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Find minimum of constrained nonlinear multivariable function
Finds the minimum of a problem specified by
b and beq are vectors, A and Aeq are matrices, c(x) and ceq(x) are functions that return vectors, and f(x) is a function that returns a scalar. f(x), c(x), and ceq(x) can be nonlinear functions.
x, lb, and ub can be passed as vectors or matrices; see Matrix Arguments.
x = fmincon(fun,x0,A,b)
x = fmincon(fun,x0,A,b,Aeq,beq)
x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub)
x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon)
x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)
x = fmincon(problem)
[x,fval] = fmincon(...)
[x,fval,exitflag] = fmincon(...)
[x,fval,exitflag,output] = fmincon(...)
[x,fval,exitflag,output,lambda] = fmincon(...)
[x,fval,exitflag,output,lambda,grad]
= fmincon(...)
[x,fval,exitflag,output,lambda,grad,hessian]
= fmincon(...)
fmincon attempts to find a constrained minimum of a scalar function of several variables starting at an initial estimate. This is generally referred to as constrained nonlinear optimization or nonlinear programming.
Note: Passing Extra Parameters explains how to pass extra parameters to the objective function and nonlinear constraint functions, if necessary. 
x = fmincon(fun,x0,A,b) starts at x0 and attempts to find a minimizer x of the function described in fun subject to the linear inequalities A*x ≤ b. x0 can be a scalar, vector, or matrix.
x = fmincon(fun,x0,A,b,Aeq,beq) minimizes fun subject to the linear equalities Aeq*x = beq and A*x ≤ b. If no inequalities exist, set A = [] and b = [].
x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub) defines a set of lower and upper bounds on the design variables in x, so that the solution is always in the range lb ≤ x ≤ ub. If no equalities exist, set Aeq = [] and beq = []. If x(i) is unbounded below, set lb(i) = Inf, and if x(i) is unbounded above, set ub(i) = Inf.
Note: If the specified input bounds for a problem are inconsistent, the output x is x0 and the output fval is []. Components of x0 that violate the bounds lb ≤ x ≤ ub are reset to the interior of the box defined by the bounds. Components that respect the bounds are not changed. 
x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon) subjects the minimization to the nonlinear inequalities c(x) or equalities ceq(x) defined in nonlcon. fmincon optimizes such that c(x) ≤ 0 and ceq(x) = 0. If no bounds exist, set lb = [] and/or ub = [].
x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options) minimizes with the optimization options specified in options. Use optimoptions to set these options. If there are no nonlinear inequality or equality constraints, set nonlcon = [].
x = fmincon(problem) finds the minimum for problem, where problem is a structure described in Input Arguments. Create the problem structure by exporting a problem from Optimization app, as described in Exporting Your Work.
[x,fval] = fmincon(...) returns the value of the objective function fun at the solution x.
[x,fval,exitflag] = fmincon(...) returns a value exitflag that describes the exit condition of fmincon.
[x,fval,exitflag,output] = fmincon(...) returns a structure output with information about the optimization.
[x,fval,exitflag,output,lambda] = fmincon(...) returns a structure lambda whose fields contain the Lagrange multipliers at the solution x.
[x,fval,exitflag,output,lambda,grad] = fmincon(...) returns the value of the gradient of fun at the solution x.
[x,fval,exitflag,output,lambda,grad,hessian] = fmincon(...) returns the value of the Hessian at the solution x. See fmincon Hessian.
Function Arguments describes the arguments passed to fmincon. Options provides the functionspecific details for the options values. This section provides functionspecific details for fun, nonlcon, and problem.
The function to be minimized. fun is a function that accepts a vector x and returns a scalar f, the objective function evaluated at x. fun can be specified as a function handle for a file: x = fmincon(@myfun,x0,A,b) where myfun is a MATLAB^{®} function such as function f = myfun(x) f = ... % Compute function value at x fun can also be a function handle for an anonymous function: x = fmincon(@(x)norm(x)^2,x0,A,b); If the gradient of fun can also be computed and the GradObj option is 'on', as set by options = optimoptions('fmincon','GradObj','on') then fun must return the gradient vector g(x) in the second output argument. If the Hessian matrix can also be computed and the Hessian option is 'on' via options = optimoptions('fmincon','Hessian','usersupplied') and the Algorithm option is trustregionreflective, fun must return the Hessian value H(x), a symmetric matrix, in a third output argument. fun can give a sparse Hessian. See Writing Objective Functions for details. If the Hessian matrix can be computed and the Algorithm option is interiorpoint, there are several ways to pass the Hessian to fmincon. For more information, see Hessian.  
