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

By Berç Rustem; Melendres Howe | Go to book overview

Chapter 4
A quasi-Newton algorithm for continuous minimax

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 subgradient. In Chapter 5, we discuss the relative merits of an algorithm based only on the former, simpler, direction.

We establish the descent property of the direction chosen by the algorithm. A step is taken using an Armijo-type stepsize strategy consistent with the direction. This ensures the monotonic decrease of the maximizing function and the convergence of the algorithm. We show that the stepsize selected by the algorithm converges to unity and that the local convergence rate is Qsuperlinear.

1 INTRODUCTION

As in Chapters 1 and 2, we consider the problem

where Y ⊂ m and f : n × Y ↦ 1. Let for all x ∈ n. We call Φ(x) the max-function. Hence, (1.1) can be written as In this chapter, we discuss a quasi-Newton algorithm to solve (1.3). The salient features of the algorithm are the generation of a descent direction based on a subgradient of f(x, ·) and an approximate Hessian, in the presence

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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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