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# fmincon

Find minimum of constrained nonlinear multivariable function

## Equation

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.

## Syntax

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(...)

## Description

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.

## Input Arguments

Function Arguments describes the arguments passed to fmincon. Options provides the function-specific details for the options values. This section provides function-specific details for fun, nonlcon, and problem.

fun

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','user-supplied') and the Algorithm option is trust-region-reflective, 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 interior-point, 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 trust-region-reflective and interior-point algorithms use sparse linear algebra. If A or Aeq is large, with relatively few nonzero entries, save running time and memory in the trust-region-reflective or interior-point algorithms by using sparse matrices.

nonlcon

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 interior-point algorithm by representing them as sparse matrices. For more information, see Nonlinear Constraints.

 Note   Because Optimization Toolbox™ functions only accept inputs of type double, user-supplied objective and nonlinear constraint functions must return outputs of type double.

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
lbVector of lower bounds
ubVector of upper bounds

nonlcon

Nonlinear constraint function

solver

'fmincon'

options

Options created with optimoptions

## Output Arguments

Function Arguments describes arguments returned by fmincon. This section provides function-specific 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 First-order 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. trust-region-reflective, interior-point, and sqp algorithms: 2 Change in x was less than options.TolX and maximum constraint violation was less than options.TolCon. trust-region-reflective algorithm only: 3 Change in the objective function value was less than options.TolFun and maximum constraint violation was less than options.TolCon. active-set 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. interior-point 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 (active-set algorithm only) constrviolation Maximum of constraint functions stepsize Length of last displacement in x (active-set and interior-point algorithms) algorithm Optimization algorithm used cgiterations Total number of PCG iterations (trust-region-reflective and interior-point algorithms) firstorderopt Measure of first-order optimality message Exit message

### Hessian

fmincon uses a Hessian as an optional input. This Hessian is the second derivatives of the Lagrangian (see Equation 3-1), namely,

 (10-1)

The various fmincon algorithms handle input Hessians differently:

• The active-set and sqp algorithms do not accept a user-supplied Hessian. They compute a quasi-Newton approximation to the Hessian of the Lagrangian.

• The trust-region-reflective algorithm can accept a user-supplied 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','trust-region-reflective','Hessian','user-supplied');`

If you do not pass a Hessian, the algorithm computes a finite-difference approximation.

• The interior-point algorithm can accept a user-supplied Hessian as a separately defined function—it is not computed in the objective function. The syntax is

`hessian = hessianfcn(x, lambda)`

hessian is an n-by-n 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 10-1. Indicate that you are supplying a Hessian by

```options = optimoptions('fmincon','Algorithm','interior-point',...
'Hessian','user-supplied','HessFcn',@hessianfcn);```

For an example, see fmincon Interior-Point Algorithm with Analytic Hessian.

The interior-point algorithm has several more options for Hessians, see Choose Input Hessian for interior-point fmincon:

• options = optimoptions('fmincon','Hessian','bfgs');

fmincon calculates the Hessian by a dense quasi-Newton approximation. This is the default.

• options = optimoptions('fmincon','Hessian','lbfgs');

fmincon calculates the Hessian by a limited-memory, large-scale quasi-Newton approximation. The default memory, 10 iterations, is used.

• options = optimoptions('fmincon','Hessian',{'lbfgs',positive integer});

fmincon calculates the Hessian by a limited-memory, large-scale quasi-Newton approximation. The positive integer specifies how many past iterations should be remembered.

fmincon calculates a Hessian-times-vector 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.

#### Hessian Multiply Function

The interior-point and trust-region-reflective algorithms allow you to supply a Hessian multiply function. This function gives the result of a Hessian-times-vector product, without computing the Hessian directly. This can save memory.

The syntax for the two algorithms differ:

• For the interior-point 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 10-1), lambda is the Lagrange multiplier (computed by fmincon), and v is a vector of size n-by-1. Set options as follows:

```options = optimoptions('fmincon','Algorithm','interior-point','Hessian','user-supplied',...
'SubproblemAlgorithm','cg','HessMult',@HessMultFcn);```

Supply the function HessMultFcn, which returns an n-by-1 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 trust-region-reflective 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','trust-region-reflective',...
'Hessian','user-supplied','HessMult',@HessMultFcn);```

For an example using a Hessian multiply function with the trust-region-reflective algorithm, see Minimization with Dense Structured Hessian, Linear Equalities.

## Options

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.

fmincon uses one of four algorithms: active-set, interior-point, sqp, or trust-region-reflective. Choose the algorithm at the command line with optimoptions. For example:

`options = optimoptions('fmincon','Algorithm','active-set');`

The default trust-region-reflective requires:

• A gradient to be supplied in the objective function

• GradObj to be set to 'on'

• Either bound constraints or linear equality constraints, but not both

If these conditions are not all satisfied, the active-set algorithm is the default. In a future release, the default algorithm will be interior-point.

The active-set algorithm is not a large-scale algorithm. See Large-Scale vs. Medium-Scale Algorithms.

