Definition of a minimum and a maximum

As interval coefficients may appear in the function $F$ we have to define what will be called a minimum or a maximum of $F$. First we assume that there is no interval coefficients in $F$ and denote by $F^m$ the minimal or maximal value of $F$ over a set of ranges defined for $X$ and an accuracy $\epsilon$ with which we want to determine the extremum. The algorithm will return an interval $F_e$ as an approximation of $F^m$ such that for a minimization problem $\underline{F_e} -\epsilon \le F^m \le \underline{F_e}+\epsilon$ and for a maximization problem $\overline{F_e} -\epsilon \le F^m \le \overline{F_e}+\epsilon$. The algorithm will also return a value $X^m$ for $X$ where the extremum occurs. If we deal with a constrained optimization problem we will have:

• $-\epsilon_f \le G_i(X^m) \le \epsilon_f$
• $H_i(X^m)\le 0$ or $0 \in H_i(X^m)$ and $\overline{H_i(X^m)}-\underline{H_i(X^m)} \le \epsilon_f$
• $K_i(X^m) \ge 0$ $0 \in K_i(X^m)$ and $\overline{K_i(X^m)}-\underline{K_i(X^m)} \le \epsilon_f$

where $\epsilon_f$ has a pre-defined value. Note that if we have constraint equation of type $G_i(X)=0$ the result of the optimization problem may be no more guaranteed as the constraint itself $G_i$ may not be verified.

If there are interval coefficients in the optimum function there is not a unique $F^m$ but according to the value of the coefficient a minimal extremum value $F^m_1$ and a maximal extremum value $F^m_2$. The algorithm will return in the lower bound of $F_e$ an approximation of $F^m_1$ and in the upper bound of $F_e$ an approximation of $F^m_2$ which verify for a minimization problem:

• $\underline{F_e} -\epsilon \le F^m_1 \le \underline{F_e}+\epsilon$
• $\overline{F_e} -\epsilon \le F^m_2 \le \overline{F_e}+\epsilon$

Note that the width of $F_e$ may now be greater than $\epsilon$.The algorithm will return also two solutions $X^m_1, X^m_2$ for $X$ corresponding respectively to the values of $F^m_1, F^m_2$. If we are dealing with a constrained optimization problem the solutions will verify the above constraint equations.

If the optimum function has no interval coefficients the algorithm may return no solution if the interval evaluation of the optimum function has a width larger than $\epsilon$. Evidently the algorithm will also return no solution if there is no solution that satisfy all the constraints.