Methods and systems for multi-objective optimization and/or wireless power delivery

1. A method for wireless power transmission from a transmitter to a plurality of receivers, the method comprising:

receiving a set of initial points in a transmission parameter space, wherein:

each point in the transmission parameter space corresponds to a respective transmission configuration of the transmitter;

a set of objective functions map from the transmission parameter space to an objective space, wherein each receiver of the plurality is associated with a different objective function of the set; and

the objective space defines a Pareto front in the objective space, wherein each point on the Pareto front is Pareto-optimal with respect to the set of objective functions;

determining a first set of Pareto points based on the set of initial points, wherein:

each Pareto point of the first set is substantially on the Pareto front; and

determining the first set of Pareto points comprises: for each initial point, determining, based on the initial point, a corresponding Pareto point of the first set, wherein determining the corresponding Pareto point comprises evaluating each objective function based on the corresponding Pareto point;

determining a set of intermediary points, comprising, for each Pareto point of the first set, determining a respective set of points based on the Pareto point, wherein the set of intermediary points comprises each respective set of points;

determining a second set of Pareto points based on the set of intermediary points, wherein:

each Pareto point of the second set is substantially on the Pareto front; and

determining the second set of Pareto points comprises: for each intermediary point, determining, based on the intermediary point, a corresponding Pareto point of the second set, wherein determining the corresponding Pareto point comprises evaluating each objective function based on the corresponding Pareto point;

based on the first and second sets of Pareto points, selecting a selected Pareto point; and

at the transmitter, wirelessly transmitting power to the plurality of receivers based on the selected Pareto point.

2. The method of claim 1 , further comprising determining the set of initial points based on a previous transmission configuration of the transmitter.

3. The method of claim 2 , wherein the initial points of the set are distributed around the previous transmission configuration in the transmission parameter space.

4. The method of claim 1 , wherein each objective function of the set is indicative of a respective power reception metric of the receiver associated with the objective function.

5. The method of claim 1 , wherein:

the transmitter comprises a plurality of transmission elements; and

the transmission parameter space comprises, for each transmission element of the plurality, at least one of: a respective phase parameter and a respective amplitude parameter.

6. The method of claim 1 , wherein:

the transmitter comprises a plurality of transmission elements;

evaluating each objective function based on a Pareto point comprises:

controlling the plurality of transmission elements to transmit based on a transmission configuration corresponding to the Pareto point;

concurrent with controlling the plurality of transmission elements to transmit based on the transmission configuration, for each receiver of the plurality, sampling a respective dataset indicative of a respective wireless power reception characteristic associated with the receiver; and

for each receiver, based on the respective dataset, determining a respective value associated with the receiver, thereby evaluating the respective objective function associated with the receiver based on the Pareto point; and

wirelessly transmitting power to the plurality of receivers based on the selected Pareto point comprises controlling the plurality of transmission elements to transmit based on a selected transmission configuration corresponding to the selected Pareto point.

7. The method of claim 6 , wherein, for each receiver, sampling the respective dataset is performed by the receiver.

8. The method of claim 1 , wherein determining a corresponding Pareto point based on the initial point comprises:

a) determining an updated point based on the initial point and the set of objective functions, comprising:

for each objective function of the set, determining a respective ascent direction in the transmission parameter space, comprising performing a local optimum search around the initial point based on the objective function;

determining a total ascent direction based on a linear combination of the respective ascent directions; and

performing a line search along a line in the transmission parameter space, comprising evaluating the set of objective functions at a plurality of test points, wherein:

the line propagates from the initial point along the total ascent direction;

the line comprises the updated point and the plurality of test points; and

the updated point Pareto-dominates each test point of the plurality; and

b) determining the Pareto point based on the updated point.

9. The method of claim 8 , wherein b) comprises iteratively repeating a) until the updated point is substantially on the Pareto front, wherein the updated point from the previous iteration of a) is used as the initial point of the current iteration of a).

