Kinetic simulated annealing optimization with entropy-based cooling rate

Figure 1.  Top row: evolution of the Shannon entropy $ H(f|f^q)(t) $ for several values of $ \alpha $, and for $ \epsilon = 10^{-2} $ (left) and $ \epsilon = 10^{-3} $ (right), we show in green the approximated evolution of the KSA algorithm with $ \epsilon = 10^{-4} $. Bottom row: evolution of the mean temperature of the system of particles $ m_1(t) = \int_{\mathbb R_+}Tg(T, t)dT $ for $ \epsilon = 10^{-2} $ (left) and $ \epsilon = 10^{-3} $ (right)

Figure 2.  The distribution $ f(x, t) $ at time $ t = 1 $ for several values of $ \alpha = 0.025, 0.05, 0.1 $ for $ \epsilon = 10^{-2} $ (left) or $ \epsilon = 10^{-3} $ (right). In dotted green, we highlight $ x^* $, corresponding to the exact minimum of the function $ \mathcal{F}(x) $

Figure 3.  Evolution of the distributions $ f(x, t) $, $ g(T, t) $ at time $ t = 1, 5, 10 $ for several values of $ \alpha = 0.025 $ (left), $ \alpha = 0.05 $ (center), and $ \alpha = 0.1 $ (right) for a fixed $ \epsilon = 10^{-3} $. In the top row, we report the evolution of $ f(x, t) $, and in the bottom row the evolution of $ g(T, t) $. We considered $ N = 10^6 $ particles both in space and temperature, $ p = 1/4 $, $ \theta = 0.05 $, and $ \sigma^2 = 0.1 $

Figure 4.  Evolution of the mean position $ m_x = \int_{\mathbb R} x f(x, t)dx $ and of its variance $ Var_x = \int_{\mathbb R} (x-m_x)^2 f(x, t)dx $ over the time interval $ [0, 50] $ and several values of the parameter $ \alpha = 0.025, 0.05, 0.1 $

Figure 5.  Top row: estimated values of $ \lambda>0 $ defined in (38) and several values of $ \alpha = 0.025, 0.05, 0.1 $. Bottom row: estimated values of $ \mathcal I_ \mathcal{F}(t) = \int_{\mathbb R} \mathcal{F}(x)(f^q(x, t)-f(x, t))dx $ for several values of $ \alpha = 0.025, 0.05, 0.1 $. We considered $ \epsilon = 10^{-2} $ (left) and $ \epsilon = 10^{-3} $ (right)

Figure 6.  Top row: evolution of the Shannon entropy $ H(f|f^q)(t) $ for several values of $ \alpha $ for $ k \in\{ 1.5, 2.0, 2.5\} $ (left). Bottom row: evolution of the mean temperature of the system of particles $ m_k(t) = \int_{\mathbb R_+}Tg^q(T, t)dT $

Figure 7.  Evolution of the distribution $ f(x, t) $ at time $ t = 1, 5, 10 $ for several values of $ \alpha = 0.025 $ (left), $ \alpha = 0.05 $ (center), $ \alpha = 0.1 $ (right), and $ k = 1.5 $ (first row), $ k = 2.0 $ (second row), $ k = 2.5 $ (third row). In all the tests we fixed $ N = 10^6 $, $ p = 1/4 $, $ \theta = 0.5 $, and $ \sigma = 0.1 $, the definition of $ \lambda[f](\cdot) $ is (46), and the initial state is (50)

Figure 8.  Evolution of the quasi-equilibrium distribution of the temperature $ g^q(T, t) $ at time $ t = 1, 5, 10 $ for several values of $ \alpha = 0.025 $ (left), $ \alpha = 0.05 $ (center), $ \alpha = 0.1 $ (right), and $ k = 1.5 $ (first row), $ k = 2.0 $ (second row), $ k = 2.5 $ (third row). We computed the generalized gamma distribution defined in (48), where the definition of $ \lambda[f](\cdot) $ is (46)

Figure 9.  Evolution of the mean position $ m_x = \int_{\mathbb R}xf(x, t)dx $ and of its variance $ Var_x = \int_{\mathbb R}(x-m_x)^2f(x, t)dx $ over the time interval $ [0, 50] $ and several values of the parameter $ \alpha = 0.025, 0.05, 0.1 $ in the case $ k = 2.5 $

Figure 10.  Top row: estimated values of $ \lambda>0 $ as in (46) for several values of $ \alpha = 0.025, 0.05, 0.1 $ and $ \kappa = 1.5 $ (left), $ \kappa = 2.0 $ (center), $ \kappa = 2.5 $ (right). Bottom row: estimated values of $ \mathcal I_ \mathcal{F}(t) $. We considered $ N = 10^6 $ particles both in space and temperature, $ p = 1/4 $, $ \theta = 0.05 $ and $ \sigma^2 = 0.1 $. The initial condition is reported in (50)