Maximizing a Nonlinear Function with Local Maxima

1
Example 8.1

• Maximizing a Nonlinear Function with Local Maxima

2
Background Information

• To see how Evolutionary Solver works, we consider
  a simple function that is difficult for the
  standard Solvers nonlinear algorithm.
• This example, analyzed in Chapter 7, is a
  function of a single variable x f(x)(x - 1)(x -
  2)(x - 3)(x - 4)(x - 5) for 1 x 5.
• We want to maximize f(x) over this range.
• However, the graph of this function shown on the
  next slide indicates that there are two local
  maxima one at around x 3.5 and the other at x
  5.
• The global maximum, the one we want, is near x
  1.5.
• How do we find it with Excel?

3
(No Transcript)
4
LocalMaximaFunction.xls

• The model is particularly simple.
• It is shown here and in this file.

5
Solution

• To set it up, enter any value in cell B6 (the
  only changing cell), enter the formula B6-1 in
  cell B7, copy this down to cell B11, and enter
  the formula PRODUCT(B7B11) in cell B12.
• We want to maximize the value in cell B12 while
  constraining the value in cell B6 to be between 1
  and 5.
• If we use the standard Solver, without checking
  the Assume Linear Model option, you can check
  that the solution depends on the starting value
  in cell B6.

6
Solution -- continued

• If you didnt have a graph of the function to
  lead you in the right direction, how would you
  know what to pick as starting points.
• Whatever starting point you use, how would you
  know whether Solver had found the global maximum?
• We now see how Evolutionary Solver can help solve
  this problem.

7
Using Evolutionary Solver

• Evolutionary Solver uses GAs to obtain good
  solutions.
• Occasionally, Evolutionary Solver drastically
  changesmutatesthe value of a changing cell.
• Usually we stop Evolutionary Solver after a
  specified time period (say 5 minutes) or when
  there has been no improvement in the target cell
  value after a given time.

8
Using Evolutionary Solver -- continued

• Following is some general information about
  Evolutionary Solver
• Evolutionary Solver usually finds a good
  solution, but there is no guarantee that it will
  find the best solution.
• Evolutionary Solver is not very efficient at
  handling constraints. The best way to handle
  constraints is to penalize a violation of a
  constraint. The penalty is then included in the
  target cell. We do not use penalties in this
  example, but we illustrate them in a later
  example.
• A good starting solutionthe values you place in
  the changing cellsusually helps Evolutionary
  Solver in its search for an optimal solution.
  However, the starting solution is not absolutely
  critical to success.

9
Using Evolutionary Solver -- continued

• Evolutionary Solver places more of a burden on
  you to specify certain parameters of the
  algorithm. These parameters are specified in
  Options dialog boxes, as we illustrate shortly.
  Unfortunately, these parameters are not very
  intuitive to most users, and some experimentation
  is necessary to find the best settings of these
  parameters for any given model. Nevertheless, if
  you use the default settings or the settings we
  suggest, they should work reasonably well.
• Much of the solution process is driven by random
  numbers that direct the search. Therefore, two
  people can get different solutions to the same
  problem. In fact, running Evolutionary Solver a
  second time can possibly yield a different
  solution! You can set a random seed parameter to
  get the same solution on two successive runs.

10
Using Evolutionary Solver -- continued

• After Evolutionary Solver has found a good
  solution, you can use the GRG Nonlinear Solver
  (the nonlinear algorithm included with the
  Premium Solver software) to try to find a
  slightly better solution. If there is no
  improvement, you can probably infer that the
  solution found by Evolutionary Solver is close to
  optimal.

11
Using Evolutionary Solver -- continued

• In general, use the following steps to implement
  Evolutionary Solver
• Specify the target cell, changing cells, and
  constraints. This is done in the usual way. The
  only difference is that you should put lower and
  upper bounds on all changing cellsin addition to
  any other constraints that might be in the model.
  At this point, the Solver dialog box should
  appear as shown on the next slide.

12
(No Transcript)
13
Using Evolutionary Solver -- continued

• Select Evolutionary Solver. Click on the Premium
  button, and then click on the dropdown list of
  available algorithms to select Evolutionary
  Solver.

14
Using Evolutionary Solver -- continued

• Solver Options. Click on the Options button to
  show the Solver Options dialog box.
• Max Time is not important. However, if you want
  Solver to run unattended, set this to a large
  value such as 10,000 seconds.
• Iterations is not important. You can leave it at
  its default value.
• Precision governs how close the constraint values
  must be to their bounds, and Convergence is
  concerned with the rate of change of the
  objective.
• Population size is the number of candidate
  solutions (chromosomes) at any point in time, and
  the default value of 100 should work well
  although we sometimes increase it to 150.
• Mutation rate governs the frequency at which
  mutations are introduced into the population of
  solutions.

15
Using Evolutionary Solver -- continued

• Evolutionary Solver uses a random mechanism to
  perform its search, but you can make it go
  through exactly the same calculations on two
  separate runs if you use the same Random Seed
  (any integer) on each run.
• The check boxes at the lower left are much like
  the ones of the same names in the standard
  Solver. However, you should check the Require
  Bounds on Variables option. This forces you to
  enter explicit upper and lower bounds on all
  changing cells which aids Evolutionary Solver in
  its search process.
• The radio buttons under Local Search are newthey
  werent available in the versionthat accompanied
  the second edition of this book. At each step of
  the GA, Evolutionary Solver performs a local
  search in the neighborhood of the best solution
  found so far (the incumbent) to find better
  nearby solutions. The options here govern the way
  this local search is performed.

16
(No Transcript)
17
Using Evolutionary Solver -- continued

• Limit Options. Click on the Limit Options button
  to bring up the dialog box.
• Max Subproblems and Max Feasible Sols both have
  to do with the length of time Evolutionary Solver
  runs before asking you (with a beep) whether you
  want to stop or continue.
• Tolerance and Max Time w/o Improvement work in
  tandem to specify when Evolutionary Solver should
  stop. Specifically, if the percentage improvement
  in the best solutions objective value is less
  than the Tolerance value for the number of
  seconds in the Max Time without Improvement box,
  Evolutionary Solver stops and declares the best
  solution so far as optimal. We recommend leaving
  Tolerance at its default value but possibly
  increasing Max Time w/o Improvement to, say, 100
  seconds.However, chances are that this will just
  let the algorithm spin its wheels at the same

18
Using Evolutionary Solver -- continued

• Solve Without Integer Constraints allows you to
  ignore integer constraints, at least temporarily,
  to get a good noninteger solution. Then you can
  recheck this box to add back the integer
  constraints

19
Using Evolutionary Solver -- continued

• Solve. From the Limit Options dialog box, back
  your way out to the beginning Solver dialog box
  and click on Solve. You can watch the progress of
  the solution process in the status bar of your
  screen. In particular, watch the Incumbent,
  which is the current best value of the target
  cell.
• For this particular model, the Evolutionary
  Solver gets to the solution shown earlier almost
  instantaneously. Then it runs for 30 seconds (the
  time we specified in the Limit Options dialog
  box) without being able to find a better
  solution, at which time it quits.

20
Using Evolutionary Solver -- continued

• Evolutionary Solver is not as likely to get stuck
  at a local optimum as the standard nonlinear
  Solver.