Algorithms for Worst-Case Design and Applications to Risk Management

By Berç Rustem; Melendres Howe | Go to book overview

(1979) and Becker et al. (1986) consider rival macroeconomic models of the same economy. A similar approach to resource allocation is discussed in Pang and Yu (1989). The robustness property is explored in Hansen et al. (1998). In finance, Howe et al. (1994, 1996), Dert and Oldenkamp (1997), Howe and Rustem (1997), Ibanez (1998) and Rustem et al. (2000) consider worst-case decisions in options pricing and portfolio optimization. In engineering, BenTal and Nemirovski (1993, 1994) discuss truss topology design under rival load scenarios. In control systems, H-control theory is essentially an equivalent minimax formulation for the uncertainties in the system and this aspect is explored in Basar and Bernhard (1991). Rival representations can be characterized either in terms of a discrete choice, such as two or more models of the same system, or as values from a continuous range, such as all the values an uncertain variable may take within an upper and lower bound.

The worst-case design or minimax problem can thus be formulated as

where x ∈ n is a column vector of real numbers, denoting the decision variables in the n-dimensional Euclidian space. The vector y represents the uncertain variables and is defined over the feasible set Y. An equivalent problem to the above formulation is given by where is the max-function. If Y is a set of continuous variables, then the problem is known as continuous minimax. An example for such a set is where and are the lower and upper bounds on the ith element of y. Algorithms for this problem are discussed in detail in Chapters 2-5.

If Y consists of a discrete set of values, the corresponding problem is known as discrete minimax. We consider equality and inequality constraints on x in particular for the case of discrete min-max. The problem is expressed as

where fi(x) is the value corresponding to the ith member of the discrete set {1; 2; …, m} over which the maximum is evaluated and g, h are vectors of equality and inequality constraints. Properties of, and algorithms for, this formulation are considered in Chapters 6 and 7.

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Algorithms for Worst-Case Design and Applications to Risk Management

• Title Page *
• Algorithms for Worst-Case Design and Applications to Risk Management *
• Contents vii
• Preface xiii
• Chapter 1 - Introduction to Minimax 1
• References 17
• Chapter 2 - A Survey of Continuous Minimax Algorithms 23
• References *
• Chapter 3 - Algorithms for Computing Saddle Points 37
• References *
• Chapter 4 - A Quasi-Newton Algorithm for Continuous Minimax 63
• References *
• Appendix A - Implementation Issues *
• Appendix B - Motivation for the Search Direction D̄ *
• 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
• Chapter 9 - Minimax and Asset Allocation Problems 247
• References *
• Chapter 10 - Asset/liability Management under Uncertainty 291
• References *
• Chapter 11 - Robust Currency Management 341
• References *
• Appendix - Currency Forecasting *
• Index 381
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