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

By Berç Rustem; Melendres Howe | Go to book overview

Chapter 2
A survey of continuous minimax algorithms

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 that f(x, y) is convex in x and concave in y continuous minimax can be formulated as a saddle point problem. This is an interesting special case for minimax problems and we discuss algorithms for computing saddle point solutions in Chapter 3. Another special case is discrete minimax and a superlinearly convergent quasi-Newton algorithm to solve this problem is discussed in Chapter 7.

1 INTRODUCTION

In this chapter, we survey algorithms for solving the continuous minimax problem introduced in Section 1.2,

A new quasi-Newton algorithm for this problem is developed and analyzed in detail in Chapter 4. Numerical experiments with a number of algorithms are discussed in Chapter 5.

Continuous minimax belongs to the general class of nonsmooth problems. The main reason for this nonsmooth character is the possible multiplicity of maximizers at any given point. The objective function has a different gradient, with respect to x corresponding to each maximizer. As such, continuous minimax problems may be solved using nonsmooth optimization methods such as subgradient and bundle methods. Subgradient methods require at least one subgradient to be evaluated at each iteration to find a direction of descent (see CN 1). Bundle methods use subgradient information from successive iterations, within a ball of radius r > 0. These methods have been developed to solve either the general class or particular types of nonsmooth

-23-

If you are trying to select text to create highlights or citations, remember that you must now click or tap on the first word, and then click or tap on the last word.
One moment ...
Default project is now your active project.
Project items
Notes

#### Cited page

Style
Citations are available only to our active members.
Buy instant access to cite pages or passages in MLA 8, MLA 7, APA and Chicago citation styles.

(Einhorn, 1992, p. 25)

(Einhorn 25)

(Einhorn 25)

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

#### Cited page

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
Settings

#### Settings

Typeface
Text size Reset View mode
Search within

Look up

#### Look up a word

• Dictionary
• Thesaurus
Please submit a word or phrase above.

Why can't I print more than one page at a time?

Help
Full screen
Items saved from this book
• Bookmarks
• Highlights & Notes
• Citations
/ 389

### How to highlight and cite specific passages

1. Click or tap the first word you want to select.
2. Click or tap the last word you want to select, and you’ll see everything in between get selected.
3. You’ll then get a menu of options like creating a highlight or a citation from that passage of text.

## Cited passage

Style
Citations are available only to our active members.
Buy instant access to cite pages or passages in MLA 8, MLA 7, APA and Chicago citation styles.

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn, 1992, p. 25).

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn 25)

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn 25)

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences."1

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

## Thanks for trying Questia!

Please continue trying out our research tools, but please note, full functionality is available only to our active members.

Your work will be lost once you leave this Web page.