidence levels many engineering problems can be cast as optimization problems subject to convex constraints that are parameterized by an uncertainty or instance parameter two main approaches are generally available to tackle constrained optimization problems in presence of uncertainty robust optimization and chance constrained optimization robust optimization is a deterministic paradigm where one seeks a solution which simultaneously satisfies all possible constraint instances in chance constrained optimization a probability distribution is instead assumed on the uncertain parameters and the constraints are enforced up to a pre specified level of probability unfortunately however both approaches lead to computationally intractable problem formulations

lagrangian solution techniques and bounds for loosely coupled mixed integer stochastic programs many production problems involve facility setups that lead to integer variables production decisions that are continuous and demands that are likely to be random while these problems can be quite difficult to solve we propose a model and an efficient solution technique for this basic class of stochastic mixed integer programs we use a set of scenarios to reflect uncertainty the resulting mathematical model is solved using lagrangian relaxation we show that the duality gap of our relaxation is bounded above by a constant that depends on the cost function and the number of branching points in the scenario tree we apply our technique to the problem of generating electric power numerical results indicate significant savings when the stochastic model is used instead of a deterministic one

approximation of a circular cylindrical shell by clough johnson flat plate finite elements in this paper we study the approximation of a right circular cylindrical shell by a nonconforming method using clough johnson flat plate finite elements compatibility conditions which have to be satisfied by the degrees of freedom at every node of the triangulation are given then we prove that convergence is insured for shallow shells when using particular families of triangulations which are practically easy to implement finally we propose a new approximation method by flat plate finite elements which assures the convergence for any kind of circular cylindrical shell

concavity and efficient points of discrete distributions in probabilistic programming we consider stochastic programming problems with probabilistic constraints involving integer valued random variables the concept of a p efficient point of a probability distribution is used to derive various equivalent problem formulations next we introduce the concept of r concave discrete probability distributions and analyse its relevance for problems under consideration these notions are used to derive lower and upper bounds for the optimal value of probabilistically constrained stochastic programming problems with discrete random variables the results are illustrated with numerical examples

applications of the method of partial inverses to convex programming decomposition a primal dual decomposition method is presented to solve the separable convex programming problem convergence to a solution and lagrange multiplier vector occurs from an arbitrary starting point the method is equivalent to the proximal point algorithm applied to a certain maximal monotone multifunction in the nonseparable case it specializes to a known method the proximal method of multipliers conditions are provided which guarantee linear convergence

a decomposition procedure based on approximate newton directions the efficient solution of large scale linear and nonlinear optimization problems may require exploiting any special structure in them in an efficient manner we describe and analyze some cases in which this special structure can be used with very little cost to obtain search directions from decomposed subproblems we also study how to correct these directions using decomposable preconditioned conjugate gradient methods to ensure local convergence in all cases the choice of appropriate preconditioners results in a natural manner from the structure in the problem finally we conduct computational experiments to compare the resulting procedures with direct methods

dade s conjecture for the simple o nan group let g be the simple o nan group it has order 29 34 5 73 11 19 31 it has schur multiplier of order 3 and aut g g is cyclic of order 2 in this paper we prove that dade s conjecture for g is true here we mean by saying the conjecture the inductive form of the conjecture see 3 according to 3 the inductive form is equivalent to the invariant projective form if the outer automorphism group has cyclic sylow q subgroups for each prime q thus for the simple o nan group it suffices to check the invariant projective form moreover if a defect group of a block is cyclic then the invariant projective form is proved to be true for this block in 4 thus it suffices to treat the primes 2 3 and 7 in this paper a b and a b denote a split and a nonsplit extension of a by b respectively we use n and dn to denote a cyclic and a dihedral group of order n respectively moreover for a prime p an elementary abelian group of order p is denoted by p for an odd prime p we denote by p1 2n an extraspecial group of order p1 2n and exponent p and by 21 2n a minus type extraspecial group of order 21 2n finally sn and an denote the symmetric group and the alternating group of degree n respectively

