Syllabus for the course: Optimization Theory and Application

Instructor: Xiaoxi Li∗

Spring, 2017

Basic information

• Title. Optimization Theory and Application

• Volume. 3 course hours by 18 weeks

• Time and classroom. Friday 9:50-12:15 at room 01-4-202.

Grading rule:

�nal grade ≈ 20% HW (about 4 assignments) + 25% mid-term exam + 55% �nal exam

Course description:

This is an introductory course in optimization theory and its applications, which consists of

two main parts: nonlinear optimization and dynamic optimization. We leave the topics as linear

programming, network optimization, integer programming etc. (optimization problems featured

a discrete structure) for the course Operations Research which takes place in the next semester.

In the static part, we work in Rn, and derive classical results in nonlinear optimization:

existence results, the �rst-order and the second-order optimality conditions (necessary, su�cient

conditions) for unconstrained problems, the Lagrange Multiplier Theory and the Karush-Kuhn-

Tucker (KKT) conditions for constrained problems.

The second part is on dynamic optimization and we discuss three classical models: dynamic

programming, calculus of variations, and optimal control. For the �rst model dynamic program-

ming which is in discrete time. We derive the classical Bellman Equation in di�erent contexts:

�rst in �nite horizon and with �nite state space case and next in in�nite horizon (discounted)

and with in�nite state space. Next we see how dynamic programming can be served as a helpful

tool in solving many other optimization models.

To study variational problems in continuous time, a comparison is �rst made from the opti-

mization in �nite dimension to that in in�nite dimensions. In this part, we shall derive the Euler-

Lagrange Equation for the calculus of variations problems, the Pontryagin Maximum (Minimum)

Principle, and the Hamilton-Jacobi-Bellman (HJB) Equation for the optimal control problems.

∗Department of Mathematical Economics and Mathematical Finance, Economics and Management School& Institute for Advanced Study, Wuhan University. Email: xli_whu@163.com. Course's web page: xiaox-ili.weebly.com/teaching.

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Along the course, applications of optimization theory in other disciplines, such as statistics,

industry design, engineering, economics, �nance etc. will be discussed. Optional topics include,

among others, the algorithmic aspect and the computer implementation of optimization prob-

lems, the breadth and the depth of which might depend on smoothness of our course progress.

Course contents:

Part I Static (Nonlinear) optimization. (≈ 5 weeks)

1. Introduction and Examples.

2. Existence Results: the Weierstrass Theorem and Beyond.

3. Unconstrained Problems: First-Order Conditions; Second-Order Conditions.

4. Constrained Problems: the Lagrange Multiplier Theory and the KKT Conditions.

5. Applications: Least-Squares Problems.

Part II Dynamic optimization.

1. Optimization in Discrete Time. (≈ 5 weeks)

(a) Dynamic Programming (DP): Principle of Optimality and Bellman Equation.

(b) Applications: the Shortest-Path Problems and others.

(c) Introduction to Stochastic DP: Markovian Decision Processes and Stochastic Games.

2. Optimization in Continuous Time. (≈ 6 weeks)

(a) Optimization in In�nite-Dimensional Space: Variational Idea and Basic Facts.

(b) Calculus of Variations: Euler-Lagrange Equation.

(c) Optimal Control: Pontryagin's Minimum Principle and the HJB Equation.

(d) Applications: Economic Growth, Principle-Agent Model.

Part III Optimization Methods. (*optional) (≈ 2 weeks)

1. The Gradient Method, Newton Method.

2. Convergence Analysis of Algorithms.

3. MATLAB Implementations.

Recommended References.

• Amir Becker. Introduction to Nonlinear Optimization: Theory, Algorithm, and Application

with MATLAB. MOS-SIAM Series on Optimization, 2014.

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• Dimitri P. Bertsekas. Dynamic Programming and Optimal Control (Volume I), 3e. Athena

Scienti�c, 2005. (we may cover only Chapter 1-3)

Complementary References.

• Dimitri P. Bertsekas. Nonlinear Programming, 2e. Athena Scienti�c, 1999.

• David G. Luenberger. Optimization by Vector Space Methods. John Wiley & Sons, Inc.,

1968.

• Daniel Liberzon. Calculus of Variations and Optimal Control Theory: A Concise Intro-

duction. Princeton University Press, 2012.

• Morton I. Kamien and Nancy L. Schwartz. Dynamic Optimization: the Calculus of Varia-

tions and Optimal Control in Economics and Management, 2e. Elsevier Science Publishing

Cn. Inc., 1991.

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