Author: YehLiang Hsu, ChiaChieh Yu,
YaoTung Wang, MingHsiu Hsu (20081211); recommended: YehLiang Hsu
(20100609).
Note: This paper is published in Journal of the Chinese Institute of
Engineers, Vol. 32, No. 5, pp. 717725, July 2009.
Uniform
luminance design of the directtype backlight unit of liquid crystal displays
ABSTRACT
Light efficiency
optimization of a directtype backlight unit (BLU) of the liquid crystal
display (LCD) is considered in this paper. The purpose of this research is to obtain
the greatest uniformity in a directtype BLU, while the brightness is
maintained in a satisfactory level.
It is difficult to find a numerical optimization algorithm that is readily applicable to this problem
because of the existence of discrete variables, implicit objective functions
and constraints, or even binary constraints. The sequential neuralnetwork
approximation (SNA) method is designed specifically for this type of
engineering design optimization problem. In this paper,
a two variable LCD BLU design example is presented first to illustrate the
design process using the SNA method. Then a four variable LCD BLU design
optimization problem is solved.
Keywords: backlight unit, light
efficiency optimization, luminance uniformity, sequential
approximation method (ME21)
I.
INTRODUCTION
The liquid
crystal display (LCD) has become the mainstream display component in various
digital consumer products from cellular phones, personal digital assistants
(PDA), notebook computers, to television sets. The backlight unit (BLU) module
provides a light source for transmission type LCD. Based on the location of the
light sources, BLU modules can be classified into two categories: edgetype BLU
and directtype BLU. The edgetype BLU module mounts the light source lamp (such
as coldcathode fluorescent lamp, CCFL) at the top or the bottom edge of the
LCD. A light guide plate is used to guide the light from edgelight into surfacelight.
One advantage of the edgetype BLU module is that it is thin. However, the
edgetype BLU is not able to produce enough brightness, and is often used for
small or midsized LCDs.
For products
requiring largesized LCDs, television sets for example, the directtype BLU
module is commonly used. The directtype BLU module places a number of CCFLs
directly behind the LCD panel in a parallel configuration. The core components
of a directtype BLU include a reflective sheet, a light source module (CCFLs
and clamps), a diffuser plate, and a prismatic sheet sandwiched between two
diffuser sheets.
Brightness and
luminance uniformity are the major performance indices of the BLU. While the
directtype BLU module provides greater brightness, the uniformity of luminance
is an important issue to be considered. Automatic optical inspection (AOI)
systems are used to measure the performance of the BLU in industry. Luminance
uniformity is a measure of how well the luminance remains constant over the
surface of the screen. “Sampled uniformity”, which compares several discrete
points on the screen to provide a quick check of the uniformity, is the
measuring scheme commonly used in industry.
Figure 1 shows
the 9 symmetrical sampling points (P1~P9) which divide the LCD screen into
small squares. The center luminance is measured at center point (P5). The “9point
uniformity” is defined by the minimum luminance measured at the 9 sampling
points divided by the maximum luminance measured at the 9 sampling points, as
shown in Eq. (1).
9Point
Uniformity (Luminance uniformity) =_{} (1)
Note that from
Eq. (1), the perfect luminance uniformity is 100%.
Fig. 1 Nine symmetrical sampling points on the LCD
screen
Brightness and
luminance uniformity depend on the shapes, locations and dimensions of the
optical components of the BLU. Researchers have been trying to change the locations
of the lamps and the shapes of the reflectors in order to improve the
brightness and to achieve uniform luminance. Chan et al. (2003) developed a special reflector for directtype BLUs to
improve the light efficiency, which achieved a 33% increase in luminance. Park
and Lim (2001) experimented with four types of reflector structures, including flat
reflectors, Vshape reflectors, Ushape reflectors and Wshape reflectors in
directtype BLUs. They found that the distance between the lamp and the diffuser
surface is proportional to the luminance uniformity.
In practice,
experimental methods are used to find a design with acceptable luminance
uniformity by LCD manufacturers. Optical simulation tools have also been used
to avoid high cost iterations needed during the experimental design process. The
use of the simulation tools helps engineers to speed up BLU design by replacing
physical mockups with virtual models created by simulation tools during the BLU
