Progressive hedging innovations for a class of stochastic mixed-integer resource allocation problems

Numerous planning problems can be formulated as multi-stage stochastic programs and many possess key discrete (integer) decision variables in one or more of the stages. Progressive hedging (PH) is a scenario-based decomposition technique that can be leveraged to solve such problems. Originally devised for problems possessing only continuous variables, PH has been successfully applied as a heuristic to solve multi-stage stochastic programs with integer variables. However, a variety of critical issues arise in practice when implementing PH for the discrete case, especially in the context of very difficult or large-scale mixed-integer problems. Failure to address these issues properly results in either non-convergence of the heuristic or unacceptably long run-times. We investigate these issues and describe algorithmic innovations in the context of a broad class of scenario-based resource allocation problem in which decision variables represent resources available at a cost and constraints enforce the need for sufficient combinations of resources. The necessity and efficacy of our techniques is empirically assessed on a two-stage stochastic network flow problem with integer variables in both stages.     

A hierarchy of relaxations for nonlinear convex generalized disjunctive programming

We propose a framework to generate alternative mixed-integer nonlinear programming formulations for disjunctive convex programs that lead to stronger relaxations. We extend the concept of “basic steps” defined for disjunctive linear programs to the nonlinear case. A basic step is an operation that takes a disjunctive set to another with fewer number of conjuncts. We show that the strength of the relaxations increases as the number of conjuncts decreases, leading to a hierarchy of relaxations. We prove that the tightest of these relaxations, allows in theory the solution of the disjunctive convex program as a nonlinear programming problem. We present a methodology to guide the generation of strong relaxations without incurring an exponential increase of the size of the reformulated mixed-integer program. Finally, we apply the theory developed to improve the computational efficiency of solution methods for nonlinear convex generalized disjunctive programs (GDP). This methodology is validated through a set of numerical examples.     

On the Glivenko-Cantelli problem in stochastic programming: Mixed-integer linear recourse

Expected recourse functions in linear two-stage stochastic programs with mixed-integer second stage are approximated by estimating the underlying probability distribution via empirical measures. Under mild conditions, almost sure uniform convergence of the empirical means to the original expected recourse function is established.     

Continuity and stability of fully random two-stage stochastic programs with mixed-integer recourse

In order to derive continuity and stability of two-stage stochastic programs with mixed-integer recourse when all coefficients in the second-stage problem are random, we first investigate the quantitative continuity of the objective function of the corresponding continuous recourse problem with random recourse matrices. Then by extending derived results to the mixed-integer recourse case, the perturbation estimate and the piece-wise lower semi-continuity of the objective function are proved. Under the framework of weak convergence for probability measure, the epi-continuity and joint continuity of the objective function are established. All these results help us to prove a qualitative stability result. The obtained results extend current results to the mixed-integer recourse with random recourse matrices which have finitely many atoms.     

Lifting inequalities: a framework for generating strong cuts for nonlinear programs

In this paper, we introduce the first generic lifting techniques for deriving strong globally valid cuts for nonlinear programs. The theory is geometric and provides insights into lifting-based cut generation procedures, yielding short proofs of earlier results in mixed-integer programming. Using convex extensions, we obtain conditions that allow for sequence-independent lifting in nonlinear settings, paving a way for efficient cut-generation procedures for nonlinear programs. This sequence-independent lifting framework also subsumes the superadditive lifting theory that has been used to generate many general-purpose, strong cuts for integer programs. We specialize our lifting results to derive facet-defining inequalities for mixed-integer bilinear knapsack sets. Finally, we demonstrate the strength of nonlinear lifting by showing that these inequalities cannot be obtained using a single round of traditional integer programming cut-generation techniques applied on a tight reformulation of the problem.      

Lift-and-project cuts for convex mixed integer nonlinear programs

We describe a computationally effective method for generating lift-and-project cuts for convex mixed-integer nonlinear programs (MINLPs). The method relies on solving a sequence of cut-generating linear programs and in the limit generates an inequality as strong as the lift-and-project cut that can be obtained from solving a cut-generating nonlinear program. Using this procedure, we are able to approximately optimize over the rank one lift-and-project closure for a variety of convex MINLP instances. The results indicate that lift-and-project cuts have the potential to close a significant portion of the integrality gap for convex MINLPs. In addition, we find that using this procedure within a branch-and-cut solver for convex MINLPs significantly reduces the total solution time for many instances. We also demonstrate that combining lift-and-project cuts with an extended formulation that exploits separability of convex functions yields significant improvements in both relaxation bounds and the time to calculate the relaxation. Overall, these results suggest that with an effective separation routine, like the one proposed here, lift-and-project cuts may be as effective for solving convex MINLPs as they have been for solving mixed-integer linear programs.     

求解凸随机规划的Monte Carlo模拟方法

基于对目标函数和约束函数的同时抽样,给出求解凸随机规划的Monte CaLrlo模拟的算法及其收敛性.将得到的结果和算法应用到以半偏差为约束的投资组合优化问题,并且给出相应的数值试验.     

