Optimizing Non-Linear Function using nloptr - Unexpected Results

Question

When I try to maximize my objective function using nloptr, it is terminating on the initial values that I have set, which I think is very unlikely to return the maximum.

For some background, I am trying to allocate 80M of investment (x) between 114 products in a way that maximizes revenue (y). Predicted revenue for each product and investment level is based off a two-part model i.e. it is the result of multiplying the predicted probability of any revenue being generated, by the predicted level of revenue, conditional on revenue being generated. Each line of the objective function represents the fitted model for each product.

library(nloptr)

# objective function
eval_f0 <- function(x){
  return(-(plogis(0.872 + 0.0000471*x[1]-1.04483)*exp(8.459+0.468*log(x[1])+1.07743) +
             plogis(0.872 + 0.0000471*x[2]-1.04483)*exp(8.459+0.468*log(x[2])+1.07743) +
             plogis(0.872 + 0.0000471*x[3]-1.04483)*exp(8.459+0.468*log(x[3])+1.07743) +
             plogis(0.872 + 0.0000471*x[4]-1.04483)*exp(8.459+0.468*log(x[4])+1.07743) +
             plogis(0.872 + 0.0000471*x[5]-1.04483)*exp(8.459+0.468*log(x[5])+1.07743) +
             plogis(0.872 + 0.0000471*x[6]-1.04483)*exp(8.459+0.468*log(x[6])+1.07743) +
             plogis(0.872 + 0.0000471*x[7]-1.04483)*exp(8.459+0.468*log(x[7])+1.07743) +
             plogis(0.872 + 0.0000471*x[8]-1.04483)*exp(8.459+0.468*log(x[8])+1.07743) +
             plogis(0.872 + 0.0000471*x[9]-1.04483)*exp(8.459+0.468*log(x[9])+1.07743) +
             plogis(0.872 + 0.0000471*x[10]-1.04483)*exp(8.459+0.468*log(x[10])+1.07743) +
             plogis(0.872 + 0.0000471*x[11]-1.04483)*exp(8.459+0.468*log(x[11])+1.07743) +
             plogis(0.872 + 0.0000471*x[12]-1.04483)*exp(8.459+0.468*log(x[12])+1.07743) +
             plogis(0.872 + 0.0000471*x[13]-1.04483)*exp(8.459+0.468*log(x[13])+1.07743) +
             plogis(0.872 + 0.0000471*x[14]-1.04483)*exp(8.459+0.468*log(x[14])+1.07743) +
             plogis(0.872 + 0.0000471*x[15]-1.04483)*exp(8.459+0.468*log(x[15])+1.07743) +
             plogis(0.872 + 0.0000471*x[16]-1.04483)*exp(8.459+0.468*log(x[16])+1.07743) +
             plogis(0.872 + 0.0000471*x[17]-1.04483)*exp(8.459+0.468*log(x[17])+1.07743) +
             plogis(0.872 + 0.0000471*x[18]-1.04483)*exp(8.459+0.468*log(x[18])+1.07743) +
             plogis(0.872 + 0.0000471*x[19]-1.04483)*exp(8.459+0.468*log(x[19])+1.07743) +
             plogis(0.872 + 0.0000471*x[20]-1.04483)*exp(8.459+0.468*log(x[20])+1.07743) +
             plogis(0.872 + 0.0000471*x[21]-1.04483)*exp(8.459+0.468*log(x[21])+1.07743) +
             plogis(0.872 + 0.0000471*x[22]-1.04483)*exp(8.459+0.468*log(x[22])+1.07743) +
             plogis(0.872 + 0.0000471*x[23]-1.04483)*exp(8.459+0.468*log(x[23])+1.07743) +
             plogis(0.872 + 0.0000471*x[24]-1.04483)*exp(8.459+0.468*log(x[24])+1.07743) +
             plogis(0.872 + 0.0000471*x[25]-1.04483)*exp(8.459+0.468*log(x[25])+1.07743) +
