Iterative Display

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, seeIterations and Function Counts.

Obtain the iterative display by usingoptimoptionswith theDisplayoption 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; for`fsolve`, the square of the norm of the function value vector
`First-order optimality`	First-order optimality measure (seeFirst-Order Optimality Measure)
`Func-count`or`F-count`	Number of function evaluations; seeIterations and Function Counts
`Iteration`or`Iter`	Iteration number; seeIterations 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

This table describes the headings specific tofgoalattain,fmincon,fminimax, andfseminf.

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 (seePreconditioned 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 (seeEquation 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, seeUpdating 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 (seeEquation 29)
`Trust-region radius`	Current trust-region radius

fminbnd and fzero

This table describes the headings specific tofminbndandfzero.

fminbnd or fzero Heading Information Displayed

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`	当前点的算法

Procedure

Procedures forfminbnd:

initial
golden(golden section search)
parabolic(parabolic interpolation)

Procedures forfzero:

initial(initial point)
search(search for an interval containing a zero)
bisection
interpolation(linear interpolation or inverse quadratic interpolation)

x

当前点的算法

fminsearch

This table describes the headings specific tofminsearch.

fminsearch Heading Information Displayed

fminsearch Heading	Information 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, seefminsearch Algorithm.

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, seefminsearch Algorithm.

fminunc

This table describes the headings specific tofminunc.

fminunc Heading	Information Displayed
`CG-iterations`	Number of conjugate gradient iterations taken in the current iteration (seePreconditioned Conjugate Gradient Method)
`Line search steplength`	Multiplicative factor that scales the search direction (seeEquation 11)

Thefminunc'quasi-newton'algorithm can issue askipped updatemessage to the right of theFirst-order optimalitycolumn. This message means thatfminuncdid 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 tofsolve.

fsolve Heading	Information Displayed
`Directional derivative`	Gradient of the function along the search direction
`Lambda`	λ_kvalue defined inLevenberg-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

This table describes the headings specific tointlinprog.

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 (%)`	$\frac{100 (b - a)}{\| b \| + 1},$ where bis the objective function value of the best integer feasible point. ais the best lower bound on the objective function value. Note Although you specify`RelativeGapTolerance`as a decimal number, the iterative display and`output.relativegap`report 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 tolinprog. 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, seePredictor-Corrector(`'interior-point'`) orMain Algorithm(`'interior-point-legacy'`) orDual-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, seePredictor-Corrector. For the definition of dual infeasibility, seePredictor-Corrector(`'interior-point'`) orMain Algorithm(`'interior-point-legacy'`) orDual-Simplex Algorithm.
`Upper Bounds {x}+s-ub`	Upper bound feasibility.{x}means thosexwith finite upper bounds. This value is ther_uresidual inInterior-Point-Legacy Linear Programming.
`Duality Gap x'z+s'w`	Duality gap (seeInterior-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 ofMain Algorithm
`Complementarity`	A measure of the Lagrange multipliers times distance from the bounds, which should be zero at a solution. See ther_cvariable inStopping Conditions.
`Time`	Time in seconds that`linprog`已经运行

lsqlin

Thelsqlin'interior-point'iterative display is inherited from thequadprogiterative display. The relationship between these functions is explained inLinear Least Squares: Interior-Point or Active-Set. For iterative display details, seequadprog.

lsqnonlin and lsqcurvefit

This table describes the headings specific tolsqnonlinandlsqcurvefit.

lsqnonlin or lsqcurvefit Heading	Information Displayed
`Directional derivative`	Gradient of the function along the search direction
`Lambda`	λ_kvalue defined inLevenberg-Marquardt Method
`Resnorm`	Value of the squared 2-norm of the residual at`x`
`Residual`	Residual vector of the function

quadprog

This table describes the headings specific toquadprog. Only the'interior-point-convex'algorithm has the iterative display.

quadprog Heading	Information Displayed
`Primal Infeas`	Primal infeasibility, defined as`max( norm(Aeqx - beq, inf), abs(min(0, min(Ax-b))) )`
`Dual Infeas`	Dual infeasibility, defined as`norm(Hx + f - Alambda_ineqlin - Aeq*lambda_eqlin, inf)`
`Complementarity`	的最大绝对值的测量滞后range multipliers of inactive inequalities, which should be zero at a solution. This quantity isginInfeasibility Detection.