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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.

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.

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.

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.

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.

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.

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 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 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.

The paper gives an algorithm for L-shaped linear programs which arise naturally in optimal control problems with state constraints and stochastic linear programs (which can be represented in this form with an infinite number of linear constraints). The first section describes a cutting hyperplane algorithm which is shown to be equivalent to a partial decomposition algorithm of the dual program. The two last sections are devoted to applications of the cutting hyperplane algorithm to a linear optimal control problem and stochastic 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 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 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 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.

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.

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.

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.

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.

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.

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.

Bibliography: l. 66

Bibliography: l. 66

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.

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.

The paper is a summary of some applications of the theory of dynamic programming to various classes of multi-stage decision problems of stochastic type.

SPX was the tenth in the series of international conferences on Stochastic Programming (SP) The first was held on the campus of Oxford University 30 years ago. Since then, the series of conferences has crossed the Atlantic Ocean several times (the last two meetings in North America were held in Ann Arbor MT (1989) and Vancouver BC (1995). Stochastic programming has emerged as one of the leading modeling paradigms for decision-making under uncertainty, and the current surge in interest in SP was reflected in the attendance at SPX. With representation from more than 25 nations, including students and faculty from a world-wide selection of universities, the conference was attended by more than 200 participants.

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)

Bibliography: p. 269-296

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.

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.

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.

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

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)

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.

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.

  Introduction: In recent years, the problem of water scarcity is becoming one of the most challenging issues with the economic development and population growth that have involved many sectors due to its importance and economic status and has received increasing attention from governments and international research organizations. This emphasizes the need for optimal allocation of mentioned resources to balance socio-economic development and save water. Therefore, the aim of this study is to develop an uncertainty-based framework for agricultural water resources allocation and calculate the amount of water shortage after allocation and also risk evaluation of agricultural water shortage. The developed framework will be applied to a real case study in the Marand basin, northwest of Iran. Perception of the amount and severity of risk on the system can be a good guide in the optimal allocation of resources and reduction of damage. Materials and Methods: Since various uncertainties exist in the interactions among many system components, optimal allocation of agricultural irrigation water resources in real field conditions is more challenging. Therefore, introduction of uncertainty into traditional optimization methods is an effective way to reflect the complexity and reality of an agricultural water resources allocation system. Among different methods, inexact two-stage stochastic programming (ITSP) has proved to be an effective technique for dealing with uncertain coefficients in water resources management problems. ITSP is incapable of reflecting random uncertainties that coexist in the objective function and constraints. Considering the risk of violating uncertain constraints and the stochastic uncertainty of agricultural irrigation water availability on the right hand side of constraints and uncertainties related to economic data such as the revenue and penalty in the objective function which are expressed as probability distributions, the CCP method and Kataoka’s criterion are introduced into the ITSP model, thus forming the uncertainty-based interactive two-stage stochastic programming (UITSP) model for supporting water resources management. A set of decision alternatives with different combinations of risk levels applied to the objective function and constraints can be generated for planning the water resources allocation system. In the next step, on the basis of results of UITSP agricultural irrigation water shortage risk evaluation can be conducted by using risk assessment indicators (reliability, resiliency, vulnerability, risk degree and consistency) and the fuzzy comprehensive evaluation method. Results and Discussion: A series of water allocation results under different flow levels and different combinations of risk levels were obtained and analyzed in detail through optimally allocating limited water resources to different irrigation areas of Marand basin. The results can help decision makers examine potential interactions between risks related to the stochastic objective function and constraints. Furthermore, a number of solutions can be obtained under different water policy scenarios, which are useful for decision makers to formulate an appropriate policy under uncertainty. The results show that the dry season, i.e., July, August and September are the peak periods of water allocation and demand in Marand basin, which in these months, despite the higher water demand, the amount of water allocation in the current situation is less, which leads to more water shortages in these months. However, the results show that by increasing the efficiency of irrigation and water allocation using the developed framework, the amount of agricultural water allocation and demand is almost balanced and in addition to reducing water shortages, it leads to control over extraction from wells. Also, the goals of the regional water organization, which is reducing the amount of water allocated in the agricultural sector, will be achieved. Comparison with actual conditions shows that the allocation of water resources using the developed framework reduces water shortages while allocation becomes more efficient. Furthermore, the net system benefits per unit water increase which will demonstrate the feasibility and applicability of the developed framework. Results of evaluation of agricultural irrigation water shortage risks indicate that the water shortage risks in the Marand basin are in the category of serious or critical risk level. Therefore, if the current trend of allocation and exploitation of water resources continues, with the population growth, climate change, increasing demand for agricultural products and changing the probability of available water in the future, the water shortage risk would increase to the unbearable risk level. The continuation of this process threatens all investments and economic foundations of this study area. Therefore, the risk of water shortage in the future should be managed by improving the water-saving technologies and also changing the cultivation pattern to drought resistant crops. Conclusion: In this study, an uncertainty-based framework for agricultural water resources allocation and risk evaluation was developed, including model optimization of agricultural water and risk evaluation of water shortage. The developed framework is capable of fully reflecting multiple uncertainties. The developed framework will be helpful for managers in gaining insights into the tradeoffs between system benefits and related risks, permitting an in-depth analysis of risks of agricultural irrigation water shortage under various scenarios. The assessment of agricultural water shortage risk based on the results of the optimization model helps decision makers to obtain in-depth analysis of agricultural irrigation water shortage risk under various scenarios. In application of the developed framework to Marand basin, series of results of agricultural water resources allocation expressed as intervals, and agricultural water shortage risk evaluation levels under different flow levels and also different combinations of risk levels are generated. Comparison between optimal results and actual conditions of agricultural irrigation water allocation demonstrates the feasibility and applicability of the developed framework. Results of evaluation of agricultural irrigation water shortage risks indicate that the water shortage risks in the Marand basin are in the category of serious or critical risk level. Therefore, effective risk management measures should be taken first for different irrigation areas of Marand basin.

