Nonlinear Programming

Nonlinear Programming - Lecture 4: Introduction to...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Lecture 4: Introduction to nonlinear programming Contents 1 Nonlinear programming 1 1.1 Unconstrained nonlinear programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Constrained nonlinear optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Convex programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Separable problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Quadratic programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Strategy for solving nonlinear programs 6 2.1 Detour into 1-dimensional problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 First order condition for higher dimensional problems . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Second order necessary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 Lagrangian Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.5 Global optimality and convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.6 Markowitz minimum variance portfolio selection . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.7 Descent methods for general problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.8 Quadratic programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.9 Equality constrained QP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.10 QP with inequality constraints: Active set method . . . . . . . . . . . . . . . . . . . . . . . . 16 2.11 Active set method: extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3 Reading assignment 20 1 Nonlinear programming A nonlinear program is an optimization problem of the following form. max/min f ( x ) , s.t. g i ( x ) b i , i G, g i ( x ) = b i , i E, g i ( x ) b i , i L, x R n , where the functions f ( x ) and g i ( x ) are allowed to be nonlinear , i.e. not linear. Note that the variables x R n , i.e. unless otherwise stated a nonlinear program assumes that the variables are continuous. Typically, one assumes more about the functions f ( x ) and g i ( x ). A frequent assumption is that all the nonlinear functions in the program are at least twice differentiable. Unlike linear programming, there is no unified theory for nonlinear programming. Nonlinear programming methods exploit various special problem structure to compute the optimal solution; therefore, it is very important to recognize and use these structure. In this section we will introduce various special cases of nonlinear programs....
View Full Document

Page1 / 20

Nonlinear Programming - Lecture 4: Introduction to...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online