# Documentation

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## 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:

• lsqlin 'trust-region-reflective' and 'active-set' algorithms

• lsqnonneg

• quadprog 'trust-region-reflective' algorithm

### Common Headings

The following table lists some common headings of iterative display.

HeadingInformation Displayed

f(x) or Fval

Current objective function value. For fsolve, the square of the norm of the function value vector.

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). For the 'trust-region' or 'trust-region-reflective' algorithms, when there are constraints Norm of step is the norm of D*s. Here s is the step, and D is a diagonal scaling matrix described in the algorithm description, trust-region subproblem section.

### Function-Specific Headings

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

For more information, see Updating the Hessian Matrix.

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

Current trust-region radius.

#### fminbnd and fzero

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

fminbnd or fzero HeadingInformation Displayed

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.

fminsearch HeadingInformation Displayed

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.

fminunc HeadingInformation Displayed

CG-iterations

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

Line search steplength

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.

fsolve HeadingInformation Displayed

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

Current trust-region radius (change in the norm of the trust-region radius)

#### intlinprog

The following table describes the headings specific to intlinprog.

intlinprog HeadingInformation Displayed

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.

 Note:   While you specify RelativeGapTolerance as a decimal number, the iterative display and output.relativegap report the gap in percentage, meaning 100 times the measured relative gap. If the exit message refers to the relative gap, this value is the measured relative gap, not a percentage.

#### linprog

The following table describes the headings specific to linprog. Each algorithm has its own iterative display.

linprog HeadingInformation Displayed

Primal Infeas A*x-b or Primal Infeas

Primal infeasibility, a measure of the constraint violations, which should be zero at a solution.

For definitions, see Predictor-Corrector ('interior-point') or Main Algorithm ('interior-point-legacy') or Dual-Simplex Algorithm.

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

Dual infeasibility, a measure of the derivative of the Lagrangian, which should be zero at a solution.

For the definition of the Lagrangian, see Predictor-Corrector. For the definition of dual infeasibility, see Predictor-Corrector ('interior-point') or Main Algorithm ('interior-point-legacy') or Dual-Simplex Algorithm.

Upper Bounds {x}+s-ub

Upper bound feasibility. {x} means those x with finite upper bounds. This is the ru residual in Interior-Point-Legacy Linear Programming.

Duality Gap x'*z+s'*w

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

Total Rel Error

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

Complementarity

A measure of the Lagrange multipliers times distance from the bounds, which should be zero at a solution. See the rc variable in Stopping Conditions.

Time

Time in seconds that linprog has been running.

#### lsqlin

The lsqlin 'interior-point' iterative display is inherited from the quadprog iterative display. The relationship between these functions is explained in Interior-Point Linear Least Squares. For iterative display details, see quadprog.

#### lsqnonlin and lsqcurvefit

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

lsqnonlin or lsqcurvefit HeadingInformation Displayed

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

#### quadprog

The following table describes the headings specific to quadprog. Only the 'interior-point-convex' algorithm has iterative display.

quadprog HeadingInformation Displayed

Primal Infeas

Primal infeasibility, defined as max( norm(Aeq*x - beq, inf), abs(min(0, min(A*x-b))) )

Dual Infeas

Dual infeasibility, defined as norm(H*x + f - A*lambda_ineqlin - Aeq*lambda_eqlin, inf)

Complementarity

A measure of the maximum absolute value of the Lagrange multipliers of inactive inequalities, which should be zero at a solution. This quantity is g in Infeasibility Detection.

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