Below is the raw output from the NEOS server, using IPOPT to solve the nonlinear program. At the end the following information was printed out: x1 = 3.16778e-09 x2 = -2 conQ = 0.105263 conM = 0.0263158 conQ.slack = -3.62321e-08 conM.slack = -4.44165e-08 c1 + conM*(M11*x1 + M12*x2) + conQ*(Q11*x1 + Q12*x2) = 7.95544e-12 c2 + conM*(M21*x1 + M22*x2) + conQ*(Q21*x1 + Q22*x2) = 5.42855e-12 c1*x1 + c2*x2 + conM*(0.5*(x1^2*M11 + 2*x1*x2*M12 + x2^2*M22) - 14) + conQ*( 0.5*(x1^2*Q11 + 2*x1*x2*Q12 + x2^2*Q22) - 6) = -2 This shows the optimal solution is x=(0,-2), the KKT multipliers are 0.105263 and 0.026318, the KKT gradient conditions hold to a tolerance of 1e-12, and the value of the Lagrangian is equal to the value of the objective function, namely -2. ************************************************************* NEOS Server Version 5.0 Job# : 1951738 Password : LpFwXuxZ Solver : nco:Ipopt:AMPL Start : 2009-10-23 15:54:07 End : 2009-10-23 15:54:20 Host : point.ie.lehigh.edu Disclaimer: This information is provided without any express or implied warranty. In particular, there is no warranty of any kind concerning the fitness of this information for any particular purpose. ************************************************************* Job 1951738 sent to point.ie.lehigh.edu password: LpFwXuxZ ---------- Begin Solver Output ----------- Executing /home/neos/neos-solvers/ipopt-ampl/ipopt-ampl-driver.py Load Avg: ( 0.0 , 0.0 , 0.03 ) File exists You are using the solver ipopt. Executing AMPL. processing data. processing commands. 2 variables, all nonlinear 2 constraints, all nonlinear; 4 nonzeros 1 linear objective; 1 nonzero. Ipopt 3.3.3: ****************************************************************************** This program contains Ipopt, a library for large-scale nonlinear optimization. Ipopt is released as open source code under the Common Public License (CPL). For more information visit http://projects.coin-or.org/Ipopt ****************************************************************************** Number of nonzeros in equality constraint Jacobian...: 0 Number of nonzeros in inequality constraint Jacobian.: 4 Number of nonzeros in Lagrangian Hessian.............: 3 Total number of variables............................: 2 variables with only lower bounds: 0 variables with lower and upper bounds: 0 variables with only upper bounds: 0 Total number of equality constraints.................: 0 Total number of inequality constraints...............: 2 inequality constraints with only lower bounds: 0 inequality constraints with lower and upper bounds: 0 inequality constraints with only upper bounds: 2 iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls 0 0.0000000e+00 0.00e+00 1.00e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0 1 -1.0596026e-01 3.34e-02 9.90e-01 -1.0 1.06e-01 - 9.97e-01 1.00e+00f 1 2 -1.0114615e+01 3.22e+02 1.27e-01 -1.7 1.00e+01 - 1.00e+00 1.00e+00f 1 3 -6.0883538e+00 1.06e+02 2.01e-01 -1.7 8.90e+00 - 1.00e+00 8.31e-01h 1 4 -3.3338134e+00 2.43e+01 2.54e-01 -1.7 2.75e+00 - 1.00e+00 1.00e+00h 1 5 -2.2656600e+00 3.52e+00 1.57e-01 -1.7 1.76e+00 - 1.00e+00 1.00e+00h 1 6 -1.9711162e+00 2.57e-01 3.52e-02 -1.7 5.91e-01 - 1.00e+00 1.00e+00h 1 7 -1.9612328e+00 6.34e-04 1.80e-04 -1.7 1.03e-01 - 1.00e+00 1.00e+00h 1 8 -1.9991901e+00 1.04e-02 6.51e-04 -3.8 6.65e-01 - 1.00e+00 1.00e+00f 1 9 -1.9996980e+00 6.36e-05 1.85e-06 -3.8 4.10e-02 - 1.00e+00 1.00e+00h 1 iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls 10 -1.9999963e+00 7.87e-07 4.91e-08 -5.7 5.79e-03 - 1.00e+00 1.00e+00h 1 11 -2.0000000e+00 1.27e-10 7.96e-12 -8.6 7.31e-05 - 1.00e+00 1.00e+00h 1 Number of Iterations....: 11 (scaled) (unscaled) Objective...............: -2.0000000049827640e+00 -2.0000000049827640e+00 Dual infeasibility......: 7.9554418608296373e-12 7.9554418608296373e-12 Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00 Complementarity.........: 2.5187084792608655e-09 2.5187084792608655e-09 Overall NLP error.......: 2.5187084792608655e-09 2.5187084792608655e-09 Number of objective function evaluations = 12 Number of objective gradient evaluations = 12 Number of equality constraint evaluations = 0 Number of inequality constraint evaluations = 12 Number of equality constraint Jacobian evaluations = 0 Number of inequality constraint Jacobian evaluations = 12 Number of Lagrangian Hessian evaluations = 11 Total CPU secs in IPOPT (w/o function evaluations) = 0.010 Total CPU secs in NLP function evaluations = 0.000 EXIT: Optimal Sol Ipopt 3.3.3: Optimal Solution Found suffix ipopt_zU_out OUT; suffix ipopt_zL_out OUT; x1 = 3.16778e-09 x2 = -2 conQ = 0.105263 conM = 0.0263158 conQ.slack = -3.62321e-08 conM.slack = -4.44165e-08 c1 + conM*(M11*x1 + M12*x2) + conQ*(Q11*x1 + Q12*x2) = 7.95544e-12 c2 + conM*(M21*x1 + M22*x2) + conQ*(Q21*x1 + Q22*x2) = 5.42855e-12 c1*x1 + c2*x2 + conM*(0.5*(x1^2*M11 + 2*x1*x2*M12 + x2^2*M22) - 14) + conQ*( 0.5*(x1^2*Q11 + 2*x1*x2*Q12 + x2^2*Q22) - 6) = -2