A, b, Aeq, beq  Linear constraint matrices A and Aeq, and their corresponding vectors b and beq, can be sparse or dense. The trustregionreflective and interiorpoint algorithms use sparse linear algebra. If A or Aeq is large, with relatively few nonzero entries, save running time and memory in the trustregionreflective or interiorpoint algorithms by using sparse matrices.  
The function that computes the nonlinear inequality constraints c(x)≤ 0 and the nonlinear equality constraints ceq(x) = 0. nonlcon accepts a vector x and returns the two vectors c and ceq. c is a vector that contains the nonlinear inequalities evaluated at x, and ceq is a vector that contains the nonlinear equalities evaluated at x. nonlcon should be specified as a function handle to a file or to an anonymous function, such as mycon: x = fmincon(@myfun,x0,A,b,Aeq,beq,lb,ub,@mycon) where mycon is a MATLAB function such as function [c,ceq] = mycon(x) c = ... % Compute nonlinear inequalities at x. ceq = ... % Compute nonlinear equalities at x. If the gradients of the constraints can also be computed and the GradConstr option is 'on', as set by options = optimoptions('fmincon','GradConstr','on') then nonlcon must also return, in the third and fourth output arguments, GC, the gradient of c(x), and GCeq, the gradient of ceq(x). GC and GCeq can be sparse or dense. If GC or GCeq is large, with relatively few nonzero entries, save running time and memory in the interiorpoint algorithm by representing them as sparse matrices. For more information, see Nonlinear Constraints.  
problem  objective  Objective function  
x0  Initial point for x  
Aineq  Matrix for linear inequality constraints  
bineq  Vector for linear inequality constraints  
Aeq  Matrix for linear equality constraints  
beq  Vector for linear equality constraints  
lb  Vector of lower bounds  
ub  Vector of upper bounds  
nonlcon  Nonlinear constraint function  
solver  'fmincon'  
options  Options created with optimoptions 
Function Arguments describes arguments returned by fmincon. This section provides functionspecific details for exitflag, lambda, and output:
exitflag  Integer identifying the reason the algorithm terminated. The following lists the values of exitflag and the corresponding reasons the algorithm terminated.  
All Algorithms:  
1  Firstorder optimality measure was less than options.TolFun, and maximum constraint violation was less than options.TolCon.  
0  Number of iterations exceeded options.MaxIter or number of function evaluations exceeded options.MaxFunEvals.  
1  Stopped by an output function or plot function.  
2  No feasible point was found.  
trustregionreflective, interiorpoint, and sqp algorithms:  
2  Change in x was less than options.TolX and maximum constraint violation was less than options.TolCon.  
trustregionreflective algorithm only:  
3  Change in the objective function value was less than options.TolFun and maximum constraint violation was less than options.TolCon.  
activeset algorithm only:  
4  Magnitude of the search direction was less than 2*options.TolX and maximum constraint violation was less than options.TolCon.  
5  Magnitude of directional derivative in search direction was less than 2*options.TolFun and maximum constraint violation was less than options.TolCon.  
interiorpoint and sqp algorithms:  
3  Objective function at current iteration went below options.ObjectiveLimit and maximum constraint violation was less than options.TolCon.  