the probabilistic set covering problem in a probabilistic set covering problem the right hand side is a random binary vector and the covering constraint has to be satisfied with some prescribed probability we analyze the structure of the set of probabilistically efficient points of binary random vectors develop methods for their enumeration and propose specialized branch and bound algorithms for probabilistic set covering problems

an adi difference scheme based on fractional trapezoidal rule for fractional integro differential equation with a weakly singular kernel abstract in this paper we propose a fast and efficient numerical method to solve the two dimensional integro differential equation with a weakly singular kernel the numerical method are considered by finite difference approach for spatial discretization and alternating direction implicit adi method in time combined with the second order fractional quadrature convolution rule introduced by lubich and the classical l1 approximation for caputo fractional derivative the detailed analysis shows that the proposed scheme is unconditionally stable and convergent with the convergence order o min 1 2 h 1 2 h 2 2 some numerical results are also given to confirm our theoretical prediction

the radical 2 subgroups of some sporadic simple groups abstract the radical p subgroups of the monster the baby monster and the harada norton group hn for p 3 5 7 are determined up to conjugacy which completes the classification problem of odd radical subgroups for all sporadic simple groups together with m kitazume s yoshiara the radical subgroups of the fischer simple groups j algebra 255 2002 22 58 and the other works

verification of dade s conjecture for janko groupj3 abstract in 3 dade made a conjecture expressing the number k b d of characters of a given defect d in a given p block b of a finite group g in terms of the corresponding numbers k b d for blocks b of certain p local subgroups of g several different forms of this conjecture are given in 5 dade claims that the most complicated form of this conjecture called the inductive conjecture 5 8 in 5 will hold for all finite groups if it holds for all covering groups of finite simple groups in this paper we verify the inductive conjecture for all covering groups of the third janko group j 3 in the notation of the atlas 1 this is one step in the inductive proof of the conjecture for all finite groups certain properties of j 3 simplify our task the schur multiplier of j 3 is cyclic of order 3 see 1 p 82 hence there are just two covering groups of j 3 namely j 3 itself and a central extension 3 j 3 of j 3 by a cyclic group z of order 3 we treat these two covering groups separately the outer automorphism group out j 3 of j 3 is cyclic of order 2 see 1 p 82 in this case dade affirms in 5 section 6 that the inductive conjecture for j 3 is equivalent to the much weaker invariant conjecture 2 5 in 5 furthermore dade has proved in 6 that this invariant conjecture holds for all blocks with cyclic defect groups the sylow p subgroups of j 3 are cyclic of order p for all primes dividing j 3 except 2 and 3 so we only need to verify the invariant conjecture for the two primes p 2 and p 3 we do that in theorem 2 10 1 and theorem 3 6 1 below the group out 3 j 3 z of outer automorphisms of 3 j 3 centralizing z is trivial see 1 p 82 in this case dade affirms that the inductive conjecture is equivalent to the projective conjecture 4 4 in 5 again dade has shown in 6 that this projective conjecture holds for all blocks with cyclic defect groups so we only need to verify the projective conjecture for p 2 and p 3 we do that in theorem 4 4 2 and theorem 4 3 1 for the above reasons the four theorems 2 10 1 3 6 1 4 4 2 and 4 3 1 below are sufficient to prove the inductive form of dade s conjecture for all covering groups of j 3 this paper is my ph d thesis it would not have been possible without the help and infinite patience of my advisor everett dade

information relaxations and duality in stochastic dynamic programs we describe a general technique for determining upper bounds on maximal values or lower bounds on minimal costs in stochastic dynamic programs in this approach we relax the nonanticipativity constraints that require decisions to depend only on the information available at the time a decision is made and impose a penalty that punishes violations of nonanticipativity in applications the hope is that this relaxed version of the problem will be simpler to solve than the original dynamic program the upper bounds provided by this dual approach complement lower bounds on values that may be found by simulating with heuristic policies we describe the theory underlying this dual approach and establish weak duality strong duality and complementary slackness results that are analogous to the duality results of linear programming we also study properties of good penalties finally we demonstrate the use of this dual approach in an adaptive inventory control problem with an unknown and changing demand distribution and in valuing options with stochastic volatilities and interest rates these are complex problems of significant practical interest that are quite difficult to solve to optimality in these examples our dual approach requires relatively little additional computation and leads to tight bounds on the optimal values

