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Optimality functions in nonlinear programming conveniently measure, in some sense, the distance between a candidate solution and a stationary point. They may also provide guidance towards the development of implementable algorithms. In this paper, we use an optimality function to construct procedures for validation analysis in stochastic programs with nonlinear, possibly nonconvex, expected value functions as both objective and constraint functions. We construct an estimator of the optimality function and examine its consistency, bias, and asymptotic distribution. The estimator leads to confidence intervals for the value of the optimality function at a candidate solution and, hence, provides a quantitative measure of solution quality. We also construct an implementable algorithm for solving smooth stochastic programs based on sample average approximations and the optimality function estimator. Preliminary numerical tests illustrate the proposed algorithm and validation analysis procedures.

Agricultural sustainability under climate change is a major challenge in semi-arid countries, mainly because of over-exploited water resources. This article explores short- and long-term consequences of farmers’ adaptation decisions on groundwater resource use, under several climate change scenarios. We model farmer decisions on crop choice, investment in irrigation and water application rates, using a stochastic dynamic programming model with embedded year and season decision stages. Several sources of risk are considered that may impact farmer decisions, with poor rainfall affecting crop yield and market prices, while driving crop and borewell failure probabilities. We further investigate the performance of water management policies for groundwater resource conservation. This is achieved through policy simulations from a calibrated version of the stochastic dynamic model, using data from a field survey in the Berambadi watershed, Karnataka state, southern India. The most relevant and novel aspect of our model is the joint consideration of (i) investment decisions about irrigation over a long-term horizon and with the probability of borewell failure, (ii) several water management policies, and (iii) detailed farmers’ water practices and the representation of crop choice for each agricultural season with crop failure.

To model combinatorial decision problems involving uncertainty and probability, we introduce scenario based stochastic constraint programming. Stochastic constraint programs contain both decision variables, which we can set, and stochastic variables, which follow a discrete probability distribution. We provide a semantics for stochastic constraint programs based on scenario trees. Using this semantics, we can compile stochastic constraint programs down into conventional (non-stochastic) constraint programs. This allows us to exploit the full power of existing constraint solvers. We have implemented this framework for decision making under uncertainty in stochastic OPL, a language which is based on the OPL constraint modelling language [Hentenryck et al., 1999]. To illustrate the potential of this framework, we model a wide range of problems in areas as diverse as portfolio diversification, agricultural planning and production/inventory management.

This paper formed part of a preliminary research report for a risk consultancy and academic research. Stochastic Programming models provide a powerful paradigm for decision making under uncertainty. In these models the uncertainties are represented by a discrete scenario tree and the quality of the solutions obtained is governed by the quality of the scenarios generated. We propose a new technique to generate scenarios based on Gaussian Mixture Hidden Markov Modelling. We show that our approach explicitly captures important time varying dynamics of stochastic processes (such as autoregression and jumps) as well as non-Gaussian distribution characteristics (such as skewness and kurtosis). Our scenario generation method enables richer robustness and scenario analysis through exploiting the tractable properties of Markov models and Gaussian mixture distributions. We demonstrate the benefits of our scenario generation method by conducting numerical experiments on FTSE-100 data.

Mixed-Integer Programming has traditionally been restricted to deterministic models. Recent research has opened the door to stochastic optimization models, which are typically dynamic in nature. This project lays the foundation for stochastic dynamic mixed-integer and linear programming (SD-MIP). This project has produced several new ideas in connection with a) convexification of two-stage mixed-integer sets and b) multi-stage (including two-stage) stochastic linear programming. Together a) and b) provide the foundations for SD-MIP problems. From new concepts and algorithms to applications and software, this project has made significant breakthroughs in all aspects. This report provides a synopsis of both theoretical and computational results. As a preview, we mention that currently available deterministic MIP solvers, as powerful as they are known to be, are unable to solve SD-MIP models of modest size within an hour of computing. In contrast, our decomposition approach provides provably optimal solutions within the hour time-limit.

Mixed-Integer Programming has traditionally been restricted to deterministic models. Recent research has opened the door to stochastic optimization models, which are typically dynamic in nature. This project lays the foundation for stochastic dynamic mixed-integer and linear programming (SD-MIP). This project has produced several new ideas in connection with a) convexification of two-stage mixed-integer sets and b) multi-stage (including two-stage) stochastic linear programming. Together a) and b) provide the foundations for SD-MIP problems. From new concepts and algorithms to applications and software, this project has made significant breakthroughs in all aspects. This report provides a synopsis of both theoretical and computational results. As a preview, we mention that currently available deterministic MIP solvers, as powerful as they are known to be, are unable to solve SD-MIP models of modest size within an hour of computing. In contrast, our decomposition approach provides provably optimal solutions within the hour time-limit.