             plogis(0.872 + 0.0000471*x[26]-1.04483)*exp(8.459+0.468*log(x[26])+1.07743) +
             plogis(0.872 + 0.0000471*x[27]-1.04483)*exp(8.459+0.468*log(x[27])+1.07743) +
             plogis(0.872 + 0.0000471*x[28]-1.04483)*exp(8.459+0.468*log(x[28])+1.07743) +
             plogis(0.872 + 0.0000471*x[29]-1.04483)*exp(8.459+0.468*log(x[29])+1.07743) +
             plogis(0.872 + 0.0000471*x[30]-1.04483)*exp(8.459+0.468*log(x[30])+1.07743) +
             plogis(0.872 + 0.0000471*x[31]-1.04483)*exp(8.459+0.468*log(x[31])+1.07743) +
             plogis(0.872 + 0.0000471*x[32]-1.04483)*exp(8.459+0.468*log(x[32])+1.07743) +
             plogis(0.872 + 0.0000471*x[33]-1.04483)*exp(8.459+0.468*log(x[33])+1.07743) +
             plogis(0.872 + 0.0000471*x[34]-1.04483)*exp(8.459+0.468*log(x[34])+1.07743) +
             plogis(0.872 + 0.0000471*x[35]-1.04483+0.289)*exp(8.459+0.468*log(x[35])+1.07743-0.83) +
             plogis(0.872 + 0.0000471*x[36]-1.04483+0.289)*exp(8.459+0.468*log(x[36])+1.07743-0.83) +
             plogis(0.872 + 0.0000471*x[37]-1.04483+0.289)*exp(8.459+0.468*log(x[37])+1.07743-0.83) +
             plogis(0.872 + 0.0000471*x[38]-1.04483+0.289)*exp(8.459+0.468*log(x[38])+1.07743-0.83) +
             plogis(0.872 + 0.0000471*x[39]-1.04483+0.289)*exp(8.459+0.468*log(x[39])+1.07743-0.83) +
             plogis(0.872 + 0.0000471*x[40]-1.04483+0.289)*exp(8.459+0.468*log(x[40])+1.07743-0.83) +
             plogis(0.872 + 0.0000471*x[41]-1.04483+0.289)*exp(8.459+0.468*log(x[41])+1.07743-0.83) +
             plogis(0.872 + 0.0000471*x[42]-1.04483+0.289)*exp(8.459+0.468*log(x[42])+1.07743-0.83) +
             plogis(0.872 + 0.0000471*x[43]-1.04483+0.289)*exp(8.459+0.468*log(x[43])+1.07743-0.83) +
             plogis(0.872 + 0.0000471*x[44]-1.04483+0.289)*exp(8.459+0.468*log(x[44])+1.07743-0.83) +
             plogis(0.872 + 0.0000471*x[45]-1.04483+0.289)*exp(8.459+0.468*log(x[45])+1.07743-0.83) +
             plogis(0.872 + 0.0000471*x[46]-1.04483+0.289)*exp(8.459+0.468*log(x[46])+1.07743-0.83) +
             plogis(0.872 + 0.0000471*x[47]-1.04483+0.289)*exp(8.459+0.468*log(x[47])+1.07743-0.83) +
             plogis(0.872 + 0.0000471*x[48]-1.04483+0.289)*exp(8.459+0.468*log(x[48])+1.07743-0.83) +
             plogis(0.872 + 0.0000471*x[49]-1.04483+0.289)*exp(8.459+0.468*log(x[49])+1.07743-0.83) +
             plogis(0.872 + 0.0000471*x[50]-1.04483+0.289)*exp(8.459+0.468*log(x[50])+1.07743-0.83) +
             plogis(0.872 + 0.0000471*x[51]-1.04483+0.289)*exp(8.459+0.468*log(x[51])+1.07743-0.83) +
             plogis(0.872 + 0.0000471*x[52]-1.04483+0.289)*exp(8.459+0.468*log(x[52])+1.07743-0.83) +
             plogis(0.872 + 0.0000471*x[53]-1.04483+0.289)*exp(8.459+0.468*log(x[53])+1.07743-0.83) +
             plogis(0.872 + 0.0000471*x[54]-1.04483+0.289)*exp(8.459+0.468*log(x[54])+1.07743-0.83) +
             plogis(0.872 + 0.0000471*x[55]-1.04483+0.289)*exp(8.459+0.468*log(x[55])+1.07743-0.83) +
             plogis(0.872 + 0.0000471*x[56]-1.04483+0.289)*exp(8.459+0.468*log(x[56])+1.07743-0.83) +
