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We consider a dynamic vehicle routing problem with time windows and stochastic customers (DS-VRPTW), such that customers may request for services as vehicles have already started their tours. To solve this problem, the goal is to provide a decision rule for choosing, at each time step, the next action to perform in light of known requests and probabilistic knowledge on requests likelihood. We introduce a new decision rule, called Global Stochastic Assessment (GSA) rule for the DS-VRPTW, and we compare it with existing decision rules, such as MSA. In particular, we show that GSA fully integrates nonanticipativity constraints so that it leads to better decisions in our stochastic context. We describe a new heuristic approach for efficiently approximating our GSA rule. We introduce a new waiting strategy. Experiments on dynamic and stochastic benchmarks, which include instances of different degrees of dynamism, show that not only our approach is competitive with state-of-the-art methods, but also enables to compute meaningful offline solutions to fully dynamic problems where absolutely no a priori customer request is provided.

We consider a dynamic vehicle routing problem with time windows and stochastic customers (DS-VRPTW), such that customers may request for services as vehicles have already started their tours. To solve this problem, the goal is to provide a decision rule for choosing, at each time step, the next action to perform in light of known requests and probabilistic knowledge on requests likelihood. We introduce a new decision rule, called Global Stochastic Assessment (GSA) rule for the DS-VRPTW, and we compare it with existing decision rules, such as MSA. In particular, we show that GSA fully integrates nonanticipativity constraints so that it leads to better decisions in our stochastic context. We describe a new heuristic approach for efficiently approximating our GSA rule. We introduce a new waiting strategy. Experiments on dynamic and stochastic benchmarks, which include instances of different degrees of dynamism, show that not only our approach is competitive with state-of-the-art methods, but also enables to compute meaningful offline solutions to fully dynamic problems where absolutely no a priori customer request is provided.

Use of Chance-Constrained Programming to Account for Stochastic Variation in the A-Matrix of Large-Scale Linear Programs: A Forestry Application

Use of Chance-Constrained Programming to Account for Stochastic Variation in the A-Matrix of Large-Scale Linear Programs: A Forestry Application

Use of Chance-Constrained Programming to Account for Stochastic Variation in the A-Matrix of Large-Scale Linear Programs: A Forestry Application

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Traditional Linear Genetic Programming (LGP) algorithms are based only on the selection mechanism to guide the search. Genetic operators combine or mutate random portions of the individuals, without knowing if the result will lead to a fitter individual. Probabilistic Model Building Genetic Programming (PMB-GP) methods were proposed to overcome this issue through a probability model that captures the structure of the fit individuals and use it to sample new individuals. This work proposes the use of LGP with a Stochastic Context-Free Grammar (SCFG), that has a probability distribution that is updated according to selected individuals. We proposed a method for adapting the grammar into the linear representation of LGP. Tests performed with the proposed probabilistic method, and with two hybrid approaches, on several symbolic regression benchmark problems show that the results are statistically better than the obtained by the traditional LGP.

Traditional Linear Genetic Programming (LGP) algorithms are based only on the selection mechanism to guide the search. Genetic operators combine or mutate random portions of the individuals, without knowing if the result will lead to a fitter individual. Probabilistic Model Building Genetic Programming (PMB-GP) methods were proposed to overcome this issue through a probability model that captures the structure of the fit individuals and use it to sample new individuals. This work proposes the use of LGP with a Stochastic Context-Free Grammar (SCFG), that has a probability distribution that is updated according to selected individuals. We proposed a method for adapting the grammar into the linear representation of LGP. Tests performed with the proposed probabilistic method, and with two hybrid approaches, on several symbolic regression benchmark problems show that the results are statistically better than the obtained by the traditional LGP.

We propose a first-order method for stochastic strongly convex optimization that attains $O(1/n)$ rate of convergence, analysis show that the proposed method is simple, easily to implement, and in worst case, asymptotically four times faster than its peers. We derive this method from several intuitive observations that are generalized from existing first order optimization methods.

We propose a first-order method for stochastic strongly convex optimization that attains $O(1/n)$ rate of convergence, analysis show that the proposed method is simple, easily to implement, and in worst case, asymptotically four times faster than its peers. We derive this method from several intuitive observations that are generalized from existing first order optimization methods.

