Output from FIND_GLOBAL_MIN on 05/06/2012 at 22:14:10. Version for the system is: March 13, 2009 Codelist file name is: ex3_1_4G.CDL Box data file name is: ex3_1_4.DT1 Initial box: [ 0.00 , 2.00 ], [ 0.00 , 0.100E+05 ] [ 0.00 , 3.00 ] 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: [ 0.499 , 0.501 ], [ 0.00 , 0.150E-02 ] [ 3.00 , 3.00 ] PHI: [ -4.00 , -3.99 ] Box contains the following approximate root: 0.500 , -0.998E-08, 3.00 OBJECTIVE ENCLOSURE AT APPROXIMATE ROOT: [ -4.00 , -4.00 ] Unknown = T Contains_root = F Fritz John multiplier U0: [ 0.128 , 0.443 ] Fritz John multipliers U: [ 0.129 , 0.163 ], [ 0.00 , 0.430 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] [ 0.00 , 1.00 ], [ 0.431E-01, 0.223 ] [ 0.00 , 1.00 ], [ 0.317 , 0.539 ] INEQ_CERT_FEASIBLE: F T T T T F T F NIN_POSS_BINDING: 3 ------------------------------------------------- Box no.: 2 Box coordinates: [ 2.00 , 2.00 ], [ 0.00 , 0.150E-02 ] [ 0.00 , 0.150E-02 ] PHI: [ -4.00 , -3.99 ] Box contains the following approximate root: 2.00 , -0.999E-08, -0.800E-08 OBJECTIVE ENCLOSURE AT APPROXIMATE ROOT: [ -4.00 , -4.00 ] Unknown = T Contains_root = F Fritz John multiplier U0: [ 0.285 , 0.313 ] Fritz John multipliers U: [ 0.621E-01, 0.144 ], [ 0.00 , 1.00 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] [ 0.293 , 0.381 ], [ 0.162 , 0.251 ] [ 0.00 , 0.431 ], [ 0.00 , 1.00 ] INEQ_CERT_FEASIBLE: F T T T F F T T NIN_POSS_BINDING: 3 ------------------------------------------------- Box no.: 3 Box coordinates: [ 0.625 , 0.626 ], [ 0.374 , 0.376 ] [ 3.00 , 3.00 ] PHI: [ -3.88 , -3.87 ] Box contains the following approximate root: 0.626 , -0.998E-08, 3.00 OBJECTIVE ENCLOSURE AT APPROXIMATE ROOT: [ -4.25 , -4.25 ] Unknown = T Contains_root = F Fritz John multiplier U0: [ 0.290 , 0.561 ] Fritz John multipliers U: [ 0.118 , 0.189 ], [ 0.00 , 0.923E-01 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] [ 0.00 , 1.00 ], [ 0.229 , 0.552 ] INEQ_CERT_FEASIBLE: F F T T T T T F NIN_POSS_BINDING: 3 ------------------------------------------------- Box no.: 4 Box coordinates: [ 0.624 , 0.625 ], [ 0.374 , 0.375 ] [ 3.00 , 3.00 ] PHI: [ -3.88 , -3.87 ] Box contains the following approximate root: 0.625 , 0.375 , 3.00 OBJECTIVE ENCLOSURE AT APPROXIMATE ROOT: [ -3.87 , -3.87 ] Unknown = T Contains_root = F Fritz John multiplier U0: [ 0.290 , 0.561 ] Fritz John multipliers U: [ 0.118 , 0.189 ], [ 0.00 , 0.923E-01 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] [ 0.00 , 1.00 ], [ 0.229 , 0.552 ] INEQ_CERT_FEASIBLE: F F T T T T T F NIN_POSS_BINDING: 3 ------------------------------------------------- Box no.: 5 Box coordinates: [ 0.624 , 0.625 ], [ 0.375 , 0.377 ] [ 3.00 , 3.00 ] PHI: [ -3.88 , -3.87 ] Box contains the following approximate root: 0.625 , -0.998E-08, 3.00 OBJECTIVE ENCLOSURE AT APPROXIMATE ROOT: [ -4.25 , -4.25 ] Unknown = T Contains_root = F Fritz John multiplier U0: [ 0.290 , 0.561 ] Fritz John multipliers U: [ 0.118 , 0.189 ], [ 0.00 , 0.923E-01 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] [ 0.00 , 1.00 ], [ 0.229 , 0.552 ] INEQ_CERT_FEASIBLE: F F T T T T T F NIN_POSS_BINDING: 3 ------------------------------------------------- Box no.: 6 Box coordinates: [ 0.623 , 0.624 ], [ 0.375 , 0.376 ] [ 3.00 , 3.00 ] PHI: [ -3.87 , -3.87 ] Box contains the following approximate root: 0.624 , 0.376 , 3.00 OBJECTIVE ENCLOSURE AT APPROXIMATE ROOT: [ -3.87 , -3.87 ] Unknown = T Contains_root = F Fritz John multiplier