Multistart Methods for Global Optimization

Solver’s multistart methods for global optimization can overcome some of the limitations of the GRG Solving method alone, but they are not a panacea. The multistart methods will automatically run the GRG method from a number of starting points and will display the best of several locally optimal solutions found, as the probable globally optimal solution. Because the starting points are selected at random and then “clustered” together, they will provide a reasonable degree of “coverage” of the space enclosed by the bounds on the variables. The tighter the variable bounds you specify and the longer Solver runs, the better the coverage.

However, the performance of the multistart methods is generally limited by the performance of the GRG method on the subproblems. If the GRG method stops prematurely due to slow convergence, or fails to find a feasible point on a given run, the multistart method can improve upon this only by finding another starting point from which the GRG method can find a feasible solution, or a better locally optimal solution, by following a different path into the same region.

If the GRG method reaches the same locally optimal solution on many different runs initiated by the multistart method, this will tend to decrease a Bayesian estimate of the number of locally optimal solutions in the problem, causing the multistart method to stop relatively quickly. In many cases this indicates that the globally optimal solution has been found – but you should always inspect and think about the solution, and consider whether you should run the GRG method manually from starting points selected based on your knowledge of the problem.

Nonlinear problems with integer constraints are solved by a Branch and Bound method that runs the GRG method on a series of subproblems. If the GRG method stops prematurely due to slow convergence, or fails to find a feasible point on a given run, this may prevent the Branch & Bound method from finding the true integer optimal solution; though in most cases – given enough time – a good integer solution can be found.

Note that, when the GRG Nonlinear Solving method is selected in the dropdown list in the Solver Parameters dialog, the Generalized Reduced Gradient algorithm is used to solve the problem – even if it is actually a linear model that could be solved by the (faster and more reliable) Simplex LP method. The GRG method will usually find the optimal solution to a linear problem, but occasionally you will receive a Solver Result Message indicating some uncertainty about the status of the solution – especially if the model is poorly scaled. So you should always ensure that you have selected the right Solving method for your problem.

Problems with Poorly Scaled Models

A poorly scaled model is one that computes values of the objective, constraints, or intermediate results that differ by several orders of magnitude. A classic example is a financial model that computes a currency amount in millions or billions and a return or risk measure in fractions of a percent. Because of the finite precision of computer arithmetic, when these values of very different magnitudes (or others derived from them) are added, subtracted, or compared – in the user’s model or in Solver’s own calculations – the result will be accurate to only a few significant digits. After many such steps, Solver may detect or suffer from “numerical instability.”

The effects of poor scaling in a large, complex optimization model can be among the most difficult problems to identify and resolve. It can cause Solver to return messages such as “Solver could not find a feasible solution,” “Solver could not improve the current solution,” or even “The linearity conditions required by this LP Solver are not satisfied,” with results that are suboptimal or otherwise very different from your expectations. The effects may not be apparent to you, given the initial values of the variables, but when Solver explores Trial Solutions with very large or small values for the variables, the effects will be greatly magnified.

Dealing with Poor Scaling

The Solver Options dialog includes a Use Automatic Scaling check box. When this box is selected (it is selected by default), Solver rescales the values of the objective and constraint functions internally in order to minimize the effects of poor scaling. But this can only help with Solver’s own calculations – it cannot help with poorly scaled results that arise in the middle of your Excel formulas. Further, in some poorly scaled models, automatic scaling can make things worse rather than better – so you may wish to try clearing this check box and re-solving.

The best way to avoid scaling problems is to carefully choose the “units” implicitly used in your model so that all computed results are within a few orders of magnitude of each other. For example, instead of expressing currency amounts in single units, you could express them in units of (say) millions, if appropriate for your model, so the actual numbers computed on your worksheet may range from perhaps 1 to 1,000.

NOTES

 You can interrupt the solution process by pressing ESC. Microsoft Office Excel recalculates the worksheet with the last values that are found for the decision variable cells, and displays the Show Trial Solution dialog.

 To create a report that is based on your solution after Solver finishes, you can click a report type in the Reports box and then click OK. The report is created on a new worksheet in your workbook. If Solver doesn't find a solution, either certain diagnostic reports or no reports are available.

 To save your decision variable cell values as a scenario that you can display later, click Save Scenario in the Solver Results dialog box, and then type a name for the scenario in the Scenario Name box.