# Multistart

Nonlinear optimization problems are often nonconvex due to the
objective function, constraint functions, or both. When this is true, there
may be many points that satisfy the local optimality conditions.
Default Knitro behavior is to return the
first locally optimal point found. Knitro offers a simple
*multi-start* feature that searches for a better optimal point by
restarting Knitro from different initial points. The feature is
enabled by setting `ms_enable`

= 1.

Note

In many cases the user would like to obtain the global optimum to the optimization problem; that is, the local optimum with the very best objective function value. Knitro cannot guarantee that multi-start will find the global optimum. In general, the global optimum can only be found with special knowledge of the objective and constraint functions; for example, the functions may need to be bounded by other piece-wise convex functions. Knitro executes with very little information about functional form. Although no guarantee can be made, the probability of finding a better local solution improves if more start points are tried.

## Multistart algorithm

The multi-start procedure generates new start points by randomly selecting
components of *x* that satisfy lower and upper bounds on the variables.
Knitro finds a local optimum from each start point using the same
problem definition and user options.
The final solution returned from `KTR_solve()`

is the local optimum
with the best objective function value if any local optima have been found.
If no local optimum has been found, Knitro will return the best
feasible solution estimate it found. If no feasible solution estimate has
been found, Knitro will return the least infeasible point.

## Parallel multistart

The multistart procedure can run in parallel on shared memory
multi-processor machines by setting `par_numthreads`

greater
than 1. See Parallelism for more details on controlling
parallel performance in Knitro.

When the multistart procedure is run in parallel, Knitro will produce the same sequence of initial points and solves that you see when running multistart sequentially (though, perhaps, not in the same order).

Therefore, as long as you run multistart to completion (`ms_terminate`

=0)
and use the deterministic option (`ms_deterministic`

=1),
you should visit
the same initial points encountered when running multistart sequentially,
and get the same final solution. By default `ms_terminate`

=0
and `ms_deterministic`

=1 so that the parallel multistart produces
the same solution as the sequential multistart.

However, if `ms_deterministic`

=0, or `ms_terminate`

>0,
there is no guarantee that the final solution reported by multistart will
be the same when run in parallel compared to the solution when run sequentially,
and even the parallel solution may change when run at different times.

The option `par_msnumthreads`

can be used to set the number
of threads used by the multistart procedure. For instance, if
`par_numthreads`

=16 and `par_msnumthreads`

=8,
Knitro will run 8 solves in parallel and each solve will be
allocated 2 threads.

## Multistart output

For multistart, you can have output from each local solve written to a file
named `knitro_ms_x.log`

where “x” is the solve number by setting
the option `ms_outsub=1`

.

## Multistart options

The multi-start option is convenient for conducting a simple search for
a better solution point. Search time is improved if the variable bounds
are made as tight as possible, confining the search to a region where a
good solution is likely to be found. The user can restrict the multi-start
search region without altering bounds by using the options
`ms_maxbndrange`

and `ms_startptrange`

. The other multi-start
options are the following.

Option | Meaning |
---|---|

`ms_deterministic` |
Control whether to use deterministic multistart |

`ms_enable` |
Enable multistart |

`ms_maxbndrange` |
Maximum unbounded variable range for multistart |

`ms_maxsolves` |
Maximum Knitro solves for multistart |

`ms_maxtime_cpu` |
Maximum CPU time for multistart, in seconds |

`ms_maxtime_real` |
Maximum real time for multistart, in seconds |

`ms_num_to_save` |
Feasible points to save from multistart |

`ms_outsub` |
Can write each solve to a file (parallel only) |

`ms_savetol` |
Tol for feasible points being equal |

`ms_seed` |
Initial seed for generating random start points |

`ms_startptrange` |
Maximum variable range for multistart |

`ms_terminate` |
Termination condition for multistart |

The number of start points tried by multi-start is specified with the
option `ms_maxsolves`

. By default, Knitro will try
*min(200, 10*n)*, where *n* is the number of variables in the problem.
Users may override the default by setting `ms_maxsolves`

to
a specific value.

