Specifications

Multi-Objective Optimization
Multi-objective optimization is concerned
with the minimization of multiple objective
functions that are subject to a set of con-
straints. The Optimization Toolbox provides
functions for solving two formulations of
multi-objective optimization problems: goal
attainment and minimax.
The goal attainment problem involves reduc-
ing the value of a linear or nonlinear vector
function to attain the goal values given in a
goal vector. The relative importance of the
goals is indicated using a weight vector. The
goal attainment problem may also be subject
to linear and nonlinear constraints.
The minimax problem involves minimizing
the worst-case value of a set of multivariate
functions, possibly subject to linear and non-
linear constraints.
The Optimization Toolbox transforms both
types of multi-objective problems into stan-
dard constrained optimization problems and
then solves them using a sequential quadratic
programming approach.
Nonlinear Least Squares, Data Fitting, and
Nonlinear Equations
The Optimization Toolbox can solve
nonlinear least squares problems,
data fitting problems, and systems of
nonlinear equations.
The toolbox uses three methods for
solving nonlinear least squares problems:
trust-region, Levenberg-Marquardt, and
Gauss-Newton.
Trust-region is used for unconstrained and
bound constrained problems.
Levenberg-Marquardt is a line search
method whose search direction is a cross
between the Gauss-Newton and steepest
descent directions.
Gauss-Newton is a line search method that
chooses a search direction based on the
solution to a linear least squares problem.
The toolbox also includes a specialized inter-
face for data-fitting problems to find the
member of a family of nonlinear functions that
best fits a set of data points. The toolbox uses
the same methods for data-fitting problems as
it uses for nonlinear least-squares problems.
The Optimization Toolbox implements a
dogleg trust region method for solving a
system of nonlinear equations where there are
as many equations as unknowns. The toolbox
can also solve this problem using either the
trust-region, the Levenberg-Marquandt, or
the Gauss-Newton method.
A user-defined output function (top) plots the
current iterate at each algorithm iteration (left).
The Optimization Toolbox also provides details for
each iteration (bottom).