Mixed-Integer Programming has traditionally been restricted to deterministic models. Recent research has opened the door to stochastic optimization models, which are typically dynamic in nature. This project lays the foundation for stochastic dynamic mixed-integer and linear programming (SD-MIP). This project has produced several new ideas in connection with a) convexification of two-stage mixed-integer sets and b) multi-stage (including two-stage) stochastic linear programming. Together a) and b) provide the foundations for SD-MIP problems. From new concepts and algorithms to applications and software, this project has made significant breakthroughs in all aspects. This report provides a synopsis of both theoretical and computational results. As a preview, we mention that currently available deterministic MIP solvers, as powerful as they are known to be, are unable to solve SD-MIP models of modest size within an hour of computing. In contrast, our decomposition approach provides provably optimal solutions within the hour time-limit.

Mixed-Integer Programming has traditionally been restricted to deterministic models. Recent research has opened the door to stochastic optimization models, which are typically dynamic in nature. This project lays the foundation for stochastic dynamic mixed-integer and linear programming (SD-MIP). This project has produced several new ideas in connection with a) convexification of two-stage mixed-integer sets and b) multi-stage (including two-stage) stochastic linear programming. Together a) and b) provide the foundations for SD-MIP problems. From new concepts and algorithms to applications and software, this project has made significant breakthroughs in all aspects. This report provides a synopsis of both theoretical and computational results. As a preview, we mention that currently available deterministic MIP solvers, as powerful as they are known to be, are unable to solve SD-MIP models of modest size within an hour of computing. In contrast, our decomposition approach provides provably optimal solutions within the hour time-limit.

We extend the proof of the dynamic programming principle (DPP) for standard stochastic optimal control problems driven by general L\'{e}vy noises. Under appropriate assumptions, it is shown that the DPP still holds when the state process fails to have any moments at all.

This document considers nonlinear programming problems with stochastic constraints. The Lagrangian corresponding to such problems has a stochastic part, which in this work is replaced by its certainty equivalent (in the sense of expected utility theory). It is shown that the deterministic surrogate problem thus obtained, contains a penalty function which penalized violation of the constraints in the mean. The dual problem is studied (for problems with stochastic righthand sides in the constraints) and a comprehensive duality theory is developed by introducing a new certainty equivalent concept, which possesses, for arbitrary utility functions, some of the properties that the classical certainty equivalent retains only for the exponential utility. Additional keywords: Minmax theorems; Convex functions; and Risk aversion. (Author)

We are concerned with the linear-quadratic optimal stochastic control problem with random coefficients. Under suitable conditions, we prove that the value field $V(t,x,\omega), (t,x,\omega)\in [0,T]\times R^n\times \Omega$, is quadratic in $x$, and has the following form: $V(t,x)=\langle K_tx, x\rangle$ where $K$ is an essentially bounded nonnegative symmetric matrix-valued adapted processes. Using the dynamic programming principle (DPP), we prove that $K$ is a continuous semi-martingale of the form $$K_t=K_0+\int_0^t \, dk_s+\sum_{i=1}^d\int_0^tL_s^i\, dW_s^i, \quad t\in [0,T]$$ with $k$ being a continuous process of bounded variation and $$E\left[\left(\int_0^T|L_s|^2\, ds\right)^p\right]

This research investigates a special class of stochastic linear programs known as two-stage stochastic linear programming with relatively complete and fixed recourse. These models characterize a two-phase process where the first- stage decision (itself subject to a separate set of first-stage linear constraints) allocates a set of resources to the second-stage linear program prior to the realization of random variables affecting second-stage resource availability. Since the second-stage decision deterministically follows both first-stage allocation and random variable realization, the first-stage variables constitute the only true decision. The expected cost of the two-stage recourse problem is also a piecewise convex function of the first-stage decision variables, thus allowing a global optimal solution that minimizes the total expected cost.