U0: [ 0.290 , 0.561 ] Fritz John multipliers U: [ 0.118 , 0.189 ], [ 0.00 , 0.923E-01 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] [ 0.00 , 1.00 ], [ 0.229 , 0.552 ] INEQ_CERT_FEASIBLE: F F T T T T T F NIN_POSS_BINDING: 3 ------------------------------------------------- Box no.: 7 Box coordinates: [ 0.622 , 0.624 ], [ 0.376 , 0.378 ] [ 3.00 , 3.00 ] PHI: [ -3.87 , -3.86 ] Box contains the following approximate root: 0.500 , -0.998E-08, 3.00 OBJECTIVE ENCLOSURE AT APPROXIMATE ROOT: [ -4.00 , -4.00 ] Unknown = T Contains_root = F Fritz John multiplier U0: [ 0.290 , 0.561 ] Fritz John multipliers U: [ 0.118 , 0.189 ], [ 0.00 , 0.923E-01 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] [ 0.00 , 1.00 ], [ 0.229 , 0.552 ] INEQ_CERT_FEASIBLE: F F T T T T T F NIN_POSS_BINDING: 3 ------------------------------------------------- Box no.: 8 Box coordinates: [ 2.00 , 2.00 ], [ 1.26 , 1.27 ] [ 0.735 , 0.737 ] PHI: [ -3.47 , -3.46 ] Box contains the following approximate root: 2.00 , -0.999E-08, -0.800E-08 OBJECTIVE ENCLOSURE AT APPROXIMATE ROOT: [ -4.00 , -4.00 ] Unknown = T Contains_root = F Fritz John multiplier U0: [ 0.531 , 0.531 ] Fritz John multipliers U: [ 0.116 , 0.116 ], [ 0.00 , 0.486E-09 ] [ 0.00 , 0.522 ], [ 0.00 , 1.00 ] [ 0.354 , 0.354 ], [ 0.00 , 1.00 ] [ 0.00 , 0.940 ], [ 0.00 , 1.00 ] INEQ_CERT_FEASIBLE: F F T T F T T T NIN_POSS_BINDING: 3 ------------------------------------------------- Box no.: 9 Box coordinates: [ 1.29 , 1.29 ], [ 1.03 , 1.04 ] [ 1.68 , 1.68 ] PHI: [ -3.22 , -3.21 ] Box contains the following approximate root: 2.00 , -0.999E-08, -0.800E-08 OBJECTIVE ENCLOSURE AT APPROXIMATE ROOT: [ -4.00 , -4.00 ] Unknown = T Contains_root = F Fritz John multiplier U0: [ 0.696 , 0.696 ] Fritz John multipliers U: [ 0.177 , 0.177 ], [ 0.127 , 0.127 ] [ 0.00 , 0.624 ], [ 0.00 , 1.00 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] INEQ_CERT_FEASIBLE: F F T T T T T T NIN_POSS_BINDING: 2 ------------------------------------------------- Box no.: 10 Box coordinates: [ 0.848 , 0.850 ], [ 0.00 , 0.150E-02 ] [ 1.50 , 1.50 ] PHI: [ -3.20 , -3.19 ] Box contains the following approximate root: 2.00 , -0.999E-08, -0.800E-08 OBJECTIVE ENCLOSURE AT APPROXIMATE ROOT: [ -4.00 , -4.00 ] Unknown = T Contains_root = F Fritz John multiplier U0: [ 0.643 , 0.643 ] Fritz John multipliers U: [ 0.178 , 0.178 ], [ 0.00 , 1.00 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] [ 0.00 , 1.00 ], [ 0.178 , 0.178 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] INEQ_CERT_FEASIBLE: F T T T T F T T NIN_POSS_BINDING: 2 ------------------------------------------------- Box no.: 11 Box coordinates: [ 1.06 , 1.07 ], [ 0.499 , 0.501 ] [ 1.50 , 1.50 ] PHI: [ -3.13 , -3.12 ] Box contains the following approximate root: 2.00 , -0.999E-08, -0.800E-08 OBJECTIVE ENCLOSURE AT APPROXIMATE ROOT: [ -4.00 , -4.00 ] Unknown = T Contains_root = F Fritz John multiplier U0: [ 0.789 , 0.789 ] Fritz John multipliers U: [ 0.211 , 0.211 ], [ 0.00 , 0.855 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] [ 0.00 , 1.00 ], [ 0.00 , 1.00 ] INEQ_CERT_FEASIBLE: F T T T T T T T NIN_POSS_BINDING: 1 ------------------------------------------------- THERE WERE NO BOXES CORRESPONDING TO VERIFIED FEASIBLE POINTS. ALGORITHM COMPLETED WITH LESS THAN THE MAXIMUM NUMBER, 500000 OF BOXES. Number of bisections: 597 No. dense interval residual evaluations -- gradient code list: 30398 Number of orig. system inverse