Using interval techniques to solve a parametric multi-objective optimization problem

1. A computer implemented method for solving a parametric multi-objective optimization problem in a combined design space and parameter space using interval techniques, wherein the design space contains design-space variables which are fixed for a selected design, and wherein the parameter space contains parameters which are variables for the selected design, the method comprising:

receiving a design-optimization problem at a computer system, wherein the problem is specified by multiple-objective functions which are to be optimized in the combined design space and parameter space;

initializing, using the computer system, a design-variable box spanning the design space;

performing, using the computer system, an interval optimization process on the parameter space by subdividing the design-variable box in the design space into design-variable sub-boxes, and:

determining, using the computer system, a parametric Pareto front for a design-variable sub-box using an interval optimization technique;

comparing, using the computer system, a set of parametric Pareto fronts associated with a set of design-variable sub-boxes to determine which parametric Pareto fronts are certainly dominated by other parametric Pareto fronts;

eliminating, using the computer system, the design-variable sub-boxes associated which the parametric Pareto fronts which are certainly dominated by other parametric Pareto fronts; and

if predetermined stopping criteria are not met:

subdividing remaining design-variable sub-boxes; and

repeating the determining, comparing and eliminating steps;

otherwise:

producing, using the computer system, an optimized solution for the design-optimization problem from the remaining design-variable sub-boxes and the associated parametric Pareto fronts.

2. The method of claim 1 , wherein prior to performing the interval optimization process on the parameter space, the method further comprises:

determining a global Pareto front for the combined design space and parameter space by solving a nonparametric multiple-objective optimization problem, wherein the parameters are treated as additional design variables; and

wherein the method is terminated if,

the global Pareto front results from a single point in the combined design space and parameter space;

the global Pareto front results from a single point in the design space but different points in the parameter space; or

the global Pareto front results from a single point in the parameter space but different points in the design space.

3. The method of claim 2 , wherein if the global Pareto front results in a single value for a variable in the combined design space and parameter space, the method further comprises removing the variable from the optimization problem.

4. The method of claim 2 , wherein the global Pareto front forms an upper bound for the parametric multiple-objective optimization problem.

5. The method of claim 1 , wherein determining the parametric Pareto front for the design-variable sub-box involves:

subdividing the parameter space into parameter-space sub-boxes; and

performing an iterative interval optimization process on the parameter-space sub-boxes until predetermining stopping criteria are met.

6. The method of claim 5 , wherein performing the iterative interval optimization process involves:

evaluating the multiple-objective functions on the parameter-space sub-boxes;

applying a set of criteria to eliminate some of the parameter-space sub-boxes; and

subdividing remaining parameter-space sub-boxes.

7. The method of claim 6 , wherein the set of criteria to eliminate some of the parameter-space sub-boxes includes:

a direct comparison of the evaluation results on the multiple-objective functions; and

a gradient technique.

8. The method of claim 5 , wherein the predetermined stopping criteria include:

the change during successive evaluations on the multiple-objective functions as a result of further subdividing the remaining sub-boxes is less than a predetermined amount;

a predetermined maximum number of iterations is reached; or

the largest size of any remaining sub-box is below a predetermined value.

9. The method of claim 1 , wherein a parametric Pareto front for a design-variable sub-box is represented by a finite set of boxes which contains the actual parametric Pareto front for the parametric multi-objective optimization problem.

10. The method of claim 9 , wherein a parametric Pareto front P(A) associated with a design-variable sub-box A certainly dominates a parametric Pareto front P(B) associated with a design-variable sub-box B, if:

for every box VεS(B) there is at least one box UεS(A) such that:

F i (A; U)≦F i (B; V ), wherein i=1, . . . , n; and

there is at least one combination of VεS(B) and UεS(A) such that: F i (A; U)<F i (B; V) for some iε{1, . . . , n},

wherein S(A) and S(B) are the Pareto optimal sets associated with the parametric Pareto fronts P(A) and P(B), respectively, and wherein F i are the set of multiple-objective functions.

11. The method claim 1 , wherein the method further comprises applying a set of subjective elimination criteria to further eliminate remaining parametric Pareto fronts and the associated design-variable sub-boxes.

12. A computer-readable storage medium storing instructions that when executed by a computer cause the computer to perform a method for solving a parametric multi-objective optimization problem in a combined design space and parameter space using interval techniques, wherein the design space contains design-space variables which are fixed for a selected design, and wherein the parameter space contains parameters which are variables for the selected design, the method comprising:

receiving a design-optimization problem at a computer system, wherein the problem is specified by multiple-objective functions which are to be optimized in the combined design space and parameter space;

initiating a design-variable box spanning the design space;

performing an interval optimization process on the parameter space by subdividing the design-variable box in the design space into design-variable sub-boxes, and:

determining a parametric Pareto front for a design-variable sub-box using an interval optimization technique;

comparing a set of parametric Pareto fronts associated with a set of design-variable sub-boxes to determine which parametric Pareto fronts are certainly dominated by other parametric Pareto fronts;

eliminating the design-variable sub-boxes associated which the parametric Pareto fronts which are certainly dominated by other parametric Pareto fronts; and

if predetermined stopping criteria are not met:

subdividing remaining design-variable sub-boxes; and

repeating the determining, comparing and eliminating steps;

otherwise:

producing an optimized solution for the design-optimization problem from the remaining design-variable sub-boxes and the associated parametric Pareto fronts.