On Convergence of One-Step Schemes for Weak Solutions of Quantum Stochastic Differential Equations

Several one-step schemes for computing weak solutions of Lipschitzian quantum stochastic differential equations (QSDE) driven by certain operator-valued stochastic processes associated with creation, annihilation and gauge operators of quantum field theory are introduced and studied. This is accomplished within the framework of the Hudson–Parthasarathy formulation of quantum stochastic calculus and subject to the matrix elements of solution being sufficiently differentiable. Results concerning convergence of these schemes in the topology of the locally convex space of solution are presented. It is shown that the Euler–Maruyama scheme,with respect to weak convergence criteria for Itô stochastic differential equation is a special case of Euler schemes in this framework. Numerical examples are given.     

Global Optimization of 0-1 Hyperbolic Programs

We develop eight different mixed-integer convex programming reformulations of 0-1 hyperbolic programs. We obtain analytical results on the relative tightness of these formulations and propose a branch and bound algorithm for 0-1 hyperbolic programs. The main feature of the algorithm is that it reformulates the problem at every node of the search tree. We demonstrate that this algorithm has a superior convergence behavior than directly solving the relaxation derived at the root node. The algorithm is used to solve a discrete p-choice facility location problem for locating ten restaurants in the city of Edmonton.The research was supported in part by NSF awards DMII 95-02722 and BES 98-73586 to NVS.     

General Information

In this paper, we propose and analyze an SQP-type method for solving linearly constrained convex minimization problems where the objective functions are too complex to be evaluated exactly. Some basic results for global convergence and local superlinear convergence are obtained according to the properties of the approximation sequence. We illustrate the applicability of our approach by proposing a new method for solving two-stage stochastic programs with fixed recourse.     

Quantitative stability of mixed-integer two-stage quadratic stochastic programs

For our introduced mixed-integer quadratic stochastic program with fixed recourse matrices, random recourse costs, technology matrix and right-hand sides, we study quantitative stability properties of its optimal value function and optimal solution set when the underlying probability distribution is perturbed with respect to an appropriate probability metric. To this end, we first establish various Lipschitz continuity results about the value function and optimal solutions of mixed-integer parametric quadratic programs with parameters in the linear part of the objective function and in the right-hand sides of linear constraints. The obtained results extend earlier results about quantitative stability properties of stochastic integer programming and stability results for mixed-integer parametric quadratic programs.     

Continuity of parametric mixed-integer quadratic programs and its application to stability analysis of two-stage quadratic stochastic programs with mixed-integer recourse

For mixed-integer quadratic program where all coefficients in the objective function and the right-hand sides of constraints vary simultaneously, we show locally Lipschitz continuity of its optimal value function, and derive the corresponding global estimation; furthermore, we also obtain quantitative estimation about the change of its optimal solutions. Applying these results to two-stage quadratic stochastic program with mixed-integer recourse, we establish quantitative stability of the optimal value function and the optimal solution set with respect to the Fortet-Mourier probability metric, when the underlying probability distribution is perturbed. The obtained results generalize available results on continuity properties of mixed-integer quadratic programs and extend current results on quantitative stability of two-stage quadratic stochastic programs with mixed-integer recourse.     

A Reformulation-Linearization Technique (RLT) for semi-infinite and convex programs under mixed 0-1 and general discrete restrictions

The Reformulation-Linearization Technique (RLT) provides a hierarchy of relaxations spanning the spectrum from the continuous relaxation to the convex hull representation for linear 0-1 mixed-integer and general mixed-discrete programs. We show in this paper that this result holds identically for semi-infinite programs of this type. As a consequence, we extend the RLT methodology to describe a construct for generating a hierarchy of relaxations leading to the convex hull representation for bounded 0-1 mixed-integer and general mixed-discrete convex programs, using an equivalent semi-infinite linearized representation for such problems as an intermediate stepping stone in the analysis. For particular use in practice, we provide specialized forms of the resulting first-level RLT formulation for such mixed 0-1 and discrete convex programs, and illustrate these forms through two examples.     

Stochastic programming with integer variables

 Including integer variables into traditional stochastic linear programs has considerable implications for structural analysis and algorithm design. Starting from mean-risk approaches with different risk measures we identify corresponding two- and multi-stage stochastic integer programs that are large-scale block-structured mixed-integer linear programs if the underlying probability distributions are discrete. We highlight the role of mixed-integer value functions for structure and stability of stochastic integer programs. When applied to the block structures in stochastic integer programming, well known algorithmic principles such as branch-and-bound, Lagrangian relaxation, or cutting plane methods open up new directions of research. We review existing results in the field and indicate departure points for their extension. Received: December 2, 2002 / Accepted: April 23, 2003 Published online: May 28, 2003 Mathematics Subject Classification (2000): 90C15, 90C11, 90C06, 90C57     

A Class of stochastic programs with decision dependent uncertainty

We address a class of problems where decisions have to be optimized over a time horizon given that the future is uncertain and that the optimization decisions influence the time of information discovery for a subset of the uncertain parameters. The standard approach to formulate stochastic programs is based on the assumption that the stochastic process is independent of the optimization decisions, which is not true for the class of problems under consideration. We present a hybrid mixed-integer disjunctive programming formulation for the stochastic program corresponding to this class of problems and hence extend the stochastic programming framework. A set of theoretical properties that lead to reduction in the size of the model is identified. A Lagrangean duality based branch and bound algorithm is also presented. Financial support from the ExxonMobil Upstream Research Company is gratefully acknowledged.     