design process. However, design modification to improve luminance uniformity is
still based on the experience of the engineer, and is tested in a “trial and
error” manner using the simulation tools.
Optimization
processes combining optimization algorithms with optical simulation tools have
been introduced into edgetype BLU design to achieve a uniform luminance. Cheng
and Pedram (2004) proposed a technique aims at conserving power by reducing the
backlight illumination while retaining the image fidelity through preservation
of the image contrast. This concurrent brightness and contrast scaling
optimization problem was solved subject to constraint on contrast distortion. Choi
et al. (2004) considered an ink
density optimization problem in order to produce a uniform distribution of
luminaries on the front face of BLU. The ink density pattern was represented by
a 3^{rd}degree polynomial function. Four coefficients of the
polynomial function were the design variables. The NelderMead direct search
method was used in this study to solve the optimization problem. Cassarly and Irving
(2005) developed a hybrid optimization approach that combines a mesh feedback
optimization method with a classical optimizer to design the extraction pattern
for an edgetype BLU. The design variables of their work include density, size
and spacing of extraction patterns on a light guided plate. Park and Ko (2007)
optimized the emitting structure of multichanneltype flat fluorescent lamps
(FFLs) combined by a lenticularlenspatterned diffuser plate by the ray
tracing technique for best luminance uniformity.
In this paper, a
uniform luminance design optimization problem for directtype LCD BLU under maximum
brightness constraints is considered. Fig. 2 shows the critical dimensions of a
BLU in top view. The design parameters C
(thickness of the diffuser plate), D
(diameter of the lamps), E (thickness
of the inner diffuser sheet), F
(thickness of the prismatic sheet), G
(thickness of the outer diffuser sheet), LU (size of the display), and LQ
(number of lamps) are defined by the manufacturers. The design variables a
(angle of the reflector), b (distance between a lamp and one side of the
reflector), h (height of the enclosure), lb (width of the bottom
side of the reflector), p (distance between two lamps), and lp (distance
between a lamp and bottom side of the reflector) are to be decided by the
designer. Design variables p and lb can be expressed in terms of
other parameters and variables as:
_{}, (2)
_{}. (3)
Thus there are
only 4 independent design variables a, b, h, and lp.
Variables p and lb can be obtained using Eqs. (2) and (3) after
the optimal values of the 4 independent design variables a, b, h,
and lp are obtained.
Fig. 2 Critical dimensions of a BLU in top view
The design goals
for the BLU are to achieve higher luminance uniformity and to satisfy the
luminance requirement of the manufacturer (for example, luminance at each of
the 9 sampling points has to be higher than 9,000 lux in this case). This design
problem can be formulated as follows:
max. Uniformity(x)
s.t. Brightness(x)=1, (4)
where x is
the vector of design variables described above. In industry, the design
variables are often discrete variables. The function Uniformity(x) is the 9point uniformity defined
in Eq. (1), and the Brightness(x)
is the constraint on the luminance of the 9 sampling points of LCD. Both
functions are “implicit functions,” that is, the functions cannot be expressed in
an analytical function in terms of design variables. Both functions are evaluated
by the optical simulation tools.
In Eq. (4), the
constraint Brightness(x) can be expressed as 9 inequality
constraints for the 9 sampling points. Instead of using 9 implicit constraints,
this constraint is expressed in a simplified, binary form of “passorfail,” “feasibleorinfeasible”
in Eq. (4). If the current design x satisfies the brightness
requirement of the LCD (luminance at each of the 9 sampling points is higher
than 9,000 lux), then Brightness(x)=1, otherwise Brightness(x)=0.
This binary constraint “Brightness(x)=1” alone defines the feasible domain of the
optimization problem. This type of binary constraint is useful in many
engineering design applications where the final design has to pass certain critical
tests. The results of the tests are often simply “passfail”, without precise
function values.
The Sequential
NeuralNetwork Approximation (SNA) method was developed by the authors (Hsu et. al, 2001; Hsu et. al, 2003) to handle engineering design optimization problems
with discrete variables, implicit functions and binary constraints. The SNA
method was also applied to reduce the weight of aluminum disc wheels under