basis functions for general hsieh clough tocher triangles complete or reduced in this note we give the basis functions shape functions for hsieh clough tocher triangles complete or reduced in order to take the best advantage of symmetry and for simplicity we derive these functions for a general triangle by using simultaneously the area co ordinates and the so called eccentricity parameters we conclude with some remarks concerning the implementation

distributional efficiency in multiobjective stochastic linear programming several concepts of distributional efficiency are proposed for the multiobjective stochastic linear programming mslp problem in contexts where the probability distribution of random parameters is known and the decision maker dm has an unknown multi attribute utility function belonging to a given glass u we present a general efficient set the u admissible solutions and two subsets the u unanimous and u advocated solutions the latter being particularly relevant to the case of a single dm we show how advocated solutions can be generated and or tested when u is the class of non decreasing additive concave functions

bdf adi orthogonal spline collocation scheme for the fractional integro differential equation with two weakly singular kernels abstract in this paper a second order backward differentiation formula bdf alternatingdirection implicit adi orthogonal spline collocation osc method is presented for the fractional integro differential equation with two different weakly singular kernels the integral terms are approximated by the second order fractional quadrature rule suggested by lubich the convergence of the bdf adi osc method in l 2 norm is proved besides we provide the numerical experiments to demonstrate the results of theoretical analysis and show the accuracy and the effectiveness of the bdf adi osc method numerical experiments also exhibit the optimal error estimates

a polyhedral study on chance constrained program with random right hand side the essential structure of the mixed integer programming formulation for chance constrained program ccp is the intersection of multiple mixing sets with a 0 1 knapsack to improve our computational capacity on ccp an underlying substructure the single mixing set with a 0 1 knapsack has received substantial attentions recently in this study we consider a ccp problem with stochastic right hand side under a finite discrete distribution we first present a family of strong inequalities that subsumes known facet defining ones for that single mixing set due to the flexibility of our generalized inequalities we develop a new separation heuristic that has a complexity much less than existing one and guarantees generated cutting planes are facet defining for the polyhedron of ccp then we study lifting and superadditive lifting on knapsack cover inequalities and provide an implementable procedure on deriving another family of strong inequalities for the single mixing set finally different from the traditional approach that aggregates original constraints to investigate polyhedral implications due to their interactions we propose a novel blending procedure that produces strong valid inequalities for ccp by integrating those derived from individual mixing sets we show that under certain conditions they are the first type of facet defining inequalities describing intersection of multiple mixing sets and design an efficient separation heuristic for implementation in the computational experiments we perform a systematic study and illustrate the efficiency of the proposed inequalities on solving chance constrained static probabilistic lot sizing problems

complexity of stochastic dual dynamic programming stochastic dual dynamic programming is a cutting plane type algorithm for multi stage stochastic optimization originated about 30 years ago in spite of its popularity in practice there does not exist any analysis on the convergence rates of this method in this paper we first establish the number of iterations i e iteration complexity required by a basic dual dynamic programming method for solving single scenario multi stage optimization problems by introducing novel mathematical tools including the saturation of search points we then refine these basic tools and establish the iteration complexity for an explorative dual dynamic programing method proposed herein and the classic stochastic dual dynamic programming method for solving more general multi stage stochastic optimization problems under the standard stage wise independence assumption our results indicate that the complexity of these methods mildly increases with the number of stages t in fact linearly dependent on t for discounted problems therefore they are efficient for strategic decision making which involves a large number of stages but with a relatively small number of decision variables in each stage without explicitly discretizing the state and action spaces these methods might also be pertinent to the related reinforcement learning and stochastic control areas

