The "classic" method for solving these problems is called Branch and Bound. This method begins by finding the optimal solution to the "relaxation" of the problem without the integer constraints via standard linear or nonlinear optimization methods. If in this solution, the decision variables with integer constraints have integer values, then no further work is required. If one or more integer variables have non-integral solutions, the Branch and Bound method chooses one such variable and "branches," creating two new subproblems where the value of that variable is more tightly constrained.
These subproblems are solved and the process is repeated, until a solution that satisfies all of the integer constraints is found. Alternative methods, such as genetic and evolutionary algorithms, randomly generate candidate solutions that satisfy the integer constraints.
Such initial solutions are usually far from optimal, but these methods then transform existing solutions into new candidate solutions, through methods such as integer- or permutation-preserving mutation and crossover , that continue to satisfy the integer constraints, but may have better objective values.
This process is repeated until a sufficiently "good solution" is found. Generally, these methods are not able to "prove optimality" of the solution. Free Trial.
Search form X. This means that the list of constraints may not even fit in the memory of the computer, and, even if it did, the simplex method would then be extremely slow…. In practice, we first solve the linear relaxation.
Then, we add some linear constraints which cut this fractional solution out of the feasible polyhedron. This yields a better linear relaxation to work on! And we can then repeat this cutting plane method on this improved linear relaxation until an integer optimum is found!
This is the case illustrated below:. A key remark to be made here is that the cutting planes are interesting only locally, so a cutting plane may later become useless. Thus, in general, a right management of these cutting planes is key to solve efficiently the integer program. However, in most cases, the best strategy is still to couple cutting plane methods with the powerful branch and bound method! The branch and bound is sort of a divide and conquer method.
It consists in dividing the integer program into two much simpler ones. The key point of the branch and bound is to enable us to delete the middle grey band from the optimization problem, which makes each of the 2 obtained integer programs much easier to solve than the initial one. The optimal value of the initial integer program then equals the best of the optimal values of the 2 branched integer programs.
More often than not, one branching is not enough. Each branched integer program then gets branched itself, until the optima of the branched integer program are integer solutions… or until we find out that the branched integer program has no feasible point! This is represented by the figure below, which depicts what we call the branching tree :.
Each linear relaxation can either have an integer optimum, a fractional optimum, or no feasible solution. These cases are depicted by green, blue and red crosses. In the green case, when the optimal solution found is integer, we have found a integer solution of the initial program.
Thus, its value is a lower bound of the initial integer program! Let me now address the red case which is simpler than the blue one. Thus, the branch explored is completely useless and we can just drop it. The only chance of good news is if the optimal value of the linear relaxation, which is an upper bound of the corresponding integer program, is lower than the best lower bound we have found with green solutions. This remark corresponds to bounding and enables to greatly speed up the branch and bound method.
Then its branches still might have a better integer solution than the best one found so far… So we need to keep branching!
Let me finish this section with one last major remark. There are two extreme possibilities. On one hand, we can explore in depth the first branches we generate. The major advantage is that this enables to quickly find a feasible integer solution, which can be returned even if the algorithm stops halfway.
The drawback though will be the quality of the lower and upper bounds. Basically, searching in depth also means that you are focusing on a particular kind of solution, as you rule out most of the other possibilities. To conclude, I want to stress the extraordinary accomplishments there have been in integer programming over the last decades. Within 20 years, the speed of integer programming solvers has been increased by an astonishing factor of ,!
Can you believe that? In other words, any problem we were able to solve using a year old computer in 7 years can now be solved with this same old computer in just 1 second! It seems impossible right now to even imagine what will be possible within the next decades! Too often, people forget that we live at an unbelievable time where new technologies improve at an exponential rate. As I try to make sense of that, many of the political debates seem deprecated to me… or about to be deprecated!
By working on increasingly faithful approximations of the integer program, the actual optimum can then be found much quicker! Yet, the problems these technics address still suffer the curse of dimensionality. Finally, two hot topics in integer programming like, in any field of applied mathematics, are parallelization and machine learning. This article shows the construction of the dual and its interpretation, as well as major results. In particular, matching of primal and dual bases will be dealt, before presenting the issue of degeneracy and its dual interpretation.
An intuitive approach is given. But that's not all. We present an important variant called the dual simplex. Finally, we'll explain its main default, that is, when facing degeneracy. They are a very interesting class of problems that include assignment problems, which have a very large range of applications.
Additional results have been found for variants which include the introduction of boys and girls' preferences, and cases where people cannot be categorized into two groups. Not only is it a very theoretical question in computer science and mathematics, but it also has major implications to real world. Its resolution could revolutionize the world. My PhD supervisor effectively took great advantage of these tricks and founded companies with it.
This article explains the tricks. When there are integer constraints on only some of the variables, the problem is called a mixed-integer program MIP.
Example integer programming problems include portfolio optimization in finance, optimal dispatch of generating units unit commitment in energy production, design optimization in engineering, and scheduling and routing in transportation and supply chain applications. This is the most general form of integer programming and is called a mixed-integer nonlinear program MINLP.
Many problems can be formulated with only linear objectives and constraints. In this case, the integer program is called a mixed-integer linear program MILP and is written as:. Solving MILPs typically requires using a combination of techniques to narrow the solution space, find integer-feasible solutions, and discard portions of the solution space that do not contain better integer-feasible solutions. Common techniques for integer programming include:. Some MINLPs can be solved by adapting these integer programming techniques to nonlinear functions or by linearizing the nonlinear functions and solving a sequence of MILPs.
When the nonlinear functions can only be evaluated at integral points, other techniques are needed. Two algorithms applicable to these types of integer programs are implemented in Global Optimization Toolbox:.
0コメント