Non-linear programming

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Class date(s): 04 February 2015
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Resources

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Date Class number Topic Slides/handouts for class Video file References and Notes
04 February 05A
  • Why consider unconstrained, single-variable problems
  • Newton's method review to solve these problems

Handout from class

Video
09 February 06A
  • Newton's method reviewed again for unconstrained, single-variable problems
  • Using finite differences instead in Newton's method
  • Multivariate unconstrained optimization

Handout from class

Video
11 February 06B
  • Unconstrained single-variable optimization using gradient search
  • Unconstrained multivariate optimization using gradient search
  • Understanding the line search problem

Handout from class

Video
16 to 27 February 07

Reading week break and midterm

02 March 08A
  • Unconstrained optimization in two variables review
  • Contrasting it back to the single variable case
  • Extending to the multidimensional Newton's method

Handout from class

Video
04 March 08B
  • Examples on the multidimensional Newton's method
  • Quasi Newton method in multiple dimensions
  • Positive and negative definiteness of the Hessian

Handout from class

Video

Code used in class (see below)

09 March 09A
  • Constrained nonlinear optimization introduction
  • Model formulation (convert a problem to mathematics)

Handout from class

Video
11 March 09B

Guest lecture

Handout from class

Video
16 March 10A
  • Convexity, concavity
  • Guarantees on when problems are globally optimal

Handout from class (continued with handout 09A)

Video
18 March 10B
  • Lagrange multiplier method for constrained optimization
  • Interpretation of the Lagrange multiplier constraints

Handout from class

Video
23 March 11A
  • The Nelder-Mead method (several of you are using it in your projects)
  • Practice with using the Nelder Mead method. Optimize this system: http://yint.org/nm

Handout from class

Video



Taking full Newton's steps to solve the class example

clear all;
close all;
clc;
[X1,X2] = meshgrid(-0.5:0.1:6, 0:0.01:9);
Z = func(X1,X2);
contour(X1, X2, Z)
hold on
grid on
 
x = [1,3]';
plot(x(1), x(2), 'o')
text(x(1)+0.2, x(2), '0')
 
for k = 1:10
   slope = -first_deriv(x)
   step = hessian(x)\slope;   % Solves the Ax=b problem, as x = A\b
   x = x + step;
   plot(x(1), x(2), '*')
   text(x(1)+0.1, x(2), num2str(k))
end

func.m

function y = func(x1,x2)
  y = 4.*x1.*x2 - 5.*(x1-2).^4 - 3.*(x2-5).^4;

first_deriv.m

function y = first_deriv(x)
  y = [4*x(2) - 20*(x(1)-2)^3; 
       4*x(1) - 12*(x(2)-5)^3];

hessian.m

function y = hessian(x)
  y = [-60*(x(1)-2)^2, 4; 
        4, -36*(x(2)-5)^2];