Algorithms for Worst-Case Design and Applications to Risk Management

By Berç Rustem; Melendres Howe | Go to book overview

Chapter 6
Minimax as a robust strategy for discrete rival scenarios

The discrete minimax problem arises when the worst-case is to be determined over a discrete set. The latter is characterized by a discrete number of scenarios. Minimax is thus the best strategy in view of the worst-case scenario.

In the presence of a discrete set of rival decision models, forecasts or scenarios purporting to describe the same system, the optimal decision needs to take account of all possible representations. The minimax problem arises when statistical or economic analysis cannot rule out all but one of the rival possibilities. We then need to consider the optimal strategy corresponding to the worst case. Optimality is no longer determined by a single scenario but by all scenarios simultaneously.

In this chapter, we discuss the discrete minimax problem and the robust character of its solution. We consider nonlinear equality and inequality constraints and use an augmented Lagrangian formulation to characterize the problem. The solution algorithm is discussed in Chapter 7.


1 INTRODUCTION TO RIVAL MODELS AND FORECAST
SCENARIOS

Forecasting with rival models is usually resolved by some form of forecast pooling (see, e.g., Fuhrer and Haltmaier, 1986; Granger and Newbold, 1977; Lawrence et al., 1986; Makridakis and Winkler, 1983). In policy optimization, a similar approach leads to the pooling of objective functions derived from the rival models (see Rustem, 1987, 1994). Chow (1979) initially formulated a robust policy approach for two rival economic models. This approach obtains the optimal policy based only on one model and evaluates its effect if the second model turns out to actually represent the system. A “payoff matrix” is constructed and the strategy chosen is the optimal strategy based on the model that inflicts the lesser damage when implemented on the rival model. There is a discrete choice set of policy strategies. Each member of the set is an optimal policy derived using only one of the models as the true representation of the economy.

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Algorithms for Worst-Case Design and Applications to Risk Management
Table of contents

Table of contents

  • Title Page *
  • Algorithms for Worst-Case Design and Applications to Risk Management *
  • Contents vii
  • Preface xiii
  • Chapter 1 - Introduction to Minimax 1
  • References 17
  • Comments and Notes *
  • Chapter 2 - A Survey of Continuous Minimax Algorithms 23
  • References *
  • Comments and Notes *
  • Chapter 3 - Algorithms for Computing Saddle Points 37
  • References *
  • Comments and Notes *
  • Chapter 4 - A Quasi-Newton Algorithm for Continuous Minimax 63
  • References *
  • Appendix A - Implementation Issues *
  • Appendix B - Motivation for the Search Direction D̄ *
  • Comments and Notes *
  • Chapter 5 - Numerical Experiments with Continuous Minimax Algorithms 93
  • References 119
  • Chapter 6 - Minimax as a Robust Strategy for Discrete Rival Scenarios 121
  • References *
  • Chapter 7 - Discrete Minimax Algorithm for Equality and Inequality Constrained Models 139
  • References *
  • Chapter 8 - A Continuous Minimax Strategy for Options Hedging 179
  • References *
  • Appendix A - Weighting Hedge Recommendations, Variant B* *
  • Appendix B - Numerical Examples 237
  • Comments and Notes 244
  • Chapter 9 - Minimax and Asset Allocation Problems 247
  • References *
  • Comments and Notes *
  • Chapter 10 - Asset/liability Management under Uncertainty 291
  • References *
  • Comments and Notes *
  • Chapter 11 - Robust Currency Management 341
  • References *
  • Appendix - Currency Forecasting *
  • Comments and Notes *
  • Index 381
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