Iterative Display

• Introduction

• Common Headings

• Function-Specific Headings

Introduction

The 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. The table appears in the MATLAB^® Command Window when you run solvers with appropriate options. For more information about iterations, see Iterations and Function Counts.

Obtain the iterative display by using optimoptions 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-05 
    4          12             -1             1             0 

Local minimum found.

Optimization completed because the size of the gradient is less than
the value of the optimality tolerance.

The iterative display is available for all solvers except:

• lsqlin 'trust-region-reflective' algorithm

• lsqnonneg

• quadprog 'trust-region-reflective' algorithm

Common Headings

This table lists some common headings of iterative display.

Heading	Information Displayed
f(x) or Fval	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). For the 'trust-region' and 'trust-region-reflective' algorithms, when constraints exist, Norm of step is the norm of D*s. Here, s is the step and D is a diagonal scaling matrix described in the trust-region subproblem section of the algorithm description.

Function-Specific Headings

The tables in this section describe headings of the iterative display whose meaning is specific to the optimization function you are using.

• fgoalattain, fmincon, fminimax, and fseminf

• fminbnd and fzero

• fminsearch

• fminunc

• fsolve

• intlinprog

• linprog

• lsqlin

• lsqnonlin and lsqcurvefit

• quadprog

fgoalattain, fmincon, fminimax, and fseminf

This table describes the headings specific to fgoalattain, fmincon, fminimax, and fseminf.

fgoalattain, fmincon, fminimax, or fseminf Heading	Information 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 29)
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 — The solver detected and removed dependent (redundant) equality constraints. 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 poorly conditioned.
Steplength	Multiplicative factor that scales the search direction (see Equation 29)
Trust-region radius	Current trust-region radius

fminbnd and fzero

This table describes the headings specific to fminbnd and fzero.

fminbnd or fzero Heading	Information 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

This table describes the headings specific to fminsearch.

fminsearch Heading	Information Displayed
Procedure	Simplex procedure at the current iteration. Procedures include: initial simplex expand reflect shrink contract inside contract outside For details, see fminsearch Algorithm.

fminunc

This table describes the headings specific to fminunc.

fminunc Heading	Information 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 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

This table describes the headings specific to fsolve.

fsolve Heading	Information Displayed
||f(x)||^2	Current objective function value, the squared Euclidean norm of the objective function vector
Lambda	λk value defined in Levenberg-Marquardt Method
Trust-region radius	Current trust-region radius (change in the norm of the trust-region radius)

intlinprog

This table describes the headings specific to intlinprog.

intlinprog Heading	Information 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 value is an upper bound for the final objective function value
relative gap (%)	$$\frac{100(b-a)}{\left|b\right|+1},$$ 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 Although you specify RelativeGapTolerance as a decimal number, the iterative display reports the gap as a 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

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

linprog Heading	Information 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-Legacy 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-Legacy Algorithm.
Upper Bounds {x}+s-ub	Upper bound feasibility. {x} means those x with finite upper bounds. This value 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 the problem has 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 Linear Least Squares: Interior-Point or Active-Set. For iterative display details, see quadprog. The sole difference in the iterative display is lsqlin displays a column titled Resnorm, instead of the quadprog title f(x).

lsqnonlin and lsqcurvefit

This table describes the headings specific to lsqnonlin and lsqcurvefit.

lsqnonlin or lsqcurvefit Heading	Information Displayed
Lambda	λk value defined in Levenberg-Marquardt Method
Resnorm	Value of the squared 2-norm of the residual at x
Feasibility	Maximum constraint violation, where satisfied inequality constraints count as 0 ('interior-point' algorithm)

quadprog

This table describes the headings specific to quadprog.

quadprog Heading	Information 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.
Feasibility	Maximum constraint violation, where satisfied inequality constraints count as 0 ('active-set' algorithm)