Sampling-based Approximation Algorithms for Multi-stage Stochastic Optimization

Stochastic optimization problems provide a means to model uncertainty in the input data where the uncertainty is modeled by a probability distribution over the possible realizations of the data. We consider a broad class of these problems, called {it multi-stage stochastic programming problems with recourse}, where the uncertainty evolves through a series of stages and one take decisions in each stage in response to the new information learned. These problems are often computationally quite difficult with even very specialized (sub)problems being #P-complete. We obtain the first fully polynomial randomized approximation scheme (FPRAS) for a broad class of multi-stage stochastic linear programming problems with any constant number of stages, without placing any restrictions on the underlying probability distribution or on the cost structure of the input. For any fixed $k$, for a rich class of $k$-stage stochastic linear programs (LPs), we show that, for any probability distribution, for any $epsilon>0$, one can compute, with high probability, a solution with expected cost at most $(1+e)$ times the optimal expected cost, in time polynomial in the input size, $frac{1}{epsilon}$, and a parameter $lambda$ that is an upper bound on the cost-inflation over successive stages. Moreover, the algorithm analyzed is a simple and intuitive algorithm that is often used in practice, the {it sample average approximation} (SAA) method. In this method, one draws certain samples from the underlying distribution, constructs an approximate distribution from these samples, and solves the stochastic problem given by this approximate distribution. This is the first result establishing that the SAA method yields near-optimal solutions for (a class of) multi-stage programs with a polynomial number of samples. As a corollary of this FPRAS, by adapting a generic rounding technique of Shmoys and Swamy, we also obtain the first approximation algorithms for the analogous class of multi-stage stochastic integer programs, which includes the multi-stage versions of the set cover, vertex cover, multicut on trees, facility location, and multicommodity flow problems

Sampling based progressive hedging algorithms for stochastic programming problems

Many real-world optimization problems have parameter uncertainty. For instances where the uncertainties can be estimated to a certain degree, stochastic programming (SP) methodologies are used to identify robust plans. Despite advances in SP, it is still a challenge to solve real world stochastic programming problems, in part due to the exponentially increasing number of scenarios. For two-stage and multi-stage problems, the number of scenarios increases exponentially with the number of uncertain parameters, and for multi-stage problems also with the number of decision stages. In the case of large scale mixed integer stochastic problem instances, there are usually two common approaches: approximation methods and decomposition methods. Most common sampling-based approximation (SAA) SP technique is the Monte Carlo sampling-based method. The Progressive Hedging Algorithm (PHA) on the other hand can optimally solve large problems through the decomposition into smaller problem instances. The SAA, while effectively used in many applications, can lead to poor solution quality if the selected sample sizes are not sufficiently large. With larger sample sizes and multi-stage SPs, however, the SAA method is not practical due to the significant computational effort required. In contrast, PHA suffers from the need to solve many sub-problems iteratively which is computationally expensive. In this dissertation, we develop novel SP algorithms integrating sampling based SAA and decomposition based PHA SP methods. The proposed integrated methods are novel in that they marry the complementary aspects of PHA and SAA in terms of exactness and computational efficiency. Further, the developed methods are practical in that they allow the analyst to calibrate the tradeoff between the exactness and speed of attaining a solution. We demonstrate the effectiveness of the developed integrated approaches, Sampling Based Progressive Hedging Algorithm (SBPHA) and Discarding SBPHA (d-SBPHA), over the pure strategies (i.e. SAA or PHA) as well as other commonly used SP methods through extensive experimentation. In addition, we develop alternative hybridization strategies and present results of extensive experiments for these strategies under different uncertainty models. The validation of the methods is demonstrated through Capacitated Reliable facility Location Problem (CRFLP) and Multi-stage stochastic lot-sizing problems

Towards Distributed Two-Stage Stochastic Optimization

The weighted vertex cover problem is concerned with selecting a subset of the vertices that covers a target set of edges with the objective of minimizing the total cost of the selected vertices. We consider a variant of this classic combinatorial optimization problem where the target edge set is not fully known; rather, it is characterized by a probability distribution. Adhering to the model of two-stage stochastic optimization, the execution is divided into two stages so that in the first stage, the decision maker selects some of the vertices based on the probabilistic forecast of the target edge set. Then, in the second stage, the edges in the target set are revealed and in order to cover them, the decision maker can augment the vertex subset selected in the first stage with additional vertices. However, in the second stage, the vertex cost increases by some inflation factor, so the second stage selection becomes more expensive. The current paper studies the two-stage stochastic vertex cover problem in the realm of distributed graph algorithms, where the decision making process (in both stages) is distributed among the vertices of the graph. By combining the stochastic optimization toolbox with recent advances in distributed algorithms for weighted vertex cover, we develop an algorithm that runs in time O(log (?) / ?), sends O(m) messages in total, and guarantees to approximate the optimal solution within a (3 + ?)-ratio, where m is the number of edges in the graph, ? is its maximum degree, and 0 < ? < 1 is a performance parameter

Approximation algorithms for stochastic and risk-averse optimization

We present improved approximation algorithms in stochastic optimization. We prove that the multi-stage stochastic versions of covering integer programs (such as set cover and vertex cover) admit essentially the same approximation algorithms as their standard (non-stochastic) counterparts; this improves upon work of Swamy \& Shmoys which shows an approximability that depends multiplicatively on the number of stages. We also present approximation algorithms for facility location and some of its variants in the $2$-stage recourse model, improving on previous approximation guarantees. We give a $2.2975$-approximation algorithm in the standard polynomial-scenario model and an algorithm with an expected per-scenario $2.4957$-approximation guarantee, which is applicable to the more general black-box distribution model.Comment: Extension of a SODA'07 paper. To appear in SIAM J. Discrete Mat

Software tools for stochastic programming: A Stochastic Programming Integrated Environment (SPInE)

SP models combine the paradigm of dynamic linear programming with modelling of random parameters, providing optimal decisions which hedge against future uncertainties. Advances in hardware as well as software techniques and solution methods have made SP a viable optimisation tool. We identify a growing need for modelling systems which support the creation and investigation of SP problems. Our SPInE system integrates a number of components which include a flexible modelling tool (based on stochastic extensions of the algebraic modelling languages AMPL and MPL), stochastic solvers, as well as special purpose scenario generators and database tools. We introduce an asset/liability management model and illustrate how SPInE can be used to create and process this model as a multistage SP application