Optimization with function evaluation

The optimization method is implemented as:

 
int Minimize_Maximize(int m,int n,
      INTEGER_VECTOR &Type_Eq,
      INTERVAL_VECTOR (* TheIntervalFunction)(int,int,INTERVAL_VECTOR &), 
      INTERVAL_VECTOR & TheDomain, 
      int Iteration,int Order,
      double epsilon,double epsilonf,double epsilone,
      int Func_Has_Interval,
      INTERVAL Optimum,
      INTERVAL_MATRIX & Solution,
      int (* Simp_Proc)(INTERVAL_VECTOR &));

the arguments being:

• m: number of unknowns
• n: number of equations, see the note 2.3.4.1
• Type_Eq: type of the equations:
  • Type_Eq(i)=-1 if equation i is a constraint equation of type $H(X) \le 0$
  • Type_Eq(i)=0 if equation i is a constraint equation of type $G(X)=0$
  • Type_Eq(i)=1 if equation i is a constraint equation of type $K(X) \ge 0$
  • Type_Eq(i)=-2 if equation i is the optimum function to be minimized
  • Type_Eq(i)=2 if equation i is the optimum function to be maximized
  • Type_Eq(i)=10 if equation i is the optimum function for which is sought the minimum and maximum
• IntervalFunction: a function which return the interval vector evaluation of the equations, see the note 2.3.4.3. This function must be written in a similar manner than for the general solving procedures. Note also that a convenient way to write the IntervalFunction procedure is to use the possibilities offered by the ALIAS-Maple (see the ALIAS-Maple manual).
• TheDomain: box in which we are looking for the extremum of the optimum function
• Iteration: the number of boxes that may be stored
• Order: a flag describing which order is used to store the new boxes, see the note 8.3.4
• epsilon: the maximal width of the solution intervals but not used. Should be set to 0.
• epsilonf: the maximal error for the equality constraints. If the problem has constraint of type $G(X)=0$ then a solution will verify $-{\tt epsilonf} \le G(X) \le {\tt epsilonf}$
• epsilone: the maximal error on the extremum value. If the extremum of the function is $G_1$ and the procedure returns the value $G$, then a minimum will verify $G-{\tt epsilone} \ge G_1$ and a maximum $G_1 \le G+{\tt epsilone}$.
• Func_Has_Interval: 1 if the optimum function has interval coefficients, 0 otherwise
• Optimum: an interval which contain the extremum value of the optimum function
• Solution: an interval matrix of size at least (2,m) which will contained the values of $X$ for which the extremum are obtained
• Simp_Proc: an optional parameter which is a simplification procedure that may be provided by the user. It takes as input a box $P$ and may:
  • either returns in $P$ a box with lower width than the initial $P$ and a return code 0 or 1
  • or indicates that there is no solution to the optimization problem in the current box, in which case the procedure returns -1
  An often efficient simplification procedure is the 2B method (see section 2.17) that may be automatically generated by the HullIConsistency procedure of ALIAS-Maple

Thus to minimize a function you have to set its Type_Eq to -2, to maximize it to set its Type_Eq to 2, while if you are looking for both a minimum and a maximum Type_Eq should be set to 10.

Remember that you may use the 3B method to improve the efficiency of this algorithm (see section 2.3.2) if you have constraint equations.

In some cases it may be interesting to determine if the minimum and maximum have same sign. This may be done by setting the flag ALIAS_Stop_Sign_Extremum either to:

• 1: the procedure will return immediately as soon as it is proven that the extremum will have opposite sign i.e. as soon as two points lead to opposite value for the function. But if the extremum have identical sign the minimum and maximum will be computed up to the accuracy epsilone
• 2: the procedure will return immediately as soon as it is proven that the extremum will have opposite sign i.e. as soon as two points lead to opposite value for the function. If the extremum have same sign the values returned by the procedure are not the minimum and maximum of the function

With the flag set to 2 the detection that the extremum will have opposite sign is faster.