Online optimization algorithms 173
Single-objective optimization: Under most circumstances, only one
objective function is optimized at a time. This is the case as long as tuning
the related control parameters does not negatively impact the machine performance in other measures. With the normalized parameters, the optimization
problem for one objective function can be expressed as
x min = arg min
x∈[0,1] n
f (x).
(7.2)
In accelerator tuning problems, the objective function, f (x), can be considered a smooth function. However, the function value evaluated through measurement, ¯
f (x), is not smooth, as there is always a random deviation in the
measurement from the true performance function,
¯
f (x) = f (x) + ξ,
(7.3)
where ξ is a random variable that denotes the measurement error.
The random error has a significant impact over the behaviors of the optimization algorithms. Many traditional algorithms assume the objective function to be smooth. Depending on the working principle of the algorithms and
the nature of the optimization problems, with noise in the objective function,
the algorithms may have slow convergence or fail to converge to the minimum. Robustness against noise is a main requirement for online optimization
algorithms.
Multi-objective optimization: There are cases when the machine performance in terms of multiple measures needs to be simultaneously optimized,
while the optimal setting for one measure may not be the best condition for
the other(s). As a gain made in one measure by adjusting the knobs could lead
to the loss in another measure, a trade-off between the performance measures
is needed in selecting the operation condition.
A common approach is to combine these measures into one objective by
assigning weights to the individual objectives,
f w =
M
i
w i f i ,
(7.4)
where M is the number of objectives and the weights, w i , i = 1–M, account
for both the scale differences of the objectives and their relative importance.
The combined objective can also be defined with normalized values
f w =
M
i
w i
f i − f
target
i
L i
,
(7.5)
where f
target
i
is the target performance for the i’th performance metric, and
L i ’s are scale constants that bring the metrics to comparable numeric values.
In the above approach the assigned weights in the definition of the combined objective function have a big impact to the optimal solution. This is
Single-objective optimization: Under most circumstances, only one
objective function is optimized at a time. This is the case as long as tuning
the related control parameters does not negatively impact the machine performance in other measures. With the normalized parameters, the optimization
problem for one objective function can be expressed as
x min = arg min
x∈[0,1] n
f (x).
(7.2)
In accelerator tuning problems, the objective function, f (x), can be considered a smooth function. However, the function value evaluated through measurement, ¯
f (x), is not smooth, as there is always a random deviation in the
measurement from the true performance function,
¯
f (x) = f (x) + ξ,
(7.3)
where ξ is a random variable that denotes the measurement error.
The random error has a significant impact over the behaviors of the optimization algorithms. Many traditional algorithms assume the objective function to be smooth. Depending on the working principle of the algorithms and
the nature of the optimization problems, with noise in the objective function,
the algorithms may have slow convergence or fail to converge to the minimum. Robustness against noise is a main requirement for online optimization
algorithms.
Multi-objective optimization: There are cases when the machine performance in terms of multiple measures needs to be simultaneously optimized,
while the optimal setting for one measure may not be the best condition for
the other(s). As a gain made in one measure by adjusting the knobs could lead
to the loss in another measure, a trade-off between the performance measures
is needed in selecting the operation condition.
A common approach is to combine these measures into one objective by
assigning weights to the individual objectives,
f w =
M
i
w i f i ,
(7.4)
where M is the number of objectives and the weights, w i , i = 1–M, account
for both the scale differences of the objectives and their relative importance.
The combined objective can also be defined with normalized values
f w =
M
i
w i
f i − f
target
i
L i
,
(7.5)
where f
target
i
is the target performance for the i’th performance metric, and
L i ’s are scale constants that bring the metrics to comparable numeric values.
In the above approach the assigned weights in the definition of the combined objective function have a big impact to the optimal solution. This is