Gradient based methods for multi-objective optimization

1. A computer system comprising:

a processor;

a display; and

a memory system, comprising instructions, which, when executed, cause the computer, system to perform a multi-objective optimization of a computational simulation model of a physical system by Pareto navigator method, the method of stepping along a Pareto frontier in such a way that a few high priority objective functions F i (X), i=1,2. . . , m + ,1 ≦m + <m are improving, where m 30 and m are finite positive integers, where said Pareto frontier is a set of points in the m−dimensional objective space those do not dominate each other with respect to said m objective functions; point A dominates point B if F i (A)≦F i (B), i=1,2. . . , m and at least for one of i F i (A)<F i (B); the method comprising the steps of:

a) determining output variables of said physical system that need to be optimized;

b) describing said output variables of said physical system as minimized objective functions F i (X), i =1,2. . . , m, considered in an n-dimensional design space X={x 1 ,x 2 , . . .,x n }, x* j ≦x j ≦x* j , j=1, . . . , n , where each x j is a design variable of said physical system, x* j ,x* j are finite numbers determining minimum and maximum values of each said design variable, the values entered by a user and based on technical characteristics of said physical system;

c) assigning a high priority status to a subset of m + <m objective functions;

d) specifying, from said Pareto frontier, a known Pareto optimal point previously found by any multi-objective optimization algorithm which is optimal with respect to all said m objective functions and needs to be improved with respect to said m + high priority objective functions;

e) estimating a gradient for each of said m + high priority objective functions on said Pareto optimal point specified in step d) by any algorithm of gradient estimation;

f) determining a direction of simultaneous improvement of all said m + high priority objective functions based on gradients specified in step e) by any algorithm which allows a determination of a direction of simultaneous improvement for several objective functions;

g) assigning a step size S 1 for improvement of said m + high priority objective functions, where said step size S 1 is a distance between start point and end point of a step in the n-dimensional design space X={x 1 ,x 2 , . . . , x n }, x* j ≦x j ≦x* j , j=1, . . . , n determined in such a way that said end point remains inside of said design space, and 0<S 1 <δ PN , where δ PN is a pre-assigned navigation approximation tolerance;

h) starting from said Pareto optimal point specified in step d) and performing a step with said step size S 1 in the direction determined in step f), and declaring a found point as an improved point, which is moved in the desired direction;

i) starting an optimization process from said improved point found in step h), and declaring said improved point as an initial current point;

j) estimating a gradient for each of said m objective functions on said initial current point specified in step i) by any algorithm of gradient estimation;

k) determining a direction of simultaneous improvement of all said m objective functions based on gradients specified in step j) by any algorithm which determines a direction of simultaneous improvement for several objective functions;

l) assigning a step size S 2 for improvement of all said m objective functions, where said step size S 2 is a distance between the start point and the end point of a step in the n-dimensional design space X={x 1 ,x 2 , . . . , x n }, x* j ≦x j ≦x* j , j=1, . . . , n determined in such a way that said end point remains inside of said design space, and 0<S 2 <δ PN , where δ PN is the pre-assigned navigation approximation tolerance;

m) starting from said initial current point and performing a step with step size S 2 in the direction specified in step k), and declaring a found point as a new point-candidate;

n) comparing said initial current point with said point-candidate;

o) if said initial current point is not dominated by said point-candidate with respect to all said m objective functions, then declaring said initial current point as Pareto optimal point, and proceed to step s);

p) assigning a new step size S 2 =S 2 *t, where t is a pre-assigned number 0<t<1;

q) if the new step size S 2 <δ a where δ a is a pre-assigned approximation tolerance then declaring said initial current point as Pareto optimal point, and proceed to step s);

r) declaring said point-candidate as a subsequent current point, and repeating steps m)-r) with the new step size S 2 and Initial current point replaced by subsequent current point;

s) interpreting said declared Pareto optimal point in terms of said physical system.

2. The computer system of claim 1 wherein the Pareto navigator method further comprising the step of sequential use of each Pareto optimal point declared as a start point for a next step along the Pareto frontier in said direction of simultaneous improvement of said high priority objective functions until a pre-assigned number of steps has been made, or said high priority objective functions have been improved over pre-assigned values.

3. The computer system of claim 2 wherein the Pareto navigator method further comprising the step of sequentially processing a plurality of known Pareto optimal points specified from said Pareto frontier.

4. The computer system of claim 1 wherein the Pareto navigator method further comprising reuse of intermediate points as part of an optimal solution, comprises the steps of:

t) saving all said intermediate points in memory where said intermediate points are points evaluated to estimate all said gradients by any method of gradients estimation;

u) comparing all saved intermediate points against each other, and against said known Pareto optimal point and said declared Pareto optimal point with respect to all objective functions;

v) filtering out all non-dominating points, declaring the rest of points as Pareto optimal points, and declaring at least two additional Pareto optimal points.