topic Re: proc optmodel WARNING: Objective function cannot be evaluated at starting point. NOTE: Did not c in Mathematical Optimization, Discrete-Event Simulation, and OR

proc optmodel WARNING: Objective function cannot be evaluated at starting point. NOTE: Did not conve

mathmannix — Wed, 24 Oct 2018 20:00:15 GMT

[rge, obviously].

I am trying to learn and use proc optmodel (replacing an older program someone else wrote with proc nlp, which didn't really work as desired anyway.)

Am I doing this at all correctly?

Here is my code:

proc optmodel;

     /*decision variables*/
	var n {1..&n};
	var denom;
	var sum_n;

	number pop {1..&n};
	number p_rate {1..&n};

	read data ced_rates into [_n_] pop p_rate;

	denom = sum{k in 1..&n} n[k] * p_rate[k];

	minimize sum_sqrs = sum {j in 1..&n.} ((n[j] * p_rate[j] / denom) - (pop[j] / &usa_population.))**2;

	sum_n = sum{k in 1..&n} n[k];

     /*problem constraints*/
	con	sum_n = &n_usa.,
		p_rate[1] * n[1] >=	&n_min_urban.,
		p_rate[2] * n[2] >=	&n_min_urban.,
		p_rate[3] * n[3] >=	&n_min_urban.,
		p_rate[4] * n[4] >=	&n_min_urban.,
		p_rate[5] * n[5] >=	&n_min_urban.,
		p_rate[6] * n[6] >=	&n_min_urban.,
		p_rate[7] * n[7] >=	&n_min_urban.,
		p_rate[8] * n[8] >=	&n_min_urban.,
		p_rate[9] * n[9] >=	&n_min_urban.,

		p_rate[10]*n[10] >=	&n_min_rural.,
		p_rate[11]*n[11] >=	&n_min_rural.,
		p_rate[12]*n[12] >=	&n_min_rural.,
		p_rate[13]*n[13] >=	&n_min_rural.,
		p_rate[14]*n[14] >=	&n_min_rural.,
		p_rate[15]*n[15] >=	&n_min_rural.,
		p_rate[16]*n[16] >=	&n_min_rural.,
		p_rate[17]*n[17] >=	&n_min_rural.,
		p_rate[18]*n[18] >=	&n_min_rural.,

		p_rate[19]*n[19] >=	&n_min_urban.,
		p_rate[20]*n[20] >=	&n_min_urban.,
		p_rate[21]*n[21] >=	&n_min_urban.,
		p_rate[22]*n[22] >=	&n_min_urban.,
		p_rate[23]*n[23] >=	&n_min_urban.,
		p_rate[24]*n[24] >=	&n_min_urban.,
		p_rate[25]*n[25] >=	&n_min_urban.,
		p_rate[26]*n[26] >=	&n_min_urban.,
		p_rate[27]*n[27] >=	&n_min_urban.,
		p_rate[28]*n[28] >=	&n_min_urban.,
		p_rate[29]*n[29] >=	&n_min_urban.,
		p_rate[30]*n[30] >=	&n_min_urban.,
		p_rate[31]*n[31] >=	&n_min_urban.,
		p_rate[32]*n[32] >=	&n_min_urban.,
		p_rate[33]*n[33] >=	&n_min_urban.,
		p_rate[34]*n[34] >=	&n_min_urban.,
		p_rate[35]*n[35] >=	&n_min_urban.,
		p_rate[36]*n[36] >=	&n_min_urban.,
		p_rate[37]*n[37] >=	&n_min_urban.,
		p_rate[38]*n[38] >=	&n_min_urban.,
		p_rate[39]*n[39] >=	&n_min_urban.,
		p_rate[40]*n[40] >=	&n_min_urban.,
		p_rate[41]*n[41] >=	&n_min_urban.;

	solve;
	print n;
	print sum_sqrs;

quit;

(I wanted the number of constraints to be dynamic (&n. = 41), but I couldn't figure out a way to do that.)

I guess the major problem is that the log shows

WARNING: Objective function cannot be evaluated at starting point.
NOTE: Did not converge.

and then a bunch of "NOTE: Division by zero at line"... lines.

Another problem is that, when the program does print out the 41 n values, they do NOT add up to the desired total, &n_usa.

(They are all given values greater than their constraint minimums, so at least that's something.)

The sum_sqrs variable is output as "."

What am I doing wrong? (Am I doing anything right, even?)

Thanks in advance for any help or constructive input.

