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Branch and cut [1] is a method of combinatorial optimization for solving integer linear programs (ILPs), that is, linear programming (LP) problems where some or all the unknowns are restricted to integer values. [2] Branch and cut involves running a branch and bound algorithm and using cutting planes to tighten
In many cases, this method allows to solve large linear programs that would otherwise be intractable. The classical example of a problem where it is successfully used is the cutting stock problem. One particular technique in linear programming which uses this kind of approach is the Dantzig–Wolfe decomposition algorithm.
Branch and price is a branch and bound method in which at each node of the search tree, columns may be added to the linear programming relaxation (LP relaxation). At the start of the algorithm, sets of columns are excluded from the LP relaxation in order to reduce the computational and memory requirements and then columns are added back to the LP relaxation as needed.
For example, the function w = z 1/2 has two branches: one where the square root comes in with a plus sign, and the other with a minus sign. A branch cut is a curve in the complex plane such that it is possible to define a single analytic branch of a multi-valued function on the plane minus that curve. Branch cuts are usually, but not always ...
The following is the skeleton of a generic branch and bound algorithm for minimizing an arbitrary objective function f. [3] To obtain an actual algorithm from this, one requires a bounding function bound, that computes lower bounds of f on nodes of the search tree, as well as a problem-specific branching rule.
An example of a simple hardware branch prediction scheme is to assume that all backward branches (i.e. to a smaller program counter) are taken (because they are part of a loop), and all forward branches (to a larger program counter) are not taken (because they leave a loop). Better branch predictors are developed and validated statistically by ...
[1] The goal of loop unwinding is to increase a program's speed by reducing or eliminating instructions that control the loop, such as pointer arithmetic and "end of loop" tests on each iteration; [2] reducing branch penalties; as well as hiding latencies, including the delay in reading data from memory. [3]
In these examples, if N < 1 then the body of loop may execute once (with I having value 1) or not at all, depending on the programming language. In many programming languages, only integers can be reliably used in a count-controlled loop. Floating-point numbers are represented imprecisely due to hardware constraints, so a loop such as