grad  Gradient at x  
hessian  Hessian at x  
lambda  Structure containing the Lagrange multipliers at the solution x (separated by constraint type). The fields of the structure are:  
lower  Lower bounds lb  
upper  Upper bounds ub  
ineqlin  Linear inequalities  
eqlin  Linear equalities  
ineqnonlin  Nonlinear inequalities  
eqnonlin  Nonlinear equalities  
output  Structure containing information about the optimization. The fields of the structure are:  
iterations  Number of iterations taken  
funcCount  Number of function evaluations  
lssteplength  Size of line search step relative to search direction (activeset algorithm only)  
constrviolation  Maximum of constraint functions  
stepsize  Length of last displacement in x (activeset and interiorpoint algorithms)  
algorithm  Optimization algorithm used  
cgiterations  Total number of PCG iterations (trustregionreflective and interiorpoint algorithms)  
firstorderopt  Measure of firstorder optimality  
message  Exit message 
fmincon uses a Hessian as an optional input. This Hessian is the second derivatives of the Lagrangian (see Equation 31), namely,
(101) 
The various fmincon algorithms handle input Hessians differently:
The activeset and sqp algorithms do not accept a usersupplied Hessian. They compute a quasiNewton approximation to the Hessian of the Lagrangian.
The trustregionreflective algorithm can accept a usersupplied Hessian as the final output of the objective function. Since this algorithm has only bounds or linear constraints, the Hessian of the Lagrangian is same as the Hessian of the objective function. See Writing Scalar Objective Functions for details on how to pass the Hessian to fmincon. Indicate that you are supplying a Hessian by
options = optimoptions('fmincon','Algorithm','trustregionreflective','Hessian','usersupplied');
If you do not pass a Hessian, the algorithm computes a finitedifference approximation.
The interiorpoint algorithm can accept a usersupplied Hessian as a separately defined function—it is not computed in the objective function. The syntax is
hessian = hessianfcn(x, lambda)
hessian is an nbyn matrix, sparse or dense, where n is the number of variables. If hessian is large and has relatively few nonzero entries, save running time and memory by representing hessian as a sparse matrix. lambda is a structure with the Lagrange multiplier vectors associated with the nonlinear constraints:
lambda.ineqnonlin lambda.eqnonlin
fmincon computes the structure lambda. hessianfcn must calculate the sums in Equation 101. Indicate that you are supplying a Hessian by
options = optimoptions('fmincon','Algorithm','interiorpoint',... 'Hessian','usersupplied','HessFcn',@hessianfcn);
For an example, see fmincon InteriorPoint Algorithm with Analytic Hessian.
The interiorpoint algorithm has several more options for Hessians, see Choose Input Hessian for interiorpoint fmincon:
options = optimoptions('fmincon','Hessian','bfgs');
fmincon calculates the Hessian by a dense quasiNewton approximation. This is the default.
options = optimoptions('fmincon','Hessian','lbfgs');
fmincon calculates the Hessian by a limitedmemory, largescale quasiNewton approximation. The default memory, 10 iterations, is used.
options = optimoptions('fmincon','Hessian',{'lbfgs',positive integer});
fmincon calculates the Hessian by a limitedmemory, largescale quasiNewton approximation. The positive integer specifies how many past iterations should be remembered.
options = optimoptions('fmincon','Hessian','findiffgrads',...
'SubproblemAlgorithm','cg','GradObj','on',...
'GradConstr','on');
fmincon calculates a Hessiantimesvector product by finite differences of the gradient(s). You must supply the gradient of the objective function, and also gradients of nonlinear constraints if they exist.
The interiorpoint and trustregionreflective algorithms allow you to supply a Hessian multiply function. This function gives the result of a Hessiantimesvector product, without computing the Hessian directly. This can save memory.
The syntax for the two algorithms differ:
For the interiorpoint algorithm, the syntax is
W = HessMultFcn(x,lambda,v);
The result W should be the product H*v, where H is the Hessian of the Lagrangian at x (see Equation 101), lambda is the Lagrange multiplier (computed by fmincon), and v is a vector of size nby1. Set options as follows:
options = optimoptions('fmincon','Algorithm','interiorpoint','Hessian','usersupplied',... 'SubproblemAlgorithm','cg','HessMult',@HessMultFcn);
Supply the function HessMultFcn, which returns an nby1 vector, where n is the number of dimensions of x. The HessMult option enables you to pass the result of multiplying the Hessian by a vector without calculating the Hessian.
