Non-Linear Problems
General approach
Non-linear Optimization
• Many objective functions, tend to be non-linear.
• Design problems for which the objective is to
minimize cost or maximize benefits minus costs
usually have cost functions with economies of
scale.
• This implies a non-linear function
Non-linear Optimization
• Various approaches
exist for solving nonlinear problems.
• One of these is to
divide the nonlinear
functions into several
linear sections
(piecewise
linearization).
• Another approach
would be Genetic
Algorithms
GA
• It is robust and computationally efficient for
many types of problems, especially those that
are highly nonlinear
• It is based on Theory of Evolution
GA Steps
• Step 1: Population Generation: A population of n chromosomes
(i.e., individuals) is generated by randomly selecting values for the
genes in the chromosomes. (I.e., randomly assign values to the
decision variables for each of a large number of alternatives.)
• Step 2: Fitness Evaluation: Evaluate the “fitness” of each
chromosome in the population. (I.e., calculate the value of the
objective function for each alternative.)
• Step 3: Test for Completion: Test to see if an end condition has
been achieved (e.g., test to see if a maximum number of
generations has been reached, etc.). If so, stop. If not, continue
with the next step.
• Step 4: Create a New Population: Apply the processes of selection,
crossover, mutation, and replacement to build a new population.
– Step 4a: Selection: Select two parent chromosomes from the present
population according to their fitness: the greater the fitness of an
individual, the greater is the chance that the individual will be selected
to be a parent and produce offspring. (I.e., select two alternatives from
the current collection of alternatives, and base that selection upon the
value of the objective function of the current alternatives.)
– Step 4b: Crossover: With a pre-selected probability, select genes from
one parent or the other to form a new individual (i.e., to form an
offspring). (I.e., use some of the decision variable values from one of
the alternatives, and some from the other, to formulate a new
alternative.)
– Step 4c: Mutation: With a pre-selected probability, cause a mutation to
happen at any given gene in the new individual (i.e., make a small
change in the value of a randomly selected decision variable). (I.e.,
make small, random changes in the values of some of the decision
variables of the new alternative.)
Selection Process
Crossover Process
Mutation Process
Example