The `ms_maxbndrange`

option applies to variables unbounded in at least one direction
(i.e., the upper or lower bound, or both, is infinite)
and keeps new start points within a total range equal to the value of
`ms_maxbndrange`

.
The `ms_startptrange`

option applies to all variables and keeps new start points within
a total range equal to the value of `ms_startptrange`

, overruling
`ms_maxbndrange`

if it is a tighter bound.
In general, use `ms_startptrange`

to limit the multi-start search
only if the initial start point supplied by the user is known to be the center
of a desired search area. Use `ms_maxbndrange`

as a surrogate bound
to limit the multi-start search when a variable is unbounded.

The `ms_num_to_save`

option
allows a specific number of distinct feasible points to be saved in a file named
`knitro_mspoints.log`

. Each point results from a Knitro solve
from a different starting point, and must satisfy the absolute and relative
feasibility tolerances. Different start points may return the same feasible
point, and the file contains only distinct points. The option
`ms_savetol`

determines that two points are distinct if their
objectives or any solution components (including Lagrange multipliers)
are separated by more than the value of
`ms_savetol`

using a relative tolerance test.
More specifically, two values *x* and *y* are considered distinct if:

The file stores points in order from best objective to worst. If objectives
are the same (as defined by `ms_savetol`

), then points are
ordered from smallest feasibility error to largest. The file can be read
manually, but conforms to a fixed property/value format for machine reading.

Instead of using multi-start to search for a global solution, a user may want to
use multi-start as a mechanism for finding any locally optimal or feasible solution
estimate of a nonconvex problem and terminate as soon as one such point is found.
The `ms_terminate`

option, provides the user more control over when to terminate
the multi-start procedure.

If `ms_terminate`

= *optimal* the multi-start procedure will stop as soon as
the first locally optimal solution is found or after `ms_maxsolves`

– whichever comes first. If `ms_terminate`

= *feasible* the multi-start
procedure will instead stop as soon as the first feasible solution estimate is found
or after `ms_maxsolves`

– whichever comes first. If
`ms_terminate`

= *maxsolves*, it will only terminate after `ms_maxsolves`

.

The option `ms_seed`

can be used to change the seed used to generate the random
initial points for multistart.

## Multistart callbacks

The multistart procedure provides two callback utilities for the callable library API.

```
int KNITRO_API KTR_set_ms_process_callback (KTR_context_ptr kc,
KTR_callback * const fnPtr);
int KNITRO_API KTR_set_ms_initpt_callback (KTR_context_ptr kc,
KTR_ms_initpt_callback * const fnPtr);
```

The first callback can be used to perform some user task after each multistart solve
and is set by calling `KTR_set_ms_process_callback()`

. You can use the second
callback to specify your own initial points for multistart instead of using the
randomly generated Knitro initial points. This callback function can be set through the
function `KTR_set_ms_initpt_callback()`

.

See the Knitro API section in the Reference Manual for details on setting these callback functions and the prototypes for these callback functions.

In the object-oriented interface, the following functions are used to set the callbacks:

```
void KTRProblem::setMSProcessCallback(KTRMSProcessCallback* MSProcessCallback);
void KTRProblem::setMSInitPtCallback(KTRMSInitptCallback* MSInitPtCallback);
```

See the Object-oriented interface reference section for details on setting these callback functions in the object-oriented interface.

## AMPL example

Let us consider again our AMPL example from Section Getting started with AMPL and run it with a different set of options:

1 2 3 4 5 | ```
ampl: reset;
ampl: option solver knitroampl;
ampl: option knitro_options "ms_enable=1 ms_num_to_save=5 ms_savetol=0.01";
ampl: model testproblem.mod;
ampl: solve;
``` |

The Knitro log printed on screen changes to reflect the results of the many solver runs that were made during the multistart procedure, and the very end of this log reads:

```
Multistart stopping, reached ms_maxsolves limit.
MULTISTART: Best locally optimal point is returned.
EXIT: Locally optimal solution found.
Final Statistics
----------------
Final objective value = 9.35999999745429e+02
Final feasibility error (abs / rel) = 1.44e-07 / 3.83e-10
Final optimality error (abs / rel) = 6.48e-07 / 4.28e-08
# of iterations = 415
# of CG iterations = 90
# of function evaluations = 545
# of gradient evaluations = 475
# of Hessian evaluations = 422
Total program time (secs) = 0.02660 ( 0.027 CPU time)
===============================================================================
Knitro 10.0.0: Locally optimal or satisfactory solution.
objective 935.9999997; feasibility error 1.44e-07
415 iterations; 545 function evaluations
```

which shows that many more functions calls were made than without
multistart. A file `knitro_mspoints.txt`

was also created,
whose content reads:

```
// Knitro 10.0.0 Multi-start Repository for feasible points.
// Each point contains information about the problem and the point.
// Points are sorted by objective value, from best to worst.
// Next feasible point.
numVars = 3
numCons = 2
objGoal = MINIMIZE
obj = 9.3600000342420878e+02
knitroStatus = 0
localSolveNumber = 1
feasibleErrorAbsolute = 0.00e+00
feasibleErrorRelative = 0.00e+00
optimalityErrorAbsolute = 2.25e-07
optimalityErrorRelative = 1.41e-08
x[0] = 2.0511214409048425e-07
x[1] = 4.1077619358921463e-08
x[2] = 7.9999996834308824e+00
lambda[0] = -4.5247620510168322e-08
lambda[1] = 2.2857143915699769e+00
lambda[2] = -1.0285715141992103e+01
lambda[3] = -3.2000001143071813e+01
lambda[4] = -2.1985040913238130e-07
// Next feasible point.
numVars = 3
numCons = 2
objGoal = MINIMIZE
obj = 9.5100000269458542e+02
knitroStatus = 0
localSolveNumber = 2
feasibleErrorAbsolute = 0.00e+00
feasibleErrorRelative = 0.00e+00
optimalityErrorAbsolute = 3.67e-07
optimalityErrorRelative = 2.62e-08
x[0] = 6.9999996377946481e+00
x[1] = 7.4479065893720198e-08
x[2] = 2.6499084231411754e-07
lambda[0] = -6.3891336872934633e-08
lambda[1] = 1.7500001368019027e+00
lambda[2] = -2.1791026695882249e-07
lambda[3] = -1.7500002055167382e+01
lambda[4] = -5.2500010586300956e+00
```

In addition to the known solution with value 936 that we had already found with a single solver run, we discover another local minimum with value 951 that we had never seen before. In this case, the new solution is not as good as the first one, but sometimes running the multistart algorithm significantly improves the objective function value with respect to single-run optimization.

## MATLAB example

In order to run the multistart algorithm in MATLAB, we must pass
the relevant set of options to Knitro via the Knitro options file.
Let us create a simple text file named `knitro.opt`

with the
following content:

```
ms_enable 1
ms_num_to_save 5
ms_savetol 0.01
hessopt 2
```

(the last line *hessopt 2* is necessary to remind Knitro to use approximate
second derivatives, since we are not providing the exact hessian).
Then let us run our MATLAB example from Section MATLAB example
again, where the call to *knitromatlab* has been replaced with:

```
knitromatlab(@obj, x0, A, b, Aeq, beq, lb, ub, @nlcon, [], options, 'knitro.opt');
```

and where the `knitro.opt`

file was placed in the current directory so that
MATLAB can find it. The Knitro log looks simlar to what we observed with AMPL.

## C example

The C example can also be easily modified to enable
multistart by adding the following lines before the
call to `KTR_solve()`

:

```
// multistart
if (KTR_set_int_param_by_name (kc, "ms_enable", 1) != 0)
exit( -1 );
if (KTR_set_int_param_by_name (kc, "ms_num_to_save", 5) != 0)
exit( -1 );
if (KTR_set_double_param_by_name (kc, "ms_savetol", 0.01) != 0)
exit( -1 );
```

Again, running this example we get a Knitro log that looks similar to what we observed with AMPL.

## Object-oriented example

The object-oriented example can be modified to enable
multistart by adding the following lines before the
call to `solver.solve()`

:

```
// multistart
solver.setParam("ms_enable", 1);
solver.setParam("ms_num_to_save", 5);
solver.setParam("ms_savetol", 0.01);
```

Again, running this example we get a Knitro log that looks similar to the trace obtained with AMPL.