Nonlinear Inequality Constrained Example. If inequality constraints are added to Eq. , the resulting problem can be solved by the fmincon function. Optimization Toolbox. Genetic Algorithm and Direct Search Toolbox. Function handles. GUI. Homework. Optimization in Matlab. Kevin Carlberg. MATLAB (MAtrix LABboratory) is a numerical computing environment and fourth- [x,fval,exitflag,output] = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options);.

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Passing Extra Parameters explains how to pass extra parameters to the objective function and nonlinear constraint functions, if necessary. If the specified input bounds for a problem are inconsistent, fmincon throws an error.

In this case, output x is x0 and fval is []. For the ‘trust-region-reflective’ algorithm, fmincon sets violating components to the interior of the bound region. For other algorithms, fmincon sets violating components to the closest bound.

Components that respect the bounds are not changed. See Iterations Can Violate Constraints. Use optimoptions to set these options. Create the problem structure by exporting a problem from Optimization app, as described in Exporting Your Work. Find the minimum value of Rosenbrock’s function when there is a linear inequality constraint.

Set the objective function fun to be Rosenbrock’s function. Rosenbrock’s function is well-known to be difficult to minimize. It has its minimum objective value of 0 at the point 1,1. For more information, see Solve a Constrained Nonlinear Problem. Find the minimum value starting from the point [-1,2]constrained to have. Notice that this constraint means that the solution will not be at the unconstrained solution 1,1because at that point.

Find the minimum value of Rosenbrock’s function when there are both a linear inequality constraint and a linear equality constraint. Find the minimum value starting from the point [0. Find the minimum of an objective function in the presence of bound constraints. Try an initial point in the middle of the region.

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Find the minimum of funsubject to the bound constraints. To see which solution is better, see Obtain the Objective Function Value. Find the point where Rosenbrock’s function is minimized within a circle, also subject to bound ttuorial. Look within the region. To observe the fmincon solution process, set the Display option to ‘iter’.

Also, try the ‘sqp’ algorithm, which is sometimes faster or more accurate than the default ‘interior-point’ algorithm. Find the minimum of Rosenbrock’s function on the unit disk.

First create a function that represents the nonlinear constraint. Save this as a file named unitdisk.

Include gradient evaluation in the objective function for faster or more reliable computations. Include the gradient evaluation as a conditionalized output in the objective function file. For details, see Including Gradients and Hessians.

The objective function is Rosenbrock’s function. Save this code as a file named rosenbrockwithgrad. Solve the same problem as in Nondefault Options using a problem structure instead of separate arguments. Create the options and a problem structure.

See problem for the field names and required fields. The iterative display and solution are the same as in Nondefault Options. Call fmincon with the fval output to obtain the value of the objective function at the solution. The Bound Constraints example shows two solutions. Run the example requesting the fval output as well as the solution. The first solution x has a lower local minimum objective function value. To easily examine the quality of a solution, request the exitflag and output outputs.

Set up the fmiincon of minimizing Rosenbrock’s function on the unit disk. Call fmincon using the fvalexitflagand output outputs. The exitflag value 1 indicates that the solution is a local minimum. The output structure reports several statistics about the solution process. In particular, it gives the number of iterations in output.

Set up the problem of minimizing Rosenbrock’s function on the unit disk. The grad output gives the value of the gradient of the objective function at the solution x.

The hessian output is described in fmincon Hessian. Function tutoriao minimize, specified as a function handle or function name. You can also specify fun as a function handle for an anonymous function:. If you can also compute the Hessian matrix and the HessianFcn option is set to ‘objective’ via optimoptions and the Algorithm option is ‘trust-region-reflective’fun must return the Hessian value H x fincon, a symmetric matrix, in a third output argument.

See Hessian for fminunc trust-region or fmincon trust-region-reflective algorithms for details. If you can also compute the Hessian matrix and the Algorithm option is set to ‘interior-point’there is a different way to pass the Hessian to fmincon. For more information, see Hessian for fmincon interior-point algorithm.

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

This can save memory. See Hessian Multiply Function. Initial point, specified as a real vector or real array. Solvers use the number of elements in, and size of, x0 to determine the number and size of variables that fun accepts.

Linear inequality constraints, specified as a real matrix. A is an M -by- N matrix, where M is the number of inequalities, and N is the number of variables number of elements in x0. For large problems, pass A as a sparse matrix.

Linear inequality constraints, specified as a real vector. If you pass b as a row vector, solvers internally convert b to the column vector b: For large problems, pass b as a sparse vector. Linear equality constraints, specified as a real matrix. Aeq is an Me -by- N matrix, where Me is the number of equalities, and N is the number of variables number of elements in x0.

### Find minimum of constrained nonlinear multivariable function – MATLAB fmincon

For large problems, pass Aeq as a sparse matrix. Linear equality constraints, specified as a real vector. If you pass beq as a row vector, solvers internally convert beq to the column vector beq: For large problems, pass beq as a sparse vector.

Lower bounds, specified as a real vector or real array. If the number of elements in x0 is equal to the number of elements in lbthen lb specifies that. If there are fewer elements in lb than in x0solvers issue dmincon warning. Upper bounds, specified as a real vector or real array. If the number of elements in x0 is equal to the number of elements in ubthen ub specifies that.

If there are fewer elements in ub than in x0solvers issue a warning. Nonlinear constraints, specified as a tutoiral handle or function name. 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.

Optimization options, specified as the output of optimoptions or a structure such as optimset returns. Some options apply to all algorithms, and others are relevant for particular algorithms.

See Optimization Options Reference for detailed information. Some options are absent from the optimoptions display.

### Tutorial for the Optimization Toolboxâ„˘ – MATLAB & Simulink Example

These options appear in italics in the following table. For details, see View Options. For information on choosing the algorithm, see Choosing the Algorithm.

SpecifyObjectiveGradient to be set to true.