1. Lecture 8 – Nonlinear Programming Models
Topics
General formulations
Local vs. global solutions
Solution characteristics
Convexity and convex programming
Examples

2. Nonlinear Optimization
In LP ... the objective function & constraints are linear and the problems are “easy” to solve.
Many real-world engineering and business problems have nonlinear elements and are hard to solve.

3. General NLP Minimize f(x) s.t. gi(x) (, , =) bi, i = 1,…,m
x = (x1,…,xn) is the n-dimensional vector of decision variables
f (x) is the objective function
gi(x) are the constraint functions
bi are fixed known constants

4. “decreasing efficiencies”
Examples of NLPs 4
Example 1
Max f (x) =
3x1 + 2x2
s.t. x1 + x2 £ 1, x1 ³ 0, x2 unrestricted 2
Example 2
Max f (x) = e c 1 x e 2 … n
s.t. Ax = b, x ³ 0 n
Example 3
Min å
j =1
fj (xj )
s.t. Ax = b, x ³ 0
where each fj(xj ) is of the form
Problems with
“decreasing efficiencies”
fj(xj) xj
Examples 2 and 3 can be reformulated as LPs

5. NLP Graphical Solution Method
Max f(x1, x2) = x1x2
s.t. 4x1 + x2 £ 8
x1 ³ 0, x2 ³ 0 x2 8
f(x) = 2
f(x) = 1 2 x1
Optimal solution will lie on the line g(x) = 4x1 + x2 – 8 = 0.

6. Solution Characteristics
Gradient of f (x) = f (x1, x2)  (f/x1, f/x2)T
This gives f/x1 = x2, f/x2 = x1
and g/x1 = 4, g/x2 = 1
At optimality we have f (x1, x2) = g (x1, x2)
or x1* = 1 and x2* = 4
Solution is not a vertex of feasible region. For this particular problem the solution is on the boundary of the feasible region. This is not always the case.
In a more general case, f (x1, x2) = g (x1, x2) with   0. (In this case,  = 1.)

7. Nonconvex Function
global
max
stationary
point
f(x)
local
max
local
min
local
min x
Let S  n be the set of feasible solutions to an NLP.
Definition: A global minimum is any x0  S such than
f (x0)  f (x)
for all feasible x not equal to x0.

8. Function with Unique Global Minimum at x = (1, –3)
If g1 = x1 ³ 0 and g2 = x2 ³ 0, what is the optimum ?
At (1, 0), f(x1, x2) = 1g1(x1, x2) + 2g1(x1, x2)
or (0, 6) = 1(1, 0) + 2(0, 1), 1 ³ 0, 2 ³ 0
so 1 = 0 and 2 = 6

9. Function with Multiple Maxima and Minima
Min { f (x)= sin(x) : 0  x  5p}

10. Constrained Function with Unique Global Maximum and Unique Global Minimum

11. Convexity
Convex function: If you draw a straight line between any two points on f (x) the line will be above or on the line.
Concave function: If f (x) is convex than – f (x) is concave.
d2 f (x)
dx2
≥ 0 for all x
Convexity condition for
univariate f :
Linear functions are both convex and concave.

12. Definition of Convexity
Let x1 and x2 be two points in S  n. A function f (x) is convex if and only if
f (lx1 + (1–l)x2) ≤ lf (x1) + (1–l)f (x2)
for all 0 < l < 1. It is strictly convex if the inequality sign ≤ is replaced with the sign <.
1-dimensional example

13. Nonconvex -- Nonconave Function
f(x) x

14. Theoretical Result for Convex Functions
A positively weighted sum of convex functions is convex:
if fk(x) k =1,…,m are convex and 1,…,m ³ 0,
then f (x) = å ak fk(x) is convex. m
k =1 d 2 f dx 1
. . . n
Hessian of f at x : s2f (x) = .

15. Determining Convexity
One-Dimensional Functions:
A function f (x) Î C 1 is convex if and only if it is underestimated by linear extrapolation; i.e.,
f (x2) ≥ f (x1) + (df (x1)/dx)(x2 – x1) for all x1 and x2.
x x2
f(x)
A function f (x)  C 2 is convex if and only if its second derivative is nonnegative.
d2f (x)/dx2 ≥ 0 for all x
If the inequality is strict (>), the function is strictly convex.

16. Multiple Dimensional Functions
f (x) is convex if only if f (x2) ≥ f (x1) + Tf (x1)(x2 – x1) for all x1 and x2.
Definition: The Hessian matrix H(x) associated with f (x) is the n  n symmetric matrix of second partial derivatives of f (x) with respect to the components of x.
When f (x) is quadratic, H(x) has only constant terms; when f (x) is linear, H(x) does not exist.
Example: f (x) = 3(x1)2 + 4(x2)3 – 5x1x2 + 4x1

17. Properties of the Hessian
How can we use Hessian to determine whether or not f(x) is convex?
H(x) is positive definite if and only if xTHx > 0 for all x  0.
H(x) is positive semi-definite if and only if xTHx ≥ 0 for all x and there exists and x  0 such that xTHx = 0.
H(x) is indefinite if and only if xTHx > 0 for some x, and xTHx < 0 for some other x.

18. Multiple Dimensional Functions and Convexity
f (x) is strictly convex (convex) if its associated Hessian matrix H(x) is positive definite (semi- definite) for all x.
f (x) is neither convex nor concave if its associated Hessian matrix H(x) is indefinite
The terms negative definite and negative-semi definite are also appropriate for the Hessian and provide symmetric results for concave functions. Recall that a function f (x) is concave if –f (x) is convex.

19. Testing for Definiteness
Let Hessian, H =
, where hij = 2f (x)/xixj
Definition: The ith leading principal submatrix of H is the matrix formed taking the intersection of its first i rows and i columns. Let Hi be the value of the corresponding determinant:

20. Rules for Definiteness
H is positive definite if and only if the determinants of all the leading principal submatrices are positive; i.e., Hi > 0 for i = 1,…,n.
H is negative definite if and only if H1 < 0 and the remaining leading principal determinants alternate in sign:
H2 > 0, H3 < 0, H4 > 0, . . .
Positive-semidefinite and negative semi-definiteness require that all principal submatrices satisfy the above conditions for the particular case.

21. Quadratic Functions Example 1: f (x) = 3x1x2 + x12 + 3x22
so H1 = 2 and H2 = 12 – 9 = 3
Conclusion  f (x) is convex because H(x) is positive definite.

22. Quadratic Functions Example 2: f (x) = 24x1x2 + 9x12 + 16x22
so H1 = 18 and H2 = 576 – 576 = 0
Thus H is positive semi-definite (determinants of all submatrices are nonnegative) so f (x) is convex.
Note, xTHx = 2(3x1 + 4x2)2 ≥ 0. For x1 = -4, x2 = 3, we get xTHx = 0.

23. Nonquadratic Functions
Example 3: f (x) = (x2 – x12)2 + (1 – x1)2
Thus the Hessian depends on the point under consideration:
At x = (1, 1), which is positive definite.
At x = (0, 1), which is indefinite.
Thus f(x) is not convex although it is strictly convex near (1, 1).

24. What You Should Know About Nonlinear Programming
How to identify the decision variables.
How to write constraints.
How to identify a convex function.
The difference between a local and global solution.