This paper deals with the stochastic control of nonlinear systems in the presence of state and control constraints, for uncertain discrete-time dynamics in finite dimensional spaces. In the deterministic case, the viability kernel is known to play a basic role for the analysis of such problems and the design of viable control feedbacks. In the present paper, we show how a stochastic viability kernel and viable feedbacks relying on probability (or chance) constraints can be defined and computed by a dynamic programming equation. An example illustrates most of the assertions.

Differential dynamic programming (DDP) has been demonstrated as a viable approach to low-thrust trajectory optimization, namely with the recent success of NASA's Dawn mission. The Dawn trajectory was designed with the DDP-based Static/Dynamic Optimal Control algorithm used in the Mystic software.1 Another recently developed method, Hybrid Differential Dynamic Programming (HDDP),2, 3 is a variant of the standard DDP formulation that leverages both first-order and second-order state transition matrices in addition to nonlinear programming (NLP) techniques. Areas of improvement over standard DDP include constraint handling, convergence properties, continuous dynamics, and multi-phase capability. DDP is a gradient based method and will converge to a solution nearby an initial guess. In this study, monotonic basin hopping (MBH) is employed as a stochastic search method to overcome this limitation, by augmenting the HDDP algorithm for a wider search of the solution space.

Differential dynamic programming (DDP) has been demonstrated as a viable approach to low-thrust trajectory optimization, namely with the recent success of NASA's Dawn mission. The Dawn trajectory was designed with the DDP-based Static/Dynamic Optimal Control algorithm used in the Mystic software.1 Another recently developed method, Hybrid Differential Dynamic Programming (HDDP),2, 3 is a variant of the standard DDP formulation that leverages both first-order and second-order state transition matrices in addition to nonlinear programming (NLP) techniques. Areas of improvement over standard DDP include constraint handling, convergence properties, continuous dynamics, and multi-phase capability. DDP is a gradient based method and will converge to a solution nearby an initial guess. In this study, monotonic basin hopping (MBH) is employed as a stochastic search method to overcome this limitation, by augmenting the HDDP algorithm for a wider search of the solution space.

Differential dynamic programming (DDP) has been demonstrated as a viable approach to low-thrust trajectory optimization, namely with the recent success of NASA's Dawn mission. The Dawn trajectory was designed with the DDP-based Static/Dynamic Optimal Control algorithm used in the Mystic software.1 Another recently developed method, Hybrid Differential Dynamic Programming (HDDP),2, 3 is a variant of the standard DDP formulation that leverages both first-order and second-order state transition matrices in addition to nonlinear programming (NLP) techniques. Areas of improvement over standard DDP include constraint handling, convergence properties, continuous dynamics, and multi-phase capability. DDP is a gradient based method and will converge to a solution nearby an initial guess. In this study, monotonic basin hopping (MBH) is employed as a stochastic search method to overcome this limitation, by augmenting the HDDP algorithm for a wider search of the solution space.

Differential dynamic programming (DDP) has been demonstrated as a viable approach to low-thrust trajectory optimization, namely with the recent success of NASA's Dawn mission. The Dawn trajectory was designed with the DDP-based Static/Dynamic Optimal Control algorithm used in the Mystic software.1 Another recently developed method, Hybrid Differential Dynamic Programming (HDDP),2, 3 is a variant of the standard DDP formulation that leverages both first-order and second-order state transition matrices in addition to nonlinear programming (NLP) techniques. Areas of improvement over standard DDP include constraint handling, convergence properties, continuous dynamics, and multi-phase capability. DDP is a gradient based method and will converge to a solution nearby an initial guess. In this study, monotonic basin hopping (MBH) is employed as a stochastic search method to overcome this limitation, by augmenting the HDDP algorithm for a wider search of the solution space.

Differential dynamic programming (DDP) has been demonstrated as a viable approach to low-thrust trajectory optimization, namely with the recent success of NASA's Dawn mission. The Dawn trajectory was designed with the DDP-based Static/Dynamic Optimal Control algorithm used in the Mystic software.1 Another recently developed method, Hybrid Differential Dynamic Programming (HDDP),2, 3 is a variant of the standard DDP formulation that leverages both first-order and second-order state transition matrices in addition to nonlinear programming (NLP) techniques. Areas of improvement over standard DDP include constraint handling, convergence properties, continuous dynamics, and multi-phase capability. DDP is a gradient based method and will converge to a solution nearby an initial guess. In this study, monotonic basin hopping (MBH) is employed as a stochastic search method to overcome this limitation, by augmenting the HDDP algorithm for a wider search of the solution space.