midpoint preconditioner rows: 15267 Number of orig. system C-LP preconditioner rows: 2882 Number of solutions for a component in the expanded system: 105289 Total number of forward_substitutions: 104578 Number of Gauss--Seidel steps on the dense system: 20934 Number point dense residual evaluations, gradient codelist: 11159 Number of gradient evaluations from a gradient code list: 4595 Total number of dense slope matrix evaluations: 56117 Total number second-order interval evaluations of the original function: 2309 Total number dense interval constraint evaluations: 668176 Total number dense interval constraint gradient component evaluations: 1405224 Total number dense point constraint gradient component evaluations: 1468368 Total number dense interval reduced gradient evaluations: 72046 Total number of calls to FRITZ_JOHN_RESIDUALS: 18084 Number of times a box was rejected because the gradient or reduced gradient did not contain zero: 148 Average number of overall loop iterations in each call to the reduced interval Newton method): 16.5 Number of times a box was rejected in the interval Newton method due to an empty intersection: 311 Number of times the interval Newton method made a coordinate interval smaller: 20528 Number of times a pivoting preconditioner made a coordinate interval smaller or rejected a coordinate: 16020 Number of times a pivoting preconditioner was successful after the first sweep: 15651 Number of times a midpoint matrix was factored: 2572 Total number of times the reduced interval Newton method was tried: 1101 Number of times an inverse midpoint preconditioner led to improvement or rejection: 3530 Number of times a C LP preconditioner led to improvement or rejection: 1289 N_C_LP_INFEASIBLE = 514 N_LPF_INF_OR_UNB = 514 Number of possible splits as detected by the pivoting preconditioner: 17936 Total time spent in the LP filter (creating and solving the LP): 2.94 Total time spent in subsit (constraint propagation): 0.280E-01 Total time spent in reduced_interval_Newton (iteration to reduce the box): 2.40 Total time spent actually solving the linear relaxations: 2.67 Total time spent doing linear algebra (preconditioners and solution processes): 0.968 Total time spent running the approximate optimizer: 0.624 LIST_BOOKKEEPING_TIME: 0.320E-01 FUNCTION_EVALUATION_TIME (in forward_substitution): 0.656 Time spent setting up pivoting preconditioners: 0.252 Time spent computing pivoting preconditioners: 0.392 Time spent computing LP preconditioners: 0.360E-01 Time spent computing inverse midpoint preconditioners: 0.360E-01 Number of times the approximate solver was called: 256 Number Fritz-John matrix evaluations: 17936 Number of times SUBSIT decreased one or more coordinate widths: 127 Number of times SUBSIT rejected a box: 92 Number times a box was rejected due infeasible inequality constraints: 295 Total number of boxes processed in loop: 1196 BEST_ESTIMATE: -3.12 Overall CPU time: 16.8 CPU time in PEEL_BOUNDARY: 0.00 CPU time in REDUCED_INTERVAL_NEWTON: 2.40 =================================================== =================================================== Number of boxes in the list with proven feasible points: 0 Number of boxes in the list of other small boxes: 11 Number of unfathomed boxes: 0 Interval hull of the small unverified boxes: [ 0.499 , 2.00 ], [ 0.00 , 1.27 ] [ 0.00 , 3.00 ] Rigorously verified bounds on the optimum, provided an optimum exists: [ -4.00 , -3.12 ]