an integer programming approach for linear programs with probabilistic constraints linear programs with joint probabilistic constraints pclp are difficult to solve because the feasible region is not convex we consider a special case of pclp in which only the right hand side is random and this random vector has a finite distribution we give a mixed integer programming formulation for this special case and study the relaxation corresponding to a single row of the probabilistic constraint we obtain two strengthened formulations as a byproduct of this analysis we obtain new results for the previously studied mixing set subject to an additional knapsack inequality we present computational results which indicate that by using our strengthened formulations instances that are considerably larger than have been considered before can be solved to optimality

convex approximations of chance constrained programs we consider a chance constrained problem where one seeks to minimize a convex objective over solutions satisfying with a given close to one probability a system of randomly perturbed convex constraints this problem may happen to be computationally intractable our goal is to build its computationally tractable approximation i e an efficiently solvable deterministic optimization program with the feasible set contained in the chance constrained problem we construct a general class of such convex conservative approximations of the corresponding chance constrained problem moreover under the assumptions that the constraints are affine in the perturbations and the entries in the perturbation vector are independent of each other random variables we build a large deviation type approximation referred to as bernstein approximation of the chance constrained problem this approximation is convex and efficiently solvable we propose a simulation based scheme for bounding the optimal value in the chance constrained problem and report numerical experiments aimed at comparing the bernstein and well known scenario approximation approaches finally we extend our construction to the case of ambiguous chance constrained problems where the random perturbations are independent with the collection of distributions known to belong to a given convex compact set rather than to be known exactly while the chance constraint should be satisfied for every distribution given by this set

risk minimization regret minimization and progressive hedging algorithms this paper begins with a study on the dual representations of risk and regret measures and their impact on modeling multistage decision making under uncertainty a relationship between risk envelopes and regret envelopes is established by using the lagrangian duality theory such a relationship opens a door to a decomposition scheme called progressive hedging for solving multistage risk minimization and regret minimization problems in particular the classical progressive hedging algorithm is modified in order to handle a new class of linkage constraints that arises from reformulations and other applications of risk and regret minimization problems numerical results are provided to show the efficiency of the progressive hedging algorithms

eventual convexity of probability constraints with elliptical distributions probability constraints are often employed to intuitively define safety of given decisions in optimization problems they simply express that a given system of inequalities depending on a decision vector and a random vector is satisfied with high enough probability it is known that even if this system is convex in the decision vector the associated probability constraint is not convex in general in this paper we show that some degree of convexity is still preserved for the large class of elliptical random vectors encompassing for example gaussian or student random vectors more precisely our main result establishes that under mild assumptions eventual convexity holds i e the probability constraint is convex when the safety level is large enough we also provide tools to compute a concrete convexity certificate from nominal problem data our results are illustrated on several examples including the situation of polyhedral systems with random technology matrices and arbitrary covariance structure

scenarios and policy aggregation in optimization under uncertainty a common approach in coping with multiperiod optimization problems under uncertainty where statistical information is not really enough to support a stochastic programming model has been to set up and analyze a number of scenarios the aim then is to identify trends and essential features on which a robust decision policy can be based this paper develops for the first time a rigorous algorithmic procedure for determining such a policy in response to any weighting of the scenarios the scenarios are bundled at various levels to reflect the availability of information and iterative adjustments are made to the decision policy to adapt to this structure and remove the dependence on hindsight

bfc a branch and fix coordination algorithmic framework for solving some types of stochastic pure and mixed 0 1 programs we present a framework for solving some types of 0 1 multi stage scheduling planning problems under uncertainty in the objective function coe cients and the right hand side a scenario analysis scheme with full recourse is used the solution o ered for each scenario group at each stage takes into account all scenarios but without subordinating to any of them the constraints are modelled by a splitting variables representation via scenarios so a 0 1 model for each scenario is consid ered plus the non anticipativity constraints that equate the 0 1 variables from the same group of scenarios in each stage the mathematical representation of the model is very amenable for the proposed framework to deal with the 0 1 character of the variables a branch and fix coordi nation approach is introduced for coordinating the selection of the branching nodes and branching variables in the scenario subproblems to be jointly optimized some computational experience is reported for di erent types of problems