The replacement of nuclear power with renewable energy sources is the main theme of Taiwanese energy policy. While nuclear power provides approximately 16% of total electricity supply, investigation of whether renewable energy could generate sufficient electricity to replace nuclear power becomes an important research question. Bioenergy, whose development is highly dependent on stable supply of agricultural commodities and residuals, is of particular interest to Taiwanese government because a significant amount of cropland is currently available. However, change in past climate is evidenced to alter regional temperature and precipitation, resulting in non-neglectable influences on agricultural activities and crop production, and consequently on bioenergy production. To explore how climate change plays a role in bioenergy development, this study incorporates multiple climate change scenarios and develops a two-stage stochastic programming model to analyze Taiwan's bioenergy development under different market conditions. The results show that under a small climate-induced crop yield change, net bioenergy production will not change a lot, while land use and agricultural resource allocation could vary considerably. In addition, at higher GHG prices, ethanol will not be produced and all feedstocks will be used in pyrolysis electricity, providing approximately 1.58% of total energy demand. The result also indicates that a desired level of carbon emission can be achieved, but high fluctuation of government expenses on supporting policies may occur when climate impacts are uncertain. Based on our findings, bioenergy alone is not able to provide enough electricity if nuclear power plants are shut down, and collaborative development of other renewable energy such wind power and solar energy may be required.

This document studies the dual problem corresponding to a linear program in which the stochastic objective function is replaced by its expected utility, and discusses its relevance as a penalty method to a stochastically constrained dual linear program.

This paper presents a method to approximately solve stochastic optimal control problems in which the cost function and the system dynamics are polynomial. For stochastic systems with polynomial dynamics, the moments of the state can be expressed as a, possibly infinite, system of deterministic linear ordinary differential equations. By casting the problem as a deterministic control problem in moment space, semidefinite programming is used to find a lower bound on the optimal solution. The constraints in the semidefinite program are imposed by the ordinary differential equations for moment dynamics and semidefiniteness of the outer product of moments. From the solution to the semidefinite program, an approximate optimal control strategy can be constructed using a least squares method. In the linear quadratic case, the method gives an exact solution to the optimal control problem. In more complex problems, an infinite number of moment differential equations would be required to compute the optimal control law. In this case, we give a procedure to increase the size of the semidefinite program, leading to increasingly accurate approximations to the true optimal control strategy.

We analyze a zero-sum stochastic differential game between two competing players who can choose unbounded controls. The payoffs of the game are defined through backward stochastic differential equations. We prove that each player's priority value satisfies a weak dynamic programming principle and thus solves the associated fully non-linear partial differential equation in the viscosity sense.

Stochastic algorithms are efficient approaches to solving machine learning and optimization problems. In this paper, we propose a general framework called Splash for parallelizing stochastic algorithms on multi-node distributed systems. Splash consists of a programming interface and an execution engine. Using the programming interface, the user develops sequential stochastic algorithms without concerning any detail about distributed computing. The algorithm is then automatically parallelized by a communication-efficient execution engine. We provide theoretical justifications on the optimal rate of convergence for parallelizing stochastic gradient descent. Splash is built on top of Apache Spark. The real-data experiments on logistic regression, collaborative filtering and topic modeling verify that Splash yields order-of-magnitude speedup over single-thread stochastic algorithms and over state-of-the-art implementations on Spark.

Measuring and managing risk has become crucial in modern decision making under stochastic uncertainty. In two-stage stochastic programming, mean risk models are essentially defined by a parametric recourse problem and a quantification of risk. From the perspective of qualitative robustness theory, we discuss sufficient conditions for continuity of the resulting objective functions with respect to perturbation of the underlying probability measure. Our approach covers a fairly comprehensive class of both stochastic-programming related risk measures and relevant recourse models. Not only this unifies previous approaches but also extends known stability results for two-stage stochastic programs to models with mixed-integer quadratic recourse and mixed-integer convex recourse, respectively.

We introduce the online stochastic Convex Programming (CP) problem, a very general version of stochastic online problems which allows arbitrary concave objectives and convex feasibility constraints. Many well-studied problems like online stochastic packing and covering, online stochastic matching with concave returns, etc. form a special case of online stochastic CP. We present fast algorithms for these problems, which achieve near-optimal regret guarantees for both the i.i.d. and the random permutation models of stochastic inputs. When applied to the special case online packing, our ideas yield a simpler and faster primal-dual algorithm for this well studied problem, which achieves the optimal competitive ratio. Our techniques make explicit the connection of primal-dual paradigm and online learning to online stochastic CP.

