# Algorithms for Worst-Case Design and Applications to Risk Management

Berç Rustem
Melendres Howe
Pages: 408
https://www.jstor.org/stable/j.ctt7s093

1. Front Matter
(pp. i-vi)
(pp. vii-xii)
3. Preface
(pp. xiii-xvi)
Berç Rustem and Melendres Howe
4. Chapter 1 Introduction to minimax
(pp. 1-22)

We consider the problem of minimizing a nondifferentiable function, defined by the maximum of an inner function. We refer to this objective function as themax-function. In practical applications of minimax, the max-function takes the form of a maximized error, or disutility, function. For example, portfolio selection models in finance can be formulated in a scenario-based framework where the max-function takes the form of a maximized risk measure across all given scenarios. To solve the minimax problem, algorithms requiring derivative information cannot be used directly and the usual methods that do not require gradients are inadequate for this purpose. Instead...

5. Chapter 2 A survey of continuous minimax algorithms
(pp. 23-36)

We consider several continuous minimax algorithm models. All of these base their progress on gradient information. While some are implementable, others require substantial further development to be of practical use. In Chapter 4, we introduce and analyze in detail a quasi-Newton algorithm that builds upon some of the models introduced in the present chapter. In Chapter 5, we consider numerical experiments with a number of algorithms to justify empirically a simplified quasi-Newton algorithm.

Under the special assumption thatf(x,y) is convex inxand concave iny, continuous minimax can be formulated as a saddle point problem. This is an...

6. Chapter 3 Algorithms for computing saddle points
(pp. 37-62)

Consider the basic continuous minimax problem when the underlying functionf(x,y) is convex inxand concave iny. Then, there is a unique solution to minimax which can be computed using specialized algorithms. Saddle point solutions like this are also used by the decision maker to assess the worst-case strategy of an opponent and compute the optimal response to the worst case. In the present context, the opponent can be interpreted as nature choosing the worst-case value of the uncertainty. The solution is an equilibrium strategy which ensures an optimal response to the worst case. Neither the decision maker...

7. Chapter 4 A quasi-Newton algorithm for continuous minimax
(pp. 63-92)

In this chapter, we develop the algorithm models considered in Chapter 2 to consider a fast algorithm for the continuous minimax problem. This extends the steepest descent approach of Panin (1981) and the convex combination rule for subgradients of Kiwiel (1987) to quasi-Newton search directions, conditional on an approximate maximizer.

In effect, we evaluate the choice between two alternative directions. The first is relatively easy to compute and is based on an augmented maximization to ensure that the multiplicity of maximizers does not result in an inferior search direction. The second involves a quadratic suproblem to determine the minimum norm...

8. Chapter 5 Numerical experiments with continuous minimax algorithms
(pp. 93-120)

In this chapter, we consider the implementation of continuous minimax algorithms. The first is the gradient-based algorithm due to Kiwiel (1987), discussed in Section 2.4, and the second is the quasi-Newton algorithm in Chapter 4, and a simplified variant derived from this algorithm.

An important complication of continuous minimax is the presence of multiple maximizers. The contribution of these maximizers to the definition of the direction of search of an algorithm is often difficult to evaluate. We consider several strategies suggested by these algorithms for this purpose. The utility of adopting a simplified direction is evaluated. In addition, the general...

9. Chapter 6 Minimax as a robust strategy for discrete rival scenarios
(pp. 121-138)

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...

10. Chapter 7 Discrete minimax algorithm for nonlinear equality and inequality constrained models
(pp. 139-178)

In this chapter, we consider a sequential quadratic programming algorithm for the discrete minimax problem. As described in Chapter 6, the constraints are used to formulate an augmented Lagrangian. The algorithm involves a sequential quadratic programming subproblem, an adaptive penalty parameter selection rule to regulate the emphasis on constraint satisfaction, an Armijo-type stepsize strategy, convergent to unit steps, that ensures progress towards optimality and feasibility of the constraints.

The algorithm is formulated for general nonlinear constrained problems in which the objective and the constraints are twice continuously differentiable. In the case of linear constraints, the algorithm simplifies considerably. The global...

11. Chapter 8 A continuous minimax strategy for options hedging
(pp. 179-246)

In this chapter, we consider how minimax can provide a robust hedging strategy for written call options. The contingent nature of the liability behind an option makes it important to address the management of this kind of liability. We formulate a minimax hedging strategy that minimizes the effect of a predefined worst-case scenario, mainly in terms of bounds on the underlying source of uncertainty, that is, the future price of the asset that underlies the option. We then identify variants, including multiperiod strategies, and discuss their performance relative to a standard strategy referred to as delta hedging. We also look...

12. Chapter 9 Minimax and asset allocation problems
(pp. 247-290)

In this chapter, we consider potential uses of minimax in the context of portfolio asset allocation, with specific illustrations for bond portfolios. We demonstrate that the issue of mis-forecasting can be appropriately addressed within the minimax framework. An asset allocation based on minimax has robustness properties that cushion the performance of the portfolio against the occurrence of predefined worst-case scenarios. There is a guaranteed performance which improves when the worst-case scenario fails to materialize. A number of minimax asset allocation techniques are discussed, all applicable to stocks, bonds or currencies. These are considered in the context of meanvariance optimization and...

13. Chapter 10 Asset/liability management under uncertainty
(pp. 291-340)

In this chapter, we present robust methods for solving Asset/Liability Management (ALM) problems. Whereas Chapter 9 concentrates on assets-only optimization, this chapter explores the difficulty of simultaneously optimizing both the asset and the liability sides of a portfolio. Alternative minimax formulations with differing objective functions are presented, depending on the targets or objectives of the ALM portfolios. We illustrate the robustness property of the minimax formulation when the liability structure of an ALM portfolio is sensitive to shifts in yield curves. It can be shown that the minimax solution, as compared to standard immunization, provides the least deterioration in the...

14. Chapter 11 Robust currency management
(pp. 341-380)

In this chapter we discuss currency management. We consider the strategic point of view where a currency benchmark is identified. We also consider the tactical point of view where the frequent re-balancing of a portfolio’s currency hedge is managed in order for the portfolio to benefit from short- and medium-term currency fluctuations. Currency benchmark identification is important in finding the long-term optimal hedge ratio that a portfolio should adopt in order to minimize the negative impact of any currency depreciation in the medium and long term. We discuss benchmark identification, first, for a pure currency portfolio as this simplifies and...

15. Index
(pp. 381-389)