Evaluation of the physical annealing strategy for simulated annealing: A function-based analysis in the landscape paradigm

The effectiveness of the actual annealing strategy in finite-time optimization by simulated annealing (SA) is analyzed by focusing on the search function of the relaxation dynamics observed in the multimodal landscape of the cost function. The rate-cycling experiment, which was introduced in the previous study [M. Hasegawa, Phys. Rev. E 83, 036708 (2011)] to examine the role of the relaxation dynamics in optimization, and the temperature-cycling experiment, which was developed for a laboratory experiment on relaxation-related phenomena, are conducted on two types of random traveling salesman problems (TSPs). In each experiment, the SA search starting from a quenched solution is performed systematically under a nonmonotonic temperature control used in the actual heat treatment of metals and glasses. The results show that, as in the previous monotonic cooling from a random solution, the optimizing ability is enhanced by allocating a lot of time to the search performed near an effective intermediate temperature irrespective of the annealing technique. In this productive phase, the relaxation dynamics successfully function as an optimizer and the relevant characteristics analogous to the stabilization phenomenon and the acceleration of relaxation, which are observed in glass-forming materials, play favorable roles in the present optimization. This nonmonotonic approach also has the advantage of a wider operation range of the effective relaxation dynamics, and in conclusion, the actual annealing strategy is useful and more workable than the conventional slow-cooling strategy, at least for the present TSPs. Further discussion is given of an illuminating aspect of computational physics analysis in the optimization algorithm research.

Figure 1

An example of the temperature schedule $\Theta \left(t\right)$ used in the rate-cycling experiment; depicted is the case for h-annealing (${\Theta }_{\text{s}}={\Theta }_{\text{e}}=-4$, ${\Theta }_{\text{c}}=-2.0$, $\Delta \Theta =1.0$, $\lambda =10$, and $t_{\text{e}}={10}^6$). See text for symbols.

Figure 2

An example of the temperature schedule used in the temperature-cycling experiment (${\Theta }_{\text{c}}=-2.0$, $\Delta \Theta =1.0$, $M=2$, and $n_{\text{e}}=10L={10}^6$). See text for symbols.

Figure 3

Optimization performance $f(y\left(x^{*}\right))$ (RE-TSP with $N=316$; ${\Theta }_{\text{s}}={\Theta }_{\text{e}}=-4$, $\Delta \Theta =1.0$, $n_{\text{e}}={10}^7$, and $I=2^5$). Results are averaged over $I$ search processes and plotted against the target logarithmic temperature ${\Theta }_{\text{c}}$.

Figure 4

Hidden search dynamics $f(y\left(x_n\right))$ (RE-TSP with $N=316$; ${\Theta }_{\text{s}}={\Theta }_{\text{e}}=-4$, $\Delta \Theta =1.0$, $\lambda =100$, $n_{\text{e}}={10}^7$, and $I=2^5$). Points are the results at some representative time steps. All results are averages over $I$ search processes.

Figure 5

Actual [(a) $f\left(x_n\right)$] and hidden [(b) $f(y\left(x_n\right))$] search dynamics (RE-TSP with $N=316$; ${\Theta }_{\text{c}}=-2.1$, $\Delta \Theta =1.0$, $\lambda =100$, $n_{\text{e}}={10}^7$, and $I=2^5$). Depicted are the results for h-annealing (${\Theta }_{\text{s}}={\Theta }_{\text{e}}=-4$), c-annealing (${\Theta }_{\text{s}}=1$ and ${\Theta }_{\text{e}}=-4$), and the reference SA (c-annealing with $\lambda =1$; ${\Theta }_{\text{s}}=1$ and ${\Theta }_{\text{e}}=-4$). Note that the abscissa represents the logarithmic temperature $\Theta$ instead of the time step $n$. Points are the results at some representative time steps and are connected by line segments along the search trajectory. Arrows indicate the direction of the time evolution for h-annealing. All results are averages over $I$ search processes.

Figure 6

Actual [(a) $f\left(x_n\right)$] and hidden [(b) $f(y\left(x_n\right))$] search dynamics (RE-TSP with $N=316$; ${\Theta }_{\text{c}}=-2.4$, $\Delta \Theta =1.4$, $\lambda =100$, $n_{\text{e}}={10}^6$, and $I=2^5$). Other details are the same as in Fig. 5.

Figure 7

Actual [(a) $f\left(x_n\right)$] and hidden [(b) $f(y\left(x_n\right))$] search dynamics (RE-TSP with $N=316$; ${\Theta }_{\text{c}}=-2.2$, $\Delta \Theta =0.6$, $\lambda =100$, $n_{\text{e}}={10}^5$, and $I=2^5$). Other details are the same as in Fig. 5.

Figure 8

Optimization performance $f(y\left(x^{*}\right))$ (RE-TSP with $N=316$; $M=2$, $n_{\text{e}}=10L={10}^7$, and $I=2^5$). Results are averaged over $I$ search processes and plotted against the target logarithmic temperature ${\Theta }_{\text{c}}$.

Figure 9

Hidden search dynamics $f(y\left(x_n\right))$ (RE-TSP with $N=316$; $n_{\text{e}}=10L={10}^7$ and $I=2^5$). The result for the no-cycling case at the lower temperature (${\Theta }_{\text{c}}-\frac{1}{2}\Delta \Theta$) is superimposed; the two behaviors are the same in the first $L$(=${10}^6$) steps. (a) ${\Theta }_{\text{c}}=-1.4$, $\Delta \Theta =1.0$, and $M=1$; (b) ${\Theta }_{\text{c}}=-1.9$, $\Delta \Theta =0.2$, and $M=2$; (c) ${\Theta }_{\text{c}}=-1.9$, $\Delta \Theta =1.8$, and $M=2$; (d) ${\Theta }_{\text{c}}=-2.7$, $\Delta \Theta =1.4$, and $M=4$. All results are averages over $I$ search processes.