             plogis(0.872 + 0.0000471*x[57]-1.04483+0.289)*exp(8.459+0.468*log(x[57])+1.07743-0.83) +
             plogis(0.872 + 0.0000471*x[58]-1.04483+0.289)*exp(8.459+0.468*log(x[58])+1.07743-0.83) +
             plogis(0.872 + 0.0000471*x[59]-1.04483+0.289)*exp(8.459+0.468*log(x[59])+1.07743-0.83) +
             plogis(0.872 + 0.0000471*x[60]-1.04483+0.289)*exp(8.459+0.468*log(x[60])+1.07743-0.83) +
             plogis(0.872 + 0.0000471*x[61]-1.04483+0.289)*exp(8.459+0.468*log(x[61])+1.07743-0.83) +
             plogis(0.872 + 0.0000471*x[62]-1.04483+0.289)*exp(8.459+0.468*log(x[62])+1.07743-0.83) +
             plogis(0.872 + 0.0000471*x[63]-1.04483+0.289)*exp(8.459+0.468*log(x[63])+1.07743-0.83) +
             plogis(0.872 + 0.0000471*x[64]-1.04483+0.289)*exp(8.459+0.468*log(x[64])+1.07743-0.83) +
             plogis(0.872 + 0.0000471*x[65]-1.04483+0.289)*exp(8.459+0.468*log(x[65])+1.07743-0.83) +
             plogis(0.872 + 0.0000471*x[66]-1.04483+0.289)*exp(8.459+0.468*log(x[66])+1.07743-0.83) +
             plogis(0.872 + 0.0000471*x[67]-1.04483+0.289)*exp(8.459+0.468*log(x[67])+1.07743-0.83) +
             plogis(0.872 + 0.0000471*x[68]-1.04483+0.289)*exp(8.459+0.468*log(x[68])+1.07743-0.83) +
             plogis(0.872 + 0.0000471*x[69]-1.04483+1.081)*exp(8.459+0.468*log(x[69])+1.07743-0.637) +
             plogis(0.872 + 0.0000471*x[70]-1.04483+1.081)*exp(8.459+0.468*log(x[70])+1.07743-0.637) +
             plogis(0.872 + 0.0000471*x[71]-1.04483+1.081)*exp(8.459+0.468*log(x[71])+1.07743-0.637) +
             plogis(0.872 + 0.0000471*x[72]-1.04483+1.081)*exp(8.459+0.468*log(x[72])+1.07743-0.637) +
             plogis(0.872 + 0.0000471*x[73]-1.04483+1.081)*exp(8.459+0.468*log(x[73])+1.07743-0.637) +
             plogis(0.872 + 0.0000471*x[74]-1.04483+1.081)*exp(8.459+0.468*log(x[74])+1.07743-0.637) +
             plogis(0.872 + 0.0000471*x[75]-1.04483+1.081)*exp(8.459+0.468*log(x[75])+1.07743-0.637) +
             plogis(0.872 + 0.0000471*x[76]-1.04483+1.081)*exp(8.459+0.468*log(x[76])+1.07743-0.637) +
             plogis(0.872 + 0.0000471*x[77]-1.04483+1.081)*exp(8.459+0.468*log(x[77])+1.07743-0.637) +
             plogis(0.872 + 0.0000471*x[78]-1.04483+1.081)*exp(8.459+0.468*log(x[78])+1.07743-0.637) +
             plogis(0.872 + 0.0000471*x[79]-1.04483+1.081)*exp(8.459+0.468*log(x[79])+1.07743-0.637) +
             plogis(0.872 + 0.0000471*x[80]-1.04483+1.081)*exp(8.459+0.468*log(x[80])+1.07743-0.637) +
             plogis(0.872 + 0.0000471*x[81]-1.04483+1.081)*exp(8.459+0.468*log(x[81])+1.07743-0.637) +
             plogis(0.872 + 0.0000471*x[82]-1.04483+1.081)*exp(8.459+0.468*log(x[82])+1.07743-0.637) +
             plogis(0.872 + 0.0000471*x[83]-1.04483+1.081)*exp(8.459+0.468*log(x[83])+1.07743-0.637) +
             plogis(0.872 + 0.0000471*x[84]-1.04483+1.081)*exp(8.459+0.468*log(x[84])+1.07743-0.637) +
             plogis(0.872 + 0.0000471*x[85]-1.04483+1.081)*exp(8.459+0.468*log(x[85])+1.07743-0.637) +
             plogis(0.872 + 0.0000471*x[86]-1.04483+1.081)*exp(8.459+0.468*log(x[86])+1.07743-0.637) +