The trustregionreflective algorithm does not involve lambda:
W = HessMultFcn(H,v);
The result W = H*v. fmincon passes H as the value returned in the third output of the objective function (see Writing Scalar Objective Functions). fmincon also passes v, a vector or matrix with n rows. The number of columns in v can vary, so write HessMultFcn to accept an arbitrary number of columns. H does not have to be the Hessian; rather, it can be anything that enables you to calculate W = H*v.
Set options as follows:
options = optimoptions('fmincon','Algorithm','trustregionreflective',... 'Hessian','usersupplied','HessMult',@HessMultFcn);
For an example using a Hessian multiply function with the trustregionreflective algorithm, see Minimization with Dense Structured Hessian, Linear Equalities.
Optimization options used by fmincon. Some options apply to all algorithms, and others are relevant for particular algorithms. Use optimoptions to set or change the values in options. See Optimization Options Reference for detailed information.
All four algorithms use these options:
Algorithm  Choose the optimization algorithm:
For information on choosing the algorithm, see Choosing the Algorithm. The trustregionreflective algorithm requires:
If you select the 'trustregionreflective' algorithm and these conditions are not all satisfied, fmincon throws an error. The 'activeset' and 'sqp' algorithms are not largescale. See LargeScale vs. MediumScale Algorithms. 
DerivativeCheck  Compare usersupplied derivatives (gradients of objective or constraints) to finitedifferencing derivatives. The choices are 'on' or the default, 'off'. 
Diagnostics  Display diagnostic information about the function to be minimized or solved. The choices are 'on' or the default, 'off'. 
DiffMaxChange  Maximum change in variables for finitedifference gradients (a positive scalar). The default is Inf. 
DiffMinChange  Minimum change in variables for finitedifference gradients (a positive scalar). The default is 0. 
Display  Level of display:

FinDiffRelStep  Scalar or vector step size factor. When you set FinDiffRelStep to a vector v, forward finite differences delta are delta = v.*sign(x).*max(abs(x),TypicalX); and central finite differences are delta = v.*max(abs(x),TypicalX); Scalar FinDiffRelStep expands to a vector. The default is sqrt(eps) for forward finite differences, and eps^(1/3) for central finite differences. 
FinDiffType  Finite differences, used to estimate gradients, are either 'forward' (default), or 'central' (centered). 'central' takes twice as many function evaluations but should be more accurate. fmincon is careful to obey bounds when estimating both types of finite differences. So, for example, it could take a backward, rather than a forward, difference to avoid evaluating at a point outside bounds. However, for the interiorpoint algorithm, 'central' differences might violate bounds during their evaluation if the AlwaysHonorConstraints option is set to 'none'. 
FunValCheck  Check whether objective function and constraints values are valid. 'on' displays an error when the objective function or constraints return a value that is complex, Inf, or NaN. The default, 'off', displays no error. 
GradConstr  Gradient for nonlinear constraint functions defined by the user. When set to 'on', fmincon expects the constraint function to have four outputs, as described in nonlcon in the Input Arguments section. When set to the default, 'off', gradients of the nonlinear constraints are estimated by finite differences. The trustregionreflective algorithm does not accept nonlinear constraints. 
GradObj  Gradient for the objective function defined by the user. See the preceding description of fun to see how to define the gradient in fun. Set to 'on' to have fmincon use a userdefined gradient of the objective function. The default, 'off', causes fmincon to estimate gradients using finite differences. You must provide the gradient, and set GradObj to 'on', to use the trustregionreflective method. 
MaxFunEvals  Maximum number of function evaluations allowed, a positive integer. The default value for all algorithms except interiorpoint is 100*numberOfVariables; for the interiorpoint algorithm the default is 3000. 
MaxIter  Maximum number of iterations allowed, a positive integer. The default value for all algorithms except interiorpoint is 400; for the interiorpoint algorithm the default is 1000. 
OutputFcn  Specify one or more userdefined functions that an optimization function calls at each iteration, either as a function handle or as a cell array of function handles. The default is none ([]). See Output Function. 
PlotFcns  Plots various measures of progress while the algorithm executes, select from predefined plots or write your own. Pass a function handle or a cell array of function handles. The default is none ([]).
For information on writing a custom plot function, see Plot Functions. 
TolCon  Tolerance on the constraint violation, a positive scalar. The default is 1e6. 
