LocalSolver wins against MILP, MIQP, and CP solvers on QAPLIB // 01 Oct 2013

The Quadratic Assignment Problem (QAP) is one of fundamental combinatorial optimization problems in the branch of optimization or operations research, originally coming from facility location applications. Indeed, the problem models the following real-life problem. There are a set of n facilities and a set of n locations. For each pair of locations, a distance is specified and for each pair of facilities a weight or flow is specified (e.g., the amount of supplies transported between the two facilities). The problem is to assign all facilities to different locations with the goal of minimizing the sum of the distances multiplied by the corresponding flows. Intuitively, the cost function encourages factories with high flows between each other to be placed close together. The problem statement resembles that of the assignment problem, except that the cost function is expressed in terms of quadratic inequalities, hence the name. For more details, we invite the reader to have a look to the QAPLIB webpage.

A few months ago, a lot of discussions were raised by a paper presenting an attempt in using the power of quantum computing systems (actually D-Wave) to solve combinatorial optimization problems. The main benchmark used by the authors of this paper is the Quadratic Assignment Problem (QAP).

Beyond its computational hardness, this problem is in essence nonlinear: the decisions are boolean (0-1) and the objective contains quadratic terms (some products of decision variables like "x_i * x_j"). Then, it is not really suited to model and solve using Mixed-Integer Linear Programming (MILP) solvers, as well as using Constraint Programming (CP) solvers. One can imagine that Mixed-Integer Quadratic Programming (MIQP and MIQCQP) solvers could be better in solving these kinds of problems. Unfortunately, it is not the case: some evidences of the bad results of MILP, MIQP, and CP solvers are given here. In this article, the main reason invoked to explain why these general-purpose model-and-run solvers provide bad solutions to QAP instances is the following: they not only try to find good feasible solutions, but they spend a fair amount of time trying to prove optimality. This explanation is not clear since proving optimality of a solution implies you have this solution, which is not the case here. Worst, on many instances, these solvers do not find any feasible solution after hours of computation.

The reason why MILP solvers are not able to tackle QAP instances effectively is rather simple. Since the model is inherently quadratic (that is, nonlinear), the linear relaxation is not good. In particular not enough to cut the exponential tree search, as well as to find feasible solutions using rounding heuristics. Relying on several complementary optimization techniques embedded in a global neighborhood strategy, LocalSolver is able to adapt: if the linear relaxation brings nothing, then LocalSolver don't use it. Instead, LocalSolver follows a local search approach based on an incremental, direct evaluation of the constraints as well as of the objective.

Now, have a look to the QAP model written in LocalSolver, or more precisely using the modeling and scripting language (LSP) coming with LocalSolver. No need of linearizing when using LocalSolver, the QAP can be modeled in a straightforward way, as follows:

function model() {
  x[i in 1..n][j in 1..n] <- bool();
  for[i in 1..n]
    constraint sum[j in 1..n](x[i][j]) == 1;
  for[j in 1..n]
    constraint sum[i in 1..n](x[i][j]) == 1;
  obj <- sum[i in 1..n][j in 1..n][k in 1..n][l in 1..n]
             ( A[i][j] * B[k][l] * x[i][k] * x[j][l] );
  minimize obj;

Then we present the results obtained by LocalSolver 3.1 on the QAPLIB. The results are compared to the state of the art, obtained through dedicated, highly-tuned approaches combining different optimization techniques.

  • Lower objective is better
  • 5 minutes of running time for LocalSolver (and not for the state of the art)
  • Standard computer: Intel Core i7-820QM (4 cores, 1.73 GHz, 6 GB RAM, 8 MB cache)
  • Default settings
  • No need of linearizing or tuning in LocalSolver: the QAP model is written as is
Instances    n     Decisions  Expressions   LocalSolver 3.1  State of the art  Gap 
esc32a       32    1,024      124,299       136              130               5%  
esc32b       32    1,024      180,872       188              168               12% 
esc32c       32    1,024      219,153       642              642               0%  
esc32d       32    1,024      150,922       200              200               0%  
esc32e       32    1,024      11,148        2                2                 0%  
esc32g       32    1,024      16,138        6                6                 0%  
esc32h       32    1,024      235,791       444              438               1%  
kra30a       30     900       288,166       92,130           88,900            4%  
kra30b       30     900       288,166       93,020           91,420            2%  
kra32        32    1,024      328,558       94,440           88,700            6%  
lipa30a      30     900       732,736       13,399           13,178            2%  
lipa30b      30     900       731,258       177,255          151,426           17% 
lipa40a      40    1,600      2,374,581     31,998           31,538            1%  
lipa40b      40    1,600      2,368,797     565,314          476,581           19% 
lipa50a      50    2,500      5,885,226     62,859           62,093            1%  
lipa50b      50    2,500      5,876,130     1,431,681        1,210,244         18% 
nug30        30     900       510,877       6,238            6,124             2%  
sko42        42    1,764      2,078,709     16,282           15,812            3%  
sko49        49    2,401      3,817,588     24,146           23,386            3%  
ste36a       36    1,296      435,141       10,488           9,526             10% 
ste36b       36    1,296      435,628       18,066           15,852            14% 
ste36c       36    1,296      435,822       8,612,782        8,239,110         5%  
tai30a       30     900       745,196       1,914,966        1,818,146         5%  
tai30b       30     900       484,316       638,762,469      637,117,113       0%  
tai35a       35    1,225      1,384,849     2,515,194        2,422,002         4%  
tai35b       35    1,225      737,819       287,227,702      283,315,445       1%  
tai40a       40    1,600      2,382,379     3,284,344        3,139,370         5%  
tai40b       40    1,600      1,341,640     689,131,580      637,250,948       8%  
tai50a       50    2,500      5,905,491     5,230,902        4,938,796         6%  
tai50b       50    2,500      2,911,505     481,156,669      458,821,517       5%  
tho30        30     900       379,599       153,286          149,936           2%  
tho40        40    1,600      976,311       249,326          240,516           4%  
wil50        50    2,500      5,387,864     49,526           48,816            1%  
average                                                                        5%

The LSP file and the QAP instances are available in our Example Tour. You can download a ZIP file containing all files here. If you need a trial license to run the benchmark: create your account in a minute, download LocalSolver for your preferred platform, and follow the instructions to ask for a free full-version LocalSolver Trial license.

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