             plogis(0.872 + 0.0000471*x[87]-1.04483+1.081)*exp(8.459+0.468*log(x[87])+1.07743-0.637) +
             plogis(0.872 + 0.0000471*x[88]-1.04483+1.081)*exp(8.459+0.468*log(x[88])+1.07743-0.637) +
             plogis(0.872 + 0.0000471*x[89]-1.04483+1.081)*exp(8.459+0.468*log(x[89])+1.07743-0.637) +
             plogis(0.872 + 0.0000471*x[90]-1.04483+1.081)*exp(8.459+0.468*log(x[90])+1.07743-0.637) +
             plogis(0.872 + 0.0000471*x[91]-1.04483+1.081)*exp(8.459+0.468*log(x[91])+1.07743-0.637) +
             plogis(0.872 + 0.0000471*x[92]-1.04483+1.081)*exp(8.459+0.468*log(x[92])+1.07743-0.637) +
             plogis(0.872 + 0.0000471*x[93]-1.04483+1.081)*exp(8.459+0.468*log(x[93])+1.07743-0.637) +
             plogis(0.872 + 0.0000471*x[94]-1.04483+1.081)*exp(8.459+0.468*log(x[94])+1.07743-0.637) +
             plogis(0.872 + 0.0000471*x[95]-1.04483+1.081)*exp(8.459+0.468*log(x[95])+1.07743-0.637) +
             plogis(0.872 + 0.0000471*x[96]-1.04483+1.081)*exp(8.459+0.468*log(x[96])+1.07743-0.637) +
             plogis(0.872 + 0.0000471*x[97]-1.04483+1.081)*exp(8.459+0.468*log(x[97])+1.07743-0.637) +
             plogis(0.872 + 0.0000471*x[98]-1.04483+1.081)*exp(8.459+0.468*log(x[98])+1.07743-0.637) +
             plogis(0.872 + 0.0000471*x[99]-1.04483+1.081)*exp(8.459+0.468*log(x[99])+1.07743-0.637) +
             plogis(0.872 + 0.0000471*x[100]-1.04483+1.081)*exp(8.459+0.468*log(x[100])+1.07743-0.637) +
             plogis(0.872 + 0.0000471*x[101]-1.04483+1.081)*exp(8.459+0.468*log(x[101])+1.07743-0.637) +
             plogis(0.872 + 0.0000471*x[102]-1.04483+1.081)*exp(8.459+0.468*log(x[102])+1.07743-0.637) +
             plogis(0.872 + 0.0000471*x[103]-1.04483-1.978)*exp(8.459+0.468*log(x[103])+1.07743+0.336) +
             plogis(0.872 + 0.0000471*x[104]-1.04483-1.978)*exp(8.459+0.468*log(x[104])+1.07743+0.336) +
             plogis(0.872 + 0.0000471*x[105]-1.04483-1.978)*exp(8.459+0.468*log(x[105])+1.07743+0.336) +
             plogis(0.872 + 0.0000471*x[106]-1.04483-1.978)*exp(8.459+0.468*log(x[106])+1.07743+0.336) +
             plogis(0.872 + 0.0000471*x[107]-1.04483+0.289-1.978)*exp(8.459+0.468*log(x[107])+1.07743-0.83+0.336) +
             plogis(0.872 + 0.0000471*x[108]-1.04483+0.289-1.978)*exp(8.459+0.468*log(x[108])+1.07743-0.83+0.336) +
             plogis(0.872 + 0.0000471*x[109]-1.04483+0.289-1.978)*exp(8.459+0.468*log(x[109])+1.07743-0.83+0.336) +
             plogis(0.872 + 0.0000471*x[110]-1.04483+0.289-1.978)*exp(8.459+0.468*log(x[110])+1.07743-0.83+0.336) +
             plogis(0.872 + 0.0000471*x[111]-1.04483+1.081-1.978)*exp(8.459+0.468*log(x[111])+1.07743-0.637+0.336) +
             plogis(0.872 + 0.0000471*x[112]-1.04483+1.081-1.978)*exp(8.459+0.468*log(x[112])+1.07743-0.637+0.336) +
             plogis(0.872 + 0.0000471*x[113]-1.04483+1.081-1.978)*exp(8.459+0.468*log(x[113])+1.07743-0.637+0.336) +
             plogis(0.872 + 0.0000471*x[114]-1.04483+1.081-1.978)*exp(8.459+0.468*log(x[114])+1.07743-0.637+0.336)) )
}

