Output from FIND_GLOBAL_MIN on 05/05/2012 at 03:42:57. Version for the system is: March 13, 2009 Codelist file name is: ex14_1_1G.CDL Box data file name is: ex14_1_1.DT1 Initial box: [ -5.00 , 5.00 ], [ -5.00 , 5.00 ] [ -0.100E+05, 0.100E+05 ] BOUND_CONSTRAINT: F F F F F F --------------------------------------- CONFIGURATION VALUES: EPS_DOMAIN: 0.1000D-07 MAXITR: 500000 SMALLEST_LIST_BOX_SIZE = 0.0000D+00 A_PRIORI_UPPER_BOUND (on global optimum): 0.180+309 MAX_CPU_SECONDS: 0.720E+04 MAX_LP_PRE: 10000000 ALSO_PRINT_TO_TERMINAL F NO_ABSOLUTE_VALUE_IN_MINIMAX F MAX_PT_SOLVER_ITER 3000 MAX_SMALL_BOXES 2000 MAX_BEFORE_AMALGAMATE 200 DO_INTERVAL_NEWTON: T QUADRATIC: T FULL_SPACE: F VERY_GOOD_INITIAL_GUESS: F USE_SUBSIT: T OUTPUT UNIT: 7 PRINT_LENGTH: 3 USE_INTRINSIC_PRINTING: T PHI_MUST_CONVERGE: T EQ_CNS_MUST_CONVERGE: T INEQ_CNS_MUST_CONVERGE: T ALLOW_EPSILON_APPROXIMATE: F USES_INTERMEDIATE_VARIABLES: F PHI_THICKNESS_FACTOR: 0.500 EQ_CNS_THICKNESS_FACTOR: 0.500 INEQ_CNS_THICKNESS_FACTOR: 0.500 PHI_MUST_CONVERGE: T EQ_CNS_MUST_CONVERGE: T INEQ_CNS_MUST_CONVERGE: T PHI_CONVERGENCE_FACTOR: 0.100E-13 EQ_CNS_CONVERGENCE_FACTOR: 0.100E-13 INEQ_CNS_CONVERGENCE_FACTOR: 0.100E-13 CONTINUITY_ACROSS_BRANCHES: F SINGULAR_EXPANSION_FACTOR: 10.0 HEURISTIC PARAMETER ALPHA: 0.500 APPROX_OPT_BEFORE_BISECTION: F APPROX_OPTIMIZER_TYPE 7 USE_LP: T ITERATE__LP: F EPS_LP_FIT: 1.00000000000000002E-002 USE_EPPERLY_SPLIT: 0 PRINTING_IN_SPLIT 0 USE_REDUCED_SPACE: F REDUCED_IN_BISECTION: T USE_TAYLOR_EQUALITY_CONSTRAINTS F USE_TAYLOR_INEQ_CONSTRAINTS F USE_TAYLOR_OBJECTIVE F USE_TAYLOR_EQ_CNS_GRD F USE_TAYLOR_GRAD F USE_TAYLOR_INEQ_CNS_GRD F USE_TAYLOR_REDUCED_INEWTON F COSY_POLYNOMIAL_ORDER 5 LEAST_SQUARES_FUNCTIONS: F NONLINEAR_SYSTEM: F UNCONSTRAINED_MINIMAX: F NO_ABSOLUTE_VALUE_IN_MINIMAX: F DO_INFEASIBILITY_CHECK: T DO_PIVOTING: T DO_INV_MID: T TRY_C_LP_HEURISTIC: 10000000000.000000 REUSE_PRECONDITIONERS: T ORDERED_LIST_IN_COMPLEMENTATION 1 DO_PROBE: F DO_PROBE_TESTS_3_AND_4: F USE_INEQ_PERTURB_FOR_FEAS: F DO_SPLITS_IN_SUBSIT F PRINTING_IN_VALIDATE_FJ: 0 PRINT_SUBSIT: 0 ALSO_PRINT_TO_TERMINAL F C-LP is used for computing C-LP preconditioners. UNCONSTRAINED_MINIMAX F NO_ABSOLUTE_VALUE_IN_MINIMAX F MINIMAX_FORMULATION_2 T C_LP_DENSE, Manuel Novoa's special routine, was used to compute LP preconditioners. LIST OF SMALL BOXES: Box no.: 1 Box coordinates: [ 3.58 , 3.59 ], [ -1.85 , -1.85 ] [ -0.100E-02, 0.999E-03 ] PHI: [ -0.100E-02, 0.999E-03 ] Box contains the following approximate root: 3.58 , -1.85 , -0.996E-08 OBJECTIVE ENCLOSURE AT APPROXIMATE ROOT: [ -0.996E-08, -0.996E-08 ] Unknown = T Contains_root = F Fritz John multiplier U0: [ 0.500 , 0.500 ] Fritz John multipliers U: [ 0.00 , 0.251 ], [ 0.00 , 0.251 ] [ 0.00 , 0.252 ], [ 0.00 , 0.252 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] INEQ_CERT_FEASIBLE: F F F F T T T T NIN_POSS_BINDING: 4 ------------------------------------------------- Box