fatigue constraints (Hsu and Hsu, 2001). The SNA method is revised here to
handle the LCD BLU optimization problem. In the previous version, the objective
function was treated as an explicit function and the neural network was only
used to simulate the boundary of the feasible domain of the optimization
problem. In the revised SNA method, two backpropagation neural networks are
trained first to simulate the objective function and the feasible domain formed
by the implicit constraints using a few representative training data samples. A
sample of training data consists of three pieces of information: a design point
(i.e., a set of values for the design variables), objective function value at
this design point, and whether this design point is feasible or infeasible. A
search algorithm then searches for the “optimal point” of the simulated objective
function in the feasible domain simulated by the neural network.
This new design
point is checked against the true implicit constraints to see whether it is
feasible, and the new training data sample is then added to the training set.
The neural network is trained again with this added training data, in the hope
that the network will better approximate the objective function and the
boundary of the feasible domain of the exact optimization problem. Then the
search algorithm searches for the “optimal point” of the new simulated
objective function in the new approximated feasible domain again. This sequential
approximation process continues in an iterative manner, until the same “optimal
design point” is obtained repeatedly and no new training point is generated. A
restart strategy is employed to start the search process from different
feasible initial points, so that the method may have a better chance to reach a
global optimum.
In this paper,
the optical simulation software Speos
(Optis, 2008) is used to simulate the brightness and uniformity of the LCD. A
two variable LCD BLU design example is presented first to illustrate the design
process using the SNA method. Then a four variable LCD BLU design optimization problem
is solved.
II. OPTIMIZING LIGHT EFFICIENCY OF A BACKLIGHT UNIT
The basic idea
of sequential approximation methods is to use a simple subproblem to
approximate the hard, exact problem. The solution point of the simple
subproblem is then used to form a better approximation to the hard, exact problem
for the next iteration. In an iterative manner, it is expected that the
solution point of the simple approximate problem will get closer to the optimum
point of the hard exact problem.
In the SNA
method, the optimization problem in Eq. (4) is approximated by an approximate
problem below:
max. Uniformity_{NN}(x)
s.t. Brightness_{NN}(x)=1, (5)
where the Uniformity_{NN}(x) and Brightness_{NN}(x)
are the neural network simulations for Uniformity(x) and Brightness(x),
respectively. Note that Uniformity_{NN}(x) simulates the function values of Uniformity(x), while Brightness_{NN}(x) simulates the feasible domain formed
by Brightness(x)=1.
In this paper,
the “real” optimization model in Eq. (4) is denoted M_{real}, and the
approximate model simulated by neural network in Eq. (5) is denoted M_{NN}.
M_{NN }is a “simple” model, because once the training of the neural
networks is completed the computational cost of evaluating Uniformity_{NN}(x)
and Brightness_{NN}(x) is much less than that of evaluating the Uniformity(x) and
Brightness(x)._{}
In the SNA
method, a set of initial training points is used to train the neural network so
that the neural network can simulate a rough map of the whole design domain.
The set of initial training points should reasonably represent the whole design
domain. Orthogonal arrays are used here because they are geometrically balanced
in coverage of the experiment region with just a few representative
experiments.
As described in
Section I, there are 4 independent variables in this problem: lp, b, h, and a. For demonstration purposes, variables lp and b are assumed to
be fixed in this section. Variables h
and a have 9 possible discrete
values:
h = {12.2, 14.2, 16.2,
18.2, 20.2, 22.2, 24.2, 26.2, 28.2} mm
a = {30, 35, 40, 45, 50,
55, 60, 65, 70} degrees
The ranges of
the design variables were obtained from the engineering experience of the LCD
manufacturer. In this example, the ranges of both variables were divided into 8
equal sections. Other discrete values (e.g., uneven discrete values) are also
possible.
Table 1 shows
the initial training points determined by a threelevel orthogonal array. The values of Uniformity(x) and Brightness(x) are
evaluated by the optical simulation software Speos.
Table 1 The initial training points of the
2variable example
h (mm)

