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Advantages of electric vehicles (EV) include reduction of greenhouse gas and other emissions, energy security, and fuel economy. The societal benefits of large-scale adoption of EVs cannot be realized without adequate deployment of publicly accessible charging stations. We propose a two-stage stochastic programming model to determine the optimal network of charging stations for a community considering uncertainties in arrival and dwell time of vehicles, battery state of charge of arriving vehicles, walkable range and charging preferences of drivers, demand during weekdays and weekends, and rate of adoption of EVs within a community. We conducted studies using sample average approximation (SAA) method which asymptotically converges to an optimal solution for a two-stage stochastic problem, however it is computationally expensive for large-scale instances. Therefore, we developed a heuristic to produce near to optimal solutions quickly for our data instances. We conducted computational experiments using various publicly available data sources, and benefits of the solutions are evaluated both quantitatively and qualitatively for a given community.

Advantages of electric vehicles (EV) include reduction of greenhouse gas and other emissions, energy security, and fuel economy. The societal benefits of large-scale adoption of EVs cannot be realized without adequate deployment of publicly accessible charging stations. We propose a two-stage stochastic programming model to determine the optimal network of charging stations for a community considering uncertainties in arrival and dwell time of vehicles, battery state of charge of arriving vehicles, walkable range and charging preferences of drivers, demand during weekdays and weekends, and rate of adoption of EVs within a community. We conducted studies using sample average approximation (SAA) method which asymptotically converges to an optimal solution for a two-stage stochastic problem, however it is computationally expensive for large-scale instances. Therefore, we developed a heuristic to produce near to optimal solutions quickly for our data instances. We conducted computational experiments using various publicly available data sources, and benefits of the solutions are evaluated both quantitatively and qualitatively for a given community.

Advantages of electric vehicles (EV) include reduction of greenhouse gas and other emissions, energy security, and fuel economy. The societal benefits of large-scale adoption of EVs cannot be realized without adequate deployment of publicly accessible charging stations. We propose a two-stage stochastic programming model to determine the optimal network of charging stations for a community considering uncertainties in arrival and dwell time of vehicles, battery state of charge of arriving vehicles, walkable range and charging preferences of drivers, demand during weekdays and weekends, and rate of adoption of EVs within a community. We conducted studies using sample average approximation (SAA) method which asymptotically converges to an optimal solution for a two-stage stochastic problem, however it is computationally expensive for large-scale instances. Therefore, we developed a heuristic to produce near to optimal solutions quickly for our data instances. We conducted computational experiments using various publicly available data sources, and benefits of the solutions are evaluated both quantitatively and qualitatively for a given community.

We use a rank one Gaussian perturbation to derive a smooth stochastic approximation of the maximum eigenvalue function. We then combine this smoothing result with an optimal smooth stochastic optimization algorithm to produce an efficient method for solving maximum eigenvalue minimization problems. We show that the complexity of this new method is lower than that of deterministic smoothing algorithms in certain precision/dimension regimes.

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.

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.

The ubiquitous computing system is a paradigm shift from personal computing to physical integration. This study focuses on the deterministic and stochastic service provider model to provide sub-services to computing nodes to minimize rejection values. This deterministic service provider model aims to reduce the cost of sending data from one place to another by considering the processing capacity at each node and the demand for each sub-service. At the same time, stochastic service provider aims to optimize service provision in a stochastic environment where parameters such as demand and capacity may change randomly. The novelties of this research are the deterministic and stochastic service provider models and algorithms with mixed integer programming (MIP). The test results show that the solution found meets all the constraints and the smallest objective function value. Stochastic modeling minimizes denial of service problems during wireless sensor network (WSN) distribution. The model resented is the ability of wireless sensors to establish connections between distributed computing nodes. Stochastic modeling minimizes denial of service problems during WSN distribution.

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.

