Towards distributed two-stage stochastic optimization

We consider two- and multistage versions of stochastic combinatorial optimization problems with recourse: in this framework, the instance for the combinatorial optimization problem is drawn from a known probability distribution $\pi$ and is only revealed to the algorithm over two (or multiple) stages. At each stage, on receiving some more information about the instance, the algorithm is allowed to build some partial solution. Since the costs of elements increase with each passing stage, there is a natural tension between waiting for later stages, to gain more information about the instance, and purchasing elements in earlier stages, to take advantages of lower costs. We provide approximation algorithms for stochastic combinatorial optimization problems (such as the Steiner tree problem, the Steiner network problem, and the vertex cover problem) by means of a simple sampling-based algorithm. In every stage, our algorithm samples the probability distribution of the requirements and constructs a partial solution to serve the resulting sample. We show that if one can construct cost-sharing functions associated with the algorithms used to construct these partial solutions, then this strategy results in provable approximation guarantees for the overall stochastic optimization problem. We also extend this approach to provide an approximation algorithm for the stochastic version of the uncapacitated facility location problem, a problem that does not fit into the simpler framework of our main model.

We consider a two-stage mixed integer stochastic optimization problem and show that a static robust solution is a good approximation to the fully adaptable two-stage solution for the stochastic problem under fairly general assumptions on the uncertainty set and the probability distribution. In particular, we show that if the right-hand side of the constraints is uncertain and belongs to a symmetric uncertainty set (such as hypercube, ellipsoid or norm ball) and the probability measure is also symmetric, then the cost of the optimal fixed solution to the corresponding robust problem is at most twice the optimal expected cost of the two-stage stochastic problem. Furthermore, we show that the bound is tight for symmetric uncertainty sets and can be arbitrarily large if the uncertainty set is not symmetric. We refer to the ratio of the optimal cost of the robust problem and the optimal cost of the two-stage stochastic problem as the stochasticity gap. We also extend the bound on the stochasticity gap for another class of uncertainty sets referred to as positive. If both the objective coefficients and right-hand side are uncertain, we show that the stochasticity gap can be arbitrarily large even if the uncertainty set and the probability measure are both symmetric. However, we prove that the adaptability gap (ratio of optimal cost of the robust problem and the optimal cost of a two-stage fully adaptable problem) is at most four even if both the objective coefficients and the right-hand side of the constraints are uncertain and belong to a symmetric uncertainty set. The bound holds for the class of positive uncertainty sets as well. Moreover, if the uncertainty set is a hypercube (special case of a symmetric set), the adaptability gap is one under an even more general model of uncertainty where the constraint coefficients are also uncertain.

This paper presents an investigation on the computational complexity of stochastic optimization problems. We discuss a scenario-based model which captures the important classes of two-stage stochastic combinatorial optimization, two-stage stochastic linear programming, and two-stage stochastic integer linear programming. This model can also be used to handle chance constraints, which are used in many stochastic optimization problems. We derive general upper bounds for the complexity of computational problems related to this model, which hold under very mild conditions. Additionally, we show that these upper bounds are matched for some stochastic combinatorial optimization problems arising in the field of transportation and logistics.KeywordsStochastic combinatorial optimizationComputational complexityChance constraintsStochastic vehicle routing

A new scheme to cope with two-stage stochastic optimization problems uses a risk measure as the objective function of the recourse action, where the risk measure is defined as the worst-case expected values over a set of constrained distributions. This paper develops an approach to deal with the case where both the first and second stage objective functions are convex linear-quadratic. It is shown that under a standard set of regularity assumptions, this two-stage quadratic stochastic optimization problem with measures of risk is equivalent to a conic optimization problem that can be solved in polynomial time.

This paper proposes a data-driven optimization method to solve the integrated energy and reserve dispatch problem with variable and correlated renewable energy generation. The proposed method applies the kernel density estimation to establish an ambiguity set of continuous multivariate probability distributions and the optimization model for the integrated dispatch is formulated as a combination of stochastic and robust optimization problems. First, a risk-averse two-stage stochastic optimization model is formulated to hedge the distributional uncertainty. Next, the second-stage worst case expectation is evaluated, using the equivalent model reformulation, as a combination of conditional value-at-risk (CVaR) and the extreme cost in the worst case scenario. The CVaR is calculated using a scenario-based stochastic optimization problem. After describing the wind power correlation in ellipsoidal uncertainty sets, the robust optimization problem for finding the worst case cost is cast into a mixed-integer second-order cone programming problem. Finally, the column-and-constraint generation method is employed to solve the proposed risk-averse two-stage problem. The proposed method is tested on the 6-bus and IEEE 118-bus systems and validated by comparing the results with those of conventional stochastic and robust optimization methods.