A new decision-theoretic approach to Nonlinear Programming Problems with stochastic constraints is introduced. The Stochastic Program (SP) is replaced by a Deterministic Program (DP) in which a term is added to the objective function to penalize solutions which are not feasible in the mean. The special feature of the author's approach is the choice of the penalty function P sub E, which is given in terms if the relative entropy functional, and is accordingly called entropic penalty. It is shown that P sub E has properties which make it suitable to treat stochastic programs. Some of these properties are derived via a dual representation independent. The dual representation is also used to express the Deterministric Problem (DP) as a saddle function problem. For problems in which the randomness occurs in the rhs of the constraints, it shown that the dual problem of (DP) is equivalent to Expected Utility Maximization of the classical Lagrangian dual function of (SP), with the utility being of the constant-risk-aversion type. Finally, mean-variance approximations of P sub E and the induced Approximate Deterministic Program are considered.

The study was concerned with the essential question of how to address multiperiod inventory problems characterized by not unrealistic conditions for which modeling and solution procedures have not been developed. The research placed special emphasis on the application of deterministic mixed-integer programming models to multiperiod inventory problems characterized by changing costs and beta-distributed demands. The special concern for the mixed-integer programming model was prompted by the realization that, among all of the easy- to-use deterministic inventory models, the mixed-integer programming formulation is the only model that is amenable to the additional constraints and multiple- objective criteria that coincide with broadly conceived statements of inventory control. Through the combined usage of mixed-integer programming and computer simulation techniques, a method was developed whereby a first-period reorder policy that minimizes expected total inventory cost over a multiperiod planning horizon can be identified with a nominal investment in computer processing time. The analysis led to the conclusion that first-period policies obtained by using the mixed-integer programming model with expectations as periodic demand inputs are generally adequate under the conditions specified in the research and compare favorably with policies obtained from commonly used inventory models. (Author)

This thesis proposes new methods, based on dynamic programming, for solving certain single-stage and multi-stage integer stochastic knapsack problems. These problems model stochastic portfolio optimization problems (SPOPs) which assume deterministic unit weight, and normally distributed unit return with known mean and variance for each item type. Given an initial wealth, the objective is to select a portfolio that maximizes the probability of achieving or exceeding a specified final return threshold; the multi-stage problem allows revisions of the portfolio at regular time intervals. An exact method is developed to solve a single-stage SPOP with independence of returns among item types. For a problem from the literature with 11 item types, this method obtains an optimal solution in a fraction of a second on a laptop computer. An approximation method, based on discretization of possible wealth values, is developed to solve a multi-stage SPOP with inter- and intra-stage independence of returns among item types. Running on a desktop computer, this approximation method solves a 3-stage problem with 6 item types in under 12 minutes. With finer discretization in a 3-stage problem with 8 item types, the solution time is about 46 minutes.

This work seeks to understand the connections between learning and decision making under uncertainty. Specifically, we ask that question: when we are going to use learned models within the loop of a larger decision making process, how should we alter the learning procedure or somehow tune the learning to the specific needs of the actual decision making task? To answer this question, we developed a theory of task based model learning, learning models tuned not (just) for predictive accuracy, but to optimize the closed loop performance of a decision making procedure (specifically, those based on stochastic optimization) that uses these models as an intermediate step. Training such models requires that we differentiate through an optimization problem, for which we developed the theory and implementations. On several tasks, we show that such learning substantially outperforms traditional learning processes, where the learning and decision making stages are separate.

This work seeks to understand the connections between learning and decision making under uncertainty. Specifically, we ask that question: when we are going to use learned models within the loop of a larger decision making process, how should we alter the learning procedure or somehow tune the learning to the specific needs of the actual decision making task? To answer this question, we developed a theory of task based model learning, learning models tuned not (just) for predictive accuracy, but to optimize the closed loop performance of a decision making procedure (specifically, those based on stochastic optimization) that uses these models as an intermediate step. Training such models requires that we differentiate through an optimization problem, for which we developed the theory and implementations. On several tasks, we show that such learning substantially outperforms traditional learning processes, where the learning and decision making stages are separate.

