GLPK:   The Gnu Linear Programming Kit
1 Typed Racket
2 The Linear Programming problem
lp-solve
2.1 An Example
mip-solve
2.2 An Integer Example
Fail  Code
Solution  Status
3 GLPK Error Handling
7.3

GLPK: The Gnu Linear Programming Kit

John Clements <[email protected]>

This collection provides a simple racket interface to the Gnu Linear Programming Kit, by Andrew O. Makhorin, allowing you to solve linear optimization problems.

This package does not include that library; you’ll need to install it yourself, using the package manager of your choice, or by building it from source.

Note that you may need to configure Racket’s search path to allow it to find the installed library; search the raco documentation for 'lib-search-dirs.

The GLPK library comes with many bells and whistles, including dual simplex, Mixed Integer Programming (MIP), and other related problems.

Here’s a list, taken from the GLPK documentation:

Right now, this library supports two of these modes: basic primal simplex solving, and mixed integer programming (MIP) using the branch-and-cut method. It’s not hard to extend this library on an as-needed basis, to support more of GLPK’s functionality.

It’s worth mentioning that GLPK does have one "interesting" design choice; when you supply arguments to GLPK that don’t make sense—for instance, trying to supply a name for a column that doesn’t exist—it simply prints a message to stderr and then calls exit(1). That is: your racket (or DrRacket) process will simply halt. This is not the behavior I think I would have chosen, but we’re stuck with it. I believe I have designed and implemented the lp-solve function in such a way that this should not be possible. Nevertheless, it’s something that users of the library should be aware of. [See section GLPK Error Handling

1 Typed Racket

The outward-facing modules of this library are written using Typed Racket, and should play nicely with the rest of your Typed Racket code.

2 The Linear Programming problem

The linear programming problem can be formulated as follows: given a set of linear constraints over a set of variables, and a function to be maximized (or minimized), find the set of values for the variables that maximize (or minimize) the function.

Okay, so what kind of constraints are possible? Well, each constraint consists of a single equality, of the form

a = \sum_{i} K_i x_i

... where the K_i are real numbers, and a is what’s called an "auxiliary" variable. These auxiliary variables must occur only once each, on the left-hand side of the corresponding constraint.

The other variables, the x_i, are "structural" variables, or "problem" variables, corresponding to unknown quantities in the problem statement.

The objective function is a linear combination of structural variables. It may be either maximized or minimized, as you like.

Along with these constraints, each variable, both structural and auxiliary, comes with a pair of (possibly infinite) bounds. So, for instance, you can specify that auxiliary variable b ranges between -10 and 143.2.

An example appears below.

 (require glpk) package: GLPK

procedure

(lp-solve objective 
  direction 
  constraints 
  bounds 
  #:terminal-output terminal-output?) 
  
(list/c symbol?
        (or/c symbol? false?)
        (list/c flonum? (listof (list/c symbol? flonum?))))
  objective : objective?
  direction : (or/c 'max 'min)
  constraints : (listof constraint?)
  bounds : (listof bound?)
  terminal-output? : boolean?
Solves a linear programming problem.

Both the objective and the constraints make use of a "linear combination" form:

lin-comb? = (listof (list/c real? symbol?))

... representing a linear combination of structural variables.

The objective function includes a constant term and a linear combination of structural variables:

objective? = (pair/c real? lin-comb?)

The constraints each include the name of an auxiliary variable and a linear combination of structural variables:

constraint? = (pair/c symbol? lin-comb?)

Finally, the set of bounds provides bounds for both the auxiliary and structural variables. Each bound contains the name of a variable, and a low and high boundary. The low boundary can be 'neginf, indicating no lower bound, and the high boundary can be 'posinf, indicating no upper bound. The lower and upper bound can be equal, indicating that the corresponding variable is fixed.

You must provide bounds for every auxiliary and structural variable.

bound?    = (list/c symbol? lo-bound? hi-bound?)
lo-bound? = (or/c 'neginf real?)
hi-bound? = (or/c 'posinf real?)

You may specify #:terminal-output as true to obtain output on (yes actually) stdout. This is included for completeness, but is generally more useful in mixed integer programming, described below.

The result is a list containing a symbol that indicates either success ('good) or one of two kinds of failure, and then a symbol which in case of failure conveys information about the nature of the failure, and then either a solution or false. The reason for this "flattened" structure which can carry both failure and success simultaneously is that for certain kinds of failure in MIP (see below), we wish to provide both failure and a best-so-far solution.

A solution is represented as a list containing the optimal value of the objective function, along with a list of lists mapping structural variables to the values that produce that optimal value.

When the first symbol in the result is 'bad-result, the second element is a FailCode (definition below). When the first symbol in the result is 'bad-status, the second element is a SolutionStatus (also defined below).

Yikes! Let’s see an example.

2.1 An Example

Okay, let’s say you’re trying to figure out whether you have enough food for your picnic. In particular, you’re buying hamburgers, slices of bread, and pickles. You have three kinds of guests: Children, Adults, and Chickens.