Stochastic optimization has established itself as a major method to handle uncertainty in various optimization problems by modeling the uncertainty by a probability distribution over possible realizations. Traditionally, the main focus in stochastic optimization has been various stochastic mathematical programming (such as linear programming, convex programming). In recent years, there has been a surge of interest in stochastic combinatorial optimization problems from the theoretical computer science community. In this article, we survey some of the recent results on various stochastic versions of classical combinatorial optimization problems. Since most problems in this domain are NP-hard (or #P-hard, or even PSPACE-hard), we focus on the results which provide polynomial time approximation algorithms with provable approximation guarantees. Our discussions are centered around a few representative problems, such as stochastic knapsack, stochastic matching, multi-armed bandit etc. We use these examples to introduce several popular stochastic models, such as the fixed-set model, 2-stage stochastic optimization model, stochastic adaptive probing model etc, as well as some useful techniques for designing approximation algorithms for stochastic combinatorial optimization problems, including the linear programming relaxation approach, boosted sampling, content resolution schemes, Poisson approximation etc. We also provide some open research questions along the way. Our purpose is to provide readers a quick glimpse to the models, problems, and techniques in this area, and hopefully inspire new contributions .

The emerging distributed satellite cluster network (DSCN) holds great promise in various practical fields, including earth observation, disaster rescue, and tracking of forest fires. In the DSCN environment, it is essential to achieve the best data delivery performance by coordinating multi-dimensional heterogeneous and dynamic resources. However, in real-world applications, the distribution of long-term data arrival is not often fully known. Motivated by this fact, we propose a distributionally robust two-stage stochastic optimization framework with considering the dynamic network resources and the partially known distribution information of long-term data arrival. Aiming at maximizing the total network reward, we formulate a two-stage stochastic flow optimization problem based on the extended time expanded graph. Then, we introduce an ambiguity set for the uncertain distribution of the long-term random data arrival inspired by the idea from the distributionally robust optimization. On the basis of the proposed ambiguity set, we further propose a data arrival distribution robust two-stage recourse (DADR-TR) algorithm by converting the original stochastic optimization problem into a deterministic cone optimization problem, which is computationally tractable. The extensive simulations have been conducted to evaluate the impact of various network parameters on the algorithm performance and further validate that the proposed DADR-TR algorithm can achieve high data delivery performance without full distribution information of the long-term data arrival.

For the modern power systems with intensive renewable energy sources integration, the uncertainty analysis of its performance become necessary for desired decision making. An appropriate choice of modelling and handling methodology, for uncertain parameters associated with the renewable energy, remains one of major concerns for the power system engineers. This paper presents a stochastic optimization framework for some critical decisions making on reactive power dispatch in two consecutive stages by incorporating both generation outputs of wind farms along with load demand uncertainties. The proposed method is generic for optimal reactive power dispatch in any modern wind-integrated power system. A methodology based on scenario-based analysis is adopted for handling these uncertainties. Moreover, a hybrid fuzzy evolutionary algorithm (HFEA) is applied to obtain high-quality solutions for this complex two-stage stochastic optimization problem. Furthermore, the proposed HFEA-based stochastic optimization approach is tested on IEEE 14-bus power system with two distinct wind farms. The simulation results prove the robustness of HFEA for generating feasible and optimal solutions for all the considered scenarios.

In this paper, we update the theory of deterministic linear semi-infinite programming, mainly with the dual characterizations of the constraint qualifications, which play a crucial role in optimality and duality. From this theory, we obtain new results on conic and two-stage stochastic linear optimization. Specifically, for conic linear optimization problems, we characterize the existence of feasible solutions and some geometric properties of the feasible set, and we also provide theorems on optimality and duality. Analogously, regarding stochastic optimization problems, we study the semi-infinite reformulation of a problem-based scenario reduction problem in two-stage stochastic linear programming, providing a sufficient condition for the existence of feasible solutions as well as optimality and duality theorems to its non-combinatorial part.

We study the design of approximation algorithms for stochastic combinatorial optimization problems. We formulate the problems in the framework of two-stage stochastic optimization, and provide nearly tight approximations. Our problems range from the simple (shortest path, vertex cover, bin packing) to complex (facility location, set cover), and contain representatives with different approximation ratios.

