EML4552 - Engineering Design Systems II (Senior Design Project)

1
EML4552 - Engineering Design Systems II(Senior
Design Project)

• Optimization Theory and
• Optimum Design
• Unconstrained Optimization
• (Lagrange Multipliers)

Hyman Chapter 10
2
Unconstrained Optimization

• In 1-D the optimum is determined by

3
Unconstrained Optimization

• The condition for a local optimum can be extended
  to multi-dimensions

x2
x1
4
Unconstrained Optimization

• Condition for local optimum in unconstrained
  problem
• However, most optimization problems are
  constrained

5
Optimization

• Minimize (Maximize) an Objective Function of
  certain Variables subject to Constraints

6
Lagrange Multipliers

• An analytical approach for solving constrained
  optimization problems
• Particularly suited for problems in which the
  objective function and the constraints can be
  expressed analytical (even if highly non-linear)
• Could be numerically implemented for more general
  cases
• Will present the method through a simple example,
  it can be generalized for more complex problems

7
Lagrange Multiplers Example

• Determine the dimensions of a rectangular storage
  container to minimize fabrication costs, the
  container will hold a volume V, and be made of
  steel in the bottom (at a cost of S /unit
  surface), and wood on the side (at a cost of W
  /unit surface)

8
Lagrange Multipliers Example

• In principle, we could solve for z in terms
  of x and y. Substitute back into the equation for
  cost to obtain C(x,y) and then apply the
  condition dC/dxdC/dy0
• This method, although correct in principle, could
  be very complex if we had many variables and
  constraints, or when the equations involved are
  difficult to solve (or involve numerical models)
• A more general method is needed to approach
  constrained optimization problems.

9
Lagrange Multipliers Example

• Rewrite the constraint
• Define the Lagrangian as
• Notice that we have added zero to the objective
  function

10
Lagrange Multipliers Example

• Have turned a 3-D constrained problem into a 4-D
  unconstrained problem

11
Lagrange Multipliers Example

• The solution to the set of 4 equations in 4
  unknowns is the optimum we seek. We need to solve
  the system, in this case

12
Lagrange Multipliers Example

• Substituting

13
Lagrange Multipliers Example

• Solutions
• xy means the optimum occurs when the bottom of
  the container is square
• (the second solution can be shown to be the same
  condition xy)

14
Lagrange Multipliers Example

• Substituting

15
Lagrange Multipliers General Case
16
Lagrange Multipliers General Case
17
Lagrange Multipliers General Case
18
Lagrange Multipliers General Case
19
Other Optimization Methods

• Step 1 Convert a constrained optimization
  problem into an unconstrained problem by use of
  penalty functions

20
Other Optimization Methods

• Step 2 Use a search method to obtain the
  optimum (numerical probing of the objective
  function)
• Random search
• Directed search
• Hybrid search
• Combination of methods (decomposition,
  sequential application, etc.)

21
Search Methods

• The challenge is to create an efficient search
  method that at the same time ensures we find the
  global optimum and not just a local optimum
• Random search
• Steepest descent
• Simplex (polyhedron) search
• Genetic algorithm
• Simulated annealing