mathematical programming approaches for generating p efficient points abstract probabilistically constrained problems in which the random variables are finitely distributed are non convex in general and hard to solve the p efficiency concept has been widely used to develop efficient methods to solve such problems those methods require the generation of p efficient points pleps and use an enumeration scheme to identify pleps in this paper we consider a random vector characterized by a finite set of scenarios and generate pleps by solving a mixed integer programming mip problem we solve this computationally challenging mip problem with a new mathematical programming framework it involves solving a series of increasingly tighter outer approximations and employs as algorithmic techniques a bundle preprocessing method strengthening valid inequalities and a fixing strategy the method is exact resp heuristic and ensures the generation of pleps resp quasi pleps if the fixing strategy is not resp is employed and it can be used to generate multiple pleps to the best of our knowledge generating a set of pleps using an optimization based approach and developing effective methods for the application of the p efficiency concept to the random variables described by a finite set of scenarios are novel we present extensive numerical results that highlight the computational efficiency and effectiveness of the overall framework and of each of the specific algorithmic techniques

mip reformulations of the probabilistic set covering problem in this paper we address the following probabilistic version psc of the set covering problem min cx mathbb p ax ge xi ge p x in 0 1 n where a is a 0 1 matrix xi is a random 0 1 vector and p in 0 1 is the threshold probability level we introduce the concepts of p inefficiency and polarity cuts while the former is aimed at deriving an equivalent mip reformulation of psc the latter is used as a strengthening device to obtain a stronger formulation simplifications of the mip model which result when one of the following conditions hold are briefly discussed a is a balanced matrix a has the circular ones property the components of xi are pairwise independent the distribution function of xi is a stationary distribution or has the disjunctive shattering property we corroborate our theoretical findings by an extensive computational experiment on a test bed consisting of almost 10 000 probabilistic instances this test bed was created using deterministic instances from the literature and consists of probabilistic variants of the set covering model and capacitated versions of facility location warehouse location and k median models our computational results show that our procedure is orders of magnitude faster than any of the existing approaches to solve psc and in many cases can reduce hours of computing time to a fraction of a second

a decomposition method and its application to convex programming a method is proposed for minimizing a function of the form q x d f1 x fm x where q is definite quadratic and fi are proper closed convex the key feature of the method is its capability of reducing the problem to a sequence of subproblems of the form min q x z fi x the method is an extension of the successive projection method given in han s p 1988 a successive projection method math programming 40 1 14 and has some similar features as the partial inverse method of spingarn spingarn j e 1983 partial inverse of a monotone operator appl math optim 10 247 265 in combination with the proximal point algorithm martinet b 1970 regularisation d inequations variationelles par approximations successives rev francaise inform rech oper 154 159 martinet b 1972 determination approchee d un point fixe d une application pseudo contractante c r acad sci paris 274 163 165 rockafellar r t 1976 monotone operators and the proximal point algorithm siam j control

probabilistic programming with discrete distributions and precedence constrained knapsack polyhedra we consider stochastic programming problems with probabilistic constraints involving random variables with discrete distributions they can be reformulated as large scale mixed integer programming problems with knapsack constraints using specific properties of stochastic programming problems and bounds on the probability of the union of events we develop new valid inequalities for these mixed integer programming problems we also develop methods for lifting these inequalities these procedures are used in a general iterative algorithm for solving probabilistically constrained problems the results are illustrated with a numerical example