We propose a first-order method for stochastic strongly convex optimization that attains $O(1/n)$ rate of convergence, analysis show that the proposed method is simple, easily to implement, and in worst case, asymptotically four times faster than its peers. We derive this method from several intuitive observations that are generalized from existing first order optimization methods.

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.

In this paper we analyze the effect of two modelling approaches for supply planning problems under uncertainty: two-stage stochastic programming (SP) and robust optimization (RO). The comparison between the two approaches is performed through a scenario-based framework methodology, which can be applied to any optimization problem affected by uncertainty. For SP we compute the minimum expected cost based on the specific probability distribution of the uncertain parameters related to a set of scenarios. For RO we consider static approaches where random parameters belong to box or ellipsoidal uncertainty sets in compliance with the data used to generate SP scenarios. Dynamic approaches for RO, via the concept of adjustable robust counterpart, are also considered. The efficiency of the methodology has been illustrated for a supply planning problem to optimize vehicle-renting and procurement transportation activities involving uncertainty on demands and on buying costs for extra-vehicles. Numerical experiments through the scenario-based framework allow a fair comparison in real case instances. Advantages and disadvantages of RO and SP are discussed.

In this paper we analyze the effect of two modelling approaches for supply planning problems under uncertainty: two-stage stochastic programming (SP) and robust optimization (RO). The comparison between the two approaches is performed through a scenario-based framework methodology, which can be applied to any optimization problem affected by uncertainty. For SP we compute the minimum expected cost based on the specific probability distribution of the uncertain parameters related to a set of scenarios. For RO we consider static approaches where random parameters belong to box or ellipsoidal uncertainty sets in compliance with the data used to generate SP scenarios. Dynamic approaches for RO, via the concept of adjustable robust counterpart, are also considered. The efficiency of the methodology has been illustrated for a supply planning problem to optimize vehicle-renting and procurement transportation activities involving uncertainty on demands and on buying costs for extra-vehicles. Numerical experiments through the scenario-based framework allow a fair comparison in real case instances. Advantages and disadvantages of RO and SP are discussed.

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.

The shortage of water resources and the increasing competition among water users have highlighted the importance of the water allocation problem. Water availability is crucial for water resource allocation and changes frequently, leading to the necessity to predict available water. This paper develops a framework aimed to plan regional water allocations under different representative concentration pathways (RCP) scenarios using an interval parameter conditional value-at-risk (CVaR) two-stage stochastic programming model. This framework combines prediction and optimization to reflect climate change, the uncertainty of water system and the coordination between water resources allocation and risk. The feasibility and practicality of the framework are demonstrated by its application in a real-world case study in the Lower Songhua River Basin in northeast China. Comparison between the results of the developed model and actual conditions show that 11.61 × 108 m3 volume of water supply can be saved after optimization, indicating that the developed model tends to allocate water in a more efficient way. The ratio of surface water to groundwater is reduced from 2:1 to 1.62:1. The proposed model has practical relevance for saving water and alleviating groundwater overexploitation. The approach is applicable to most areas with severe water shortages and groundwater overexploitation, and decision makers can determine the appropriate options for water resources allocation based on risk preferences and actual conditions.

The shortage of water resources and the increasing competition among water users have highlighted the importance of the water allocation problem. Water availability is crucial for water resource allocation and changes frequently, leading to the necessity to predict available water. This paper develops a framework aimed to plan regional water allocations under different representative concentration pathways (RCP) scenarios using an interval parameter conditional value-at-risk (CVaR) two-stage stochastic programming model. This framework combines prediction and optimization to reflect climate change, the uncertainty of water system and the coordination between water resources allocation and risk. The feasibility and practicality of the framework are demonstrated by its application in a real-world case study in the Lower Songhua River Basin in northeast China. Comparison between the results of the developed model and actual conditions show that 11.61 × 108 m3 volume of water supply can be saved after optimization, indicating that the developed model tends to allocate water in a more efficient way. The ratio of surface water to groundwater is reduced from 2:1 to 1.62:1. The proposed model has practical relevance for saving water and alleviating groundwater overexploitation. The approach is applicable to most areas with severe water shortages and groundwater overexploitation, and decision makers can determine the appropriate options for water resources allocation based on risk preferences and actual conditions.

