Developed by Richard Bellman (USA) during
the early 1950s.
Dynamic programming is an optimization
method suitable for staged processes.
The result of the optimization is the optimal
function rather than
With constrained optimization, a multitude of
search methods have been developed -some specific to a given problem.
Example methods for constrained
Conversion to unc
One approach to satisfying both
From trial point, move toward first constraint
When arrive at the first constraint, move
Efficiency of the search may be a significant
Contour lines: With two independent
variables, the dependent variable generates
a surface where a particular value for the
In contrast to optimization by Lagrange Multipliers,
y behavior of the objective function is used to
arrive at the critical point.
There are two categories that most search meth
IMPLEMENTING INTERIOR POINT METHODS IN THE CLASSROOM
Gerry M. Klein
121 Electrical Engineering Bldg.
Department of Industrial Engineering
University of Missouri-Columbia
Columbia, Missouri 65211
In 1984 a new method for sol
With two independent variables, the gradient vector
y is normal, or perpendicular to the line y =
constant (which are contour lines in two
With three indepe
1. Maxima and minima: The problem in differential calculus is that of
maximum and minimum values for y = f(x)
2. Function involves one independen
Objective function: y = 3x12 + x22 (minimize)
Constraint: = x12x2 8 = 0
x 2 = 0.1
Starting point from Example 1 already worked is
x1 = 2.392
x2 = 1.410
Find: Begin search to m
Apparently Constrained Problems
Again, problem is to seek y(x) that minimizes
g( y , x )
h( y , x ) H
where g, h, and H are known.
Solving constrained dynamic pro
'In mun. . a
L312 Chapter 10. Minimization or Maximization of Functions:]
REFERENCES AND FURTHER READING:
Polak, E. 1971, Computational Methods in Optimization (New York: Aca.
demic Press), pp. 56.
Jacobs, David AiH., ed. 1977, The State
Box value in
box value in variable box of *
same row of
box value of same
x1 x 2 39
Visualization of problem 12-2
x 2 32
x1 0.5 x 2 27
Box value in
box value in
Connected to every LP problem, which is
called the primal, there is another LP problem
called its dual. These two LP problems are
very closely related to each other.
If the optimal solution to an
Generally, the limitation value (left most column in
tableau where the controlling constraint is determined)
which is the smallest ratio id