no.: 2 Box coordinates: [ 3.38 , 3.39 ], [ 0.729E-01, 0.749E-01 ] [ -0.100E-02, 0.999E-03 ] PHI: [ -0.100E-02, 0.999E-03 ] Box contains the following approximate root: 3.39 , 0.739E-01, -0.996E-08 OBJECTIVE ENCLOSURE AT APPROXIMATE ROOT: [ -0.996E-08, -0.996E-08 ] Unknown = T Contains_root = F Fritz John multiplier U0: [ 0.500 , 0.500 ] Fritz John multipliers U: [ 0.00 , 0.255 ], [ 0.00 , 0.255 ] [ 0.00 , 0.254 ], [ 0.00 , 0.254 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] INEQ_CERT_FEASIBLE: F F F F T T T T NIN_POSS_BINDING: 4 ------------------------------------------------- Box no.: 3 Box coordinates: [ -3.78 , -3.78 ], [ -3.29 , -3.28 ] [ -0.100E-02, 0.999E-03 ] PHI: [ -0.100E-02, 0.999E-03 ] Box contains the following approximate root: -3.78 , -3.28 , -0.996E-08 OBJECTIVE ENCLOSURE AT APPROXIMATE ROOT: [ -0.996E-08, -0.996E-08 ] Unknown = T Contains_root = F Fritz John multiplier U0: [ 0.500 , 0.500 ] Fritz John multipliers U: [ 0.00 , 0.256 ], [ 0.00 , 0.256 ] [ 0.00 , 0.259 ], [ 0.00 , 0.259 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] INEQ_CERT_FEASIBLE: F F F F T T T T NIN_POSS_BINDING: 4 ------------------------------------------------- Box no.: 4 Box coordinates: [ -2.81 , -2.80 ], [ 3.13 , 3.13 ] [ -0.100E-02, 0.100E-02 ] PHI: [ -0.100E-02, 0.100E-02 ] Box contains the following approximate root: -2.81 , 3.13 , -0.996E-08 OBJECTIVE ENCLOSURE AT APPROXIMATE ROOT: [ -0.996E-08, -0.996E-08 ] Unknown = T Contains_root = F Fritz John multiplier U0: [ 0.500 , 0.500 ] Fritz John multipliers U: [ 0.00 , 0.260 ], [ 0.00 , 0.260 ] [ 0.00 , 0.253 ], [ 0.00 , 0.253 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] INEQ_CERT_FEASIBLE: F F F F T T T T NIN_POSS_BINDING: 4 ------------------------------------------------- Box no.: 5 Box coordinates: [ 3.00 , 3.00 ], [ 2.00 , 2.00 ] [ -0.100E-02, 0.100E-02 ] PHI: [ -0.100E-02, 0.100E-02 ] Box contains the following approximate root: 3.00 , 2.00 , -0.996E-08 OBJECTIVE ENCLOSURE AT APPROXIMATE ROOT: [ -0.996E-08, -0.996E-08 ] Unknown = T Contains_root = F Fritz John multiplier U0: [ 0.500 , 0.500 ] Fritz John multipliers U: [ 0.00 , 0.256 ], [ 0.00 , 0.256 ] [ 0.00 , 0.251 ], [ 0.00 , 0.251 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] INEQ_CERT_FEASIBLE: F F F F T T T T NIN_POSS_BINDING: 4 ------------------------------------------------- Box no.: 6 Box coordinates: [ -3.08 , -3.07 ], [ -0.824E-01, -0.804E-01 ] [ -0.100E-02, 0.100E-02 ] PHI: [ -0.100E-02, 0.100E-02 ] Box contains the following approximate root: -3.07 , -0.814E-01, -0.996E-08 OBJECTIVE ENCLOSURE AT APPROXIMATE ROOT: [ -0.996E-08, -0.996E-08 ] Unknown = T Contains_root = F Fritz John multiplier U0: [ 0.500 , 0.500 ] Fritz John multipliers U: [ 0.00 , 0.257 ], [ 0.00 , 0.257 ] [ 0.00 , 0.255 ], [ 0.00 , 0.255 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] INEQ_CERT_FEASIBLE: F F F F T T T T NIN_POSS_BINDING: 