TolFun  Termination tolerance on the function value, a positive scalar. The default is 1e6. 
TolX  Termination tolerance on x, a positive scalar. The default value for all algorithms except 'interiorpoint' is 1e6; for the 'interiorpoint' algorithm the default is 1e10. 
TypicalX  Typical x values. The number of elements in TypicalX is equal to the number of elements in x0, the starting point. The default value is ones(numberofvariables,1). fmincon uses TypicalX for scaling finite differences for gradient estimation. The 'trustregionreflective' algorithm uses TypicalX only for the DerivativeCheck option. 
UseParallel  When true, estimate gradients in parallel. Disable by setting to the default, false. trustregionreflective requires a gradient in the objective, so UseParallel does not apply. See Parallel Computing. 
The 'trustregionreflective' algorithm uses these options:
Hessian  If 'on' or 'usersupplied', fmincon uses a userdefined Hessian (defined in fun), or Hessian information (when using HessMult), for the objective function. If 'off' (default), fmincon approximates the Hessian using finite differences.  
HessMult  Function handle for Hessian multiply function. For largescale structured problems, this function computes the Hessian matrix product H*Y without actually forming H. The function is of the form W = hmfun(Hinfo,Y) where Hinfo contains a matrix used to compute H*Y. The first argument must be the same as the third argument returned by the objective function fun, for example: [f,g,Hinfo] = fun(x) Y is a matrix that has the same number of rows as there are dimensions in the problem. W = H*Y, although H is not formed explicitly. fmincon uses Hinfo to compute the preconditioner. See Passing Extra Parameters for information on how to supply values for any additional parameters that hmfun needs. See Minimization with Dense Structured Hessian, Linear Equalities for an example.  
HessPattern  Sparsity pattern of the Hessian for finite differencing. Set HessPattern(i,j) = 1 when you can have ∂^{2}fun/∂x(i)∂x(j) ≠ 0. Otherwise, set HessPattern(i,j) = 0. Use HessPattern when it is inconvenient to compute the Hessian matrix H in fun, but you can determine (say, by inspection) when the ith component of the gradient of fun depends on x(j). fmincon can approximate H via sparse finite differences (of the gradient) if you provide the sparsity structure of H — i.e., locations of the nonzeros — as the value for HessPattern. In the worst case, when the structure is unknown, do not set HessPattern. The default behavior is as if HessPattern is a dense matrix of ones. Then fmincon computes a full finitedifference approximation in each iteration. This can be very expensive for large problems, so it is usually better to determine the sparsity structure.  
MaxPCGIter  Maximum number of PCG (preconditioned conjugate gradient) iterations, a positive scalar. The default is max(1,floor(numberOfVariables/2)). For more information, see Preconditioned Conjugate Gradient Method.  
PrecondBandWidth  Upper bandwidth of preconditioner for PCG, a nonnegative integer. By default, diagonal preconditioning is used (upper bandwidth of 0). For some problems, increasing the bandwidth reduces the number of PCG iterations. Setting PrecondBandWidth to Inf uses a direct factorization (Cholesky) rather than the conjugate gradients (CG). The direct factorization is computationally more expensive than CG, but produces a better quality step towards the solution.  
TolPCG  Termination tolerance on the PCG iteration, a positive scalar. The default is 0.1. 
The 'activeset' algorithm uses these options:
MaxSQPIter  Maximum number of SQP iterations allowed, a positive integer. The default is 10*max(numberOfVariables, numberOfInequalities + numberOfBounds). 
RelLineSrchBnd  Relative bound (a real nonnegative scalar value) on the line search step length such that the total displacement in x satisfies Δx(i) ≤ relLineSrchBnd· max(x(i),typicalx(i)). This option provides control over the magnitude of the displacements in x for cases in which the solver takes steps that are considered too large. The default is no bounds ([]). 
RelLineSrchBndDuration  Number of iterations for which the bound specified in RelLineSrchBnd should be active (default is 1). 
TolConSQP  Termination tolerance on inner iteration SQP constraint violation, a positive scalar. The default is 1e6. 
