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B  Result - the Output Result Structure

The results of the optimization attempts are stored in a structure array named Result . The currently defined fields in the structure are shown in Table B. The use of structure arrays make advanced result presentation and statistics possible. Results from many runs may be collected in an array of structures, making postprocessing on all results easy.

When running global optimization, output results are also stored in mat-files, to enable fast restart (warm start) of the solver. It is seldom the case that one knows that the solver actually converged for a particular problem. Therefore one does restarts until the optimum does not change, and one is satisfied with the results. The information stored in the mat-file glbSave.mat by the solver glbSolve  is shown in Table 24. The information stored in the mat-file glcSave.mat by the solver glcSolve  is shown in Table 25. Different information is stored when using glbFast and glcFast, see the solver reference.

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Table 24: Information stored in the mat-file glbSave.mat by the solver glbSolve . Used for automatic restarts.

Variable	Description
C	Matrix with all rectangle centerpoints, in [0,1]-space.
D	Vector with distances from centerpoint to the vertices.
DMin	Row vector of minimum function value for each distance.
DSort	Row vector of all different distances, sorted.
E	Computed tolerance in rectangle selection.
F	Vector with function values.
L	Matrix with all rectangle side lengths in each dimension.
Name	Name of the problem. Used for security if doing warm start.
glbfMin	Best function value found at a feasible point.
iMin	The index in D which has lowest function value, i.e. the rectangle which minimizes (F − glbfMin + E)./D where E = max(EpsGlob*abs(glbfMin),1E−8).

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Table 25: Information stored in the mat-file glcSave.mat by the solver glcSolve . Used for automatic restarts.

Variable	Description
C	Matrix with all rectangle centerpoints.
D	Vector with distances from centerpoint to the vertices.
F	Vector with function values.
G	Matrix with constraint values for each point.
Name	Name of the problem. Used for security if doing warm start.
Split	Split(i,j) is the number of splits along dimension i of rectangle j.
T	T(i) is the number of times rectangle i has been trisected.
fMinEQ	sum(abs(infeasibilities)) for minimum points, 0 if no equalities.
fMinIdx	Indices of the currently best points.
feasible	Flag indicating if a feasible point has been found.
glcf_min	Best function value found at a feasible point.
iL	iL(i,j) is the lower bound for rectangle j in integer dimension I(i).
iU	iU(i,j) is the upper bound for rectangle j in integer dimension I(i).
ignoreidx	Rectangles to be ignored in the rectangle selection procedure.
s	s(j) is the sum of observed rates of change for constraint j.
s_0	s_0 is used as s(0).
t	t(i) is the total number of splits along dimension i.

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Information stored in the optimization result structure Result .
Field	Description
Name	Problem name.
P	Problem number.
probType	TOMLAB  problem type, according to Table 2.2.
Solver	Solver used.
SolverAlgorithm	Solver algorithm used.
solvType	TOMLAB  solver type.
ExitFlag	0 if convergence to local min. Otherwise errors.
ExitText	Text string describing the result of the optimization.
Inform	Information parameter, type of convergence.
CPUtime	CPU time used in seconds.
REALtime	Real time elapsed in seconds.
Iter	Number of major iterations.
MinorIter	Number of minor iterations (for some solvers).
maxTri	Maximum rectangle size.
FuncEv	Number of function evaluations needed.
GradEv	Number of gradient evaluations needed.
HessEv	Number of Hessian evaluations needed.
ConstrEv	Number of constraint evaluations needed.
ConJacEv	Number of constraint Jacobian evaluations needed.
ConHessEv	Number of nonlinear constraint Hessian evaluations needed.
ResEv	Number of residual evaluations needed (least squares).
JacEv	Number of Jacobian evaluations needed (least squares).
x_k	Optimal point.
f_k	Function value at optimum.
g_k	Gradient value at optimum.
B_k	Quasi-Newton approximation of the Hessian at optimum.
H_k	Hessian value at optimum.
y_k	Dual parameters.
v_k	Lagrange multipliers for constraints on variables, linear and nonlinear constraints.
r_k	Residual vector at optimum.
J_k	Jacobian matrix at optimum.
Ax	Value of linear constraints at optimum.
c_k	Value of nonlinear constraints at optimum.
cJac	Constraint Jacobian at optimum.
x_0	Starting point.
f_0	Function value at start i.e. f(x_0).
c_0	Value of nonlinear constraints at start.
Ax0	Value of linear constraints at start.
xState	State of each variable, described in Table 26.
bState	State of each linear constraint, described in Table 27.
cState	State of each general constraint, described in Table 28.
p_dx	Matrix where each column is a search direction.
alphaV	Matrix where row i stores the step lengths tried for the i:th iteration.
x_min	Lowest x-values in optimization. Used for plotting.
x_max	Highest x-values in optimization. Used for plotting.
LS	Structure with statistical information for least squares problems, see Table 29.
F_X	F_X is a global matrix with rows: [iter_no f(x)].
SepLS	General result variable with fields z  and Jz . Used when running separable nonlinear least squares problems.
QP	Structure with special fields for QP problems. Used for warm starts, see Table 16.
SOL	Structure with some of the fields in the Prob.SOL  structure, the ones needed to do a warm start of a SOL solver, see Table 21. The routine WarmDefSOL  moves the relevant fields back to Prob.SOL  for the subsequent call.
DUNDEE	Structure with special result fields from TOMLAB /MINLP solvers.
plotData	Structure with plotting parameters.
Prob	Problem structure, see Table A. Please note that certain solvers that do reformulations of the problem, e.g. L1Solve , infSolve  and slsSolve , return the Prob  structure of the reformulated problem in this field, not the original one.

The field xState  describes the state of each of the variables. In Table 26 the different values are described. The different conditions for linear constraints are defined by the state variable in field bState . In Table 27 the different values are described.

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Table 26: The state variable xState  for the variable.

Value	Description
0	A free variable.
1	Variable on lower bound.
2	Variable on upper bound.
3	Variable is fixed, lower bound is equal to upper bound.

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Table 27: The state variable bState  for each linear constraint.

Value	Description
0	Inactive constraint.
1	Linear constraint on lower bound.
2	Linear constraint on upper bound.
3	Linear equality constraint.

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Table 28: The state variable cState  for each nonlinear constraint.

Value	Description
0	Inactive constraint.
1	Nonlinear constraint on lower bound.
2	Nonlinear constraint on upper bound.
3	Nonlinear equality constraint.

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Table 29: Information stored in the structure Result.LS .

Field	Description
SSQ	rkT rk.
Covar	Covariance matrix (inverse of JkT Jk).
sigma2	Estimate of squared standard deviation.
Corr	Correlation matrix (normalized covariance matrix).
StdDev	Estimated standard deviation in parameters.
x	The optimal point x_k.
ConfLim	95% confidence limit (roughly) assuming normal distribution
	of errors.
CoeffVar	Coefficients of variation of estimates.

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