conmin_module Module

Small number

Large number

Minimum relative change in the objective function to indicate convergence. If in ITRM consecutive iterations, ABS(1.0-OBJ(J-1)/OBJ(J))<DELFUN and the current design is feasible (all G(J)<=ABS(CT)), the minimization process is terminated. If the current design is infeasible (some G(J)>ABS(CT)), five iterations are required to terminate and this situation indicates that a feasible design may not exist.

Default value = 0.001 times the initial function value. Same as DELFUN except comparison is on absolute change in the objective function, ABS(OBJ(J)-OBJ(J-1)), instead of relative change.

Not used if NFDG = 0. Relative change in decision variable X(I) in calculating finite difference gradients. For example, FDCH = 0.01 corresponds to a finite difference step of one percent of the value of the decision variable.

Not used if NFDG = 0. Minimum absolute step in finite difference gradient calculations. FDCHM applies to the unscaled variable values.

Not used if NCON = NSIDE = 0. Constraint thickness parameter. If CT<=G(J)<=ABS(CT), G(J) is defined as active. If G(J)>ABS(CT), G(J) is said to be violated. If G(J)<CT, G(J) is not active. CT is sequentially reduced in magnitude during the optimization process. If ABS(CT) is very small, one or more constraints may be active on one iteration and inactive on the next, only to become active again on a subsequent iteration. This is often referred to as "zigzagging" between constraints. A wide initial value of the constraint thickness is desirable for highly nonlinear problems so that when a constraint becomes active it tends to remain active, thus reducing the zigzagging problem. The default value is usually adequate.

Not used if NCON = NSIDE = 0. Minimum absolute value of CT considered in the optimization process. CTMIN may be considered as "numerical zero" since it may not be meaningful to compare numbers smaller than CTMIN. The value of CTMIN is chosen to indicate that satisfaction of a constraint within this tolerance is acceptable. The default value is usually adequate.

Not used if NCON = NSIDE = 0. Constraint thickness parameter for linear and side constraints. CTL is smaller in magnitude than CT because the zigzagging problem is avoided with linear and side constraints. The default value is usually adequate.

Not used if NCON = NSIDE = 0. Minimum absolute value of CTL considered in the optimization process. The default value is usually adequate.

the maximum fractional change in any component of X as an initial estimate for ALPHA in the one-dimensional search. That is, the initial ALPHA will be such that no component of X is changed by more than this amount. This only applies to those X(i) of magnitude greater than 0.1. If an optimization run shows numerous ALPHA = 0 results for the one-dimensional search, it may help to try ALPHAX less than the default. ALPHAX is changed by CONMIN depending on the progress of the optimization.

the fractional change attempted as a first step in the one-dimensional search and is based on a linear approximation. ABOBJ1 is updated during the optimization, depending on progress. The initial step in the one-dimensional search is taken as the amount necessary to change OBJ by ABOBJ1*ABS(OBJ) or to change some X(i) by ALPHAX*ABS( X(i) ), whichever is less.

Not used if NCON = NSIDE = 0. Mean value of the push-off factor in the method of feasible directions. A larger value of THETA is desirable if the constraints, G(J), are known to be highly nonlinear, and a smaller value may be used if all G(J) are known to be nearly linear. The actual value of the push-off factor used in the program is a quadratic function of each G(J), varying from 0.0 for G(J) = CT to 4.0*THETA for G(J) = ABS(CT). A value of THETA = 0.0 is used in the program for constraints which are identified by the user to be strictly linear. THETA is called a "push-off" factor because it pushes the design away from the active constraints into the feasible region. The default value is usually adequate.

Value of objective function for the current decision variables, X(I), I = 1, NDV contained in vector X. Calculate OBJ if INFO = 1 or INFO = 2.

Number of decision variables, X(I), contained in vector X.

Number of constraint functions, G(J). NCON may be zero.

Side constraint parameter. NSIDE = 0 signifies that the variables X(I) do not have lower or upper bounds. NSIDE>0 signifies that all variables X(I) have lower and upper bounds defined by VLB(I) and VUB(I) respectively. If one or more variables are not bounded while others are, the values of the lower and upper bounds on the unbounded variables must be taken as very large negative and positive values respectively (i.e., VLB(I) = -1.0E+10, VUB(I) = 1.0E+10).

Unit number for printing. Should be non-zero.

Print control. All printing is done on unit number iunit.

the finite difference gradient parameter:

Scaling control parameter. The decision variables will be scaled linearly.

Not used if NCON = NSIDE = 0. Linear objective function identifier. If the objective, OBJ, is specifically known to be a strictly linear function of the decision variables, X(I), set LINOBJ = 1. If OBJ is a general nonlinear function, set LINOBJ = 0.

Maximum number of iterations in the minimization process. If NFDG==0 each iteration requires one set of gradient computations (INFO = 3 or 4) and approximately three function evaluations (INFO = 1 or 2). If NFDG>0 each iteration requires approximately NDV + 3 function evaluations (INFO = 1 or 2).

Number of consecutive iterations to indicate convergence by relative or absolute changes, DELFUN or DABFUN.

Default value = NDV + 1. Conjugate direction restart parameter. If the function is currently unconstrained, (all G(J)<CT or NCON = NSIDE = 0), Fletcher-Reeves conjugate direction method will be restarted with a steepest descent direction every ICNDIR iterations. If ICNDIR = 1 only steepest descent will be used.

Reverse communication flag. CONMIN executes according to the parameter IGOTO which must be initialized to zero.