This paper proposes a general duality framework for the problem of minimizing a convex integral functional over a space of stochastic processes adapted to a given filtration. The framework unifies many well-known duality frameworks from operations research and mathematical finance. The unification allows the extension of some useful techniques from these two fields to a much wider class of problems. In particular, combining certain finite-dimensional techniques from convex analysis with measure theoretic techniques from mathematical finance, we are able to close the duality gap in some situations where traditional topological arguments fail.

In this paper, we consider two sequential decision making problems with a convexity structure, namely an energy storage optimization task and a multi-product assembly example. We formulate these problems in the stochastic programming framework and discuss an approximate dynamic programming technique for their solutions. As the cost-to-go functions are convex in these cases, we use max-affine estimates for their approximations. To train such a max-affine estimate, we provide a new convex regression algorithm, and evaluate it empirically for these planning scenarios.

The energetic optimization problem, e.g., searching for the optimal switch- ing protocol of certain system parameters to minimize the input work, has been extensively studied by stochastic thermodynamics. In current work, we study this problem numerically with iterative dynamic programming. The model systems under investigation are toy actuators consisting of spring-linked beads with loading force imposed on both ending beads. For the simplest case, i.e., a one-spring actuator driven by tuning the stiffness of the spring, we compare the optimal control protocol of the stiffness for both the overdamped and the underdamped situations, and discuss how inertial effects alter the irreversibility of the driven process and thus modify the optimal protocol. Then, we study the systems with multiple degrees of freedom by constructing oligomer actuators, in which the harmonic interaction between the two ending beads is tuned externally. With the same rated output work, actuators of different constructions demand different minimal input work, reflecting the influence of the internal degrees of freedom on the performance of the actuators.

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. In contrast with previous results established for constant stopping times we allow arbitrary stopping times and randomized ones as well. There is no assumption about solvability of the the Isaacs equation in any sense (classical or viscosity). The zeroth-order "coefficient" and the "free" term are only assumed to be measurable in the space variable. We also prove that value functions are uniquely determined by the functions defining the corresponding Isaacs equations and thus stochastic games with the same Isaacs equation have the same value functions.

Book Source: Digital Library of India Item 2015.195244 dc.contributor.author: Ram Shanker Sachan dc.date.accessioned: 2015-07-08T05:50:44Z dc.date.available: 2015-07-08T05:50:44Z dc.date.digitalpublicationdate: 0000-00-00 dc.identifier.barcode: 1990010093963 dc.identifier.origpath: /rawdataupload/upload/0093/963 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/195244 dc.description.scanningcentre: IIIT, Allahabad dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 202 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher: Iit Kanpur dc.rights: Out_of_copyright dc.source.library: I I T Kanpur dc.subject.classification: Mathematics dc.title: Some Aspects Of Stochastic Nonlinear Programming Problems

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 paper develops a unified methodology for probabilistic analysis and optimal control design for jump diffusion processes defined by polynomials. For such systems, the evolution of the moments of the state can be described via a system of linear ordinary differential equations. Typically, however, the moments are not closed and an infinite system of equations is required to compute statistical moments exactly. Existing methods for stochastic analysis, known as closure methods, focus on approximating this infinite system of equations with a finite dimensional system. This work develops an alternative approach in which the higher order terms, which are approximated in closure methods, are viewed as inputs to a finite-dimensional linear control system. Under this interpretation, upper and lower bounds of statistical moments can be computed via convex linear optimal control problems with semidefinite constraints. For analysis of steady-state distributions, this optimal control problem reduces to a static semidefinite program. These same optimization problems extend automatically to stochastic optimal control problems. For minimization problems, the methodology leads to guaranteed lower bounds on the true optimal value. Furthermore, we show how an approximate optimal control strategy can be constructed from the solution of the semidefinite program. The results are illustrated using numerous examples.

