Introduction - Nonlinear Programming - Lecture Slides, Slides of Computer Science

Download Introduction - Nonlinear Programming - Lecture Slides and more Slides Computer Science in PDF only on Docsity!

6.252 NONLINEAR PROGRAMMING

LECTURE 1: INTRODUCTION

LECTURE OUTLINE

• Nonlinear Programming
• Application Contexts
• Characterization Issue
• Computation Issue
• Duality
• Organization

NONLINEAR PROGRAMMING

min f (x), x∈X† where

• f :
  n†^ →
  is a continuous (and usually differ- entiable) function of n variables
• X =
  n†^ or X is a subset of
  n†^ with a “continu- ous” character.
• If X =
  n, the problem is called unconstrained
• If f is linear and X is polyhedral, the problem is a linear programming problem. Otherwise it is a nonlinear programming problem
• Linear and nonlinear programming have tradi- tionally been treated separately. Their method- ologies have gradually come closer.

APPLICATIONS OF NONLINEAR PROGRAMMING•

• Data networks – Routing
• Production planning
• Resource allocation
• Computer-aided design
• Solution of equilibrium models
• Data analysis and least squares formulations
• Modeling human or organizational behavior

CHARACTERIZATION PROBLEM

• Unconstrained problems

− Zero 1st order variation along all directions

• Constrained problems

− Nonnegative 1st order variation along all fea- sible directions

• Equality constraints

− Zero 1st order variation along all directions on the constraint surface − Lagrange multiplier theory

• Sensitivity

POST-OPTIMAL ANALYSIS

• Sensitivity
• Role of Lagrange multipliers as prices

DUALITY

• Min-common point problem / max-intercept prob- lem duality

0 0

Min Common Point

Max Intercept Point Max Intercept Point

Min Common Point S S

(a) (b) Illustration of the optimal values of the min common point and max intercept point problems. In (a), the two optimal values are not equal. In (b), the set S, when “extended upwards” along the nth axis, yields the set

S¯^ = {x†¯ | for some x†∈ S, ¯x (^) n† ≥ xn, ¯xi† = xi, i †= 1,... , n †− 1 }

which is convex. As a result, the two optimal values are equal. This fact, when suitably formalized, is the basis for some of the most important duality results.