Representation

We use a small margin of tolerance $\epsilon$ to represent a true numeric zero. $\epsilon$ can be the computer's floating-point precision, or double precision, since high precision is required in our problem. Additionally, the uncertain polynomial in real applications may contain numerous parameters. In such uncertain parameter optimization problems, a long bit string is required in order to extend the precision and represent the entire range of each parameter. Thus the binary chromosome representation traditionally used in GAs shows some deficiencies since a long chromosome usually results in poor performance for GAs.

In order to overcome the drawback of the binary representation, we coded the GA chromosome as a vector of real numbers with double precision. Each parameter q_i, i=1,...,n, is initialized within the pre-specified domain [q_i^-, q_i⁺]. The magnitude of the root $z^{\prime}$ is initially selected to be within [0, 1], while the angle $\theta$ is in the interval $[0, \pi]$. Compared with binary representation, real numbers are better capable of representing our desired domain. The precision only depends on the machine we use and the accuracy depends on the user's requirements. We define the chromosome for the uncertain polynomial $p(z^{\prime}, {\bf q})$ as:

$$C_i^t = \langle q_1, ..., q_k, ..., q_n , z, \theta \rangle, \;\; 1 \le k \le n$$ (5)

$$C_i^{t+1} = \langle q_1, ..., q^{\prime}_k, ..., q_n , z^{\prime}, \theta^{\prime} \rangle, \;\; k \in (1,...,n)$$ (6)

where C_i^t is the chromosome of population i in tth generation. C_i^t+1 is a new chromosome for the next generation after genetic selection. The $q^{\prime}_k$, $z^{\prime}$ and $\theta^{\prime}$ in C_i^t+1 satisfy

$$q^{\prime}_k= \left\{ \begin{array} {r@{\; : \;}l} {q_k + \triang... ...}=0 \\ {q_k - \triangle (t, q_k - q_k^-)} & flip\{0;1\}=1 \end{array} \right .$$ (7)

$$z^{\prime} = \left\{ \begin{array} {r@{\quad: \quad}l} {z + \tria... ...ip\{0;1\}=0 \\ {z - \triangle ( t, z -0)} & flip\{0;1\}=1 \end{array} \right .$$ (8)

$$\theta^{\prime} = \left\{ \begin{array} {r@{\quad: \quad}l} {\the... ...0 \\ {\theta - \triangle ( t, \theta -0)} & flip\{0;1\}=1 \end{array} \right .$$ (9)