10. The method of claim 1 , further comprising determining the set of initial points based on a generator point, comprising:

for each of a plurality of objective functions of the set, determining a respective generator ascent direction in the transmission parameter space, comprising performing a local optimum search around the generator point based on the objective function;

for each initial point of the set, determining a respective displacement direction, the initial point is displaced from the generator point along the displacement direction, wherein each displacement direction is a different linear combination of the respective generator ascent directions.

11. A method for wireless power transmission from a transmitter to a plurality of receivers, the method comprising:

a) receiving an initial point in a transmission parameter space, wherein:

each point in the transmission parameter space corresponds to a respective transmission configuration of the transmitter;

a set of objective functions map from the transmission parameter space to an objective space, wherein each receiver of the plurality is associated with a different objective function of the set; and

the objective space defines a Pareto front, wherein each point on the Pareto front is Pareto-optimal with respect to the set of objective functions;

b) determining an updated point based on the initial point and the set of objective functions, comprising:

for each objective function of the set, determining a respective ascent direction in the transmission parameter space, comprising performing a local optimum search around the initial point based on the objective function;

determining a total ascent direction based on a linear combination of the respective ascent directions; and

performing a line search along a line in the transmission parameter space, comprising evaluating the set of objective functions at a plurality of test points, wherein:

the line propagates from the initial point along the total ascent direction;

the line comprises the updated point and the plurality of test points; and

the updated point Pareto-dominates each test point of the plurality;

c) based on the updated point, determining a Pareto point substantially on the Pareto front; and

d) at the transmitter, wirelessly transmitting power to the plurality of receivers based on the Pareto point.

12. The method of claim 11 , further comprising, before receiving the initial point, determining the initial point based on a generator Pareto point, wherein the initial point is within a threshold distance of the generator Pareto point in the transmission parameter space.

13. The method of claim 11 , wherein each objective function of the set is indicative of a respective power reception metric of the receiver associated with the objective function.

14. The method of claim 11 , wherein:

the transmitter comprises a plurality of transmission elements; and

the transmission parameter space comprises, for each transmission element of the plurality, at least one of: a respective phase parameter and a respective amplitude parameter.

15. The method of claim 11 , wherein:

the ascent directions span an ascent cone in the transmission parameter space;

determining the total ascent direction is performed such that the total ascent direction is Pareto-dominant, with respect to a set of ascent objective functions, over all other directions of the ascent cone; and

the set of ascent objective functions comprises, for each objective function of the set of objective functions, a respective derivative, with respect to distance from the initial point, of the objective function.

16. The method of claim 15 , wherein:

the set of objective functions consists of N objective functions;

the ascent directions are represented by d i , wherein the total ascent direction is equal to

∑

i

=

1

N

⁢

α

i

⁢

d

i

,

wherein α i >0 and

∑

i

=

1

N

⁢

α

i

=

1

;

determining the total ascent direction comprises solving a linear programming problem, wherein, for each k∈{1, . . . , N}, an objective of the linear programming problem is to maximize α k , subject to the constraints:

∑

i

=

1

N

⁢

α

i

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d

i

·

d

j

≥

0

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∀

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N

;

and

∑

i

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N

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α

i

≤

1

,

where

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∀

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…

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,

N

.

17. The method of claim 11 , wherein determining the Pareto point based on the updated point comprises iteratively repeating b) until the updated point is substantially on the Pareto front, wherein the updated point from the previous iteration of b) is used as the initial point of the current iteration of b).

18. The method of claim 11 , wherein performing the local optimum search comprises using a gradient-free optimization algorithm.

19. The method of claim 18 , wherein the gradient-free optimization algorithm comprises at least one of: a Nelder-Mead algorithm, an adaptive meshing algorithm, and a particle swarm optimization algorithm.

20. The method of claim 11 , wherein performing the local optimum search comprises evaluating each objective function based on a transmission configuration, comprising:

controlling the transmitter to transmit based on the transmission configuration;

concurrent with controlling the transmitter to transmit based on the transmission configuration, for each receiver of the plurality, sampling a respective dataset indicative of a respective wireless power reception characteristic associated with the receiver; and

for each receiver, based on the respective dataset, determining a respective value associated with the receiver, thereby evaluating the respective objective function associated with the receiver based on the transmission configuration.