Math 432/532
Homework
20 pts.
Due Feb. 3, 2012
1. For the following functions, all critical points are listed. Identify all local and global maxima and minima, along with saddle points. (a) f (x, y) = x3 + y 3 - 3x - 12y + 20 Critical points: (1, 2), (1,
Math 432/532
Optimization
Spring 2012
Time and place: Instructor: Office hours: Phone number: Text(s):
Monday, Wednesday, and Friday, 4:10-5:00 Bachelor Hall, Room 219 Doug Ward (email [email protected]), 204 Bachelor Hall MWF 10-11, 2-3; TR 11-1 or by ap
Math 432/532
Spring 2012
Announcement of March 14 test
Our second test will be on Wednesday, March 14. It will cover the material from our last three homework assignments. The main topics are these: The definitions of convex and concave functions and the
Math 432/532
Homework
15 points
Due March 23, 2012
1. Let f (x1 , x2 ) = 2x1 4 + x2 2 - 4x1 x2 + 5x2 . (a) Compute the first two terms x(1) , x(2) of the Newton's method sequence for finding a critical point of f with initial point x(0) = (0, 0). (b) Comp
Math 432/532
Homework
15 points
Due March 2, 2012
1. Find the equation of the line that gives the best "least-squares fit" to the data points (-2, 12), (-1, 11), (0, 8), (1, 5), (2, 2), (3, -3). 2. Find the point in the set cfw_(x1 , x2 , x3 ) | x1 + 3x2
Math 432/532
Homework
15 points
Due February 17, 2012
1. Which of the following functions are convex on the specified convex sets? Which are strictly convex? (a) f (x1 , x2 ) = 5x1 2 + x2 2 + 2x1 x2 - x1 + 2x2 + 3 on D = R2 . (b) f (x1 , x2 ) = x1 2 /2 +
Math 432/532
Homework
15 pts.
Due January 20, 2012
432 students: Solve any 5 of the 6 problems. 532 students: Solve all 6 problems. 1. Find the local and global minimizers and maximizers of the following functions, applying the theorems discussed in class
8. LINEAR PROGRAMMING
Linear programming is the most widely used class of constrained optimization problems. Its special structure provides for strong theoretical results and efficient algorithms.
8.1 Linear Programming Formulations
We begin this chapter
7. GENERAL NONLINEAR PROGRAMMING
In this chapter we develop necessary and sufficient conditions for optimality for the general nonlinear programming problem (NLP) minimize f (x) subject to gi (x) = 0, for i = 1, . . . , m, gi (x) 0, for i = m + 1, . . . ,
6. LINEAR LEAST SQUARES AND RELATED PROBLEMS
Linear least squares constitutes one of the most important classes of optimization problems in modern society, primarily because of its central role in statistical data analysis. Within optimization itself, lin
5. ITERATIVE METHODS FOR OPTIMIZATION
Our first step in solving an optimization problem is to use some form of necessary condition to identify candidates for optimality. For unconstrained problems, this means finding the critical points of the objective f
4. CONVEXITY AND OPTIMIZATION
Convexity plays a natural role in optimization. It has tremendous intuitive appeal and provides powerful and elegant approaches to a wide variety of pure and applied problems.
4.1 Convex Sets and Functions
Definition 4.1.1 (C
3. EXISTENCE OF GLOBAL OPTIMIZERS
This chapter presents theoretical tools for demonstrating the existence of global minimizers or maximizers. For practical purposes, such tools enable us to verify global optimality simply by comparing the function values
2. UNCONSTRAINED OPTIMIZATION
In this chapter, we develop principles for identifying and validating candidates for optimality in problems with no explicit constraints: min f (x) over all x S, where S is typically the whole space Rn or some simply describe
MTH 432/532 LECTURE NOTES Spring 2009
INTRODUCTION TO LINEAR AND NONLINEAR PROGRAMMING
S. Wright
Department of Mathematics & Statistics Miami University, Oxford, OH
c 2008, 2009
Table of Contents
1. INTRODUCTION 1.1 Terminology 1.2 Some Facts about Calcul