陆吾生华东师大讲义笔记.pdfVIP

Notes for ECNU 2007 Short Course October 2007 0. Introduction • Purpose of the course: to offer the students a concise introduction to modern unconstrained and constrained optimization methods, algorithms, and computer software; functional optimization and applications will also be briefly addressed. • Background mathematics includes first-year calculus and linear algebra, some knowledge of calculus of variations and PDE are helpful but not critical. • Familiarity with MATLAB is important. • Most of the distributed course material (Chapters 1, 2, 4, 5, 7, 9, 10, 12, 13, 14, 16, and Appendix A) are from the book Andreas Antoniou and Wu-Sheng Lu, Practical Optimization: Algorithms and Engineering Applications, Springer, New York, March 2007. The book has a web site: http://www.ece.uvic.ca/~optimization/ I. Numerical Optimization and Applications 1.1 Unconstrained Optimization (Chaps. 1, 2, 4, 5, 7, 9) • Basic optimization problem: mimimize f (x ) for x ∈R n • Assumption:f (x) is twice continuously differentiable. • An example: solve a system of algebraic equationsp i(x) = 0 for i = 1, 2, …, m. • Basic structure of a numerical algorithm for minimizingf (x) Step 1: choose an initial point x0, set a convergence tolerance ε, and set a counter k = 0. Step 2: determine a search direction d for reducingf (x) from point x . k k Step 3: determine a step size αk such that f (xk +αd k ) is minimized for α≥0 , and construct x x +αd k +1 k k k Step 4: if α