# constraint function
eval_g0 <- function(x)
{
  return(x[1]+x[2]+x[3]+x[4]+x[5]+x[6]+x[7]+x[8]+x[9]+x[10]+x[11]+x[12]+x[13]+x[14]+x[15]+x[16]+x[17]+x[18]+x[19]+x[20]+x[21]+x[22]+x[23]+x[24]+x[25]+x[26]+x[27]+x[28]+x[29]+x[30]+x[31]+x[32]+x[33]+x[34]+x[35]+x[36]+x[37]+x[38]+x[39]+x[40]+x[41]+x[42]+x[43]+x[44]+x[45]+x[46]+x[47]+x[48]+x[49]+x[50]+x[51]+x[52]+x[53]+x[54]+x[55]+x[56]+x[57]+x[58]+x[59]+x[60]+x[61]+x[62]+x[63]+x[64]+x[65]+x[66]+x[67]+x[68]+x[69]+x[70]+x[71]+x[72]+x[73]+x[74]+x[75]+x[76]+x[77]+x[78]+x[79]+x[80]+x[81]+x[82]+x[83]+x[84]+x[85]+x[86]+x[87]+x[88]+x[89]+x[90]+x[91]+x[92]+x[93]+x[94]+x[95]+x[96]+x[97]+x[98]+x[99]+x[100]+x[101]+x[102]+x[103]+x[104]+x[105]+x[106]+x[107]+x[108]+x[109]+x[110]+x[111]+x[112]+x[113]+x[114]-80000)
}