### All Algorithms

All four algorithms use these options:

### Trust-Region-Reflective Algorithm

The trust-region-reflective algorithm uses these options:

Hessian

If 'on' or 'user-supplied', fmincon uses a user-defined 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 large-scale 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.

 Note   Hessian must be set to 'on' or 'user-supplied' for fmincon to pass Hinfo from fun to hmfun.

See Minimization with Dense Structured Hessian, Linear Equalities for an example.

HessPattern

Sparsity pattern of the Hessian for finite differencing. If it is not convenient to compute the sparse Hessian matrix H in fun, the trust-region-reflective method in fmincon can approximate H via sparse finite differences (of the gradient) provided that you supply the sparsity structure of H—i.e., locations of the nonzeros—as the value for HessPattern. In the worst case, if the structure is unknown, you can set HessPattern to be a dense matrix and a full finite-difference approximation is computed at each iteration (this is the default). This can be very expensive for large problems, so it is usually worth the effort 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.

### Active-Set Algorithm

The active-set 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 1e-6.

### Interior-Point Algorithm

The interior-point 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 user-supplied Hessian (see Hessian). This is used when the Hessian option is set to 'user-supplied'. Hessian Chooses how fmincon calculates the Hessian (see Hessian). The choices are:'bfgs' (default)'fin-diff-grads''lbfgs'{'lbfgs',Positive Integer}'user-supplied' HessMult Handle to a user-supplied function that gives a Hessian-times-vector product (see Hessian). This is used when the Hessian option is set to 'user-supplied'. 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 'obj-and-constr' 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, 'ldl-factorization', 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 1e-10.

### SQP Algorithm

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 'obj-and-constr' causes the algorithm to normalize all constraints and the objective function. Disable by setting to the default 'none'.

## Examples

Find values of x that minimize f(x) = –x1x2x3, starting at the point x = [10;10;10], subject to the constraints:

0 ≤ x1 + 2x2 + 2x3 ≤ 72.

1. 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);```
2. Rewrite the constraints as both less than or equal to a constant,

x1–2x2–2x3 ≤ 0
x1 + 2x2 + 2x3≤ 72

3. 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];```
4. 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);```
5. After 11 iterations, 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*x-b
ans =
-72
0```

## Notes

### Trust-Region-Reflective Optimization

To use the trust-region-reflective 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 trust-region-reflective algorithm. If the preceding conditions are not met, fmincon reverts to the active-set algorithm.

fmincon returns a warning if you do not provide a gradient and the Algorithm option is trust-region-reflective. 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 trust-region-reflective 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 finite-difference 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 least-squares solution with matrix

where RT is the Cholesky factor of the preconditioner. Therefore, there is a potential conflict between choosing an effective preconditioner and minimizing fill in .

### Active-Set Optimization

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.

## Limitations

fmincon is a gradient-based 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 trust-region-reflective 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
large-scale method.
Use equality constraints and the medium-scale
```

There are two different syntaxes for passing a Hessian, and there are two different syntaxes for passing a HessMult function; one for trust-region-reflective, and another for interior-point.

For trust-region-reflective, 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 interior-point, 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.

Trust-Region-Reflective Coverage and Requirements

Must provide gradient for f(x) in fun.

• Provide sparsity structure of the Hessian or compute the Hessian in fun.

• The Hessian should be sparse.

• Aeq should be sparse.

expand all

### Trust-Region-Reflective Optimization

The trust-region-reflective algorithm is a subspace trust-region method and is based on the interior-reflective Newton method described in [3] and [4]. Each iteration involves the approximate solution of a large linear system using the method of preconditioned conjugate gradients (PCG). See the trust-region and preconditioned conjugate gradient method descriptions in fmincon Trust Region Reflective Algorithm.

### Active-Set Optimization

fmincon uses a sequential quadratic programming (SQP) method. In this method, the function solves a quadratic programming (QP) subproblem at each iteration. fmincon updates an estimate of the Hessian of the Lagrangian at each iteration using the BFGS formula (see fminunc and references [7] and [8]).

fmincon performs a line search using a merit function similar to that proposed by [6], [7], and [8]. The QP subproblem is solved using an active set strategy similar to that described in [5]. fmincon Active Set Algorithm describes this algorithm in detail.

### Interior-Point Optimization

This algorithm is described in fmincon Interior Point Algorithm. There is more extensive description in [1], [41], and [9].

### SQP Optimization

The fmincon 'sqp' algorithm is similar to the 'active-set' algorithm described in Active-Set Optimization. fmincon SQP Algorithm describes the main differences. In summary, these differences are:

## References

[1] Byrd, R.H., J. C. Gilbert, and J. Nocedal, "A Trust Region Method Based on Interior Point Techniques for Nonlinear Programming," Mathematical Programming, Vol 89, No. 1, pp. 149–185, 2000.

[2] Byrd, R.H., Mary E. Hribar, and Jorge Nocedal, "An Interior Point Algorithm for Large-Scale 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.

[4] Coleman, T.F. and Y. Li, "On the Convergence of Reflective Newton Methods for Large-Scale Nonlinear Minimization Subject to Bounds," Mathematical Programming, Vol. 67, Number 2, pp. 189–224, 1994.

[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.