The 'interiorpoint' algorithm uses these options:
AlwaysHonorConstraints  The default 'bounds' ensures that bound constraints are satisfied at every iteration. Disable by setting to 'none'. 
HessFcn  Function handle to a usersupplied Hessian (see Hessian). This is used when the Hessian option is set to 'usersupplied'. 
Hessian  Chooses how fmincon calculates the Hessian (see Hessian). The choices are:

HessMult  Handle to a usersupplied function that gives a Hessiantimesvector product (see Hessian). This is used when the Hessian option is set to 'usersupplied'. 
InitBarrierParam  Initial barrier value, a positive scalar. Sometimes it might help to try a value above the default 0.1, especially if the objective or constraint functions are large. 
InitTrustRegionRadius  Initial radius of the trust region, a positive scalar. On badly scaled problems it might help to choose a value smaller than the default , where n is the number of variables. 
MaxProjCGIter  A tolerance (stopping criterion) for the number of projected conjugate gradient iterations; this is an inner iteration, not the number of iterations of the algorithm. This positive integer has a default value of 2*(numberOfVariables  numberOfEqualities). 
ObjectiveLimit  A tolerance (stopping criterion) that is a scalar. If the objective function value goes below ObjectiveLimit and the iterate is feasible, the iterations halt, since the problem is presumably unbounded. The default value is 1e20. 
ScaleProblem  'objandconstr' causes the algorithm to normalize all constraints and the objective function. Disable by setting to the default 'none'. 
SubproblemAlgorithm  Determines how the iteration step is calculated. The default, 'ldlfactorization', is usually faster than 'cg' (conjugate gradient), though 'cg' might be faster for large problems with dense Hessians. 
TolProjCG  A relative tolerance (stopping criterion) for projected conjugate gradient algorithm; this is for an inner iteration, not the algorithm iteration. This positive scalar has a default of 0.01. 
TolProjCGAbs  Absolute tolerance (stopping criterion) for projected conjugate gradient algorithm; this is for an inner iteration, not the algorithm iteration. This positive scalar has a default of 1e10. 
The 'sqp' algorithm uses these options:
ObjectiveLimit  A tolerance (stopping criterion) that is a scalar. If the objective function value goes below ObjectiveLimit and the iterate is feasible, the iterations halt, since the problem is presumably unbounded. The default value is 1e20. 
ScaleProblem  'objandconstr' causes the algorithm to normalize all constraints and the objective function. Disable by setting to the default 'none'. 
Find values of x that minimize f(x) = –x_{1}x_{2}x_{3}, starting at the point x = [10;10;10], subject to the constraints:
0 ≤ x_{1} + 2x_{2} + 2x_{3} ≤ 72.
Write a file that returns a scalar value f of the objective function evaluated at x:
function f = myfun(x) f = x(1) * x(2) * x(3);
Rewrite the constraints as both less than or equal to a constant,
–x_{1}–2x_{2}–2x_{3} ≤
0
x_{1} + 2x_{2} +
2x_{3}≤ 72
Since both constraints are linear, formulate them as the matrix inequality A·x ≤ b, where
A = [1 2 2; ... 1 2 2]; b = [0;72];
Supply a starting point and invoke an optimization routine:
x0 = [10;10;10]; % Starting guess at the solution [x,fval] = fmincon(@myfun,x0,A,b);
After fmincon stops, the solution is
x x = 24.0000 12.0000 12.0000
where the function value is
fval fval = 3.4560e+03
and linear inequality constraints evaluate to be less than or equal to 0:
A*xb ans = 72.0000 0.0000
To use the trustregionreflective algorithm, you must
Supply the gradient of the objective function in fun.
Set GradObj to 'on' in options.
Specify the feasible region using one, but not both, of the following types of constraints:
Upper and lower bounds constraints
Linear equality constraints, in which the equality constraint matrix Aeq cannot have more rows than columns
You cannot use inequality constraints with the trustregionreflective algorithm. If the preceding conditions are not met, fmincon reverts to the activeset algorithm.
fmincon returns a warning if you do not provide a gradient and the Algorithm option is 'trustregionreflective'. fmincon permits an approximate gradient to be supplied, but this option is not recommended; the numerical behavior of most optimization methods is considerably more robust when the true gradient is used.