approximation of shell geometry for nonlinear analysis abstract a major difficulty in the approximation of shell structures is the definition of the medium surface except for the particular case where it is known from analytical formula one has only reduced information like the positions of vertices and the normal vector to the medium surface at these points the situation is worse for geometrically nonlinear problems like arbitrary euler lagrange formulations where the geometry is continuously updated the goal of this paper is then to define a mathematical analysis of shell model approximations based on an updating of the geometry the first step is to define an approximation of all the quantities necessary for a shell formulation for instance it is possible to construct an approximation of the curvature tensor based on a linear interpolation of the unit normal on each plane element then the derivatives are piecewise constant one can guess that this is an improvement if the method is compared to the classical flat element method the second step is to define a finite element scheme where the shell geometry is approximated by the method mentioned previously as a matter of fact there is a major difficulty because the third order derivatives of the mapping which defines the medium surface of the shell are necessary in a consistent shell model like the koiter one derivatives of the curvature operator which are not approximated in the method using a mixed finite element technique enables one to overcome this problem but obviously the convergence when the mesh size tends to zero requires that these third order derivatives are bounded

stopping rules for a class of sampling based stochastic programming algorithms monte carlo sampling based algorithms hold much promise for solving stochastic programs with many scenarios a critical component of such algorithms is a stopping criterion to ensure the quality of the solution in this paper we develop a stopping rule theory for a class of algorithms that estimate bounds on the optimal objective function value by sampling we provide rules for selecting sample sizes and terminating the algorithm under which asymptotic validity of confidence intervals for the quality of the proposed solution can be verified empirical coverage results are given for a simple example

stochastic dual dynamic integer programming multistage stochastic integer programming msip combines the difficulty of uncertainty dynamics and non convexity and constitutes a class of extremely challenging problems a common formulation for these problems is a dynamic programming formulation involving nested cost to go functions in the linear setting the cost to go functions are convex polyhedral and decomposition algorithms such as nested benders decomposition and its stochastic variant stochastic dual dynamic programming sddp which proceed by iteratively approximating these functions by cuts or linear inequalities have been established as effective approaches however it is difficult to directly adapt these algorithms to msip due to the nonconvexity of integer programming value functions in this paper we propose an extension to sddp called stochastic dual dynamic integer programming sddip for solving msip problems with binary state variables the crucial component of the algorithm is a new reformulation of the subproblems in each stage and a new class of cuts termed lagrangian cuts derived from a lagrangian relaxation of a specific reformulation of the subproblems in each stage where local copies of state variables are introduced we show that the lagrangian cuts satisfy a tightness condition and provide a rigorous proof of the finite convergence of sddip with probability one we show that under fairly reasonable assumptions an msip problem with general state variables can be approximated by one with binary state variables to desired precision with only a modest increase in problem size thus our proposed sddip approach is applicable to very general classes of msip problems extensive computational experiments on three classes of real world problems namely electric generation expansion financial portfolio management and network revenue management show that the proposed methodology is very effective in solving large scale multistage stochastic integer optimization problems

approximation of general arch problems by straight beam elements in this paper we approximate the solution of a problem of a general arch by a nonconforming method using straight beam elements and taking into account numerical integration compatibility conditions which have to be satisfied at the mesh points are given these conditions ensure for this method the same order of convergence as usual conforming finite element methods

asynchronous level bundle methods in this paper we consider nonsmooth convex optimization problems with additive structure featuring independent oracles black boxes working in parallel existing methods for solving these distributed problems in a general form are synchronous in the sense that they wait for the responses of all the oracles before performing a new iteration in this paper we propose level bundle methods handling asynchronous oracles these methods require original upper bounds using upper models or scarce coordinations to deal with asynchronicity we prove their convergence using variational analysis techniques and illustrate their practical performance on a lagrangian decomposition problem

progressive hedging as a meta heuristic applied to stochastic lot sizing abstract in a great many situations the data for optimization problems cannot be known with certainty and furthermore the decision process will take place in multiple time stages as the uncertainties are resolved this gives rise to a need for stochastic programming sp methods that create solutions that are hedged against future uncertainty the progressive hedging algorithm pha of rockafellar and wets is a general method for sp we cast the pha in a meta heuristic framework with the sub problems generated for each scenario solved heuristically rather than using an approximate search algorithm for the exact problem as is typically the case in the meta heuristic literature we use an algorithm for sub problems that is exact in its usual context but serves as a heuristic for our meta heuristic computational results reported for stochastic lot sizing problems demonstrate that the method is effective