# Analysis and Improvement of Fireworks Algorithm
## 3 Analysis and Improvement of Fireworks Algorithm
### Opposition-Based Learning Population Initialization
Global Convergence Analysis of IFWA**

[PERSON] studied the convergence of the fireworks algorithm, the main conclusions are as follows [2]:

**Lemma 1**. : _The random process of FWA (\(\{\varepsilon(t)\}_{t=0}^{\infty}\)) is an absorbing Markov random process [2]._

**Definition 6**. : Given an absorption Markov process (\(\{\varepsilon(t)\}_{t=0}^{\infty}\)) and an optimization state space (\(Y^{\star}{\subset}\)\(\gamma\)), \(\lambda(t)=P[\varepsilon(t){\in}Y^{\star}]\) represents the probability of reaching the optimal state at the \(t\) time. If \(\lim\limits_{t\rightarrow\infty}\lambda(t)=1\), \(\{\varepsilon(t)\}_{t=0}^{\infty}\) convergence [2]. **Theorem 1**. : _Given an absorption Markov process (\(\{\varepsilon(t)\}_{t=0}^{\infty}\)) and an optimization state space (\(Y^{\star}{\subset}\)\(Y\)), for\(\forall t\). If \(P\{\varepsilon(t)\in Y^{\star}|\varepsilon(t-1)\
otin Y^{\star}\}\geq d\geq 0\) and \(P\{\varepsilon(t)\in Y^{\star}|\varepsilon(t-1)\in Y^{\star}\}=1\), \(P\{\varepsilon(t)\in Y^{\star}\}\geq 1-\left(1-d\right)^{t}\)[2]._

Based on this, the global convergence of IFWA is given as follows:

IFWA contains a \(t\)-distribution mutation, which is assumed to be a random variation for simplicity. **Theorem 2**. : _Given an absorption Markov process (\(\{\epsilon(t)\}^{\infty}{}_{t=0}\)) of IFWA and an optimization state space (Y*\(\subset\)Y). The conclusion as follows:_

\[\lim_{t\rightarrow\infty}\lambda(t)=1 \tag{19}\]

**Proof of Theorem 2. ** Assume P\({}_{t}\) is the probability that a firework from the non-optimal region to the optimal region \(R_{best}\) in IFWA, under the action of \(t\)-distribution mutation:

\[P_{t}=\frac{\
u(R_{best})\times N}{\
u(S)} \tag{20}\]

where \(\
u(S)\) is the Lebesgue measure of the problem space \(S\); \(N\) is the number of fireworks. Obviously, \(\
u(R_{best})>0\), so \(P_{t}\)? 0;

Based on random Markov process of IFWA, the conclusion as follows:

\[\lambda(t)=P\{\varepsilon(t)\in Y^{*}|\varepsilon(t-1)\
otin Y^{*}\}=P_{t}+P_ {e} \tag{21}\]

where \(P_{e}\) is the probability that a firework from the non-optimal region to the optimal region \(R_{best}\) in IFWA, under the action of explosion. From Equation (21), the conclusion is as follows:

\[P\{\varepsilon(t)\in Y^{*}|\varepsilon(t-1)\
otin Y^{*}\}\geq P_{t}\geq 0 \tag{22}\]

Because the iterative process of the IFWA retains the optimal firework, that is, if the optimal firework is the global solution in the last iterative process, the optimal firework must be the global solution in the current iterative process. The conclusion as follows:

\[{x_{l}}^{k}={X_{l}}^{k} \tag{14}\]0

And because the Markov process of IFWA is an absorbing Markov process, the condition of theorem 1 is satisfied, so have the follow conclusion:

\[{x_{l}}^{k}={X_{l}}^{k} \tag{14}\]1

that is, \(\lim_{t\rightarrow\infty}P\{\varepsilon(t)\in Y^{*}\}=\lim_{t \rightarrow\infty}(1-(1-P_{t})^{t})=1\)

Based on Definition 6, the Markov process of IFWA will converge to the optimal state. ## 5 Simulation Results and Analysis

To assess the performance of IFWA, it is compared with FWA [8], EFWA [10], dynFWA [12] and SPSO2011 [22]. ### Simulation Settings

Similar to FWA, the number of fireworks in IFWA is set to 5; and the number of elite opposition-based sparks is also set to 5; but in IFWA, the maximum number of sparks each generation is set to 200. The reduction and amplification factors of IFWA are set to 0.9 and 1.2 based on experience. The explosion amplitude is initialized to the size of the search space to keep the high exploratory ability at the beginning of the algorithm. FWA parameters set in accordance with [1]. EFWA parameters set in accordance with [10]. dynFWA parameters set in accordance with [12]. SPSO2011 parameters set in accordance with [22]. In the experiment, the function of each algorithm is repeated 51 times, and the final results after the 300,000 function evaluations are presented.