*4.3.3. Global optimization with reannealing*

14 Will-be-set-by-IN-TECH

Boltzman -1.961759 24.40 Exponential -1.927647 18.51 Fast -1.786238 22.43

**Table 4.** Mean values of differential amplifier after local optimization over the results obtained by global

The mean values found for the free variables after the local search are shown in Tab. 5 . Comparing to the previous values provided by the global optimization, it is possible to note the great improvement of the Exponential temperature schedule, whose mean *W* and

Temperature schedule *W* (*μm*) *L* (*μm*)

Boltzman 8.07 19.99 Exponential 9.68 20.30 Fast 20.12 21.59

Optimum value 8.00 20.00

Optimal Results versus Execution Time

0 5 10 15 20 25 30 35 40 45 50

Time (s)

**Figure 7.** Optimal results versus execution time for the global optimization of a differential amplifier,

**Table 5.** Mean *W* and *L* values achieved by local optimization procedure of the differential amplifier

*<sup>c</sup>* Execution time (*s*)

Temperature schedule *f* ∗

optimization shown in Tab. 2.

0

100

200

300

400

500

Optimal Results

600

700

800

900

*L* approached very near to the global optimum.

over the results obtained by global optimization shown in tab. 3.

TBoltz + Local TExp + Local TFast + Local

considering 3 different temperature schedule functions - global and local.

TBoltz TExp TFast

For the analysis of the influence of reannealing in the optimization process, we performed some experiments executing Simulated Annealing with reannealing intervals of 200, 450, 700 and 950 iterations. Again, 1000 executions were done in order to guarantee a statistical analysis for the three temperature schedule functions described before.

Fig. 9 shows the relation between the number of optimal solutions found by Boltzman schedule function versus the execution time for reannealing intervals from 200 to infinite (*i.e.*, no reannealing). Reannealing interval affects the number of optimal solutions in this case. As the interval decrease, the number of optimal solutions decrease too. The best configuration is with no reannealing, demonstrating that it is not interesting to use reannealing with *TBoltz*. It happens because the temperature decreases slowly at the beginning of the annealing process. With the reannealing, the temperature increases for higher values before the search in the design space reaches a path trending to the optimal solution.

When the temperature schedule function is modified to Exponential, the behavior is opposite. As the reannealing interval decreases, more optimal solutions are found. Fig. 10 shows the relation between optimal solutions found and execution time for this temperature schedule configuration.

The same occurs for the Fast temperature schedule function, shown in Fig. 11. As the reannealing interval diminishes, the number of optimal solutions increases. This behavior is maintained for ever small intervals. A high improvement in the number of optimal solutions is obtained for reannealing intervals in the order of 100 iterations, as shown in Fig. 12. As the temperature decreases very fast, the reannealing allows to avoid local minima. Thus, it increases the chances of finding the correct path to the optimum solution. Also, we can observe the existence of an optimum value for the reannealing interval which returns the maximum number of optimal solutions.

#### *4.3.4. Analysis of state generation function*

The variation of the state generation function is also a factor that can change the convergence of the Simulated Annealing algorithm. Two of these functions are analyzed here: Boltzman and Fast. The combinations of temperature schedule function and state generation function produce distinct results for the synthesis of the differential amplifier. Fig. 13 shows

0 10 20 30 40 50 60 70 80 90 100

0 20 40 60 80 100

Time (s)

**Figure 10.** Optimal results versus execution time for the global optimization of a differential amplifier

with Exponential temperature schedule function and different reannealing intervals.

R.I.=200 R.I.=450 R.I.=700 R.I.=950 R.I.=inf

> R.I.=200 R.I.=450 R.I.=700 R.I.=950 R.I.=inf

Optimal Results versus Execution Time

Simulated Annealing to Improve Analog Integrated Circuit Design: Trade-Off s and Implementation Issues 277

Time (s)

Optimal Results versus Execution Time

**Figure 9.** Optimal results versus execution time for the global optimization of a differential amplifier

with Boltzman temperature schedule function and different reannealing intervals.

0

0

50

100

150

200

Optimal Results

250

300

350

50

100

150

200

250

Optimal Results

300

350

400

450

(b) Global search with Simulated Annealing followed by local search with Interior Point Algorithm.

**Figure 8.** Frequency histograms of the final cost found by the optimization process for three different temperature schedule functions: Boltzman, Exponential and Fast. Obs.: x-scales are different in each chart for better visualization purpose.

the number of optimal solutions returned by the algorithm after 1000 executions for 6 combinations.

We can notice that there are a great improvement in the quality of the solutions using Boltzman temperature schedule together with Boltzman state generation function. This is the best combination, according to that was theoretical predicted in Section 2.

16 Will-be-set-by-IN-TECH

TExp

−2 −1.8 −1.6 −1.4 −1.2 <sup>0</sup>

Cost Function (fc

TExp + Local

−2 −1.8 −1.6 −1.4 −1.2 <sup>0</sup>

Cost Function (fc

(b) Global search with Simulated Annealing followed by local search with Interior Point Algorithm.

the number of optimal solutions returned by the algorithm after 1000 executions for 6

We can notice that there are a great improvement in the quality of the solutions using Boltzman temperature schedule together with Boltzman state generation function. This is the best

**Figure 8.** Frequency histograms of the final cost found by the optimization process for three different temperature schedule functions: Boltzman, Exponential and Fast. Obs.: x-scales are different in each

)

(a) Global search with Simulated Annealing.

