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This example shows how to use an output function to plot and store the history of the iterations for a nonlinear problem. This history includes the evaluated points, the search directions that the solver uses to generate points, and the objective function values at the evaluated points.

For the solver-based approach to this example, see Output Functions for Optimization Toolbox™.

Plot functions have the same syntax as output functions, so this example also applies to plot functions.

For both the solver-based approach and the problem-based approach, write the output function according to the solver-based approach. In the solver-based approach, you use a single vector variable, usually denoted `x`

, instead of a collection of optimization variables of various sizes. So to write an output function for the problem-based approach, you must understand the correspondence between your optimization variables and the single solver-based `x`

. To map between optimization variables and `x`

, use `varindex`

. In this example, to avoid confusion with an optimization variable named `x`

, use "`in"`

as the vector variable name.

The problem is to minimize the following function of variables `x`

and `y`

:

$$f=\mathrm{exp}(x)(4{x}^{2}+2{y}^{2}+4xy+2y+1).$$

In addition, the problem has two nonlinear constraints:

$$\begin{array}{l}x+y-xy\ge 1.5\\ xy\ge -10.\end{array}$$

To set up the problem in the problem-based approach, define optimization variables and an optimization problem object.

x = optimvar('x'); y = optimvar('y'); prob = optimproblem;

Define the objective function as an expression in the optimization variables.

f = exp(x)*(4*x^2 + 2*y^2 + 4*x*y + 2*y + 1);

Include the objective function in `prob`

.

prob.Objective = f;

To include the nonlinear constraints, create optimization constraint expressions.

cons1 = x + y - x*y >= 1.5; cons2 = x*y >= -10; prob.Constraints.cons1 = cons1; prob.Constraints.cons2 = cons2;

Because this is a nonlinear problem, you must include an initial point structure `x0`

. Use `x0.x = –1`

and `x0.y = 1`

.

x0.x = -1; x0.y = 1;

The `outfun`

output function records a history of the points generated by `fmincon`

during its iterations. The output function also plots the points and keeps a separate history of the search directions for the `sqp`

algorithm. The search direction is a vector from the previous point to the next point that `fmincon`

tries. During its final step, the output function saves the history in workspace variables, and saves a history of the objective function values at each iterative step.

For the required syntax of optimization output functions, see Output Function and Plot Function Syntax.

An output function takes a single vector variable as an input. But the current problem has two variables. To find the mapping between the optimization variables and the input variable, use `varindex`

.

idx = varindex(prob); idx.x

ans = 1

idx.y

ans = 2

The mapping shows that `x`

is variable 1 and `y`

is variable 2. So, if the input variable is named `in`

, then `x = in(1)`

and `y = in(2)`

.

`type outfun`

function stop = outfun(in,optimValues,state,idx) persistent history searchdir fhistory stop = false; switch state case 'init' hold on history = []; fhistory = []; searchdir = []; case 'iter' % Concatenate current point and objective function % value with history. in must be a row vector. fhistory = [fhistory; optimValues.fval]; history = [history; in(:)']; % Ensure in is a row vector % Concatenate current search direction with % searchdir. searchdir = [searchdir;... optimValues.searchdirection(:)']; plot(in(idx.x),in(idx.y),'o'); % Label points with iteration number and add title. % Add .15 to idx.x to separate label from plotted 'o' text(in(idx.x)+.15,in(idx.y),... num2str(optimValues.iteration)); title('Sequence of Points Computed by fmincon'); case 'done' hold off assignin('base','optimhistory',history); assignin('base','searchdirhistory',searchdir); assignin('base','functionhistory',fhistory); otherwise end end

Include the output function in the optimization by setting the `OutputFcn`

option. Also, set the `Algorithm`

option to use the `'sqp'`

algorithm instead of the default `'interior-point'`

algorithm. Pass `idx`

to the output function as an extra parameter in the last input. See Passing Extra Parameters.

outputfn = @(in,optimValues,state)outfun(in,optimValues,state,idx); opts = optimoptions('fmincon','Algorithm','sqp','OutputFcn',outputfn);

Run the optimization, including the output function, by using the '`Options'`

name-value pair argument.

`[sol,fval,eflag,output] = solve(prob,x0,'Options',opts)`

Solving problem using fmincon.

Local minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the value of the optimality tolerance, and constraints are satisfied to within the value of the constraint tolerance.

`sol = `*struct with fields:*
x: -9.5474
y: 1.0474

fval = 0.0236

eflag = OptimalSolution

`output = `*struct with fields:*
iterations: 10
funcCount: 22
algorithm: 'sqp'
message: '...'
constrviolation: 1.2434e-14
stepsize: 1.4785e-07
lssteplength: 1
firstorderopt: 7.1930e-10
bestfeasible: [1x1 struct]
objectivederivative: "reverse-AD"
constraintderivative: "closed-form"
solver: 'fmincon'

Examine the iteration history. Each row of the `optimhistory`

matrix represents one point. The last few points are very close, which explains why the plotted sequence shows overprinted numbers for points 8, 9, and 10.

`disp('Locations');disp(optimhistory)`

Locations -1.0000 1.0000 -1.3679 1.2500 -1.6509 1.1813 -3.5870 2.0537 -4.4574 2.2895 -5.8015 1.5531 -7.6498 1.1225 -8.5223 1.0572 -9.5463 1.0464 -9.5474 1.0474 -9.5474 1.0474

Examine the `searchdirhistory`

and `functionhistory`

arrays.

`disp('Search Directions');disp(searchdirhistory)`

Search Directions 0 0 -0.3679 0.2500 -0.2831 -0.0687 -1.9360 0.8725 -0.8704 0.2358 -1.3441 -0.7364 -2.0877 -0.6493 -0.8725 -0.0653 -1.0241 -0.0108 -0.0011 0.0010 0.0000 -0.0000

`disp('Function Values');disp(functionhistory)`

Function Values 1.8394 1.8513 1.7757 0.9839 0.6343 0.3250 0.0978 0.0517 0.0236 0.0236 0.0236

`fcn2optimexpr`

If your objective function or nonlinear constraint functions are not composed of elementary functions, you must convert the functions to optimization expressions using `fcn2optimexpr`

. See Convert Nonlinear Function to Optimization Expression. For this example, you would enter the following code:

fun = @(x,y)exp(x)*(4*x^2 + 2*y^2 + 4*x*y + 2*y + 1); f = fcn2optimexpr(fun,x,y);

For the list of supported functions, see Supported Operations for Optimization Variables and Expressions.