Re: proc optmodel WARNING: Objective function cannot be evaluated at starting point. NOTE: Did not c

RobPratt — Wed, 24 Oct 2018 20:27:11 GMT

You are probably not solving the problem that you think you are. You might find the EXPAND statement useful in debugging your model. I suspect that you intended the following two lines to be constraints:

denom = sum{k in 1..&n} n[k] * p_rate[k];
sum_n = sum{k in 1..&n} n[k];

As written, they simply update the values of denom and sum_n, but then the solver is free to change those values. Instead, I think you want:

con denomCon:
   denom = sum{k in 1..&n} n[k] * p_rate[k];
con sumCon:
   sum_n = sum{k in 1..&n} n[k];

Also, you can write your family of 41 constraints more compactly as follows:

con pCon {j in 1..&n}:
   p_rate[j] * n[j] >= if j in 10..18 then &n_min_rural. else &n_min_urban.;

Alternatively, if p_rate[j] is known to be positive, you can omit these explicit constraints and instead modify the lower bound of n[j]:

for {j in 1..&n}
   n[j].lb = (if j in 10..18 then &n_min_rural. else &n_min_urban.) / p_rate[j];

Re: proc optmodel WARNING: Objective function cannot be evaluated at starting point. NOTE: Did not c

mathmannix — Thu, 25 Oct 2018 13:01:05 GMT

OK, thank you for your help so far, but I still get the same [lack of] results:

                                                    Solution Summary

                                          Solver                           NLP
                                          Algorithm             Interior Point
                                          Objective Function          sum_sqrs
                                          Solution Status               Failed
                                          Objective Value                    .

                                          Optimality Error                   .
                                          Infeasibility                      0

                                          Iterations                         0
                                          Presolve Time                   0.00
                                          Solution Time                   0.00


                                                        sum_sqrs

                                                               .

(%n n values omitted from results)

Excerpt from Log:

NOTE: Problem generation will use 4 threads.
NOTE: Previous errors might cause the problem to be resolved incorrectly.
NOTE: The problem has 43 variables (43 free, 0 fixed).
NOTE: The problem has 43 linear constraints (0 LE, 2 EQ, 41 GE, 0 range).
NOTE: The problem has 125 linear constraint coefficients.
NOTE: The problem has 0 nonlinear constraints (0 LE, 0 EQ, 0 GE, 0 range).
NOTE: The OPTMODEL presolver removed 0 variables, 41 linear constraints, and 0 nonlinear constraints.
NOTE: The OPTMODEL presolved problem has 43 variables, 2 linear constraints, and 0 nonlinear constraints.
NOTE: The OPTMODEL presolver removed 41 linear constraint coefficients, leaving 84.
NOTE: Using analytic derivatives for objective.
NOTE: Using 2 threads for nonlinear evaluation.
NOTE: The NLP solver is called.
NOTE: The Interior Point algorithm is used.
WARNING: Objective function cannot be evaluated at starting point.
NOTE: Did not converge.
15315      print n;
15316      print sum_sqrs;
NOTE: Division by zero at line 15249 column 63.

Why is it not converging?

Re: proc optmodel WARNING: Objective function cannot be evaluated at starting point. NOTE: Did not c

RobPratt — Thu, 25 Oct 2018 14:03:02 GMT

I need more information to be able to diagnose this. But you have some error before the log that you posted:

NOTE: Previous errors might cause the problem to be resolved incorrectly.

Can you please share your code and data?

Re: proc optmodel WARNING: Objective function cannot be evaluated at starting point. NOTE: Did not c

mathmannix — Thu, 25 Oct 2018 15:36:12 GMT

Oops, I didn't realize I had copied from an erroneous run when I tried to change a little bit of the code.

Here is my current PROC OPTMODEL and the results:

proc optmodel;

     /*decision variables*/
	var n {1..&n};
	var denom;
	var sum_n;

	number pop {1..&n};
	number p_rate {1..&n};

	read data ced_rates into [_n_] pop p_rate;

	 /*objective function*/
	minimize sum_sqrs = sum {j in 1..&n.} (((n[j] * p_rate[j] / denom) - (pop[j] / &usa_population.))**2);

     /*problem constraints*/
	con denomCon:
		denom = sum{k in 1..&n} (n[k] * p_rate[k]);

	con sumCon:
		sum_n = sum{k in 1..&n} n[k],
		sum_n = &n_usa.;

	con pCon {j in 1..&n}:
		p_rate[j] * n[j] >=
			if j in 10..18 then &n_min_rural. 
				else &n_min_urban.;

	solve;
	print n;
	print sum_sqrs;

quit;

                                                 The OPTMODEL Procedure

                                                    Solution Summary

                                          Solver                           NLP
                                          Algorithm             Interior Point
                                          Objective Function          sum_sqrs
                                          Solution Status               Failed
                                          Objective Value                    .

                                          Optimality Error                   .
                                          Infeasibility                  12000

                                          Iterations                         0
                                          Presolve Time                   0.00
                                          Solution Time                   0.00


 [1] n

 1 119.271
 2 140.021
(...)
 40 135.444
 41 140.301


 sum_sqrs

 .