Number of active and violated constraints (G(J)>=CT). Calculate NAC if INFO = 4 and NFDG = 0.

Iteration number. The optimization process is iterative so that the vector of decision variables at the Kth iteration is defined by X(K) = X(K - 1) + ALPHA*S(K), where in this case K refers to the iteration number and the components X(I) are all changed simultaneously. ALPHA is defined as the move parameter and is printed if the print control IPRINT>=3. S is the move direction.

Not used if NCON = NSIDE = 0. Participation coefficient, used if a design is infeasible (one or more G(J)>ABS(CT)). PHI is a measure of how hard the design will be "pushed" towards the feasible region and is, in effect, a penalty parameter. If in a given problem, a feasible solution cannot be obtained with the default value, PHI should be increased, and the problem run again. If a feasible solution cannot be obtained with PHI = 100, it is probable that no feasible solution exists. The default value is usually adequate.

Bookkeeping information. NCAL(1) gives the number of times external routine SUB1 was called with INFO = 1. NCAL(2) gives the number of times INFO = 2. NCAL(3) gives the number of times INFO = 3 and NCAL(4) gives the number of times INFO = 4.

a function to call once per iteration to report the solution

Vector of decision variables, X(I), I = 1, NDV. The initial X-vector contains the user's best estimate of the set of optimum design variables.

Used only if NSIDE/=0. VLB(I) is the lower allowable value (lower bound) of variable X(I). If one or more variables, X(I), do not have lower bounds, the corresponding VLB(I) must be initialized to a very large negative number (say -1.0E+10).

Used only if NSIDE/=0. VUB(I) is the maximum allowable value (upper bound) of X(I). If one or more variables, X(I), do not have upper bounds, the corresponding VUB(I) must be initialized to a very large positive number (say 1.0E+10).

Not used if NCON = NSIDE = 0. Vector containing all constraint functions, G(J), J = 1, NCON for current decision variables, X. Calculate G(J), J = 1, NCON if INFO = 2.

Not used if NSCAL = 0. Vector of scaling parameters. If NSCAL>0 vector SCAL need not be initialized since SCAL will be defined in CONMIN and its associated routines. If NSCAL<0, vector SCAL is initialized in the main program, and the scaled variables X(I) = X(I)/SCAL(I). Efficiency of the optimization process can sometimes be improved if the variables are either normalized or are scaled in such a way that the partial deri- vative of the objective function, OBJ, with respect to variable X(I) is of the same order of magnitude for all X(I). SCAL(I) must be greater than zero because a negative value of SCAL(I) will result in a change of sign of X(I) and possibly yield erroneous optimization results. The decision of if, and how, the variables should be scaled is highly problem dependent, and some experimentation is desirable for any given class of problems.

Analytic gradient of the objective function for the current decision variables, X(I). DF(I) contains the partial derivative of OBJ with respect to X(I). Calculate DF(I), I = 1, NDV if INFO = 3 or INFO = 4 and if NFDG = 0 or NFDG = 2.

Not used if NCON = NSIDE = 0. Gradients of active or violated constraints, for current decision variables, X(I). A(J,I) contains the gradient of the Jth active or violated constraint, G(J), with respect to the Ith decision variable, X(I) for J = 1, NAC and I = 1, NDV. Calculate if INFO = 4 and NFDG = 0.

Move direction in the NDV-dimensional optimization space. S(I) gives the rate at which variable X(I) changes with respect to ALPHA.

Not used if NCON = NSIDE = NSCAL = 0. Used for temporary storage of constraint values G(J), J = 1, NCON and decision variables X(I), I = 1, NDV.

Not used if NCON = NSIDE = 0. Used for temporary storage of constraint values G(J), J = 1, NCON.

Not used if NCON = NSIDE = 0. Used in determining direction vector S for constrained minimization problems. Array B may be used for temporary storage in external routine SUB1.

Not used in NCON = NSIDE = 0. Used with array B in determining direction vector S for constrained minimization problems. Used for temporary storage of vector X if NSCAL/=0. routine SUB1.

Not used if NCON = 0. Linear constraint identification vector. If constraint G(J) is known to be a linear function of the decision variables, X(I), ISC(I) should be initialized to ISC(I) = 1. If constraint G(J) is nonlinear ISC(I) is initialized to ISC(I) = 0. Identification of linear constraints may improve efficiency of the optimization process and is therefore desirable, but is not essential. If G(J) is not specifically known to be linear, set ISC(I) = 0.

Identifies which constraints are active or violated. IC(J) contains the number of the Jth active or violated constraint for J = 1, NAC. For example, if G(10) is the first active or violated constraint (G(J)<CT, J = 1,9), set IC(1) = 10. Calculate if INFO = 4 and NFDG = 0.

Not used if NCON = NSIDE = 0. Used with array B in determining direction vector S for constrained minimization problems. Array MS1 may be used for temporary storage in external routine SUB1.

N1 = NDV + 2

N2 = NCON + 2*NDV

N3 = NACMX1

N4 = MAX (N3,NDV)

N5 = 2*N4

NCALC = CALCULATION CONTROL.

SLOPE = INITIAL FUNCTION SLOPE = S-TRANSPOSE TIMES DF. SLOPE MUST BE NEGATIVE.

PROPOSED MOVE PARAMETER

CALCULATION CONTROL:

may be negative

CALCULATION CONTROL:

may be negative

NER = ERROR FLAG. IF NER/=0 ON RETURN, PROCESS HAS NOT CONVERGED IN 5*NDB ITERATIONS.

VECTOR MS1 IDENTIFIES THE SET OF BASIC VARIABLES.