Ariyawansa and Zhu [3] have proposed a new class of optimization problems termed stochastic semidefinite programs (SSDPs) to handle data uncertainty in applications leading to (deterministic) semidefinite programs (DSDPs). For the case where the event space of the random variables in an SSDP is finite they have also derived a class of volumetric barrier decomposition algorithms, and proved polynomial complexity of the short-step and long-step members of the class [2]. When the event space of the random variables in an SSDP is finite, the SSDP is equivalent to a large scale DSDP with special structure. Polynomial homogeneous self-dual algorithms [11] are an important class of algorithms that have been introduced for solving (general) DSDPs. It is therefore possible to solve SSDPs by applying homogeneous self-dual algorithms to their DSDP equivalents. However, such algorithms, while polynomial, will still have high computational complexities in comparison to decomposition algorithms. In this paper, we show how the special structure in DSDP equivalents of SSDPs can be exploited to design homogeneous self-dual algorithms with computation

Ariyawansa and Zhu [3] have proposed a new class of optimization problems termed stochastic semidefinite programs (SSDPs) to handle data uncertainty in applications leading to (deterministic) semidefinite programs (DSDPs). For the case where the event space of the random variables in an SSDP is finite they have also derived a class of volumetric barrier decomposition algorithms, and proved polynomial complexity of the short-step and long-step members of the class [2]. When the event space of the random variables in an SSDP is finite, the SSDP is equivalent to a large scale DSDP with special structure. Polynomial homogeneous self-dual algorithms [11] are an important class of algorithms that have been introduced for solving (general) DSDPs. It is therefore possible to solve SSDPs by applying homogeneous self-dual algorithms to their DSDP equivalents. However, such algorithms, while polynomial, will still have high computational complexities in comparison to decomposition algorithms. In this paper, we show how the special structure in DSDP equivalents of SSDPs can be exploited to design homogeneous self-dual algorithms with computation

Few if any validated guidelines exist for making decisions about the design, media, or format of new instructional products. This study examined strings of programmed learning responses to create general guidelines for making such decisions. Using a Markov model, tables were developed relating the expected proportion of students to be in a solution state at a given accuracy level and at a given level of confidence with respect to the length of response strings.

We present a new primal-dual algorithm for computing the value of the Lagrangian dual of a stochastic mixed-integer program (SMIP) formed by relaxing its nonanticipativity constraints. This dual is widely used in decomposition methods for the solution of SMIPs. The algorithm relies on the well-known progressive hedging method, but unlike previous progressive hedging approaches for SMIP, our algorithm can be shown to converge to the optimal Lagrangian dual value. The key improvement in the new algorithm is an inner loop of optimized linearization steps, similar to those taken in the classical Frank-Wolfe method. Numerical results demonstrate that our new algorithm empirically outperforms the standard implementation of progressive hedging for obtaining bounds in SMIP.

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.

In this study, we consider two classes of multicriteria two-stage stochastic programs in finite probability spaces with multivariate risk constraints. The first-stage problem features a multivariate stochastic benchmarking constraint based on a vector-valued random variable representing multiple and possibly conflicting stochastic performance measures associated with the second-stage decisions. In particular, the aim is to ensure that the associated random outcome vector of interest is preferable to a specified benchmark with respect to the multivariate polyhedral conditional value-at-risk (CVaR) or a multivariate stochastic order relation. In this case, the classical decomposition methods cannot be used directly due to the complicating multivariate stochastic benchmarking constraints. We propose an exact unified decomposition framework for solving these two classes of optimization problems and show its finite convergence. We apply the proposed approach to a stochastic network design problem in a pre-disaster humanitarian logistics context and conduct a computational study concerning the threat of hurricanes in the Southeastern part of the United States. Our numerical results on these large-scale problems show that our proposed algorithm is computationally scalable.

In this study, we consider two classes of multicriteria two-stage stochastic programs in finite probability spaces with multivariate risk constraints. The first-stage problem features a multivariate stochastic benchmarking constraint based on a vector-valued random variable representing multiple and possibly conflicting stochastic performance measures associated with the second-stage decisions. In particular, the aim is to ensure that the associated random outcome vector of interest is preferable to a specified benchmark with respect to the multivariate polyhedral conditional value-at-risk (CVaR) or a multivariate stochastic order relation. In this case, the classical decomposition methods cannot be used directly due to the complicating multivariate stochastic benchmarking constraints. We propose an exact unified decomposition framework for solving these two classes of optimization problems and show its finite convergence. We apply the proposed approach to a stochastic network design problem in a pre-disaster humanitarian logistics context and conduct a computational study concerning the threat of hurricanes in the Southeastern part of the United States. Our numerical results on these large-scale problems show that our proposed algorithm is computationally scalable.