4 ------------------------------------------------- Box no.: 7 Box coordinates: [ 0.857E-01, 0.877E-01 ], [ 2.88 , 2.89 ] [ -0.100E-02, 0.100E-02 ] PHI: [ -0.100E-02, 0.100E-02 ] Box contains the following approximate root: 0.867E-01, 2.88 , -0.996E-08 OBJECTIVE ENCLOSURE AT APPROXIMATE ROOT: [ -0.996E-08, -0.996E-08 ] Unknown = T Contains_root = F Fritz John multiplier U0: [ 0.500 , 0.500 ] Fritz John multipliers U: [ 0.00 , 0.259 ], [ 0.00 , 0.259 ] [ 0.00 , 0.252 ], [ 0.00 , 0.252 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] INEQ_CERT_FEASIBLE: F F F F T T T T NIN_POSS_BINDING: 4 ------------------------------------------------- Box no.: 8 Box coordinates: [ -0.129 , -0.127 ], [ -1.96 , -1.95 ] [ -0.100E-02, 0.100E-02 ] PHI: [ -0.100E-02, 0.100E-02 ] Box contains the following approximate root: -0.128 , -1.95 , -0.996E-08 OBJECTIVE ENCLOSURE AT APPROXIMATE ROOT: [ -0.996E-08, -0.996E-08 ] Unknown = T Contains_root = F Fritz John multiplier U0: [ 0.500 , 0.500 ] Fritz John multipliers U: [ 0.00 , 0.252 ], [ 0.00 , 0.252 ] [ 0.00 , 0.260 ], [ 0.00 , 0.260 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] INEQ_CERT_FEASIBLE: F F F F T T T T NIN_POSS_BINDING: 4 ------------------------------------------------- Box no.: 9 Box coordinates: [ -0.272 , -0.270 ], [ -0.924 , -0.922 ] [ -0.100E-02, 0.100E-02 ] PHI: [ -0.100E-02, 0.100E-02 ] Box contains the following approximate root: -0.271 , -0.923 , -0.996E-08 OBJECTIVE ENCLOSURE AT APPROXIMATE ROOT: [ -0.996E-08, -0.996E-08 ] Unknown = T Contains_root = F Fritz John multiplier U0: [ 0.500 , 0.500 ] Fritz John multipliers U: [ 0.00 , 0.250 ], [ 0.00 , 0.250 ] [ 0.00 , 0.251 ], [ 0.00 , 0.251 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] INEQ_CERT_FEASIBLE: F F F F T T T T NIN_POSS_BINDING: 4 ------------------------------------------------- THERE WERE NO BOXES CORRESPONDING TO VERIFIED FEASIBLE POINTS. ALGORITHM COMPLETED WITH LESS THAN THE MAXIMUM NUMBER, 500000 OF BOXES. Number of bisections: 5272 No. dense interval residual evaluations -- gradient code list: 87879 Number of orig. system inverse midpoint preconditioner rows: 52378 Number of orig. system C-LP preconditioner rows: 60547 Number of solutions for a component in the expanded system: 765363 Total number of forward_substitutions: 271554 Number of Gauss--Seidel steps on the dense system: 113843 Number point dense residual evaluations, gradient codelist: 17696 Number of gradient evaluations from a gradient code list: 30734 Total number of dense slope matrix evaluations: 53363 Total number second-order interval evaluations of the original function: 16781 Total number dense interval constraint evaluations: 1001080 Total number dense interval constraint gradient component evaluations: 1623960 Total number dense point constraint gradient component evaluations: 509160 Total number dense interval reduced gradient evaluations: 45502 Total number of