In this paper, we study one kind of stochastic recursive optimal control problem with the obstacle constraints for the cost function where the cost function is described by the solution of one reflected backward stochastic differential equations. We will give the dynamic programming principle for this kind of optimal control problem and show that the value function is the unique viscosity solution of the obstacle problem for the corresponding Hamilton-Jacobi-Bellman equations.

Fashion discounters face the problem of ordering the right amount of pieces in each size of a product. The product is ordered in pre-packs containing a certain size-mix of a product. For this so-called lot-type design problem, a stochastic mixed integer linear programm was developed, in which price cuts serve as recourse action for oversupply. Our goal is to answer the question, whether the resulting supply strategy leads to a supply that is significantly more consistent with the demand for sizes compared to the original manual planning. Since the total profit is influenced by too many factors unrelated to sizes (like the popularity of the product, the weather or a changing economic situation), we suggest a comparison method which excludes many outer effects by construction. We apply the method to a real-world field study: The improvements in the size distributions of the supply are significant.

We study Graver test sets for families of linear multi-stage stochastic integer programs with varying number of scenarios. We show that these test sets can be decomposed into finitely many ``building blocks'', independent of the number of scenarios, and we give an effective procedure to compute these building blocks. The paper includes an introduction to Nash-Williams' theory of better-quasi-orderings, which is used to show termination of our algorithm. We also apply this theory to finiteness results for Hilbert functions.

Motivated by applications to multi-antenna wireless networks, we propose a distributed and asynchronous algorithm for stochastic semidefinite programming. This algorithm is a stochastic approximation of a continous- time matrix exponential scheme regularized by the addition of an entropy-like term to the problem's objective function. We show that the resulting algorithm converges almost surely to an $\varepsilon$-approximation of the optimal solution requiring only an unbiased estimate of the gradient of the problem's stochastic objective. When applied to throughput maximization in wireless multiple-input and multiple-output (MIMO) systems, the proposed algorithm retains its convergence properties under a wide array of mobility impediments such as user update asynchronicities, random delays and/or ergodically changing channels. Our theoretical analysis is complemented by extensive numerical simulations which illustrate the robustness and scalability of the proposed method in realistic network conditions.

In this work we study optimization problems subject to a failure constraint. This constraint is expressed in terms of a condition that causes failure, representing a physical or technical breakdown. We formulate the problem in terms of a probability constraint, where the level of "confidence" is a modelling parameter and has the interpretation that the probability of failure should not exceed that level. Application of the stochastic Arrow-Hurwicz algorithm poses two difficulties: one is structural and arises from the lack of convexity of the probability constraint, and the other is the estimation of the gradient of the probability constraint. We develop two gradient estimators with decreasing bias via a convolution method and a finite difference technique, respectively, and we provide a full analysis of convergence of the algorithms. Convergence results are used to tune the parameters of the numerical algorithms in order to achieve best convergence rates, and numerical results are included via an example of application in finance.

We prove the dynamic programming principe for uniformly nondegenerate stochastic differential games in the framework of time-homogeneous diffusion processes considered up to the first exit time from a domain. The zeroth-order "coefficient" and the "free" term are only assumed to be measurable. In contrast with previous results established for constant stopping times we allow arbitrary stopping times and randomized ones as well. The main assumption, which will be removed in a subsequent article, is that there exists a sufficiently regular solution of the Isaacs equation.

In this work we study the stochastic recursive control problem, in which the aggregator (or called generator) of the backward stochastic differential equation describing the running cost is continuous but not necessarily Lipschitz with respect to the first unknown variable and the control, and monotonic with respect to the first unknown variable. The dynamic programming principle and the connection between the value function and the viscosity solution of the associated Hamilton-Jacobi-Bellman equation are established in this setting by the generalized comparison theorem of backward stochastic differential equations and the stability of viscosity solutions. Finally we take the control problem of continuous-time Epstein-Zin utility with non-Lipschitz aggregator as an example to demonstrate the application of our study.

We propose a hybrid algorithmic strategy for complex stochastic optimization problems, which combines the use of scenario trees from multistage stochastic programming with machine learning techniques for learning a policy in the form of a statistical model, in the context of constrained vector-valued decisions. Such a policy allows one to run out-of-sample simulations over a large number of independent scenarios, and obtain a signal on the quality of the approximation scheme used to solve the multistage stochastic program. We propose to apply this fast simulation technique to choose the best tree from a set of scenario trees. A solution scheme is introduced, where several scenario trees with random branching structure are solved in parallel, and where the tree from which the best policy for the true problem could be learned is ultimately retained. Numerical tests show that excellent trade-offs can be achieved between run times and solution quality.