In this study, we consider two classes of multicriteria two-stage stochastic programs in finite probability spaces with multivariate risk constraints. The first-stage problem features a multivariate stochastic benchmarking constraint based on a vector-valued random variable representing multiple and possibly conflicting stochastic performance measures associated with the second-stage decisions. In particular, the aim is to ensure that the associated random outcome vector of interest is preferable to a specified benchmark with respect to the multivariate polyhedral conditional value-at-risk (CVaR) or a multivariate stochastic order relation. In this case, the classical decomposition methods cannot be used directly due to the complicating multivariate stochastic benchmarking constraints. We propose an exact unified decomposition framework for solving these two classes of optimization problems and show its finite convergence. We apply the proposed approach to a stochastic network design problem in a pre-disaster humanitarian logistics context and conduct a computational study concerning the threat of hurricanes in the Southeastern part of the United States. Our numerical results on these large-scale problems show that our proposed algorithm is computationally scalable.

In this study, we consider two classes of multicriteria two-stage stochastic programs in finite probability spaces with multivariate risk constraints. The first-stage problem features a multivariate stochastic benchmarking constraint based on a vector-valued random variable representing multiple and possibly conflicting stochastic performance measures associated with the second-stage decisions. In particular, the aim is to ensure that the associated random outcome vector of interest is preferable to a specified benchmark with respect to the multivariate polyhedral conditional value-at-risk (CVaR) or a multivariate stochastic order relation. In this case, the classical decomposition methods cannot be used directly due to the complicating multivariate stochastic benchmarking constraints. We propose an exact unified decomposition framework for solving these two classes of optimization problems and show its finite convergence. We apply the proposed approach to a stochastic network design problem in a pre-disaster humanitarian logistics context and conduct a computational study concerning the threat of hurricanes in the Southeastern part of the United States. Our numerical results on these large-scale problems show that our proposed algorithm is computationally scalable.

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.

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.

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.

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.

Airspace capacity reduction due to convective weather impedes air traffic flows and causes traffic congestion. This study presents an algorithm that reroutes flights in the presence of winds, enroute convective weather, and congested airspace based on stochastic dynamic programming. A stochastic disturbance model incorporates into the reroute design process the capacity uncertainty. A trajectory-based airspace demand model is employed for calculating current and future airspace demand. The optimal routes minimize the total expected traveling time, weather incursion, and induced congestion costs. They are compared to weather-avoidance routes calculated using deterministic dynamic programming. The stochastic reroutes have smaller deviation probability than the deterministic counterpart when both reroutes have similar total flight distance. The stochastic rerouting algorithm takes into account all convective weather fields with all severity levels while the deterministic algorithm only accounts for convective weather systems exceeding a specified level of severity. When the stochastic reroutes are compared to the actual flight routes, they have similar total flight time, and both have about 1% of travel time crossing congested enroute sectors on average. The actual flight routes induce slightly less traffic congestion than the stochastic reroutes but intercept more severe convective weather.

The theory of programming under uncertainty is extended to the case when the decision variables are elements of a Banach space. This approach leads to a very natural application of the computational techniques of mathematical programming to stochastic optimal control problems. It is shown that there exists an equivalent deterministic mathematical program whose set of feasible solutions is a convex set and whose objective function can be expressed as a convex function of the initial decision variables. In the second part, a duality theory is developed for this class of problems and some of the relations to the maximum principle for stochastic linear control problems are given.

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)

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.

The applicability of known stochastic programming models and methods for the solution of problems in classical statistics and probability is shown by a number of examples. These concern testing of hypotheses, constructing of tolerance regions, planning of optimal sampling and the Moran model for the dam. (Author)

Recent developments in the area of reinforcement learning have yielded a number of new algorithms for the prediction and control of Markovian environments. These algorithms, including the TD lambda) algorithm of Sutton (1988) and the Q-learning algorithm of Watkins (1989), can be motivated heuristically as approximations to dynamic programming (DP). In this paper we provide a rigorous proof of convergence of these DP-based learning algorithms by relating them to the powerful techniques of stochastic approximation theory via a new convergence theorem. The theorem establishes a general class of convergent algorithms to which both TD(lambda) and Q-learning belong. reinforcement learning, Stochastic approximation, Convergence, Dynamic programming.