Two-stage stochastic optimization, in which a long-term master problem is coupled with a family of short-term subproblems, plays a critical role in various application areas. However, most existing algorithms for two-stage stochastic optimization only work for special cases, and/or are based on the batch method, which requires huge memory and computational complexity. To the best of our knowledge, there still lack efficient and general two-stage online stochastic optimization algorithms. This paper proposes a two-stage online successive convex approximation (TOSCA) algorithm for general two-stage nonconvex stochastic optimization problems. At each iteration, the TOSCA algorithm first solves one short-term subproblem associated with the current realization of the system state. Then, it constructs a convex surrogate function for the objective of the long-term master problem. Finally, the long-term variables are updated by solving a convex approximation problem obtained by replacing the objective function in the long-term master problem with the convex surrogate function. We establish the almost sure convergence of the TOSCA algorithm and customize the algorithmic framework to solve three important application problems. Simulations show that the TOSCA algorithm can achieve superior performance over existing solutions.

The Stochastic Optimization Problem is well known in mathematics and operations research. Due to increasing demands for performance improvement together with reduced weight and costs the introduction of uncertainties into optimization problems in mechanical engineering is also necessary. In a number of papers this Stochastic Structural Optimization Problem is formulated and solved only within certain limiting assumptions. This paper presents a general definition of the Stochastic Structural Optimization Problem. Furthermore one solution technique based on reliability calculation is presented. With this technique the use of an augmented optimization loop based on Mathematical Programming is possible. The use of the developed tool for reliability calculations and optimization is demonstrated for complex mechanical structures.

We present two-stage stochastic risk averse optimization models for the power generation mix capacity expansion planning in the long run under uncertainty. Uncertainty is described by a set of possible scenarios in the second stage and uncertain parameters are the unit production costs of the existing power plants as well as those of the candidate plants of new technologies among which to choose, the market electricity price, the price of green certificates and the emission permits and the potential market share of the producer. The problem is expressed as a two-stage stochastic integer optimization model subject to technical constraints, market opportunities and budget constraints. First stage variables represent the number of new power plants for each candidate technology to be added to the existing generation mix every year of the planning horizon. Second stage variables are the continuous operation variables of all power plants in the generation mix along the time horizon. We solve the problem of the maximization of the net present value of the expected profits along the time horizon using both a risk neutral approach and different risk averse strategies (conditional value at risk, shortfall probability, expected shortage and first- and second-order stochastic dominance), under different hypotheses of the available budget, analysing the impact of each risk averse strategy on the expected profit. Results show that risk control strongly reduces the possibility of reaching unwanted scenarios as well as providing consistent solutions under different strategies.

In stochastic optimisation, the large number of scenarios required to faithfully represent the underlying uncertainty is often a barrier to finding efficient numerical solutions. This motivates the scenario reduction problem: by finding a smaller subset of scenarios, reduce the numerical complexity while keeping the error at an acceptable level. In this paper we propose a novel and computationally efficient methodology to tackle the scenario reduction problem for two-stage problems when the error to be minimised is the implementation error, i.e. the error incurred by implementing the solution of the reduced problem in the original problem. Specifically, we develop a problem-driven scenario clustering method that produces a partition of the scenario set. Each cluster contains a representative scenario that best reflects the optimal value of the objective function in each cluster of the partition to be identified. We demonstrate the efficiency of our method by applying it to two challenging two-stage stochastic combinatorial optimization problems: the two-stage stochastic network design problem and the two-stage facility location problem. When compared to alternative clustering methods and Monte Carlo sampling, our method is shown to clearly outperform all other methods.

Sustainable development is an everlasting theme and lasting strategy in today’s era. Low‐carbon economy is an inevitable approach to the implementation of sustainable development. Cold chain logistics has become one of the main sources of carbon emissions. However, in the research on location planning of cold chain logistics, the costs of carbon emissions have not been taken into consideration in previous studies. The two‐stage stochastic optimization (TSSO) model was established based on the comprehensive consideration of transportation costs, time penalty costs, and carbon emission costs. In this case, it is extremely difficult to deal with uncertainty in TSSO model. Therefore, this paper constructs a two‐stage robust optimization (TSRO) model using data‐driven method and robust optimization theory and verifies the validity of this model through an actual case. The application of this method to a cold chain logistics enterprise showed that the service level of logistics cannot be guaranteed by stochastic optimization model. In the TSRO model, the costs increase by 2.18% at the price of robustness, whereas logistics service level shows an upward trend (from 85.83% to 92.75%). In the TSRO model, enterprises are forced to choose a better distribution path when carbon tax increases, which not only helps enterprises save costs but also achieves low‐carbon environmental benefits.