Iterative Display - MATLAB & Simulink (original) (raw)
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 theDisplay 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'algorithmlsqnonnegquadprog'trust-region-reflective'algorithm
Common Headings
This table lists some common headings of iterative display.
| Heading | Information Displayed |
|---|---|
| f(x) orFval | Current objective function value |
| First-order optimality | First-order optimality measure (see First-Order Optimality Measure) |
| Func-count orF-count | Number of function evaluations; see Iterations and Function Counts |
| Iteration orIter | 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 as0 |
| 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 pointHessian not updatedHessian modifiedHessian modified twiceFor 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:initialgolden (golden section search)parabolic (parabolic interpolation)Procedures for fzero: initial (initial point)search (search for an interval containing a zero)bisectioninterpolation (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 simplexexpandreflectshrinkcontract insidecontract outsideFor 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 thatfminunc 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) |
| 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 (%) | 100(b−a)|b |
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 orPrimal 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'). |
| Dual Infeas A'*y+z-w-f orDual 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') . |
| 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 thequadprog 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 islsqlin displays a column titledResnorm, 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 atx |
| 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 inInfeasibility Detection. |
| Feasibility | Maximum constraint violation, where satisfied inequality constraints count as 0 ('active-set' algorithm) |