a (degree)

Uniformity(x)

Brightness(x)

12.2

30

0.67

1

12.2

50

0.87

0

12.2

70

0.95

0

20.2

30

0.48

1

20.2

50

0.76

1

20.2

70

0.89

0

28.2

30

0.33

1

28.2

50

0.68

1

28.2

70

0.86

0

After the
initial training set is generated, two backpropagation neural networks, Uniformity_{NN}(x) and Brightness_{NN}(x),
are trained to simulate objective values and the feasible domain formed by the
constraint using the initial training data.
Figure 3 shows a
typical threelayer neural network is used in the SNA method. The size of the
input layer depends on the number of variables and the number of discrete
values of each variable. In this example, there are 2 design variables, with 9
discrete values each. Thus a total of 18 neurons are used in the input layer.
Each neuron in the input layer has value 1 or 0 to represent the discrete value
each variable takes, as will be explained later. There is only a single neuron
in the output layer. In Uniformity_{NN}(x),
the output neuron takes the value of Uniformity(x), which
is between 0 and 1. In Brightness_{NN}(x), the
output neuron represents whether this design point is feasible (the
output neuron has value 1) or infeasible (the output neuron has value 0).
Fig. 3 The architecture of the neural network
The number of
neurons in the hidden layer depends on the number of variables in the input
layer. There are 12 neurons in the hidden layer in this example. The transfer
functions used in the hidden and output layer of the network are both
logsigmoid functions. The neuron in the output layer has a value range [0, 1].
For Brightness_{NN}(x), after the training
is completed, a threshold value 0.25 is applied to the output layer when
simulating the boundary of the feasible domain. In other words, given a
discrete design point in the search domain, the network always output 0 (if
output neuron’s value is less than the given threshold) or 1 (otherwise) to
indicate whether this discrete design point is feasible or infeasible.
Figure 4 shows the
neural network representation of the center point (h, a)=(20.2,
50) of the discrete search domain in Uniformity_{NN}(x)
and Brightness_{NN}(x). A cross
represents a “0” in the
node, and a circle represents a “1”
in the node. For illustration purpose, the 18 input nodes are arranged into two
rows to represent the two design variables. The first 5 nodes in the first row
have a value of 1 to show that the first variable h has the discrete
value of 20.2. In the second row, the first 5 nodes have a value of 1 to show
that the second variable a has the discrete value of 50. As shown in
Table 1, checking with Uniformity(x)
and Brightness(x), (h, a)=(20.2 50)
is a feasible design point and its objective value is 0.76.
Note that in
this representation, discrete values of the variables are embedded and
neighboring nodes have very similar input patterns. Moreover, there has to be
at least one feasible design point in the set of initial training points.
Otherwise there will be no feasible domain after the training is completed.
Fig. 4 The neural network representation of the center
point (h, a)=(20.2, 50) of the discrete search domain
The
computational effort required in neural network training is critical. If the
computational cost required in neural network training is larger than that of
evaluating the implicit constraints, then the SNA method will lose its
advantage. Here all the training data are represented in a clear 01 pattern,
which makes the training process relatively fast. A quasiNewton algorithm,
which is based on Newton’s
method but does not require calculation of second derivatives, is used for the
training. In our numerical experience, the error goal of 1e6 is usually met
within 100 epochs, even for cases with many training points.
III. SEARCHING IN THE NEURALNETWORK APPROXIMATE MODEL
Using the 9
initial training points in Table 1, Fig. 5 shows the approximation by M_{NN}
after the initial training has been completed. The 9 nodes enclosed by square
brackets are the initial training points, and the contours are approximated
objective values by Uniformity_{NN}(x).
There are 81 possible discrete combinations in the search domain. A circle
represents that the design is feasible determined by Brightness_{NN}(x), and a cross represents the design is infeasible
determined by Brightness_{NN}(x).
Fig. 5 The approximation by M_{NN} after
the initial training has been completed
A search
algorithm starts to search for the optimum point of M_{NN}. Fig. 6 is
the flow chart of this discrete search algorithm. As discussed earlier, there
has to be at least one feasible design point in the set of initial training
points, so the search algorithm can always start from a feasible design point.
If a new design point obtained from the previous iteration is a feasible design
point in M_{real}, it is used as the starting point in the current
search. If the new design point is infeasible in M_{real}, then the
same starting point used in the previous iteration is used again in the current
search.
As shown in Fig.
6, a usable set U_{j} is formed from neighboring discrete nodes
of the current search point x_{j}
(j=0 for the starting point). There
are m=3^{n}1 nodes in U_{j}, where n is the number of design variables.
Then the components in U_{j} are sorted by the “distances” to x_{j} in a descending
order, thus U_{j}={u_{1},
u_{2}, u_{3}, …, u_{m}}. Note that by “distance”,
we don’t mean the absolute distance u_{k}x_{j}. Instead, we
mean the number of variables in u_{k}
that have different discrete values from x_{j}.
Therefore this sorting is the same for all discrete points, given the number of
total design variables, and does not require any computation. The node that has
a larger “distance” to x_{j}
has a higher priority to be searched first.
Pick a node u_{k}
from U_{j}. Two conditions are checked: “if u_{k}
is feasible in M_{NN}?” and “if f(u_{k})<f(x_{j})?” If both
conditions are satisfied, u_{k} is a better feasible
point than x_{j},
and we move on to u_{k}
and start the next iteration (j=j+1). On the other hand, if either
of the two conditions is not satisfied, then we check the next node in U_{j
}(k=k+1). When we reach the end of U_{j}