)

−2 −1.5 −1 <sup>0</sup>

Cost Function (Fc

TFast + Local

−2 −1.5 −1 <sup>0</sup>

Cost Function (fc

)

)

TFast

Frequency

Frequency

−1.96 −1.95 −1.94 −1.93 <sup>0</sup>

Cost Function (fc

TBoltz + Local

−1.96 −1.95 −1.94 −1.93 <sup>0</sup>

Cost Function (fc

chart for better visualization purpose.

)

)

combination, according to that was theoretical predicted in Section 2.

Frequency

Frequency

TBoltz

combinations.

Frequency

Frequency

**Figure 9.** Optimal results versus execution time for the global optimization of a differential amplifier with Boltzman temperature schedule function and different reannealing intervals.

**Figure 10.** Optimal results versus execution time for the global optimization of a differential amplifier with Exponential temperature schedule function and different reannealing intervals.

**Figure 11.** Optimal results versus execution time for the global optimization of a differential amplifier with Fast temperature schedule function and different reannealing intervals.

0 5 10 15 20 25 30 35 40

Optimal Results versus Execution Time

Simulated Annealing to Improve Analog Integrated Circuit Design: Trade-Off s and Implementation Issues 279

Time(s)

**Figure 13.** Number of optimal results returned by the optimization process for the differential amplifier

Results presented before allow us to suppose that the temperature schedule function affects directly the quality of the solutions generated by the global optimization algorithm. The Boltzman schedule, followed by a post-processing with a local search algorithm, demonstrate best convergence to the optimal point, at the expenses of a larger execution time. This additional time, however, is not a problem if we consider that the chances of finding the optimal (or near the optimal) solution are increased. For our 2-variables problem, this additional time is irrelevant (about 10*s* for 1000 executions). For more complex circuits with dozens of variables, the execution time can be a factor of concern. It is increased exponentially with the number of free variables, since the design space grows fast with the number of free

0

100

in a larger circuit.

200

300

400

500

Optimal Results

600

700

800

900

TBoltz − GFast TExp − GFast TFast − GFast TBoltz − GBoltz TExp − GBoltz TFast − GBoltz

for different annealing functions and temperature function schedules.

*4.3.5. Analysis of best SA options for the differential amplifier*

variables. We can estimate the design space size *Ds*(*X*) as:

*Ds*(*X*) = ∏

*i*

where *xi*(*ub*) and *xi*(*lb*) are upper an lower bounds of variable *xi*, respectively, and *xi*(*step*) is the minimum step allowed for variable *xi*. It is clear that the exploration of the entire design space is hard for a problem with several free variables. An alternative, in this case, is to use the Fast temperature schedule with reannealing, which is also efficient in the design space exploration. Both Boltzman followed by local search and Fast with reannealing achieved the optimal solution in about 90% of the cases. These configurations are candidates to be tested

*xi*(*ub*) − *xi*(*lb*) *xi*(*step*)

(28)

1000

**Figure 12.** Maximum number of optimal results returned by the optimization process versus reannealing interval for Fast temperature schedule. The optimum value for the reannealing interval is near 100.

#### Optimal Results versus Execution Time

**Figure 13.** Number of optimal results returned by the optimization process for the differential amplifier for different annealing functions and temperature function schedules.

#### *4.3.5. Analysis of best SA options for the differential amplifier*

18 Will-be-set-by-IN-TECH

Optimal Results versus Execution Time

R.I.=200 R.I.=450 R.I.=700 R.I.=950 R.I.=inf

0 10 20 30 40 50 60 70 80 90 100

Time (s)

**Figure 11.** Optimal results versus execution time for the global optimization of a differential amplifier

**Figure 12.** Maximum number of optimal results returned by the optimization process versus reannealing interval for Fast temperature schedule. The optimum value for the reannealing interval is

with Fast temperature schedule function and different reannealing intervals.

Optimal Results

near 100.

Results presented before allow us to suppose that the temperature schedule function affects directly the quality of the solutions generated by the global optimization algorithm. The Boltzman schedule, followed by a post-processing with a local search algorithm, demonstrate best convergence to the optimal point, at the expenses of a larger execution time. This additional time, however, is not a problem if we consider that the chances of finding the optimal (or near the optimal) solution are increased. For our 2-variables problem, this additional time is irrelevant (about 10*s* for 1000 executions). For more complex circuits with dozens of variables, the execution time can be a factor of concern. It is increased exponentially with the number of free variables, since the design space grows fast with the number of free variables. We can estimate the design space size *Ds*(*X*) as:

$$D\_s(X) = \prod\_i \frac{\mathbb{x}\_{i(ub)} - \mathbb{x}\_{i(lb)}}{\mathbb{x}\_{i(step)}} \tag{28}$$

where *xi*(*ub*) and *xi*(*lb*) are upper an lower bounds of variable *xi*, respectively, and *xi*(*step*) is the minimum step allowed for variable *xi*. It is clear that the exploration of the entire design space is hard for a problem with several free variables. An alternative, in this case, is to use the Fast temperature schedule with reannealing, which is also efficient in the design space exploration. Both Boltzman followed by local search and Fast with reannealing achieved the optimal solution in about 90% of the cases. These configurations are candidates to be tested in a larger circuit.

20 Will-be-set-by-IN-TECH 280 Simulated Annealing – Single and Multiple Objective Problems Simulated Annealing to Improve Analog Integrated Circuit Design: Trade-Offs and Implementation Issues <sup>21</sup>
