Optimization Toolbox
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Reference

This chapter contains descriptions of the Optimization Toolbox functions, listed alphabetically. The chapter starts with tables listing the types of optimization and which functions apply. The next set of tables lists general descriptions of all the input and output arguments and the parameters in the optimization options structure, and which functions use those arguments or parameters. Information specific to a function about input arguments, output arguments, and options is listed within that function's reference pages under the Arguments section, after the function description.

Information is also available through the online Help facility.

Optimization Functions

Minimization

Function
 Purpose
fgoalattain
 Multiobjective goal attainment
fminbnd
 Scalar nonlinear minimization with bounds
fmincon
 Constrained nonlinear minimization
fminimax
 Minimax optimization
fminsearch,
fminunc
 Unconstrained nonlinear minimization
fseminf
 Semi-infinite minimization
linprog
 Linear programming
quadprog
 Quadratic programming

Equation Solving

Function
 Purpose
 
    \
Linear equation solving (see the online MATLAB Function Reference guide)
 
    fsolve
Nonlinear equation solving
 
    fzero
Scalar nonlinear equation solving

Least-Squares (Curve Fitting)

Function
 Purpose
\
 Linear least squares (see the online MATLAB Function Reference guide)
lsqlin
 Constrained linear least squares
lsqcurvefit
 Nonlinear curve fitting
lsqnonlin
 Nonlinear least squares
lsqnonneg
 Nonnegative linear least squares

Utility

Function
 Purpose
optimset, optimget
 Parameter setting

Demonstrations of Large-Scale Methods

Function
 Purpose
circustent
 Quadratic programming to find shape of a circus tent
molecule
 Molecule conformation solution using unconstrained nonlinear minimization
optdeblur
 Image deblurring using bounded linear least-squares

Demonstrations of Medium-Scale Methods

Function
 Purpose
bandemo
 Minimization of the banana function
dfildemo
 Finite-precision filter design (requires Signal Processing Toolbox)
goaldemo
 Goal attainment example
optdemo
 Menu of demonstration routines
tutdemo
 Tutorial walk-through

Function Arguments

These tables describe arguments used by Optimization Toolbox functions: the first describes input arguments, the second describes the output arguments, and the third describes the optimization options parameters structure options.

This table summarizes the input arguments. Not all functions use all arguments. See the individual reference pages for function-specific details about these arguments.