calls to FRITZ_JOHN_RESIDUALS: 12543 Number of times a box was rejected because the gradient or reduced gradient did not contain zero: 349 Number of times a box was rejected due to infeasibility in LP_FILTER: 10 Average number of overall loop iterations in each call to the reduced interval Newton method): 1.31 Number of times a box was rejected in the interval Newton method due to an empty intersection: 2865 Number of times the interval Newton method made a coordinate interval smaller: 19282 Number of times a pivoting preconditioner made a coordinate interval smaller or rejected a coordinate: 1803 Number of times a pivoting preconditioner was successful after the first sweep: 821 Number of times a midpoint matrix was factored: 6922 Total number of times the reduced interval Newton method was tried: 6682 Number of times an inverse midpoint preconditioner led to improvement or rejection: 5185 Number of times a C LP preconditioner led to improvement or rejection: 325 N_C_LP_INFEASIBLE = 2979 N_REJECT_WITH_LP filter = 10 N_LPF_INF_OR_UNB = 2979 Number of times a C LP preconditioner was not computed because the heuristic determined it was not worth it: 3054 Number of possible splits as detected by the pivoting preconditioner: 8222 Total time spent in the LP filter (creating and solving the LP): 19.9 Total time spent in subsit (constraint propagation): 0.672 Total time spent in reduced_interval_Newton (iteration to reduce the box): 3.88 Total time spent searching for "D" in the LP filter: 0.320 Total time spent actually solving the linear relaxations: 17.9 Total time spent doing linear algebra (preconditioners and solution processes): 2.86 Total time spent running the approximate optimizer: 0.200E-01 LIST_BOOKKEEPING_TIME: 0.644 FUNCTION_EVALUATION_TIME (in forward_substitution): 2.24 Time spent setting up pivoting preconditioners: 0.112 Time spent computing pivoting preconditioners: 0.256 Time spent computing LP preconditioners: 0.584 Time spent computing inverse midpoint preconditioners: 0.168 Number of times the approximate solver was called: 1986 Number Fritz-John matrix evaluations: 12194 Number of times SUBSIT decreased one or more coordinate widths: 811 Number of times SUBSIT rejected a box: 112 Number times a box was rejected due infeasible inequality constraints: 183 Total number of boxes processed in loop: 6814 BEST_ESTIMATE: 1.54 Overall CPU time: 49.9 CPU time in PEEL_BOUNDARY: 0.00 CPU time in REDUCED_INTERVAL_NEWTON: 3.88 =================================================== =================================================== Number of boxes in the list with proven feasible points: 0 Number of boxes in the list of other small boxes: 9 Number of unfathomed boxes: 0 Interval hull of the small unverified boxes: [ -3.78 , 3.59 ], [ -3.29 , 3.13 ] [ -0.100E-02, 0.100E-02 ] Rigorously verified bounds on the optimum, provided an optimum exists: [ -0.100E-02, 0.100E-02 ]