Global solution of nonlinear mixed-integer bilevel programs

An algorithm for the global optimization of nonlinear bilevel mixed-integer programs is presented, based on a recent proposal for continuous bilevel programs by Mitsos et al. (J Glob Optim 42(4):475–513, 2008). The algorithm relies on a convergent lower bound and an optional upper bound. No branching is required or performed. The lower bound is obtained by solving a mixed-integer nonlinear program, containing the constraints of the lower-level and upper-level programs; its convergence is achieved by also including a parametric upper bound to the optimal solution function of the lower-level program. This lower-level parametric upper bound is based on Slater-points of the lower-level program and subsets of the upper-level host sets for which this point remains lower-level feasible. Under suitable assumptions the KKT necessary conditions of the lower-level program can be used to tighten the lower bounding problem. The optional upper bound to the optimal solution of the bilevel program is obtained by solving an augmented upper-level problem for fixed upper-level variables. A convergence proof is given along with illustrative examples. An implementation is described and applied to a test set comprising original and literature problems. The main complication relative to the continuous case is the construction of the parametric upper bound to the lower-level optimal objective value, in particular due to the presence of upper-level integer variables. This challenge is resolved by performing interval analysis over the convex hull of the upper-level integer variables.     

Postoptimality for mean-risk stochastic mixed-integer programs and its application

The mean-risk stochastic mixed-integer programs can better model complex decision problems under uncertainty than usual stochastic (integer) programming models. In order to derive theoretical results in a numerically tractable way, the contamination technique is adopted in this paper for the postoptimality analysis of the mean-risk models with respect to changes in the scenario set, here the risk is measured by the lower partial moment. We first study the continuity of the objective function and the differentiability, with respect to the parameter contained in the contaminated distribution, of the optimal value function of the mean-risk model when the recourse cost vector, the technology matrix and the right-hand side vector in the second stage problem are all random. The postoptimality conclusions of the model are then established. The obtained results are applied to two-stage stochastic mixed-integer programs with risk objectives where the objective function is nonlinear with respect to the probability distribution. The current postoptimality results for stochastic programs are improved.     

On solving discrete two-stage stochastic programs having mixed-integer first- and second-stage variables

In this paper, we propose a decomposition-based branch-and-bound (DBAB) algorithm for solving two-stage stochastic programs having mixed-integer first- and second-stage variables. A modified Benders' decomposition method is developed, where the Benders' subproblems define lower bounding second-stage value functions of the first-stage variables that are derived by constructing a certain partial convex hull representation of the two-stage solution space. This partial convex hull is sequentially generated using a convexification scheme such as the Reformulation-Linearization Technique (RLT) or lift-and-project process, which yields valid inequalities that are reusable in the subsequent subproblems by updating the values of the first-stage variables. A branch-and-bound algorithm is designed based on a hyperrectangular partitioning process, using the established property that any resulting lower bounding Benders' master problem defined over a hyperrectangle yields the same objective value as the original stochastic program over that region if the first-stage variable solution is an extreme point of the defining hyperrectangle or the second-stage solution satisfies the binary restrictions. We prove that this algorithm converges to a global optimal solution. Some numerical examples and computational results are presented to demonstrate the efficacy of this approach.     

Decomposition with branch-and-cut approaches for two-stage stochastic mixed-integer programming

Decomposition has proved to be one of the more effective tools for the solution of large-scale problems, especially those arising in stochastic programming. A decomposition method with wide applicability is Benders' decomposition, which has been applied to both stochastic programming as well as integer programming problems. However, this method of decomposition relies on convexity of the value function of linear programming subproblems. This paper is devoted to a class of problems in which the second-stage subproblem(s) may impose integer restrictions on some variables. The value function of such integer subproblem(s) is not convex, and new approaches must be designed. In this paper, we discuss alternative decomposition methods in which the second-stage integer subproblems are solved using branch-and-cut methods. One of the main advantages of our decomposition scheme is that Stochastic Mixed-Integer Programming (SMIP) problems can be solved by dividing a large problem into smaller MIP subproblems that can be solved in parallel. This paper lays the foundation for such decomposition methods for two-stage stochastic mixed-integer programs.     

Converging multistep schemes for weak solutions of quantum stochastic differential equations

Multistep schemes for computing weak solutions of Lipschitzian quantum stochastic differential equations (QSDE) driven by certain operator-valued stochastic processes associated with the basic field operators of quantum field theory are introduced and studied. This is accomplished within the framework of the Hudson–Parthasarathy formulation of quantum stochastic calculus and subject to matrix element of solution being sufficiently differentiable. Results concerning convergence of explicit schemes of class A in the topology of the locally convex space of solution are presented.Numerical examples are given.