3.2.13 The START POINT Data Cards

The START POINT indicator card is used to announce a vector of initial estimates of the values of the unknown variables and, in the case of problems with general constraints, Lagrange multipliers. The Lagrangian function associated with (2.1)-(2.4) is the function

$$\begin{eqnarray*} l(x,\lambda) & = & \sum_{i \in I_{0}} g_i\left( \sum_{j \in J... ...left( \sum_{j \in J_i} w_{i,j} f_j (x_j) + a_i^T x - b_i \right) \end{eqnarray*}$$

where the scalars $\lambda_i$ are known as Lagrange multipliers. Good estimates of these parameters can sometimes be useful for optimization procedures (see, for example, [10]). The syntax for data following this indicator card is given in Figure 3.14.

Figure 3.14: Possible data cards for START POINT

The V, XV and ZV cards are used to define the starting value for variables. In any of these cards, the string start-name in data field 2 gives the name of a starting vector and may be up to ten characters long. Several different starting vectors may be defined in the START POINT section. The string $$$$$$$$$$ in field 3 is used for two purposes.

• It must contain the name of a variable defined in the VARIABLES section, when a starting value is to be assigned to that variable. If field 1 does not contain ZV, the optional string varbl-name in data field 5 may also contain the name of such a variable whose starting value is to be assigned.

• It may be used to assign a default value to all the starting values for variables in a particular vector. In this case field 3 must contain the string 'DEFAULT'.

  Each starting point vector is given a default value for every variable. This value is initially zero, but it may be changed. The appropriate default value applies to each variable not explicitly specified.

The M, XM and ZM cards are used to define the starting value for Lagrange multipliers. In any of those cards, the string start-name in data field 2 gives the name of a starting vector and may be up to ten characters long. Several different starting vectors may be defined in the START POINT section. The string $$$$$$$$$$ in field 3 is used for two purposes.

• It must contain the name of a group defined in the GROUPS section which is not an objective function group, when a starting value is to be assigned to the corresponding Lagrange multiplier. If field 1 does not contain ZV, the optional string multp-name in data field 5 may also contain the name of such a multiplier whose starting value is to be assigned.

• It may be used to assign a default value to all the starting values for Lagrange multipliers in a particular vector. In this case field 3 must contain the string 'DEFAULT'.

  Each starting point vector is given a default value for every Lagrange multiplier. This value is initially zero, but it may be changed. The appropriate default value applies to each multiplier not explicitly specified.

The effect of a card whose field 1 is blank or contains X or Z is similar to that of V, XV, ZV, M, XM and ZM cards, except that

• variables and Lagrange multipliers may be mixed up on cards whose field 3 does not contain `DEFAULT',
• default values are assigned to both variables and Lagrange multipliers on card whose field 3 contains `DEFAULT'. The default value for both variables and multipliers is initially zero.

If the defaults are to be changed, the change must be made on the first card naming a particular starting point vector.

Starting values for an array of variables or Lagrange multipliers may only be defined on cards whose field 1 begins with the character X or Z; on X cards two arrays may be defined on a single card. On such cards, the expanded array name in field 3 (and field 5 for X cards) must be valid and the integer indices must have been defined in a parameter assignment (see Section 3.2.3).

It remains to specify the numerical value of the default or individual starting point as appropriate. On cards whose field 1 starts with Z, the value is that previously associated with the real parameter rl-p-name or array of real parameters rl-p-name (respectively) given in field 5. On other cards, the numerical value is (or values are) specified using up to twelve characters in the string(s) numerical-vl in data field 4 (and if required field 6).

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