In networks, there are often more than one source of capacity. The capacities can be permanently or temporarily owned by the decision maker. Depending on the nature of sources, we identify the permanent capacity, spot market capacity and contract capacity. We use a scenario tree to model the uncertainty, and build a multi-stage stochastic integer program that can incorporate multiple sources and multiple types of capacities in a general network. We propose two solution methodologies for the problem. Firstly, we design an asymptotically convergent approximation algorithm. Secondly, we design a cutting plane algorithm based on Benders decomposition to find tight bounds for the problem. The numerical experiments show superb performance of the proposed algorithms compared with commercial software.

Structural design generated by traditional method, optimization method and the stochastic design concept are compared. In the traditional method, the constraints are manipulated to obtain the design and weight is back calculated. In design optimization, the weight of a structure becomes the merit function with constraints imposed on failure modes and an optimization algorithm is used to generate the solution. Stochastic design concept accounts for uncertainties in loads, material properties, and other parameters and solution is obtained by solving a design optimization problem for a specified reliability. Acceptable solutions were produced by all the three methods. The variation in the weight calculated by the methods was modest. Some variation was noticed in designs calculated by the methods. The variation may be attributed to structural indeterminacy. It is prudent to develop design by all three methods prior to its fabrication. The traditional design method can be improved when the simplified sensitivities of the behavior constraint is used. Such sensitivity can reduce design calculations and may have a potential to unify the traditional and optimization methods. Weight versus reliabilitytraced out an inverted-S-shaped graph. The center of the graph corresponded to mean valued design. A heavy design with weight approaching infinity could be produced for a near-zero rate of failure. Weight can be reduced to a small value for a most failure-prone design. Probabilistic modeling of load and material properties remained a challenge.

This paper studies several stochastic programming formulations of dynamic oligopolistic games under uncertainty. We argue that one of the models, namely Games with Probabilistic Scenarios (GPS), provides an appropriate formulation. For such games, we show that symmetric players earn greater expected profits as demand volatility increases. This result suggests that even in an increasingly volatile market players may have an incentive to participate in the market. The key to our approach is the so-called scenario formulation of stochastic programming. In addition to several modeling insights, we also discuss the application of GPS to the electricity market in Ontario, Canada. The examples presented in this paper illustrate that this approach can address dynamic games that are clearly out of reach for dynamic programming, a common approach in the literature on dynamic games.

The application of Genetic Programming to the discovery of empirical laws is often impaired by the huge size of the search space, and consequently by the computer resources needed. In many cases, the extreme demand for memory and CPU is due to the massive growth of non-coding segments, the introns. The paper presents a new program evolution framework which combines distribution-based evolution in the PBIL spirit, with grammar-based genetic programming; the information is stored as a probability distribution on the gra mmar rules, rather than in a population. Experiments on a real-world like problem show that this approach gives a practical solution to the problem of intron growth.

A question which combines statistics and linear programming considerations was first raised by G. Tintner (Econometrica 28:2, 490-5, April 60). It concerns the distribution of optimum functional values when a linear programming problem has probabilistic constraints. It is proposed to accord a chance constrained programming formulation to this kind of problem and to deal with it in a way that bears on project scheduling of the kind that is usually associated with critical path analysis, for instance, in PERT. The main focus of this paper is on the statistical distributions of the project completion (and subcompletion) times. The question of total time distributions that we deal with can therefore be given a managerial policy flavor by assuming that, ab initio, a management is considering a contract for a certain project. The task sequences are known but the times are not known except in probability. Before contracting for a target completion date--with resulting delay penalties--this management would like to know the likely distribution of total times in order to decide whether to accept an offered contract or else bargain further on the completion dates, penalty rates and progress payments and prices.

The general linear programming problem is considered in which the coefficients of the objective function to be maximized are assumed to be random variables with a known multinormal distribution. Three deterministic reformulations involve maximizing the expected value, the alpha-fractile (alpha fixed, 0 alpha 1/2), and the probability of exceeding a predetermined level of payoff, respectively. In this paper the author's previous work on 'bi- criterion programs' is applied to derive an algorithm for routinely and efficiently solving the second and third reformulations. A by-product of the calculations in each case is the tradeoff-curve between the criterion being maximized and expected payoff. The intimate relationships between all three reformulations are illuminated.