The trustregionreflective method in fmincon is in general most effective when the matrix of second derivatives, i.e., the Hessian matrix H(x), is also computed. However, evaluation of the true Hessian matrix is not required. For example, if you can supply the Hessian sparsity structure (using the HessPattern option in options), fmincon computes a sparse finitedifference approximation to H(x).
If x0 is not strictly feasible, fmincon chooses a new strictly feasible (centered) starting point.
If components of x have no upper (or lower) bounds, fmincon prefers that the corresponding components of ub (or lb) be set to Inf (or Inf for lb) as opposed to an arbitrary but very large positive (or negative in the case of lower bounds) number.
Take note of these characteristics of linearly constrained minimization:
A dense (or fairly dense) column of matrix Aeq can result in considerable fill and computational cost.
fmincon removes (numerically) linearly dependent rows in Aeq; however, this process involves repeated matrix factorizations and therefore can be costly if there are many dependencies.
Each iteration involves a sparse leastsquares solution with matrix
where R^{T} is the Cholesky factor of the preconditioner. Therefore, there is a potential conflict between choosing an effective preconditioner and minimizing fill in .
If equality constraints are present and dependent equalities are detected and removed in the quadratic subproblem, 'dependent' appears under the Procedures heading (when you ask for output by setting the Display option to'iter'). The dependent equalities are only removed when the equalities are consistent. If the system of equalities is not consistent, the subproblem is infeasible and 'infeasible' appears under the Procedures heading.
fmincon is a gradientbased method that is designed to work on problems where the objective and constraint functions are both continuous and have continuous first derivatives.
When the problem is infeasible, fmincon attempts to minimize the maximum constraint value.
The 'trustregionreflective' algorithm does not allow equal upper and lower bounds. For example, if lb(2)==ub(2), fmincon gives this error:
Equal upper and lower bounds not permitted in this largescale method. Use equality constraints and the mediumscale method instead.
There are two different syntaxes for passing a Hessian, and there are two different syntaxes for passing a HessMult function; one for trustregionreflective, and another for interiorpoint.
For trustregionreflective, the Hessian of the Lagrangian is the same as the Hessian of the objective function. You pass that Hessian as the third output of the objective function.
For interiorpoint, the Hessian of the Lagrangian involves the Lagrange multipliers and the Hessians of the nonlinear constraint functions. You pass the Hessian as a separate function that takes into account both the position x and the Lagrange multiplier structure lambda.
TrustRegionReflective Coverage and Requirements
Additional Information Needed  For Large Problems 

Must provide gradient for f(x) in fun. 

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[2] Byrd, R.H., Mary E. Hribar, and Jorge Nocedal, "An Interior Point Algorithm for LargeScale Nonlinear Programming, SIAM Journal on Optimization," SIAM Journal on Optimization, Vol 9, No. 4, pp. 877–900, 1999.
[3] Coleman, T.F. and Y. Li, "An Interior, Trust Region Approach for Nonlinear Minimization Subject to Bounds," SIAM Journal on Optimization, Vol. 6, pp. 418–445, 1996.
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[5] Gill, P.E., W. Murray, and M.H. Wright, Practical Optimization, London, Academic Press, 1981.
[6] Han, S.P., "A Globally Convergent Method for Nonlinear Programming," Vol. 22, Journal of Optimization Theory and Applications, p. 297, 1977.
[7] Powell, M.J.D., "A Fast Algorithm for Nonlinearly Constrained Optimization Calculations," Numerical Analysis, ed. G.A. Watson, Lecture Notes in Mathematics, Springer Verlag, Vol. 630, 1978.
[8] Powell, M.J.D., "The Convergence of Variable Metric Methods For Nonlinearly Constrained Optimization Calculations," Nonlinear Programming 3 (O.L. Mangasarian, R.R. Meyer, and S.M. Robinson, eds.), Academic Press, 1978.
[9] Waltz, R. A., J. L. Morales, J. Nocedal, and D. Orban, "An interior algorithm for nonlinear optimization that combines line search and trust region steps," Mathematical Programming, Vol 107, No. 3, pp. 391–408, 2006.
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