# Set optimization options.
opts <- list( "algorithm"= "NLOPT_GN_ISRES",
              "xtol_rel"= 1.0e-7,
              "maxeval"= 760000,
              "print_level" = 0 )

# Solve 
res1 <- nloptr( x0=c(500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500,500),
                eval_f=eval_f0,
                lb = c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
                ub = c(30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000,30000),
                eval_g_ineq = eval_g0,
                opts = opts)
print(res1)

The results are below:

Minimization using NLopt version 2.4.2 

NLopt solver status: 5 ( NLOPT_MAXEVAL_REACHED: Optimization stopped because maxeval (above) was 
reached. )

Number of Iterations....: 760000 
Termination conditions:  xtol_rel: 0.000000000000001    maxeval: 760000 
Number of inequality constraints:  1 
Number of equality constraints:    0 
Current value of objective function:  -4536536.34788097 
Current value of controls: 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 
100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 
100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 
100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 
100 100 100 100 100 100 100 100 100 100 100 100 100 100

Could anyone take a look at my code and let me know whether I am doing anything wrong? Do I need to specify more iterations?

Edit: As @AirSquid suggested, I tried cutting the number of options down to 6 instead and that seems to have worked. So I need to understand why it doesn't work for a larger number of options

Answer 1

This is an extremely easy linear constrained problem. A solver like IPOPT solves this in no time: 0.004 seconds, 8 iterations (I used your exact formulation, with a lower bound on the variables of 0.0001 to protect the log -- I hope I transcribed the problem correctly). Note that nloptr contains largely somewhat simple algorithms. So I suggest picking a better solver.

This is Ipopt version 3.14.6, running with linear solver ma27.

Number of nonzeros in equality constraint Jacobian...:        0
Number of nonzeros in inequality constraint Jacobian.:      114
Number of nonzeros in Lagrangian Hessian.............:      114

Total number of variables............................:      114
                     variables with only lower bounds:        0
                variables with lower and upper bounds:      114
                     variables with only upper bounds:        0
Total number of equality constraints.................:        0
Total number of inequality constraints...............:        1
        inequality constraints with only lower bounds:        0
   inequality constraints with lower and upper bounds:        0
        inequality constraints with only upper bounds:        1

iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
   0 -1.6299318e+07 0.00e+00 5.96e+01   0.0 0.00e+00    -  0.00e+00 0.00e+00   0
   1 -1.9108212e+07 0.00e+00 5.08e+01   3.0 9.26e+04    -  1.00e+00 2.46e-01f  1
   2 -1.9489289e+07 0.00e+00 4.44e+01   2.0 1.58e+03    -  1.00e+00 1.44e-01f  1
   3 -2.0776301e+07 0.00e+00 2.03e+01   1.4 9.55e+02    -  1.00e+00 1.00e+00f  1
   4 -2.0890126e+07 0.00e+00 6.06e+00  -0.1 9.27e+02    -  1.00e+00 8.42e-01f  1
   5 -2.0908516e+07 0.00e+00 5.78e-01  -1.4 5.14e+02    -  9.89e-01 1.00e+00f  1
   6 -2.0908784e+07 0.00e+00 8.26e-03  -3.0 7.22e+01    -  9.97e-01 1.00e+00f  1
   7 -2.0908784e+07 0.00e+00 6.72e-06  -4.9 1.05e+00    -  9.96e-01 1.00e+00f  1
   8 -2.0908784e+07 0.00e+00 1.54e-10 -10.7 2.17e-04    -  1.00e+00 1.00e+00f  1

Number of Iterations....: 8

                                   (scaled)                 (unscaled)
Objective...............:  -6.4238210381431347e+06   -2.0908784212199997e+07
Dual infeasibility......:   1.5371171002698247e-10    5.0031359167058148e-10
Constraint violation....:   0.0000000000000000e+00    0.0000000000000000e+00
Variable bound violation:   0.0000000000000000e+00    0.0000000000000000e+00
Complementarity.........:   5.5985347935841295e-10    1.8222574260471783e-09
Overall NLP error.......:   5.5985347935841295e-10    1.8222574260471783e-09


Number of objective function evaluations             = 9
Number of objective gradient evaluations             = 9
Number of equality constraint evaluations            = 0
Number of inequality constraint evaluations          = 9
Number of equality constraint Jacobian evaluations   = 0
Number of inequality constraint Jacobian evaluations = 9
Number of Lagrangian Hessian evaluations             = 8
Total seconds in IPOPT                               = 0.004

EXIT: Optimal Solution Found.