Each adult wants one slice of bread, a patty, and two pickles. Each child wants two slices of bread, and a patty. Each chicken wants *either* one slice of bread, one patty, or one pickle.

So, if we have 30 slices of bread, 20 patties, and 50 pickles, how many guests can we invite?

Let’s make up some variables. We’ll use k for the number of children, a for the number of adults, c_b for the chickens eating a slice of bread, c_p for the chickens eating a patty, and c_k for the chickens eating a pickle. For our auxiliary variables, we’ll use f_b for the number of slices of bread eaten, f_p for the number of patties eaten, and f_k for the number of pickles eaten.

f_b = a + 2k + c_b f_p = a + k + c_p f_k = 2a + c_k

Here are the bounds we’ll use:

0 < f_b < 30 0 < f_p < 20 0 < f_k < 50 0 < a 0 < k 0 < c_b 0 < c_p 0 < c_k

Finally, our objective function: maximize f:

f = a + k + c_b + c_p + c_k

Here’s the whole thing as a call to lp_solve:

(lp-solve
 '(0 (1 a) (1 k) (1 cb) (1 cp) (1 ck))
 'max
 '((fb (1 a) (2 k) (1 cb))
   (fp (1 a) (1 k) (1 cp))
   (fk (2 a) (1 ck)))
 '((fb 0 30)
   (fp 0 20)
   (fk 0 50)
   (a 0 posinf)
   (k 0 posinf)
   (cb 0 posinf)
   (cp 0 posinf)
   (ck 0 posinf)))

You may not be surprised by the result:

'(good #f (100.0 ((a 0.0) (k 0.0) (cb 30.0) (cp 20.0) (ck 50.0))))

In other words, you can invite a hundred guests, if they’re all chickens. Well, that’s fine, but what if we decide that kids are the best (worth 10), adults are second best (worth 8), and chickens are only worth 1/2?

Here’s the call:

(lp-solve
 '(0 (8 a) (10 k) (1/2 cb) (1/2 cp) (1/2 ck))
 'max
 '((fb (1 a) (2 k) (1 cb))
   (fp (1 a) (1 k) (1 cp))
   (fk (2 a) (1 ck)))
 '((fb 0 30)
   (fp 0 20)
   (fk 0 50)
   (a 0 posinf)
   (k 0 posinf)
   (cb 0 posinf)
   (cp 0 posinf)
   (ck 0 posinf)))

... and the result:

'(good #f (195.0 ((a 10.0) (k 10.0) (cb 0.0) (cp 0.0) (ck 30.0))))

In other words: 10 adults, 10 kids, and a pile of chickens to vacuum up the leftover pickles.

We can add arbitrary further constraints on this: each chicken must be chaperoned by an adult, each chicken must be chaperoned by an adult, no adult can chaperone both a child and a chicken.

To model this, we divide adults into adults chaperoning kinds (ak) and adults chaperoning chickens (ac). We could replace a entirely, but it’s easier just to require that a is the sum of ac and ak. Also, let’s bump up the desirability of chickens, just to get a more interesting result:

(lp-solve
 '(0 (8 a) (10 k) (2 cb) (2 cp) (2 ck))
 'max
 '((fb (1 a) (2 k) (1 cb))
   (fp (1 a) (1 k) (1 cp))
   (fk (2 a) (1 ck))
   (z (-1 a) (1 ak) (1 ac))
   (excessak (1 ak) (-1 k))
   (excessac (1 ac) (-1 cb) (-1 cp) (-1 ck)))
 '((z 0 0)
   (ak 0 posinf)
   (ac 0 posinf)
   (excessak 0 posinf)
   (excessac 0 posinf)
   (fb 0 30)
   (fp 0 20)
   (fk 0 50)
   (a 0 posinf)
   (k 0 posinf)
   (cb 0 posinf)
   (cp 0 posinf)
   (ck 0 posinf)))

The result:

'(good
  #f
  (200.0
   ((a 20.0)
    (k 0.0)
    (cb 10.000000000000002)
    (cp 0.0)
    (ck 9.999999999999998)
    (ak 0.0)
    (ac 20.0))))

That is: 20 adults, all chaperoning chickens. 10 of the chickens get bread, 10 of the chickens get pickles.

Nifty!

procedure

(mip-solve objective 
  direction 
  constraints 
  bounds 
  integer-vars 
  #:terminal-output terminal-output? 
  #:time-limit time-limit) 
  
(or/c (list/c symbol?
              (or/c symbol? #f)
              (or/c (list/c flonum? (listof (list/c symbol? flonum?)))
                    #f)))
  objective : objective?
  direction : (or/c 'max 'min)
  constraints : (listof constraint?)
  bounds : (listof bound?)
  integer-vars : (listof symbol?)
  terminal-output? : boolean?
  time-limit : (or/c false? natural?)
Performs mixed-integer programming.

The Mixed-Integer-Programming solver is an extension of the linear programming solver, and the problems that it solves are an extension of linear programming problems. Specifically, in a mixed integer programming problem, some of the solution variables can be labeled as integer variables, whose values must be integers.

This is indicated in the interface using an additional input, a list of structural variables whose values must be integers.