Table 1-1: Input Arguments 
Argument
 Description
 Used by Functions
A, b
 The matrix A and vector b are, respectively, the coefficients of the linear inequality constraints and the corresponding right-hand side vector: A.x<=b.
 fgoalattain, fmincon, fminimax, fseminf, linprog, lsqlin, quadprog
Aeq, beq
 The matrix Aeq and vector beq are, respectively, the coefficients of the linear equality constraints and the corresponding right-hand side vector: Aeq.x=beq.
 fgoalattain, fmincon, fminimax, fseminf, linprog, lsqlin, quadprog
C, d
 The matrix C and vector d are, respectively, the coefficients of the over- or under-determined linear system and the right-hand-side vector to be solved.
 lsqlin, lsqnonneg
f
 The vector of coefficients for the linear term in the linear equation f'*x or the quadratic equation x'*H*x+f'*x.
linprog, quadprog
fun
 The function to be optimized. fun may be an inline object, or the name of an M-file, built-in function, or MEX-file. See the individual function reference pages for more information on fun.
 fgoalattain, fminbnd, fmincon, fminimax, fminsearch, fminunc, fseminf, fsolve, fzero, lsqcurvefit, lsqnonlin
goal
 Vector of values that the objectives attempt to attain. The vector is the same length as the number of objectives.
 fgoalattain
H
 The matrix of coefficients for the quadratic terms in the quadratic equation x'*H*x+f'*x. H must be symmetric.
 quadprog
lb, ub
 Lower and upper bound vectors (or matrices). The arguments are normally the same size as x. However, if lb has fewer elements than x, say only m, then only the first m elements in x are bounded below; upper bounds in ub can be defined in the same manner. Unbounded variables may also be specified using -Inf (for lower bounds) or Inf (for upper bounds). For example, if lb(i) = -Inf then the variable x(i) is unbounded below.
 fgoalattain, fmincon, fminimax, fseminf, linprog, lsqcurvefit, lsqlin, lsqnonlin, quadprog
nonlcon
 The function that computes the nonlinear inequality and equality constraints. nonlcon is the name of an M-file or MEX-file. See the individual reference pages for more information on nonlcon.
 fgoalattain, fmincon, fminimax
ntheta
 The number of semi-infinite constraints.
 fseminf
options
 An optimization options parameter structure that defines parameters used by the optimization functions. For information about the parameters, see Table 1-3 or the individual function reference pages.
 All functions
P1, P2,...
 Additional arguments to be passed to fun, nonlcon (if it exists), or seminfcon (if it exists), that is, when the optimization function calls the functions fun, nonlcon, or seminfcon, the calls are
    f = feval(fun,x,P1,P2,...)
[c, ceq] = feval(nonlcon,x,P1,P2,...)
[c,ceq,K1,K2,...,Kn,s]= ... 
        feval(seminfcon,x,s,P1,P2,...)
Using this feature, the same fun (or nonlcon or seminfcon) can solve a number of similar problems with different parameters, avoiding the need to use global variables.
 fgoalattain, fminbnd, fmincon, fminimax, fminsearch, fminunc, fseminf, fsolve, fzero, lsqcurvefit, lsqnonlin
seminfcon
 The function that computes the nonlinear inequality and equality constraints and the semi-infinite constraints. seminfcon is the name of an M-file or MEX-file. See the individual function reference pages for fseminf more information on seminfcon.
 fseminf
weight
 A weighting vector to control the relative under-attainment or over-attainment of the objectives.
 fgoalattain
xdata, ydata
 The input data xdata and the observed output data ydata that is to be fit to an equation.
 lsqcurvefit
x0
 Starting point (a scalar, vector or matrix).
(For fzero, x0 can also be a two-element vector representing an interval that is known to contain a zero.)
 All functions except fminbnd
x1, x2
 The interval over which the function is minimized.
 fminbnd

Table 1-2: Output Arguments 
Argument
 Description
 Used by Functions
attainfactor
 The attainment factor at the solution x.
 fgoalattain
exitflag
 The exit condition. For the meaning of a particular value, see the function reference pages.
 All functions
fval
 The value of the objective function fun at the solution x.
 fgoalattain, fminbnd, fmincon, fminimax, fminsearch, fminunc, fseminf, fsolve, fzero, linprog, quadprog
grad
 The value of the gradient of fun at the solution x.
 fmincon, fminunc
hessian
 The value of the Hessian of fun at the solution x.
 fmincon, fminunc
jacobian
 The value of the Jacobian of fun at the solution x.
 lsqcurvefit, lsqnonlin, fsolve
lambda
 The Lagrange multipliers at the solution x. lambda is a structure where each field is for a different constraint type. For structure field names, see individual function descriptions. (For lsqnonneg, lambda is simply a vector as lsqnonneg only handles one kind of constraint.)
 fgoalattain, fmincon, fminimax, fseminf, linprog, lsqcurvefit, lsqlin, lsqnonlin, lsqnonneg, quadprog
maxfval
 max{fun(x)} at the solution x.
 fminimax
output
 An output structure that contains information about the results of the optimization. For structure field names, see individual function descriptions.
 All functions
residual
 The value of the residual at the solution x.
 lsqcurvefit, lsqlin, lsqnonlin, lsqnonneg
resnorm
 The value of the squared 2-norm of the residual at the solution x.
 lsqcurvefit, lsqlin, lsqnonlin, lsqnonneg
x
 The solution found by the optimization function. If exitflag > 0 then x is a solution; otherwise, x is the value the optimization routine was at when it terminated prematurely.
 All functions

This table describes fields in the optimization parameters structure, options. The column labeled L, M, B provides this information about each parameter:


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