We consider the infinite dimensional linear programming (inf-LP) approach for solving stochastic control problems. The inf-LP corresponding to problems with uncountable state and input spaces is in general computationally intractable. By focusing on linear systems with quadratic cost (LQG), we establish a connection between this approach and the well-known Riccati LMIs. In particular, we show that the semidefinite programs known for the LQG problem can be derived from the pair of primal and dual inf-LPs. Furthermore, we establish a connection between multi-objective and chance constraint criteria and the inf-LP formulation.

This document presents a variant of Karmarkar's projective algorithm for block angular structured linear programs, such as stochastic linear programs. By computing the projection efficiently, the authors give a worst case bound on the order of the running time that can be an order of magnitude better than that of Karmarkar's standard algorithm. A related variant is applied to the dual program, and its implications for very large-scale problems are given. Keywords: Iterations; Points (Mathematics). (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.

Probabilistic modeling and machine learning have proven to be powerful tools in many defense, industrial, and scientific computing applications. Unfortunately, their continuing adoption has been hindered because engineering with them requires PhD-level expertise. Our research in this program led to the creation of multiple open-source probabilistic programming languages. These languages achieved key program goals, such as (i) reducing the lines of code required to build state-of-the-art machine learning systems by 50x; (ii) making machine learning and data science capabilities accessible to a broader class of programmers, by providing automatic model discovery mechanisms and simple, SQL like query languages; (iii) making it possible to deploy rich generative models to solve applied problems, and thereby solve hard 3D computer vision problems with no training data; and (iv) revealing interfaces and abstractions that unify abroad set of probabilistic programming languages and enable multiple inference strategies or solvers'' to interoperate

This research investigated an alternative to the traditional approaches of optimizing a stochastic response surface subject to constraints. This research investigated the bias in the expected value of the solution, possible alternative decision variable settings, and a method to improve the solution. A three step process is presented to evaluate stochastic response surfaces subject to constraints. Step 1 is to use a traditional approach to estimate the response surface and a covariance matrix through regression. Step 2 samples the objective function of the linear program (i.e., response surface) and identifies the extreme points visited. Step 3 presents a method to estimate the optimal extreme point and present that information to a decision maker.

This paper introduces a new stochastic global optimization method targeting protein-protein docking problems, an important class of problems in computational structural biology. The method is based on finding general convex quadratic underestimators to the binding energy function that is funnel-like. Finding the optimum underestimator requires solving a semidefinite programming problem, hence the name semidefinite programming-based underestimation (SDU). The underestimator is used to bias sampling in the search region. It is established that under appropriate conditions SDU locates the global energy minimum with probability approaching one as the sample size grows. A detailed comparison of SDU with a related method of convex global underestimator (CGU), and computational results for protein-protein docking problems are provided.

Results and problems in the ordinary 'here-and-now' and 'wait-and- see' stochastic linear programming problems are described. A general formulation of the 'here-and-now' problem is presented, and an approach for solving a special kind of 'here-and-now' problem is suggested.

Decomposition and 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. These rules are applied to a multistage stochastic linear programming algorithm due to Pereira and Pinto.

Decomposition and 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. These rules are applied to a multistage stochastic linear programming algorithm due to Pereira and Pinto.

Battery Energy Storage Systems (BESS) can mitigate effects of intermittent energy production from renewable energy sources and play a critical role in peak shaving and demand charge management. To optimally size the BESS from an economic perspective, the trade-off between BESS investment costs, lifetime, and revenue from utility bill savings along with microgrid ancillary services must be taken into account. The optimal size of a BESS is solved via a stochastic optimization problem considering wholesale market pricing. A stochastic model is used to schedule arbitrage services for energy storage based on the forecasted energy market pricing while accounting for BESS cost trends, the variability of renewable energy resources, and demand prediction. The uniqueness of the approach proposed in this paper lies in the convex optimization programming framework that computes a globally optimal solution to the financial trade-off solution. The approach is illustrated by application to various realistic case studies based on pricing and demand data from the California Independent System Operator (CAISO). The case study results give insight in optimal BESS sizing from a cost perspective, based on both yearly scheduling and daily BESS operation.