What Is Optimization Toolbox?

intprog• Verifies that no better integer feasible solution is possible by solvinga series of linear programming problemsThe following sections describe the branch-and-bound method ingreater detail.BranchingThe algorithm creates a search tree by repeatedly adding constraints tothe problem, that is, "branching." At a branching step, the algorithmchooses a variable x jwhosecurrentvalueisnotanintegerandaddstheconstraint x j= 0 to form one branch and the constraint x j=1toformthe other branch. This process can be represented by a binary tree, inwhich the nodes represent the added constraints. The following pictureillustrates a complete binary tree for a problem that has three variables,x 1, x 2,andx 3. Note that, in general, the order of the variables goingdown the levels in the tree is not the usual order of their subscriptsDeciding Whether to BranchAt each node, the algorithm solves an LP-relaxation problem using theconstraints at that node and decides whether to branch or to move toanother node depending on the outcome. There are three possibilities:• If the LP-relaxation problem at the current node is infeasible orits optimal value is greater than that of the best integer point,the algorithm removes the node from the tree, after which itdoes not search any branches below that node. The algorithmthen moves to a new node according tothemethodyouspecifyinNodeSearchStrategy option.8-7