18421      solve;
NOTE: Problem generation will use 4 threads.
NOTE: The problem has 43 variables (43 free, 0 fixed).
NOTE: The problem has 44 linear constraints (0 LE, 3 EQ, 41 GE, 0 range).
NOTE: The problem has 126 linear constraint coefficients.
NOTE: The problem has 0 nonlinear constraints (0 LE, 0 EQ, 0 GE, 0 range).
NOTE: The OPTMODEL presolver removed 1 variables, 42 linear constraints, and 0 nonlinear constraints.
NOTE: The OPTMODEL presolved problem has 42 variables, 2 linear constraints, and 0 nonlinear constraints.
NOTE: The OPTMODEL presolver removed 43 linear constraint coefficients, leaving 83.
NOTE: Using analytic derivatives for objective.
NOTE: Using 2 threads for nonlinear evaluation.
NOTE: The NLP solver is called.
NOTE: The Interior Point algorithm is used.
WARNING: Objective function cannot be evaluated at starting point.
NOTE: Did not converge.
18422      print n;
18423      print sum_sqrs;
NOTE: Division by zero at line 18356 column 63.
NOTE: Division by zero at line 18356 column 63.
NOTE: Division by zero at line 18356 column 63.
NOTE: Division by zero at line 18356 column 63.
NOTE: Division by zero at line 18356 column 63.
NOTE: Division by zero at line 18356 column 63.
NOTE: Division by zero at line 18356 column 63.
NOTE: Division by zero at line 18356 column 63.
NOTE: Division by zero at line 18356 column 63.
NOTE: Division by zero at line 18356 column 63.
NOTE: Division by zero at line 18356 column 63.
NOTE: Division by zero at line 18356 column 63.
NOTE: Division by zero at line 18356 column 63.
NOTE: Division by zero at line 18356 column 63.
NOTE: Division by zero at line 18356 column 63.
NOTE: Division by zero at line 18356 column 63.
NOTE: Division by zero at line 18356 column 63.
NOTE: Division by zero at line 18356 column 63.
NOTE: Division by zero at line 18356 column 63.
NOTE: Division by zero at line 18356 column 63.
NOTE: Division by zero at line 18356 column 63.
NOTE: Division by zero at line 18356 column 63.
NOTE: Division by zero at line 18356 column 63.
NOTE: Division by zero at line 18356 column 63.
NOTE: Division by zero at line 18356 column 63.
NOTE: The maximum message limit was reached during execution of the statement block. 16 notes and warnings were not
      displayed.
18424
18425  quit;
NOTE: PROCEDURE OPTMODEL used (Total process time):
      real time           0.12 seconds
      cpu time            0.06 seconds

Re: proc optmodel WARNING: Objective function cannot be evaluated at starting point. NOTE: Did not c

RobPratt — Thu, 25 Oct 2018 15:45:34 GMT

The default initial value for each variable is 0. Because denom appears in a denominator of the objective function, you get a division by zero error. One thing you can try is initializing denom to something nonzero, say 1. You can do this during declaration:

var denom init 1;

Or with an assignment statement before the solve:

denom = 1;

If that doesn't help, please share your data set and macro variables so that I can diagnose further.

Re: proc optmodel WARNING: Objective function cannot be evaluated at starting point. NOTE: Did not c

mathmannix — Thu, 25 Oct 2018 18:34:21 GMT

The program "works". That is, it compiles and presents a "feasible" solution (one which fits the constraints I gave it), but the solution is not at all realistic. Furthermore, it is highly subjective to initial constraints (such as changing the var sum_n line from the default of 0 to 1 or 10 or 308745538). It also provides a quite different solution if I specify PRESOLVE=BASIC rather than the default, but not one I like much better.

Shouldn't the sum of squares be a convex function, with only one local minimum?

Now I'm just not sure what to do, because my program seems to work syntactically, but I just don't like its output.

Re: proc optmodel WARNING: Objective function cannot be evaluated at starting point. NOTE: Did not c

mathmannix — Thu, 25 Oct 2018 18:37:24 GMT

Unfortunately, I don't think I (should / am authorized to / legally can ?) publicly post my data set and macro variables.

Re: proc optmodel WARNING: Objective function cannot be evaluated at starting point. NOTE: Did not c

RobPratt — Thu, 25 Oct 2018 19:07:29 GMT

Yes, the sum of squares of convex functions is convex, but the expression you are squaring is not convex. As a simpler example, consider (sqrt(f(x)))^2 + (sqrt(f(x)))^2, where f(x) is any nonconvex function.

To increase the likelihood of finding a globally optimal solution for a nonconvex minimization (or nonconcave maximization) problem, you can use the MULTISTART option:

solve with nlp / multistart;

Also, you can simplify the code a bit by omitting the sum_n variable and merging two of the constraints into one:

con sumCon:
   sum{k in 1..&n} n[k] = &n_usa.;

From your log, it looks like the presolver is doing that anyway.