Algorithms for Worst-Case Design and Applications to Risk Management

By Berç Rustem; Melendres Howe | Go to book overview

Chapter 7
Discrete minimax algorithm for equality and inequality
constrained models

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 convergence of the general algorithm is shown, along with the convergence of the stepsize to unity. It is also shown that the penalty parameter does not grow indefinitely. The local Q-superlinear convergence of the algorithm is demonstrated. The algorithm for linear constraints is also discussed and the convergence properties of this special case are deduced.


1 INTRODUCTION

In this chapter, we consider an algorithm for solving the discrete minimax problem

introduced in Chapter 6. As we have already discussed, this formulation applies to decision models with a discrete set of scenarios. Throughout this chapter, the discussion relies on the equivalence of the above discrete minimax problem with a continuous minimax and a nonlinear programming problem. Thus, using the terminology introduced in Chapter 6, it is established in Lemma 6.2.1 that (1.1) is equivalent to where

-139-

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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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