Programming Exercise / ILOG CPLEX Tutorial. Your Task (1) Your Task (2)

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1 Programming Exercise / ILOG CPLEX Tutorial Andreas M. Chwatal Algorithms and Data Structures Group Institute of Computer Graphics and Algorithms Vienna University of Technology VU Fortgeschrittene Algorithmen und Datenstrukturen April 2010 Recall the k-node minimum spanning tree (k-mst) problem from the todays lecture: Given: (undirected) graph G = (V, E, w) nonnegative weighting function w(e) R Goal: Find a minimum weight tree, spanning exactly k nodes. Programming Exercise: develop a branch-and-bound and a branch-and-cut algorithm for this problem 1 2 Your Task (1) Your Task (2) Carefully read this tutorial Formulate the k-mst problem as integer linear program (ILP), based on a 1 single commodity flow formulation (SCF) 2 cycle elimination formulation (CE), or, alternatively a formulation based on directed connection cuts (DC) Implement the corresponding algorithms, using ILOG CPLEX: 1 a branch-and-bound algorithm for the SCF formulation 2 a branch-and-cut algorithm based on CE/DC using a dynamic cut-separation procedure For the implementation you should use a C++ framework (which you can find on the webpage of the course) Furthermore you can find test-instances (to described later on), which should be used for the evaluation and analysis of the algorithms Compute the results for each instance for (at least) k = 1 5 V and k = 1 2 V with both algorithms 3 4

2 Your Task (3) Create a short document (preferably in LaTeX) of no more than three pages containing: 1 Problem description, used variables 2 SCF and CE/DC formulation 3 Short description of implementation, in particular the cut-generation 4 Result table, including 1 objective function value 2 running time 3 number of branch-and-bound nodes 4 node in which optimum has been found 5 For B&C: number of separated cuts 5 Organization You can work on the exercise alone, or in groups of two You get accounts for compute/development server behemoth.ads.tuwien.ac.at after the lecture, or request by to me. You can use this server for development, testing, computing results; Accounts will be deleted at end of 2010! Contact us (me and the Sysadmin) if you need an extension. Working on other computers: Copy corresponding directories (see Makefile) and the license file to your computer; delete after course is finished. Send the document and source code per to me/us two days prior to the (oral) exam. Make sure, that a correctly working version of your program is available on an institute-server at the time of your exam. 6 What is CPLEX? CPLEX: Simplex method and C programming language Commercial optimization software package High performance integer (linear) programming linear programming quadratic programming Concert Framework Interface to CPLEX solver (C, C++, C#, Java) Enables to implement: branch-and-bound, branch-and-cut, column generation,... What does Concert do for you? Variables restricted to e.g. Z or {0, 1} branching is automatically performed on these variables branch-and-bound Preprocessing/Presolving Cut-Generation: Gomory-Fractional-Cuts, Mixed-Integer-Cuts Lot s of problem-specific cuts, e.g. Cover-Cuts, Clique-Cuts,... What can you do for Concert? User-defined cuts generate new variables (pricing)... by callback objects 7 8

3 Prerequisites This tutorial should contain all information to solve the programming exercise, but if you want to know more, have a look at our manual pages: (password required) Also have a look at: Enable the licence: Environmental variable needs to be set. Type the following lines at the command line promt: $ ILOG LICENSE FILE=/home1/share/ILOG/ilm/access.ilm $ export ILOG LICENSE FILE 9 First steps 1 Include the headerfile: #include <ilcplex/ilocplex.h> 2 Before the class definition (of the class that will build the model and start the solver) use the following macro: ILOSTLBEGIN 3 Declare/create the following objects: IloCplex cplex; // the solver object IloEnv env; // the environment object IloModel model; // the model; all constraints etc. go in here 10 Basic Program Structure try { env = IloEnv(); // creating the environment model = IloModel(env); // creating the model (w/env) // build some variables and constraints // and add them to the model model.add(...); // add the objective function model.add(ilominimize(...)); cplex = IloCplex(model); // create cplex object w/model cplex.solve(); // solve the model } catch (IloException& e) {... } catch (...) {... } 11 Data-Types Constants: IloNum, IloBool, IloInt Variables: IloNumVar, IloBoolVar, IloIntVar Example: IloBoolVar x(env, my-first-bool-var ); Arrays/Vectors: IloIntVarArray, IloNumVarArray, IloBoolVarArray Example: // create array of 5 boolean variables IloBoolVarArray y(env, 5); for (u_int i=0; i<5; i++) { stringstream myname; myname << y_ << i; y[i] = IloBoolVar(env, myname.str()); } 12

4 Constraints Constraints, equalities, inequalities are added by the IloExpr class: Example: y1 + y2 4 IloExpr myexpr(env); myexpr += y[1]; myexpr += y[2]; model.add(myexpr <= 4); myexpr.end(); // IMPORTANT It is important to call the IloExpr::end() function to free the memory Objective Function / Solver The objective function is handled similar to constraints: model.add(ilominimize(env, expr)); Now we are ready to start the solver: IloCplex cplex(model); cplex.solve(); Did we succeed...? IloAlgorithm::Status algstatus = cplex.getstatus(); if (algstatus!= IloAlgorithm::Optimal) { // something went wrong... } else { // model solved to optimality env.out() << obj. value: " << cplex.getobjvalue() << endl; } What is going on now? Cut Generation 1 Let s assume, we have defined some integer variables 2 CPLEX solves the LP-relaxation of our model 3 CPLEX then tries to separate internal cutting-planes 4 If this process is finished we still may end up with an solution containing fractional variables 5 Now it s time for branching; One particular variable is selected and two subproblems are created by rounding it up and down respectively 6 One subproblem is selected Step 2 In each node of the branch-and-bound tree we may want to add further cutting-planes in order to 1 strengthen the LP-relaxation 2 add violated inequalities, not initially included to the model (e.g. if we did not add an exponentially large set of cycle-elimination inequalities) This can be achieved by the use of special callback-classes Example: CutCallbackI for case (1) LazyConstraintCallbackI for case (2) 15 16