(k=m), there is no better feasible point in U_{j},
and x_{j} is
output as the optimum (at least locally) point.
Fig. 6 Flow chart of the search algorithm
In the LCD BLU
example discussed in this section, the feasible design point (h, a)=(20.2,
50) is used as the starting point because it has the best objective value
(0.76) among all design points in the initial training set. The search
algorithm terminates at a new design point (h, a)=(20.2, 60).
This new design point is then evaluated in M_{real}. Using the optic
simulation software Speos, this new
design point is determined infeasible, and the objective value is 0.81. Thus, a
new training point {(20.2, 60), infeasible, 0.81} is added to the set of
training points, and the neuralnetwork is trained again. After the second training
is completed, M_{NN} is updated with 10 training points. The
search algorithm then starts to search for the new optimum point in M_{NN}
again.
The iteration
continues, with one more training point added into the set of training points in
each iteration. The whole process terminates when the same design point is
obtained repeatedly and no new training point is generated. In this design case
with two design variables, the SNA method terminates after 4 iterations. Fig. 7
shows the approximate model M_{NN} trained by 13 design points,
including the 9 initial training points (nodes enclosed by square brackets in
Fig. 7) and 4 new design points obtained during the iterations (nodes enclosed
by diamond brackets in Fig. 7). Again in Fig. 7, a circle represents that the
design is feasible determined by Brightness_{NN}(x),
and a cross represents the design is infeasible determined by Brightness_{NN}(x). The optimum point found by the SNA method is
(14.2, 50), and the uniformity of luminance is 0.82.
Fig. 7 Approximate model MNN trained by 13 design
points
Figure 8 shows
the iteration history of this example. The design points enclosed by circles
are the feasible design points. Note that a total of 13 points out of the 81 (9×9)
possible combinations (16%) are evaluated to obtain this optimum design. No
gradient calculation is required. To ensure a better chance of reaching a
global optimum, the searching process is restarted from other feasible points
(12.2, 30), (20.2, 30), (28.2, 30), (28.2, 50) in the initial training set. In
this example, all restart search processes terminate at the same optimum design
point (14.2, 50) in one iteration.
Fig. 8 Iteration history of the 2variable example
IV.
LIGHT EFFICIENCY OPTIMIZATION
WITH 4 DISCRETE DESIGN VARIABLES
Figure 9
outlines the SNA method described in the previous section. The set of initial
training data S_{0} with at least one feasible point is given. A
backpropagation neural network is trained to simulate the objective function
and a rough map of the feasible domain formed by the implicit constraints using
S_{0}, and the initial optimization model M_{NN} is
formed There has to be at least one feasible point in S_{0},
such that M_{NN} has a feasible domain.
Starting from a
feasible point s,_{ }the search algorithm described in
Figure 6 is used to search for the solution point x_{j} of M_{NN} in the jth
iteration. The search algorithm has to start from a feasible design point.
There can be several feasible design points in S_{0}, and the
one with lowest objective value is used first. The new solution point x_{j} is then
evaluated to see whether it is feasible in M_{real}, and whether x_{j} = x_{j}_{1}.
If not, this new solution point x_{j}
is added to the set of training points S, and the neural network is trained again with the updated
data set. Then the search algorithm is used to search for the solution point in
the updated M_{NN} again. With the increasing number of training
points, it is expected that the feasible domain simulated by the neural network
in M_{NN} will get closer to that of the exact model M_{real},
and the solution point of M_{NN} will get closer to the optimum point.
If the feasible
solution point x_{j}
generated in the current iteration is the same as x_{j}_{1}, no new training point is
generated. The set of the training point since S, and consequently M_{NN},
is not updated. The current search has to be terminated because we will be
getting the same solution point.
As shown in Fig.
9 after the process is terminated, a restart strategy is employed to start the
search process from different feasible initial points in S_{0}. The
iterations continue until all feasible points in S_{0} have been
used as initial starting points. Finally the feasible design point with the
lowest objective value in S is output as the optimal solution.
Fig. 9 Flow chart of the SNA method
In this section,
all 4 design variables (lp, b, h, and a) are considered. Each
design variable has 9 possible discrete values:
h = {12.2, 14.2, 16.2,
18.2, 20.2, 22.2, 24.2, 26.2, 28.2} mm
b = {4.0, 6.0, 8.0, 10.0,
12.0, 14.0, 16.0, 18.0, 20.0} mm
lp = {3.5, 4.0, 4.5, 5.0,
5.5, 6.0, 6.5, 7.0, 7.5} mm
a = {30, 35, 40, 45, 50,
55, 60, 65, 70} degrees
Table 2 shows
the 9 initial training points determined by the orthogonal array. The design
point (20.2, 20.0, 7.5, 70) was used as the starting point because it has the
best objective value (0.67) among all feasible design points in the initial
training set. Following the same procedure described in the previous section,
the optimal design point was obtained after 27 iterations using the SNA method.
The optimal design point is (12.2, 16.0, 5.5, 50), and uniformity of luminance
improves to 0.85, which is higher than that in the previous section with two
design variables.
Figure 10 shows
the iteration history in this example. After the optimal design point was
found, the searching process was restarted from other feasible points (12.2,
30), (20.2, 30), (28.2, 30), (28.2, 50) in the initial training data to ensure
a better chance of reaching a global optimum. However, as shown in Fig. 10, no
better design points were found during the restart process. In the optimization
process using the SNA method, 36 training points (including 9 in the initial training, and 27 during the
iterations) out of the 6561 (9^{4}) possible combinations (0.55%) were
evaluated.
Table 2 The initial training points of the
4variable example
h (mm)