  Introduction: In recent years, the problem of water scarcity is becoming one of the most challenging issues with the economic development and population growth that have involved many sectors due to its importance and economic status and has received increasing attention from governments and international research organizations. This emphasizes the need for optimal allocation of mentioned resources to balance socio-economic development and save water. Therefore, the aim of this study is to develop an uncertainty-based framework for agricultural water resources allocation and calculate the amount of water shortage after allocation and also risk evaluation of agricultural water shortage. The developed framework will be applied to a real case study in the Marand basin, northwest of Iran. Perception of the amount and severity of risk on the system can be a good guide in the optimal allocation of resources and reduction of damage. Materials and Methods: Since various uncertainties exist in the interactions among many system components, optimal allocation of agricultural irrigation water resources in real field conditions is more challenging. Therefore, introduction of uncertainty into traditional optimization methods is an effective way to reflect the complexity and reality of an agricultural water resources allocation system. Among different methods, inexact two-stage stochastic programming (ITSP) has proved to be an effective technique for dealing with uncertain coefficients in water resources management problems. ITSP is incapable of reflecting random uncertainties that coexist in the objective function and constraints. Considering the risk of violating uncertain constraints and the stochastic uncertainty of agricultural irrigation water availability on the right hand side of constraints and uncertainties related to economic data such as the revenue and penalty in the objective function which are expressed as probability distributions, the CCP method and Kataoka’s criterion are introduced into the ITSP model, thus forming the uncertainty-based interactive two-stage stochastic programming (UITSP) model for supporting water resources management. A set of decision alternatives with different combinations of risk levels applied to the objective function and constraints can be generated for planning the water resources allocation system. In the next step, on the basis of results of UITSP agricultural irrigation water shortage risk evaluation can be conducted by using risk assessment indicators (reliability, resiliency, vulnerability, risk degree and consistency) and the fuzzy comprehensive evaluation method. Results and Discussion: A series of water allocation results under different flow levels and different combinations of risk levels were obtained and analyzed in detail through optimally allocating limited water resources to different irrigation areas of Marand basin. The results can help decision makers examine potential interactions between risks related to the stochastic objective function and constraints. Furthermore, a number of solutions can be obtained under different water policy scenarios, which are useful for decision makers to formulate an appropriate policy under uncertainty. The results show that the dry season, i.e., July, August and September are the peak periods of water allocation and demand in Marand basin, which in these months, despite the higher water demand, the amount of water allocation in the current situation is less, which leads to more water shortages in these months. However, the results show that by increasing the efficiency of irrigation and water allocation using the developed framework, the amount of agricultural water allocation and demand is almost balanced and in addition to reducing water shortages, it leads to control over extraction from wells. Also, the goals of the regional water organization, which is reducing the amount of water allocated in the agricultural sector, will be achieved. Comparison with actual conditions shows that the allocation of water resources using the developed framework reduces water shortages while allocation becomes more efficient. Furthermore, the net system benefits per unit water increase which will demonstrate the feasibility and applicability of the developed framework. Results of evaluation of agricultural irrigation water shortage risks indicate that the water shortage risks in the Marand basin are in the category of serious or critical risk level. Therefore, if the current trend of allocation and exploitation of water resources continues, with the population growth, climate change, increasing demand for agricultural products and changing the probability of available water in the future, the water shortage risk would increase to the unbearable risk level. The continuation of this process threatens all investments and economic foundations of this study area. Therefore, the risk of water shortage in the future should be managed by improving the water-saving technologies and also changing the cultivation pattern to drought resistant crops. Conclusion: In this study, an uncertainty-based framework for agricultural water resources allocation and risk evaluation was developed, including model optimization of agricultural water and risk evaluation of water shortage. The developed framework is capable of fully reflecting multiple uncertainties. The developed framework will be helpful for managers in gaining insights into the tradeoffs between system benefits and related risks, permitting an in-depth analysis of risks of agricultural irrigation water shortage under various scenarios. The assessment of agricultural water shortage risk based on the results of the optimization model helps decision makers to obtain in-depth analysis of agricultural irrigation water shortage risk under various scenarios. In application of the developed framework to Marand basin, series of results of agricultural water resources allocation expressed as intervals, and agricultural water shortage risk evaluation levels under different flow levels and also different combinations of risk levels are generated. Comparison between optimal results and actual conditions of agricultural irrigation water allocation demonstrates the feasibility and applicability of the developed framework. Results of evaluation of agricultural irrigation water shortage risks indicate that the water shortage risks in the Marand basin are in the category of serious or critical risk level. Therefore, effective risk management measures should be taken first for different irrigation areas of Marand basin.