5 Cut Generation (2) When CPLEX calls such a callback object, we first need to inspect the current values of our variables: IloNum a = getvalue(x[i]); Based on this values we can now build/update some data-structures, apply some algorithms and detect some cutting-planes (e.g. the cycle-elimination-cuts) As we already know from the lecture, we can easily separate cycle elimination cuts by performing shortest path computations. If we have found a valid cut, build the corresponding expression cut we can add it to the model... 1 globally: add(cut); 2 locally: addlocal(cut); Remark: Environment can be accessed by function getenv() within the callback-object 17 Numeric Issues In particular for the cut separation we need to take care about numeric issues. Assume we specified the constraint x1 + x2 + x3 = 3 A solution might only satisfy (x1 ± ɛ) + (x2 ± ɛ) + (x3 ± ɛ) = 3 for some small ɛ, or equivalently 3 3 ɛ x1 + x2 + x ɛ Attention: according to this issue we might iteratively add inequalities that are already met (except of the numeric error), or might not find valid cuts and thus end up with an infeasible solution (if not taking care of it)!!! The value of ɛ can be obtained by cplex.getparam(ilocplex::epint) and set by cplex.setparam(...). 18 Interpreting the output Nodes Cuts/ Node Left Objective IInf Best Int. Best Node ItCnt Gap Variable B NodeID Parent Depth User: x_[96] D x_[51] U * integral % x_[82] D % x_[82] U % x_[83] U * integral % x_[86] U % x_[96] U % x_[66] U Comments: Second line, still in root node of B&B tree; after the separation of 26 internal cuts the LP relaxation is better (higher); branching starts in line 3: node 1 (with parent 0) is the problem where (boolean) variable x [96] has been set to 0 (D = down); first integral solution in line 5 at node 22 of B&B tree, indicated with * in the first column; the best integer solution value is , whereas the LP-relaxation is ; the gap is 7.11%; we are finished when the gap is 0% (proven optimality); the last line with * is the node where the optimum has been found; 19 Debugging hints Compiling with -O2 (or even -O3) enables optimization. This should be used for the runs for creating the results (otherwise particularily the cut-generation will be too slow). For developing use -p -g which includes profiling and debugging information. Debugger: start with gdb --args./yourprogram type run to start the program type bt or backtrace to see the execution stack This will show you e.g. the line number in which the error or segmentation fault occurs. The tool valgrind can be used for the detection of memory leaks 20

6 Test instances To make life easier, the test instances already include the artificial root node 0 Hence, we are actually interested in finding a k-mst of nodes {1,..., V }!! Be careful, to handle this accordingy w.r.t. the number of connected nodes k! You can download six test instances ranging from 10 to 100 nodes from the course webpage. Compute the results for each instance for (at least) k = 1 5 V and k = 1 2 V with both algorithms Instances format First line: number of nodes Second line: number of edges Subsequent lines: edge list (source node, target node; edge weight) Example: , 4; 23; 2, 3; 93; 1, 5; 56; k-mst Framework Main: starting the program, parameter handling Instance: singleton class, responsible for reading the data instances, converting the data in basic data structures, and providing them to other classes Global: singleton class, containing global variables, parameters etc. Tools: provides various useful functions that do not fit to one particular class EdgeIO: reading the edges from the input file, writing to stdout (by providing appropriate operators << and >>) kmst BC: this class should contain the ILP-based algorithms, i.e. a branch-and-bound algorithm for the flow formulation and a branch-and-cut algorithm for the packing formulation 23 class Instance The class Instance provides rudimentary graph data structures (which should be enough to solve the exercise) Data structures: u int nnodes, nedges typedef std::pair<u int, u int> E defines an edge vector<e> edges list of edges vector<int> edgeweights corresponding edge weights vector<set<u int> > adjedges for each node the set of adj. edges Functions: set<u int> getadjnodes(u int node) for building directed variables we might want to get a direction set<u int> getoutedges(u int node) set<u int> getinedges(u int node) 24

7 Shortest Path algorithm DheaMaxFlow The class kmst BC CutCallback provides a very simple (Dijkstra-based) shortest path algorithm. It does not even use a priority queue, and thus has runtime O( V 3 ). Nevertheless, this is enough for the exercise. Usage: The arguments are the indices of the source and target node you want to compute the shortest path for Return type: struct sp result s, containing the edge list list<u int> path and the path length double length The class DheaMaxFlow provides an efficient max-flow algorithm, which can be used to compute the minimum cut. test.cpp shows how to use the algorithm src/makefile compiles test.cpp (for testing purposes) Graph/Algorithm libraries (optional) Further Remarks It is recommended to use the provided (graph) methods, as they provide all necessary functionality If you want more: code yourself ;-) use graph/algorithm libraries like LEDA or boost, which are also available on the servers be nice ;-) when working on the server (behemoth) start your programs with nice with lower priority $ nice./yourprogram... The compute server has four cores, but must fit the needs of all participants. start only one process at the same time for your final test runs make sure you get a slot for your own (no more than three other processes) 27 28

8 Questions? Good luck for the programming exercise! Have fun coding! If you have any questions, don t hesitate to ask me (us)! 29