Gerd Infanger

Monte Carlo (importance) sampling within a Benders decomposition algorithm for stochastic linear programs

This paper focuses on Benders decomposition techniques and Monte Carlo sampling (importance sampling) for solving two-stage stochastic linear programs with recourse, a method first introduced by Dantzig and Glynn [7]. The algorithm is discussed and further developed. The paper gives a complete presentation of the method as it is currently implemented. Numerical results from test problems of different areas are presented. Using small test problems, we compare the solutions obtained by the algorithm with universe solutions. We present the solutions of large-scale problems with numerous stochastic parameters, which in the deterministic formulation would have billions of constraints. The problems concern expansion planning of electric utilities with uncertainty in the availabilities of generators and transmission lines and portfolio management with uncertainty in the future returns.

Multi-stage stochastic linear programs for portfolio optimization

The paper demonstrates how multi-period portfolio optimization problems can be efficiently solved as multi-stage stochastic linear programs. A scheme based on a blending of classical Benders decomposition techniques and a special technique, called importance sampling, is used to solve this general class of multi-stochastic linear programs. We discuss the case where stochastic parameters are dependent within a period as well as between periods. Initial computational results are presented.

Cut sharing for multistage stochastic linear programs with interstage dependency

Multistage stochastic programs with interstage independent random parameters have recourse functions that do not depend on the state of the system. Decomposition-based algorithms can exploit this structure by sharing cuts (outer-linearizations of the recourse function) among different scenario subproblems at the same stage. The ability to share cuts is necessary in practical implementations of algorithms that incorporate Monte Carlo sampling within the decomposition scheme. In this paper, we provide methodology for sharing cuts in decomposition algorithms for stochastic programs that satisfy certain interstage dependency models. These techniques enable sampling-based algorithms to handle a richer class of multistage problems, and may also be used to accelerate the convergence of exact decomposition algorithms.

A Complementarity Framework for Forward Contracting Under Uncertainty

We consider a particular instance of a stochastic multi-leader multi-follower equilibrium problem in which players compete in the forward and spot markets in successive periods. Proving the existence of such equilibria has proved difficult, as has the construction of globally convergent algorithms for obtaining such points. By conjecturing a relationship between forward and spot decisions, we consider a variant of the original game and relate the equilibria of this game to a related simultaneous stochastic Nash game where forward and spot decisions are made simultaneously. We characterize the complementarity problem corresponding to the simultaneous Nash game and prove that it is indeed solvable. Moreover, we show that an equilibrium to this Nash game is a local Nash equilibrium of the conjectured variant of the multi-leader multi-follower game of interest. Numerical tests reveal that the difference between equilibrium profits between the original and constrained games are small. Under uncertainty, the equilibrium point of interest is obtainable as the solution to a stochastic mixed-complementarity problem. Based on matrix-splitting methods, a globally convergent decomposition method is suggested for such a class of problems. Computational tests show that the effort grows linearly with the number of scenarios. Further tests show that the method can address larger networks as well. Finally, some policy-based insights are drawn from utilizing the framework to model a two-settlement six-node electricity market.

Dynamic asset allocation strategies using a stochastic dynamic programming aproach

Publisher Summary A major investment decision for individual and institutional investors alike is to choose between different asset classes, i.e., equity investments and interest-bearing investments. The asset allocation decision determines the ultimate risk and return of a portfolio. The asset allocation problem is frequently addressed either through a static analysis, based on Markowitzs mean-variance model, or dynamically but often myopically through the application of analytical results for special classes of utility functions, e.g., Samuelsons fixed-mix result for constant relative risk aversion. Only recently, the full dynamic and multi-dimensional nature of the asset allocation problem could be captured through applications of stochastic dynamic programming and stochastic programming techniques. This chapter reviews the different approaches to asset allocation and presents a novel approach based on stochastic dynamic programming and Monte Carlo sampling that permits one to consider many rebalancing periods, many asset classes, dynamic cash flows, and a general representation of investor risk preference. It presents a novel approach of representing utility by directly modeling risk aversion as a function of wealth, and thus provides a general framework for representing investor preference. It shows how the optimal asset allocation depends on the investment horizon, wealth, and the investors risk preference and how it therefore changes over time depending on cash flow and the returns achieved. It demonstrates how dynamic asset allocation leads to superior results compared to static or myopic techniques. It presents examples of dynamic strategies for various typical risk preferences and multiple asset classes.

