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# NLPJOB

The solver NLPJOB solves multicriteria optimization problems by sequential quadratic programming. NLPJOB offers 15 different possibilities to transform the objective function vector into a scalar function. Depending on the selected method, additional constraints can be added.

## Multicriteria options

• Weighted sum: The scalar objective function is the weighted sum of individual objectives, i.e.,
F(X) := W1*F1(X) + W2*F2(X) + ... + WL*FL(X) ,
where W1, ..., WL are non-negative weights given by the user.

• Hierarchical optimization method: The idea is to formulate a sequence of L scalar optimization problems with respect to the individual objective functions subject to bounds on previously computed optimal values, i.e., we minimize
F(X) := FI(X) , I = 1,...,L
subject to the original and the additional constraints
FJ(X) <= (1+EJ/100)*FJ , J = 1,...,I-1 ,
where EJ is the given coefficient of relative function increment as defined by the user and where FJ is the individual minimum. It is assumed that the objective functions are ordered with respect to their importance.

• Trade-off method: One objective is selected by the user and the other ones are considered as constraints with respect to individual minima, i.e.,
F(X) := FI(X)
is minimized subject to the original and some additional constraints of the form
FJ(X) <= EJ , J=1,...,L , J <> I ,
where EJ is a bound value of the J-th objective function.

• Method of distance functions in L1-norm: A sum of absolute values of the differences of objective functions from predetermined goals Y1, ..., YL is minimized, i.e.,
F(X) := |F1(X)-Y1| + ... + |FL(X)-YL|
The goals are given by the user and their choice requires some knowledge about the ideal solution vector.

• Method of distance functions in L2-norm: A sum of squared values of the differences of objective functions from predetermined goals Y1, ..., Yl is minimized,
F(X) := (F1(X)-Y1)^2 + ... + (FL(X)-YL)^2
Again the goals are provided by the user.

• Global criterion method: The scalar function to be minimized, is the sum of relative distances of individual objectives from their known minimal values, i.e.,
F(X) := (F1(X)-F1)/|F1| + ... + (FL(X)-FL)/|FL|
where F1, ..., FL are the optimal function values obtained by minimizing F1(x), ..., FL(x) subject to original constraints.

• Global criterion method in L2-norm: The scalar function to be minimized, is the sum of squared distances of individual objectives from their known optimal values, i.e.,
F(X) := ((F1-F1(X))/F1)^2 + ... + ((FL-FL(X))/FL))^2
where F1, ..., FL are the individual optimal function values.

• Min-max method no. 1: The maximum of absolute values of all objectives is minimized, i.e.,
F(X) := MAX ( |FI(X)| , I=1,...,L )

• Min-max method no. 2: The maximum of all objectives is minimized, i.e.,
F(X) := MAX ( FI(X) , I=1,...,L )

• Min-max method no. 3: The maximum of absolute distances of objective function values from given goals Y1, ..., YL is minimized, i.e.,
F(X) := MAX ( |FI(X)-YI| , I=1,...,L )
The goals must be determined by the user.

• Min-max method no. 4: The maximum of relative distances of objective function values from ideal values is minimized, i.e.,
F(X) := MAX ( (FI(X)-FI)/|FI| , I=1,...,L )

• Min-max method no. 5: The maximum of weighted relative distances of objective function values from individual minimal values is minimized,
F(X) := MAX ( WI*(FI(X)-FI)/|FI| , I=1,...,L )
Weights must be provided by the user.

• Min-max method no. 6: The maximum of weighted objective function values is minimized, i.e.,
F(X) := MAX ( WI*FI(X) , I=1,...,L )
Weights must be provided by the user.

• Weighted global criterion method: The scalar function to be minimized, is the weighted sum of relative distances of individual objectives from their goals, i.e.,
F(X) := (F1(X)-Y1)/|Y1| + ... + (FL(X)-YL)/|YL|
The weights W1, ..., WL and the goals Y1, ..., YL must be set by the user.

• Weighted global criterion method in L2-norm: The scalar function to be minimized, is the weighted sum of squared relative distances of individual objectives from their goals, i.e.,
F(X) := ((F1(X)-Y1)/Y1)^2 + ... + ((FL(X)-YL)/YL)^2
The weights W1, ..., WL and the goals Y1, ..., YL must be set by the user.