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 Nonlinear Optimization Examples

## Control Parameters Vector

For all optimization and least-squares subroutines, the input argument par specifies a vector of parameters that control the optimization process. For the NLPFDD and NLPFEA subroutines, the par argument is defined differently. For each element of the par vector there exists a default value, and if you specify a missing value, the default is used. Table 11.5 summarizes the uses of the par argument for the optimization and least-squares subroutines.

Table 11.5: Summary of the Control Parameters Vector
 Index Description 1 specifies the singularity criterion (SINGULAR) 2 specifies the initial step length or trust-region radius 3 specifies the range for active (violated) constraints (LCEPS) 4 specifies the Lagrange multiplier threshold for constraints (LCDEACT) 5 specifies a criterion to determine linear dependence of constraints (LCSING) 6 specifies the required accuracy of the line-search algorithms (LSPRECISION) 7 reduces the line-search step size in successive iterations (DAMPSTEP) 8 specifies the number of accurate digits used in evaluating the objective function (FDIGITS) 9 specifies the number of accurate digits used in evaluating the nonlinear constraints (CDIGITS) 10 specifies a scalar factor for the diagonal of the initial Hessian (DIAHES)

• par[1] specifies the singularity criterion for the decomposition of the Hessian matrix (SINGULAR). The value must be between zero and one, and the default is par[1]=1E-8.

• par[2] specifies different features depending on the subroutine in which it is used. In the NLPNMS subroutine, it defines the size of the start simplex. For the original Nelder-Mead simplex algorithm, the default value is par[2]=1; for the COBYLA algorithm, the default is par[2]=0.5. In the NLPCG, NLPQN, and NLPHQN subroutines, the par[2] argument specifies an upper bound for the initial step length for the line search during the first five iterations. The default initial step length is par[2]=1. In the NLPTR, NLPDD, and NLPLM subroutines, the par[2] argument specifies a factor for the initial trust-region radius, . For highly nonlinear functions, the default step length or trust-region radius can result in arithmetic overflows. In that case, you can specify stepwise decreasing values of par[2], such as par[2]=1E-1, par[2]=1E-2, par[2]=1E-4, until the subroutine starts to iterate successfully.

• par[3] specifies the range (LCEPS) for active and violated linear constraints. The ith constraint is considered an active constraint if the point x(k) satisfies the condition
where LCEPS is the value of par[3] and aij and bi are defined as in the section "Parameter Constraints". Otherwise, the constraint i is either an inactive inequality or a violated inequality or equality constraint. The default is par[3]=1E-8. During the optimization process, the introduction of rounding errors can force the subroutine to increase the value of par[3] by a power of 10, but the value will never become larger than 1E-3.

• par[4] specifies a threshold (LCDEACT) for the Lagrange multiplier that decides whether an active inequality constraint must remain active or can be deactivated. For maximization, par[4] must be positive, and for minimization, par[4] must be negative. The default is
where the positive value is for maximization and the negative value is for minimization. ABSGTOL is the value of the absolute gradient criterion, and gmax(k) is the maximum absolute element of the gradient, g(k), or the projected gradient, ZT g(k).

• par[5] specifies a criterion (LCSING) used in the update of the QR decomposition that decides whether an active constraint is linearly dependent on a set of other active constraints. The default is par[5]=1E-8. As the value of par[5] increases, more active constraints are recognized as being linearly dependent. If the value of par[5] is larger than 0.1, it is reset to 0.1, and if it is negative, it is reset to zero.

• par[6] specifies the degree of accuracy (LSPRECISION) that should be obtained by the second or third line-search algorithm. This argument can be used with the NLPCG, NLPHQN, and NLPNRA algorithms and with the NLPQN algorithm if the "nlc" argument is specified. Usually, an imprecise line search is computationally inexpensive and successful, but for more difficult optimization problems, a more precise and time consuming line search may be necessary. Refer to Fletcher (1987) for details. If you have numerical problems, you should decrease the value of the par[6] argument to obtain a more precise line search. The default values are given in the following table.

 Subroutine Update Method Default value NLPCG All par[6] = 0.1 NLPHQN DBFGS par[6] = 0.1 NLPHQN DDFP par[6] = 0.06 NLPNRA No update par[6] = 0.9 NLPQN BFGS, DBFGS par[6] = 0.4 NLPQN DFP, DDFP par[6] = 0.06

• par[7] specifies a scalar factor (DAMPSTEP) that can be used to reduce the step size in each of the first five iterations. In each of these iterations, the starting step size, , can be no larger than the value of par[7] times the step size obtained by the line-search algorithm in the previous iteration. If par[7] is missing or ifpar[7]=0, which is the default, the starting step size in iteration t is computed as a function of the function change from the former iteration, f(t-1) - f(t). If the computed value is outside the interval [0.1,10.0], it is moved to the next endpoint. You can further restrict the starting step size in the first five iterations with the par[2] argument.

• par[8] specifies the number of accurate digits (FDIGITS) used to evaluate the objective function. The default is , where is the machine precision, and fractional values are permitted. This value is used to compute the step size h for finite difference derivatives and the default value for the FTOL termination criterion.

• par[9] specifies the number of accurate digits (CDIGITS) used to evaluate the nonlinear constraint functions of the "nlc" module. The default is , where is the machine precision, and fractional values are permitted. The value is used to compute the step size h for finite difference derivatives. If first-order derivatives are specified by the "jacnlc" module, the par[9] argument is ignored.

• par[10] specifies a scalar factor (DIAHES) for the diagonal of the initial Hessian approximation. This argument is available in the NLPDD, NLPHQN, and NLPQN subroutines. If the opt[7] argument is not specified, the initial Hessian approximation is a multiple of the identity matrix determined by the magnitude of the initial gradient g(x(0)). The value of the par[10] argument is used to specify par[10] ×I   for the initial Hessian in the quasi-Newton algorithm.

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