We present a problem data set for stochastic programming, and associated real world applications. The problem descriptions were collected from the literature, with emphasis on variety of problem structure and application. Each problem has a short description, mathematical problem statement, and notational reconciliation to a standard problem format. In addition, most problems have one or more corresponding data files in SMPS1 format.

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.

The multiresponse surface problem is modelled as one of multiobjective stochastic optimisation, and diverse solutions are proposed. Several crucial differences are highlighted between this approach and others that have been proposed. Finally, in a numerical example, some particular solutions are applied and described in detail.

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 consider stochastic influence maximization problems arising in social networks. In contrast to existing studies that involve greedy approximation algorithms with a 63% performance guarantee, our work focuses on solving the problem optimally. To this end, we introduce a new class of problems that we refer to as two-stage stochastic submodular optimization models. We propose a delayed constraint generation algorithm to find the optimal solution to this class of problems with a finite number of samples. The influence maximization problems of interest are special cases of this general problem class. We show that the submodularity of the influence function can be exploited to develop strong optimality cuts that are more effective than the standard optimality cuts available in the literature. Finally, we report our computational experiments with large-scale real-world datasets for two fundamental influence maximization problems, independent cascade and linear threshold, and show that our proposed algorithm outperforms the greedy algorithm.

We consider stochastic influence maximization problems arising in social networks. In contrast to existing studies that involve greedy approximation algorithms with a 63% performance guarantee, our work focuses on solving the problem optimally. To this end, we introduce a new class of problems that we refer to as two-stage stochastic submodular optimization models. We propose a delayed constraint generation algorithm to find the optimal solution to this class of problems with a finite number of samples. The influence maximization problems of interest are special cases of this general problem class. We show that the submodularity of the influence function can be exploited to develop strong optimality cuts that are more effective than the standard optimality cuts available in the literature. Finally, we report our computational experiments with large-scale real-world datasets for two fundamental influence maximization problems, independent cascade and linear threshold, and show that our proposed algorithm outperforms the greedy algorithm.

Multi-stage stochastic linear programs (MSLPs) are notoriously hard to solve in general. Linear decision rules (LDRs) yield an approximation of an MSLP by restricting the decisions at each stage to be an affine function of the observed uncertain parameters. Finding an optimal LDR is a static optimization problem that provides an upper bound on the optimal value of the MSLP, and, under certain assumptions, can be formulated as an explicit linear program. Similarly, as proposed by Kuhn, Wiesemann, and Georghiou (Math. Program., 130, 177-209, 2011) a lower bound for an MSLP can be obtained by restricting decisions in the dual of the MSLP to follow an LDR. We propose a new approximation approach for MSLPs, two-stage LDRs. The idea is to require only the state variables in an MSLP to follow an LDR, which is sufficient to obtain an approximation of an MSLP that is a two-stage stochastic linear program (2SLP). We similarly propose to apply LDR only to a subset of the variables in the dual of the MSLP, which yields a 2SLP approximation of the dual that provides a lower bound on the optimal value of the MSLP. Although solving the corresponding 2SLP approximations exactly is intractable in general, we investigate how approximate solution approaches that have been developed for solving 2SLP can be applied to solve these approximation problems, and derive statistical upper and lower bounds on the optimal value of the MSLP. In addition to potentially yielding better policies and bounds, this approach requires many fewer assumptions than are required to obtain an explicit reformulation when using the standard static LDR approach. As an illustrative example we apply our approach to a capacity expansion model, and find that the two-stage LDR policy has expected cost between 20% and 34% lower than the static LDR policy, and in the dual yields lower bounds that are between 0.1% and 3.3% better.

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.

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.

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.

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.

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.

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

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.

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 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.

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.

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.

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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.

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]

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.