It is shown how the functional equation technique of dynamic programming may be used to obtain a new computational and analytic approach to variational problems. The limited memory capacity of present-day digital computers limits the successful application of these techniques to first and second order systems at the moment, with limited application to higher order systems.

To model combinatorial decision problems involving uncertainty and probability, we extend the stochastic constraint programming framework proposed in [Walsh, 2002] along a number of important dimensions (e.g. to multiple chance constraints and to a range of new objectives). We also provide a new (but equivalent) semantics based on scenarios. Using this semantics, we can compile stochastic constraint programs down into conventional (nonstochastic) constraint programs. This allows us to exploit the full power of existing constraint solvers. We have implemented this framework for decision making under uncertainty in stochastic OPL, a language which is based on the OPL constraint modelling language [Hentenryck et al., 1999]. To illustrate the potential of this framework, we model a wide range of problems in areas as diverse as finance, agriculture and production.

The proposed project was aimed at exploring various theoretical and algorithmic issues at the intersection of three optimization areas, namely, parametric, stochastic and bilevel integer programming, as well as related applications. The main contribution of the project is development of novel algorithmic methodologies (along with the necessary theoretical foundations) for solving stochastic and bilevel integer programs built upon exploiting equivalent value function reformulations. While computational limitations exist for the proposed approaches, the preliminary results of our experiments are extremely encouraging as for several broad classes of stochastic and bilevel integer optimization problems we are able to solve instances that are among the largest instances solved in the literature. Additionally, we explore several interesting related applications including those arising in wireless sensor networks (and, possibly, other networked systems). For the considered applications we derive structural properties of the optimal policies and exploit them to develop exact solution techniques. Finally, we provide theoretical investigation of randomized restart algorithms in the context of algorithm portfolios (i.e., set of algorithms run in parallel). In particular, we provide the theoretical upper bound on the computational value of mixing randomized restart algorithms with different properties. Furthermore, the constructive proof of the main result allows us to characterize restart algorithms that are capable of forming an effective mixed algorithm portfolio.

The LP recourse problem applies to two-stage optimization problems where uncertainty in resource availability of the second stage hinders informed decision making. The recourse function affords a way to compensate "later" for an error in prediction "now." The literature provides a rich body of work on the optimization of such problems, but little research has been accomplished regarding the characterization of the surface in the local region of optimality, in particular sensitivity analysis. A decision maker faced with considerations other than the modeled objective function must be presented with a way to estimate the impact of operating at non-optimal decision variable values. This work develops and demonstrates a technique for characterizing the surface using response surface methodology. Specifically, the flexibility and utility of RSM techniques applied to this class of problems is demonstrated, and a methodology for characterizing the surface in the local region using a low-order polynomial is developed.

Multi stage stochastic programs arise in many applications from engineering whenever a set of inventories or stocks has to be valued. Such is the case in seasonal storage valuation of a set of cascaded reservoir chains in hydro management. A popular method is Stochastic Dual Dynamic Programming (SDDP), especially when the dimensionality of the problem is large and Dynamic programming no longer an option. The usual assumption of SDDP is that uncertainty is stage-wise independent, which is highly restrictive from a practical viewpoint. When possible, the usual remedy is to increase the state-space to account for some degree of dependency. In applications this may not be possible or it may increase the state space by too much. In this paper we present an alternative based on keeping a functional dependency in the SDDP - cuts related to the conditional expectations in the dynamic programming equations. Our method is based on popular methodology in mathematical finance, where it has progressively replaced scenario trees due to superior numerical performance. On a set of numerical examples, we too show the interest of this way of handling dependency in uncertainty, when combined with SDDP. Our method is readily available in the open source software package StOpt.

Two-stage stochastic mixed-integer programming (SMIP) problems with general integer variables in the second-stage are generally difficult to solve. This paper develops the theory of integer set reduction for characterizing the subset of the convex hull of feasible integer points of the second-stage subproblem which can be used for solving the SMIP. The basic idea is to consider a small enough subset of feasible integer points that is necessary for generating a valid inequality for the integer subproblem. An algorithm for obtaining such a subset based on the solution of the subproblem LP-relaxation is then devised and incorporated into the Fenchel decomposition method for SMIP. To demonstrate the performance of the new integer set reduction methodology, a computational study based on randomly generated test instances was performed. The results of the study show that integer set reduction provides significant gains in terms of generating cuts faster leading to better bounds in solving SMIPs than using a direct solver.