A Probabilistic Lower Bound for Two-Stage Stochastic Programs

In the framework of Benders decomposition for two-stage stochastic linear programs, we estimate the coefficients and right-hand sides of the cutting planes using Monte Carlo sampling. We present a new theory for estimating a lower bound for the optimal objective value and we compare (using various test problems whose true optimal value is known) the predicted versus the observed rate of coverage of the optimal objective by the lower bound confidence interval.

Simulation-based confidence bounds for two-stage stochastic programs

This paper provides a rigorous asymptotic analysis and justification of upper and lower confidence bounds proposed by Dantzig and Infanger (A probabilistic lower bound for two-stage stochastic programs, Stanford University, CA, 1995) for an iterative sampling-based decomposition algorithm, introduced by Dantzig and Glynn (Ann. Oper. Res. 22:1–21, 1990) and Infanger (Ann. Oper. Res. 39:41–67, 1992), for solving two-stage stochastic programs. The paper provides confidence bounds in the presence of both independent sampling across iterations, and when common samples are used across different iterations. Confidence bounds for sample-average approximation then follow as a special case. Extensions of the theory to cover use of variance reduction and the dropping of cuts are also presented. An extensive empirical investigation of the performance of these bounds establishes that the bounds perform reasonably on realistic problems.

5.2 Decomposition and importance sampling for stochastic linear models

Linear models that have uncertain parameters with known probability distributions are called stochastic linear models. This paper focuses on the difficulties introduced by these stochastic parameters and reviews different approaches to handle them. The following solution method uses decomposition techniques and importance sampling, and its illustration is based upon a case study of a power system with random fluctuations in demand and equipment availabilities. Numerical results are presented.

Intelligent control and optimization under uncertainty with application to hydro power

Abstract A control that makes the best change in control settings in response to inputs of sensors measuring the state of the system, we refer to as intelligent . Instead of ‘hard-wirin’ response based on protocols, priorities, and pre-selected, pre-programmed ground rules that do not necessarily produce the best changes of the control settings, we show how best rules can be generated and modified by the computer during the course of controlling the system, and how learning plays an important role in the real-time implementation of an intelligent control system. The problem of finding the best control of a system is the same as optimizing a multi-stage mathematical program under uncertainty. Our formulation allows one to take into account uncertainty of the true values that the sensors are measuring, as well as uncertainties about the system response to the changes in the control settings. A feasible solution of the system is called optimum if it maximizes the expected objective value while hedging against the myriad of possible contingencies (or taking advantage of favorable events) that may arise in the future; typically these can number in the thousands, millions, or even billions. We have developed a special approach, a composite of Benders decomposition and importance sampling, to efficiently solve the extremely large mathematical programs that model the myriads of spossible future events. The dual multistage formulation measures the impact of future (down-stream) responses, which the algorithm ‘passes pack’ up-stream to the models ‘present time’ in the form of ‘cuts’ or necessary conditions for the up-stream controls to follow in order to optimally control the system. These cuts, automatically generated and modified, form a set of general ground rules, or principles, which the computer learns, remembers, and calls upon to intelligently control the real system.

4.6. Special programming models

Conventional linear programming modeling approaches for several decades now have been extended to include variants of basic nonlinear programming algorithms. Considered here are elementary spatial programming, quadratic programming, mixed integer programming, and linear complementarity programming models.