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Which is the simplified problem in Benders decomposition?
Contains Benders cuts so far generated. Simplified problem contains remaining variables y Solve inference dual to obtain Benders cut that excludes solutions no better than current one. Fix search variables x Add Benders cut Master problem Subproblem Essence of Benders Decomposition 6 • The key to generalizing Benders is generalizing the dual.
How to minimize the cost of Benders cuts?
Minimize cost zsubject to Benders cuts Solve inference dual to obtain proof of optimality Use same proof to deduce cost bounds for other assignments, yielding Benders cut. Trial value xk that solves master Benders cut Master problem Subproblem 16 min ( , ) ( , ) f x y x y S Iteration k :t () xk z B x Logic-Based Benders • In any iteration,
Which is an application of the cutting plane method?
Convex optimization. Cutting plane methods are also applicable in nonlinear programming. The underlying principle is to approximate the feasible region of a nonlinear (convex) program by a finite set of closed half spaces and to solve a sequence of approximating linear programs .
Can a cut be added to a relaxed linear program?
If it is not, there is guaranteed to exist a linear inequality that separates the optimum from the convex hull of the true feasible set. Finding such an inequality is the separation problem, and such an inequality is a cut. A cut can be added to the relaxed linear program.
When to use subproblem and master problem in decomposition?
While the master problem provides a lower bound on the value of the problem, the subproblem is used to get an upper bound. The result of solving the subproblem for any given in the recession cone can be found, or a finding that the subproblem is infeasible. At a high level, the procedure will iteratively consider the master problem and subproblem.
Where did Jacques Benders get his name from?
This block structure often occurs in applications such as stochastic programming as the uncertainty is usually represented with scenarios. The technique is named after Jacques F. Benders .