Algorithms for Worst-Case Design and Applications to Risk Management

By Berç Rustem; Melendres Howe | Go to book overview

Preface

The conventional approach to decisions under uncertainty is based on expected value optimization. The main problem with this concept is that it neglects the worst-case effect of the uncertainty in favor of expected values. While acceptable in numerous instances, decisions based on expected value optimization may often need to be justified in view of the worst-case scenario. This is especially important if the decision to be made can be influenced by such uncertainty that, in the worst case, might have drastic consequences on the system being optimized. On the other hand, given an uncertain effect, some worst-case realizations might be so improbable that dwelling on them might result in unnecessarily pessimistic decisions. Nevertheless, even when decisions based on expected value optimization are to be implemented, the worstcase scenario does provide an appropriate benchmark indicating the risks.

This book is intended for the dual role of proposing worst-case design for robust decisions and methods and algorithms for computing the solution to quantitative decision models. Actually, very little space is devoted to justify worst-case design. This is implicit in the optimality condition of minimax, discussed in Chapter 1. In Chapter 6, the robustness of worst-case optimal strategies are considered for discrete scenarios. Subsequent chapters illustrate the property. Basically, the performance of the minimax optimal strategy is noninferior for any scenario, and better for those other than the worst case. As such, worst-case design needs no further justification as a robust strategy than a deterministic optimal strategy requires in view of suboptimal alternatives.

In the book, we consider methods for optimal decisions which take account of the worst-case eventuality of uncertain events. The robust character of minimax, mentioned above, is central to the usefulness of the strategies discussed in this book. The discrete minimax strategy ensures a guaranteed optimal performance in view of the worst case and this is assured for all scenarios: if any scenario, other than the one corresponding to the worst case is realized, performance is assured to improve. The continuous minimax strategy provides a guaranteed optimal performance in view of a continuum of scenarios. If this continuum is taken as scenarios varying between upper and lower bounds, performance is assured over the worst case defined between upper and lower bounds. As such, continuous minimax is a forecaster's dream as it provides the opportunity for specifying forecasts defined over a range,

-xiii-

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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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