13. The computer-readable storage medium of claim 12 , wherein prior to performing the interval optimization process on the parameter space, the method further comprises:

determining a global Pareto front for the combined design space and parameter space by solving a nonparametric multiple-objective optimization problem, wherein the parameters are treated as additional design variables; and

wherein the method is terminated if,

the global Pareto front results from a single point in the combined design space and parameter space;

the global Pareto front results from a single point in the design space but different points in the parameter space; or

the global Pareto front results from a single point in the parameter space but different points in the design space.

14. The computer-readable storage medium of claim 13 , wherein if the global Pareto front results in a single value for a variable in the combined design space and parameter space, the method further comprises removing the variable from the optimization problem.

15. The computer-readable storage medium of claim 13 , wherein the global Pareto front forms an upper bound for the parametric multiple-objective optimization problem.

16. The computer-readable storage medium of claim 12 , wherein determining the parametric Pareto front for the design-variable sub-box involves:

subdividing the parameter space into parameter-space sub-boxes; and

performing an iterative interval optimization process on the parameter-space sub-boxes until predetermining stopping criteria are met.

17. The computer-readable storage medium of claim 16 , wherein performing the iterative interval optimization process involves:

evaluating the multiple-objective functions on the parameter-space sub-boxes;

applying a set of criteria to eliminate some of the parameter-space sub-boxes; and

subdividing remaining parameter-space sub-boxes.

18. The computer-readable storage medium of claim 17 , wherein the set of criteria to eliminate some of the parameter-space sub-boxes includes:

a direct comparison of the evaluation results on the multiple-objective functions; and

a gradient technique.

19. The computer-readable storage medium of claim 16 , wherein the predetermined stopping criteria can include:

the change during successive evaluations on the multiple-objective functions as a result of further subdividing the remaining sub-boxes is less than a predetermined amount;

a predetermined maximum number of iterations is reached; or

the largest size of any remaining sub-box is below a predetermined value.

20. The computer-readable storage medium of claim 12 , wherein a parametric Pareto front for a design-variable sub-box is represented by a finite set of boxes which contains the actual parametric Pareto front for the parametric multi-objective optimization problem.

21. The computer-readable storage medium of claim 20 , wherein a parametric Pareto front P(A) associated with a design-variable sub-box A certainly dominates a parametric Pareto front P(B) associated with a design-variable sub-box B, iff:

for every box VεS(B) there is at least one box UεS(A) such that:

F i (A; U)≦F i (B; V), wherein i=1, . . . , n; and

there is at least one combination of VεS(B) and UεS(A) such that:

F i (A; U)<F i (B; V) for some iε{1, . . . , n},

wherein S(A) and S(B) are the Pareto optimal sets associated with the parametric Pareto fronts P(A) and P(B), respectively, and wherein F i are the set of multiple-objective functions.

22. The computer-readable storage medium of claim 12 , wherein the method further comprises applying a set of subjective elimination criteria to further eliminate remaining parametric Pareto fronts and the associated design-variable sub-boxes.

23. An apparatus that solves a parametric multi-objective optimization problem in a combined design space and parameter space using interval techniques, wherein the design space contains design-space variables which are fixed for a selected design, and wherein the parameter space contains parameters which are variables for the selected design, comprising:

a computer system with at least one processor and a memory storing a program of instructions;

a receiving mechanism configured to receive a design-optimization problem at the computer system, wherein the problem is specified by multiple-objective functions which are to be optimized in the combined design space and parameter space;

a initialization mechanism configured to initialize a design-variable box spanning the design space;

an optimization mechanism configured to perform an interval optimization process on the parameter space by subdividing the design-variable box in the design space into design-variable sub-boxes, wherein while performing the interval optimization process, the optimization mechanism is configured to:

determine a parametric Pareto front for a design-variable sub-box using an interval optimization technique;

compare a set of parametric Pareto fronts associated with a set of design-variable sub-boxes to determine which parametric Pareto fronts are certainly dominated by other parametric Pareto fronts;

eliminate the design-variable sub-boxes associated which the parametric Pareto fronts which are certainly dominated by other parametric Pareto fronts; and

if predetermined stopping criteria are not met:

subdivide remaining design-variable sub-boxes; and

repeat the determining, comparing and eliminating steps;

otherwise:

a producing mechanism configured to produce an optimized solution for the design-optimization problem from the remaining design-variable sub-boxes and the associated parametric Pareto fronts.