# Documentation

## Iterative Display

### Introduction

Iterative display is a table of statistics describing the calculations in each iteration of a solver. The statistics depend on both the solver and the solver algorithm. For more information about iterations, see Iterations and Function Counts. The table appears in the MATLAB® Command Window when you run solvers with appropriate options.

Obtain iterative display by using `optimoptions` to create options with the `Display` option set to `'iter'` or `'iter-detailed'`. For example:

```options = optimoptions(@fminunc,'Display','iter','Algorithm','quasi-newton'); [x fval exitflag output] = fminunc(@sin,0,options); First-order Iteration Func-count f(x) Step-size optimality 0 2 0 1 1 4 -0.841471 1 0.54 2 8 -1 0.484797 0.000993 3 10 -1 1 5.62e-005 4 12 -1 1 0 Local minimum found. Optimization completed because the size of the gradient is less than the default value of the function tolerance.```

You can also obtain iterative display by using the Optimization app. Select Display to command window > Level of display > `iterative` or ```iterative with detailed message```.

Iterative display is available for all solvers except:

• `linprog` `active-set` algorithm

• `lsqlin`

• `lsqnonneg`

• `quadprog` `trust-region-reflective` and `active-set` algorithms

The following table lists some common headings of iterative display.

`f(x)`

Current objective function value

`First-order optimality`

First-order optimality measure (see First-Order Optimality Measure)

`Func-count` or `F-count`

Number of function evaluations; see Iterations and Function Counts

`Iteration` or `Iter`

Iteration number; see Iterations and Function Counts

`Norm of step`

Size of the current step (size is the Euclidean norm, or 2-norm)

The following sections describe headings of iterative display whose meaning is specific to the optimization function you are using:

#### fgoalattain, fmincon, fminimax, and fseminf

The following table describes the headings specific to `fgoalattain`, `fmincon`, `fminimax`, and `fseminf`.

fgoalattain, fmincon, fminimax, or fseminf HeadingInformation Displayed

`Attainment factor`

Value of the attainment factor for `fgoalattain`.

`CG-iterations`

Number of conjugate gradient iterations taken in the current iteration (see Preconditioned Conjugate Gradient Method).

`Directional derivative`

Gradient of the objective function along the search direction.

`Feasibility`

Maximum constraint violation, where satisfied inequality constraints count as `0`.

`Line search steplength`

Multiplicative factor that scales the search direction (see Equation 6-45).

`Max constraint`

Maximum violation among all constraints, both internally constructed and user-provided; can be negative when no constraint is binding.

`Objective value`

Objective function value of the nonlinear programming reformulation of the minimax problem for `fminimax`.

`Procedure`

Hessian update procedures:

• `Infeasible start point`

• `Hessian not updated`

• `Hessian modified`

• `Hessian modified twice`

QP subproblem procedures:

• `dependent` — There are dependent (redundant) equality constraints that the solver detected and removed.

• `Infeasible` — The QP subproblem with linearized constraints is infeasible.

• `Overly constrained` — The QP subproblem with linearized constraints is infeasible.

• `Unbounded` — The QP subproblem is feasible with large negative curvature.

• `Ill-posed` — The QP subproblem search direction is too small.

• `Unreliable` — The QP subproblem seems to be ill-conditioned.

`Steplength`

Multiplicative factor that scales the search direction (see Equation 6-45).

`Trust-region radius`

#### fminbnd and fzero

The following table describes the headings specific to `fminbnd` and `fzero`.

`Procedure`

Procedures for `fminbnd`:

• `initial`

• `golden` (golden section search)

• `parabolic` (parabolic interpolation)

Procedures for `fzero`:

• `initial` (initial point)

• `search` (search for an interval containing a zero)

• `bisection`

• `interpolation` (linear interpolation or inverse quadratic interpolation)

`x`

Current point for the algorithm

#### fminsearch

The following table describes the headings specific to `fminsearch`.

`min f(x)`

Minimum function value in the current simplex.

`Procedure`

Simplex procedure at the current iteration. Procedures include:

• `initial simplex`

• `expand`

• `reflect`

• `shrink`

• `contract inside`

• `contract outside`

For details, see fminsearch Algorithm.

#### fminunc

The following table describes the headings specific to `fminunc`.

`CG-iterations`

Number of conjugate gradient iterations taken in the current iteration (see Preconditioned Conjugate Gradient Method)

`Step-size`

Multiplicative factor that scales the search direction (see Equation 6-11)

The `fminunc` `'quasi-newton'` algorithm can issue a `skipped update` message to the right of the `First-order optimality` column. This message means that `fminunc` did not update its Hessian estimate, because the resulting matrix would not have been positive definite. The message usually indicates that the objective function is not smooth at the current point.

#### fsolve

The following table describes the headings specific to `fsolve`.

`Directional derivative`

Gradient of the function along the search direction

`Lambda`

λk value defined in Levenberg-Marquardt Method

`Residual`

Residual (sum of squares) of the function

`Trust-region radius`

#### intlinprog

The following table describes the headings specific to `intlinprog`.

`nodes explored`

Cumulative number of explored nodes.

`total time (s)`

Time in seconds since `intlinprog` started.

`num int solution`

Number of integer feasible points found.

`integer fval`

Objective function value of the best integer feasible point found. This is an upper bound for the final objective function value.

`relative gap (%)`

$\frac{100\left(b-a\right)}{|b|+1},$

where

• b is the objective function value of the best integer feasible point.

• a is the best lower bound on the objective function value.

#### linprog

The following table describes the headings specific to `linprog`.

`Dual Infeas A'*y+z-w-f`

Dual infeasibility.

`Duality Gap x'*z+s'*w`

Duality gap (see Interior-Point Linear Programming) between the primal objective and the dual objective. `s` and `w` appear only in this equation if there are finite upper bounds.

`Objective f'*x`

Current objective value.

`Primal Infeas A*x-b`

Primal infeasibility.

`Total Rel Error`

Total relative error, described at the end of Main Algorithm.

#### lsqnonlin and lsqcurvefit

The following table describes the headings specific to `lsqnonlin` and `lsqcurvefit`.

`Directional derivative`

Gradient of the function along the search direction

`Lambda`

λk value defined in Levenberg-Marquardt Method

`Resnorm`

Value of the squared 2-norm of the residual at `x`

`Residual`

Residual vector of the function

The following table describes the headings specific to `quadprog`.

`Feasibility`

Maximum constraint violation, where satisfied inequality constraints count as `0`.

`Total relative error`

Total relative error is a measure of infeasibility, as defined in Total Relative Error