b (mm)

l_{p} (mm)

a (degree)

Uniformity(x)

Brightness(x)

12.2

4.0

3.5

30

0.82

0

12.2

12.0

5.5

50

0.92

0

12.2

20.0

7.5

70

0.91

0

20.2

4.0

3.5

30

0.95

0

20.2

12.0

5.5

50

0.60

1

20.2

20.0

7.5

70

0.67

1

28.2

4.0

3.5

30

0.83

0

28.2

12.0

5.5

50

0.32

1

28.2

20.0

7.5

70

0.33

1

Fig. 10 Iteration history of the 4variable
example
V.
CONCLUSION
A uniform
luminance design optimization problem for directtype LCD BLU under maximum
brightness constraints was considered in this paper. It
is difficult to find a numerical
optimization algorithm that is readily applicable
to this problem because of the existence of discrete variables, implicit
objective functions and constraints, or even binary constraints. The SNA method
is designed specifically for this type of engineering design optimization
problem. In the SNA method, the precious function evaluations of the
implicit functions (by optical simulation software Speos in this case) are accumulated to progressively form a better
approximation to the real optimization problem using the neural network. Only
one evaluation of the implicit constraints is needed in each iteration. No
sensitivity calculation is required.
In the
4variable design example presented in this paper, the uniform luminance of the
LCD BLU improves from 0.67 (the highest value among all feasible design points
in the initial training set) to 0.85, a 27% improvement. Only 36 design points out
of the 6561 (9^{4}) possible combinations, or 0.55% of the possible
discrete design points were evaluated.
In the design of
LCD BLU, optical simulation tools have been used to avoid high cost iterations needed
during the experimental design process. This paper demonstrates that, combined
with the SNA optimization method, designs with optimal luminance uniformity,
while satisfying the brightness requirements, can be obtained systematically,
with the fewest calls to the optical simulation tools. The SNA method has been
coded into a software package that is readily available for design engineers.
NOMOCLATURE
a angle
of the reflector
b distance
between a lamp and one side of the reflector
C thickness
of the diffuser plate
D diameter
of the lamps
E thickness
of the inner diffuser sheet
F thickness
of the prismatic sheet
G thickness
of the outer diffuser sheet
h height
of the enclosure
lb width of the bottom side of the reflector
lp distance
between a lamp and bottom side of the reflector
LQ number
of lamps
LU size of
the display
m the number of nodes in U_{j}
M_{real} the
real optimization model
M_{NN} the
approximate model simulated by neural network
n the
number of design variables
p distance
between two lamps
Pi symmetrical
sampling points of LCD screen
S_{0} the
set of initial training data
S a new training data
s a
feasible point in S_{0}
U_{j} a
usable set which is formed from neighboring discrete nodes of the current
search point x_{j}
u_{k}_{ }the vector of design variables in
U_{j}
x_{j} the vector of discrete design variables
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