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There are two Optimization Toolbox™ multiobjective solvers: `fgoalattain`

and `fminimax`

.

`fgoalattain`

addresses the problem of reducing a set of nonlinear functions(*F*_{i}) below a set of goals*x*. Since there are several functions*F**_{i}(*F*_{i}), it is not always clear what it means to solve this problem, especially when you cannot achieve all the goals simultaneously. Therefore, the problem is reformulated to one that is always well-defined.*x*The

*unscaled goal attainment problem*is to minimize the maximum of.*F*(_{i}*x*) –*F**_{i}There is a useful generalization of the unscaled problem. Given a set of positive weights

, the*w*_{i}*goal attainment problem*tries to findto minimize the maximum of*x*$$\frac{{F}_{i}(x)-{F}_{i}^{*}}{{w}_{i}}.$$ **(7-1)**This minimization is supposed to be accomplished while satisfying all types of constraints:

*c*(*x*) ≤ 0,*ceq*(*x*) = 0,*A·x*≤*b*,*Aeq·x*=*beq*, and*l*≤*x*≤*u*If you set all weights equal to 1 (or any other positive constant), the goal attainment problem is the same as the unscaled goal attainment problem. If the

are positive, and you set all weights as*F**_{i}, the goal attainment problem becomes minimizing the relative difference between the functions*w*=_{i}*F**_{i}(*F*_{i}) and the goals*x*.*F**_{i}In other words, the goal attainment problem is to minimize a slack variable

, defined as the maximum over*γ*of the expressions in Equation 7-1. This implies the expression that is the formal statement of the goal attainment problem:*i*$$\underset{x,\gamma}{\mathrm{min}}\gamma $$

such that

*F*(*x*) –*w*·*γ*≤*F**,*c*(*x*) ≤ 0,*ceq*(*x*) = 0,*A·x*≤*b*,*Aeq·x*=*beq*, and*l*≤*x*≤*u*`fminimax`

addresses the problem of minimizing the maximum of a set of nonlinear functions, subject to all types of constraints:$$\underset{x}{\mathrm{min}}\underset{i}{\mathrm{max}}{F}_{i}(x)$$

such that

*c*(*x*) ≤ 0,*ceq*(*x*) = 0,*A·x*≤*b*,*Aeq·x*=*beq*, and*l*≤*x*≤*u*Clearly, this problem is a special case of the unscaled goal attainment problem, with

and*F**= 0_{i}.*w*= 1_{i}

This section describes the goal attainment method of Gembicki [16]. This method uses a set of design goals, $${F}^{*}=\left\{{F}_{1}^{*},{F}_{2}^{*},\mathrm{...},{F}_{m}^{*}\right\}$$, associated with a set of objectives, * F(x)
= {F_{1}(x),F_{2}(x),...,F_{m}(x)}*.
The problem formulation allows the objectives to be under- or overachieved,
enabling the designer to be relatively imprecise about the initial
design goals. The relative degree of under- or overachievement of
the goals is controlled by a vector of weighting coefficients,

$$\underset{\gamma \in \Re ,\text{}x\in \Omega}{\text{minimize}}\gamma $$ | (7-2) |

such that $${F}_{i}(x)-{w}_{i}\gamma \le {F}_{i}^{*},\text{}i=1,\mathrm{...},m.$$

The term * w_{i}γ* introduces
an element of

The goal attainment method is represented geometrically in the figure below in two dimensions.

**Figure 7-1. Geometrical Representation of the Goal Attainment
Method**

Specification of the goals, $$\left\{{F}_{1}^{*},{F}_{2}^{*}\right\}$$,
defines the goal point, * P*. The weighting vector
defines the direction of search from

The goal attainment method has the advantage that it can be posed as a nonlinear programming problem. Characteristics of the problem can also be exploited in a nonlinear programming algorithm. In sequential quadratic programming (SQP), the choice of merit function for the line search is not easy because, in many cases, it is difficult to "define" the relative importance between improving the objective function and reducing constraint violations. This has resulted in a number of different schemes for constructing the merit function (see, for example, Schittkowski [36]). In goal attainment programming there might be a more appropriate merit function, which you can achieve by posing Equation 7-2 as the minimax problem

$$\underset{x\in {\Re}^{n}}{\text{minimize}}\text{}\underset{i}{\mathrm{max}}\left\{{\Lambda}_{i}\right\},$$ | (7-3) |

where

$${\Lambda}_{i}=\frac{{F}_{i}(x)-{F}_{i}^{*}}{{w}_{i}},\text{}i=1,\mathrm{...},m.$$

Following the argument of Brayton et al. [2] for minimax optimization using SQP, using the merit function of Equation 6-46 for the goal attainment problem of Equation 7-3 gives

$$\psi (x,\gamma )=\gamma +{\displaystyle \sum _{i=1}^{m}{r}_{i}\cdot \mathrm{max}\left\{0,{F}_{i}(x)-{w}_{i}\gamma -{F}_{i}^{*}\right\}}.$$ | (7-4) |

When the merit function of Equation 7-4 is
used as the basis of a line search procedure, then, although * ψ(x,γ)* might
decrease for a step in a given search direction, the function

`max`

Λ`max`

ΛFollowing the lines of Brayton et al. [2], a solution is therefore to set * ψ(x)* equal
to the worst case objective, i.e.,

$$\psi (x)=\underset{i}{\mathrm{max}}{\Lambda}_{i}.$$ | (7-5) |

A problem in the goal attainment method is that it is common to use a weighting coefficient equal to 0 to incorporate hard constraints. The merit function of Equation 7-5 then becomes infinite for arbitrary violations of the constraints.

To overcome this problem while still retaining the features of Equation 7-5, the merit function is combined with that of Equation 6-47, giving the following:

$$\psi (x)={\displaystyle \sum _{i=1}^{m}\{\begin{array}{ll}{r}_{i}\cdot \mathrm{max}\left\{0,{F}_{i}(x)-{w}_{i}\gamma -{F}_{i}^{*}\right\}\hfill & \text{if}{w}_{i}=0\hfill \\ \underset{i}{\mathrm{max}}{\Lambda}_{i},\text{}i=1,\mathrm{...},m\hfill & \text{otherwise}\text{.}\hfill \end{array}}$$ | (7-6) |

Another feature that can be exploited in SQP is the objective
function * γ*. From the KKT equations it can
be shown that the approximation to the Hessian of the Lagrangian,

These changes make the Hessian, * H*, indefinite.
Therefore

`1e`

-10).
This allows use of the fast converging positive definite QP method
described in Quadratic Programming Solution. The preceding modifications have been implemented in `fgoalattain`

and have been found to make
the method more robust. However, because of the rapid convergence
of the SQP method, the requirement that the merit function strictly
decrease sometimes requires more function evaluations than an implementation
of SQP using the merit function of Equation 6-46.

`fminimax`

uses a goal attainment method.
It takes goals of 0, and weights of 1. With this formulation, the
goal attainment problem becomes

$$\underset{i}{\mathrm{min}}\underset{x}{\mathrm{max}}\left(\frac{{f}_{i}(x)-goa{l}_{i}}{weigh{t}_{i}}\right)=\underset{i}{\mathrm{min}}\underset{x}{\mathrm{max}}{f}_{i}(x),$$

which is the minimax problem.

Parenthetically, you might expect `fminimax`

to
turn the multiobjective function into a single objective. The function

* f*(

is a single objective function to minimize. However, it is not differentiable, and Optimization Toolbox objectives are required to be smooth. Therefore the minimax problem is formulated as a smooth goal attainment problem.

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