Optimal Decision-Making Problem

Topics: Derivative, Optimization, Decision theory Pages: 166 (47399 words) Published: August 3, 2013

Lecture Notes on Optimization Pravin Varaiya


1 INTRODUCTION 2 OPTIMIZATION OVER AN OPEN SET 3 Optimization with equality constraints 4 Linear Programming 5 Nonlinear Programming 6 Discrete-time optimal control 7 Continuous-time linear optimal control 8 Coninuous-time optimal control 9 Dynamic programing 1 7 15 27 49 75 83 95 121




PREFACE to this edition
Notes on Optimization was published in 1971 as part of the Van Nostrand Reinhold Notes on System Sciences, edited by George L. Turin. Our aim was to publish short, accessible treatments of graduate-level material in inexpensive books (the price of a book in the series was about five dollars). The effort was successful for several years. Van Nostrand Reinhold was then purchased by a conglomerate which cancelled Notes on System Sciences because it was not sufficiently profitable. Books have since become expensive. However, the World Wide Web has again made it possible to publish cheaply. Notes on Optimization has been out of print for 20 years. However, several people have been using it as a text or as a reference in a course. They have urged me to re-publish it. The idea of making it freely available over the Web was attractive because it reaffirmed the original aim. The only obstacle was to retype the manuscript in LaTex. I thank Kate Klohe for doing just that. I would appreciate knowing if you find any mistakes in the book, or if you have suggestions for (small) changes that would improve it. Berkeley, California September, 1998 P.P. Varaiya




These Notes were developed for a ten-week course I have taught for the past three years to first-year graduate students of the University of California at Berkeley. My objective has been to present, in a compact and unified manner, the main concepts and techniques of mathematical programming and optimal control to students having diverse technical backgrounds. A reasonable knowledge of advanced calculus (up to the Implicit Function Theorem), linear algebra (linear independence, basis, matrix inverse), and linear differential equations (transition matrix, adjoint solution) is sufficient for the reader to follow the Notes. The treatment of the topics presented here is deep. Although the coverage is not encyclopedic, an understanding of this material should enable the reader to follow much of the recent technical literature on nonlinear programming, (deterministic) optimal control, and mathematical economics. The examples and exercises given in the text form an integral part of the Notes and most readers will need to attend to them before continuing further. To facilitate the use of these Notes as a textbook, I have incurred the cost of some repetition in order to make almost all chapters self-contained. However, Chapter V must be read before Chapter VI, and Chapter VII before Chapter VIII. The selection of topics, as well as their presentation, has been influenced by many of my students and colleagues, who have read and criticized earlier drafts. I would especially like to acknowledge the help of Professors M. Athans, A. Cohen, C.A. Desoer, J-P. Jacob, E. Polak, and Mr. M. Ripper. I also want to thank Mrs. Billie Vrtiak for her marvelous typing in spite of starting from a not terribly legible handwritten manuscript. Finally, I want to thank Professor G.L. Turin for his encouraging and patient editorship. Berkeley, California November, 1971 P.P. Varaiya




Chapter 1

In this chapter, we present our model of the optimal decision-making problem, illustrate decisionmaking situations by a few examples, and briefly introduce two more general models which we cannot discuss further in these Notes.

1.1 The Optimal Decision Problem
These Notes show how to arrive at an optimal decision assuming that complete information is given. The phrase complete information is given means that the following requirements are met: 1....

Bibliography: 130 [54] R.T. Rockafeller. Convex Analysis. Princeton University Press, 1970.
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