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 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.

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 consider the problem of identifying underlying community-like structures in graphs. Towards this end we study the Stochastic Block Model (SBM) on $k$-clusters: a random model on $n=km$ vertices, partitioned in $k$ equal sized clusters, with edges sampled independently across clusters with probability $q$ and within clusters with probability $p$, $p>q$. The goal is to recover the initial "hidden" partition of $[n]$. We study semidefinite programming (SDP) based algorithms in this context. In the regime $p = \frac{\alpha \log(m)}{m}$ and $q = \frac{\beta \log(m)}{m}$ we show that a certain natural SDP based algorithm solves the problem of {\em exact recovery} in the $k$-community SBM, with high probability, whenever $\sqrt{\alpha} - \sqrt{\beta} > \sqrt{1}$, as long as $k=o(\log n)$. This threshold is known to be the information theoretically optimal. We also study the case when $k=\theta(\log(n))$. In this case however we achieve recovery guarantees that no longer match the optimal condition $\sqrt{\alpha} - \sqrt{\beta} > \sqrt{1}$, thus leaving achieving optimality for this range an open question.

We consider the problem of identifying underlying community-like structures in graphs. Towards this end we study the Stochastic Block Model (SBM) on $k$-clusters: a random model on $n=km$ vertices, partitioned in $k$ equal sized clusters, with edges sampled independently across clusters with probability $q$ and within clusters with probability $p$, $p>q$. The goal is to recover the initial "hidden" partition of $[n]$. We study semidefinite programming (SDP) based algorithms in this context. In the regime $p = \frac{\alpha \log(m)}{m}$ and $q = \frac{\beta \log(m)}{m}$ we show that a certain natural SDP based algorithm solves the problem of {\em exact recovery} in the $k$-community SBM, with high probability, whenever $\sqrt{\alpha} - \sqrt{\beta} > \sqrt{1}$, as long as $k=o(\log n)$. This threshold is known to be the information theoretically optimal. We also study the case when $k=\theta(\log(n))$. In this case however we achieve recovery guarantees that no longer match the optimal condition $\sqrt{\alpha} - \sqrt{\beta} > \sqrt{1}$, thus leaving achieving optimality for this range an open question.

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.