Mixed integer programming is ... well, a lot harder than simple linear programming. In fact, the first step in MIP is to solve the corresponding linear programming problem, where the variables are all allowed to take on non-integer variables. The branch-and-cut algorithm then attempts to find a related solution where the specified structural variables have integer values.

This problem is NP-hard, so ... it can take a while. This motivates the addition of three interface elements. First, the #:time-limit argument accepts a number of milliseconds, indicating how long to run before giving up. Second (making the previous one useful), the result of a timeout includes a "best solution so far" along with the 'GLP_ETMLIM failure result. Third, the #:terminal-output option is genuinely useful in the case of mixed integer programming, because it provides information every few seconds (ten?) on the progress of the search; the gap between the best known solution and the best solution that might be possible, the number of possibilities left to explore, etc.

Unfortunately, the GLPK library is configured by default to provide this output on stdout, which isn’t terribly natural for a DrRacket program, since the output appears on the terminal that you started from DrRacket... if you actually started DrRacket from a terminal at all.

Fortunately, GLPK provides "glp_term_hook()" for exactly this purpose, so you can redirect output to a more sane location.

Unfortunately, I haven’t implemented support for this. Sorry!

In fact, I found that I’ve moved toward running my GLPK MIP problems more or less exclusively in the terminal, both in order to see the output and also because interrupting the search usually involves some rather violent process actions (control-backslash, for instance).

It would also be great to provide support for the various tuning options for the integer solver; I haven’t done this, but it would be less than an hour’s work; if you need any of these, let me know and I’d be happy to add them.

But wait... let’s have an example.

2.2 An Integer Example

Here’s a simple example: we need at least 4.5 sections of csc101 and 202. We have two instructors, smith and martinez, each of whom can teach 9 sections. Can we staff both of our classes? Yes.

Here’s the call. We’re trying to maximize their unused sections:

(mip-solve '(0 (1 smith-extra) (1 martinez-extra)) 'max
           '((csc101-offered (1 smith-csc101) (1 martinez-csc101))
             (csc202-offered (1 smith-csc202) (1 martinez-csc202))
             (smith-secns (1 smith-csc101)
                          (1 smith-csc202)
                          (1 smith-extra))
             (martinez-secns (1 martinez-csc101)
                             (1 martinez-csc202)
                             (1 martinez-extra)))
           '((csc101-offered 4.5 posinf)
             (csc202-offered 4.5 posinf)
             (smith-secns 9 9)
             (martinez-secns 9 9)
             (smith-csc101 0 posinf)
             (smith-csc202 0 posinf)
             (smith-extra 0 posinf)
             (martinez-csc101 0 posinf)
             (martinez-csc202 0 posinf)
             (martinez-extra 0 posinf))
           '(smith-csc101 smith-csc202
                          martinez-csc101 martinez-csc202))

Here’s the solution that GLPK comes up with (one of many possible equivalent ones:

'(good
     #f
     (8.0
      ((smith-csc101 4.0)
       (martinez-csc101 1.0)
       (smith-csc202 5.0)
       (martinez-csc202 0.0)
       (smith-extra 0.0)
       (martinez-extra 8.0))))

syntax

FailCode

A code that indicates the nature of a failure. Defined as follows:

(define-type FailCode
  (U 'GLP_EBADB   ;; invalid basis
     'GLP_ESING   ;; singular matrix
     'GLP_ECOND   ;; ill-conditioned matrix
     'GLP_EBOUND  ;; invalid bounds
     'GLP_EFAIL   ;; solver failed
     'GLP_EOBJLL  ;; objective lower limit reached
     'GLP_EOBJUL  ;; objective upper limit reached
     'GLP_EITLIM  ;; iteration limit exceeded
     'GLP_ETMLIM  ;; time limit exceeded
     'GLP_ENOPFS  ;; no primal feasible solution
     'GLP_ENODFS  ;; no dual feasible solution
     'GLP_EROOT   ;; root LP optimum not provided
     'GLP_ESTOP   ;; search terminated by application
     'GLP_EMIPGAP ;; relative mip gap tolerance reached
     'GLP_ENOFEAS ;; no primal/dual feasible solution
     'GLP_ENOCVG  ;; no convergence
     'GLP_EINSTAB ;; numerical instability
     'GLP_EDATA   ;; invalid data
     'GLP_ERANGE  ;; result out of range
     ))
 

A symbol that indicates the status of the solution. Defined as follows:

(define-type SolutionStatus
  (U 'GLP_UNDEF
     'GLP_FEAS
     'GLP_INFEAS
     'GLP_NOFEAS
     'GLP_OPT
     'GLP_UNBND))

3 GLPK Error Handling

EDIT: I’m now aware that GLPK includes a glp_error_hook function that allows the installation of a hook function that is called before the library halts execution; the documentation suggests the use of setjmp/longjmp to escape if a user wishes not to exit.

If I understand the internals of Racket correctly, making use of this would require separately compiling a C stub that establishes a jump buffer and uses setjmp before calling into each GLPK library function. This stub would have to be compiled for every platform separately, and I frankly don’t have enough interest to implement something like this now.... Sad face.