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The weak duality theorem says that, for each feasible solution x of the primal and each feasible solution y of the dual: c T x ≤ b T y. In other words, the objective value in each feasible solution of the dual is an upper-bound on the objective value of the primal, and objective value in each feasible solution of the primal is a lower-bound ...
The Lagrangian dual program is the program of maximizing g: max λ ≥ 0 g ( λ ) {\displaystyle \max _{\lambda \geq 0}g(\lambda )} . The optimal solution to the dual program is a lower bound for the optimal solution of the original (primal) program; this is the weak duality principle.
A basis B of the LP is called dual-optimal if the solution = is an optimal solution to the dual linear program, that is, it minimizes . In general, a primal-optimal basis is not necessarily dual-optimal, and a dual-optimal basis is not necessarily primal-optimal (in fact, the solution of a primal-optimal basis may even be unfeasible for the ...
HiGHS has implementations of the primal and dual revised simplex method for solving LP problems, based on techniques described by Hall and McKinnon (2005), [6] and Huangfu and Hall (2015, 2018). [ 7 ] [ 8 ] These include the exploitation of hyper-sparsity when solving linear systems in the simplex implementations and, for the dual simplex ...
This alternative "duality gap" quantifies the discrepancy between the value of a current feasible but suboptimal iterate for the primal problem and the value of the dual problem; the value of the dual problem is, under regularity conditions, equal to the value of the convex relaxation of the primal problem: The convex relaxation is the problem ...
Farkas' lemma is the key result underpinning the linear programming duality and has played a central role in the development of mathematical optimization (alternatively, mathematical programming). It is used amongst other things in the proof of the Karush–Kuhn–Tucker theorem in nonlinear programming. [2]
Duality theory tells us that if the primal is unbounded then the dual is infeasible by the weak duality theorem. Likewise, if the dual is unbounded, then the primal must be infeasible. However, it is possible for both the dual and the primal to be infeasible. See dual linear program for details and several more examples.
GNU Linear Programming Kit with C API. HiGHS: MIT: linear programming (LP), mixed integer programming (MIP), and convex quadratic programming (QP). [1] IPOPT: EPL (was CPL) large scale nonlinear optimizer for continuous systems (requires gradient), C++ (formerly Fortran and C). It became a part of COIN-OR. [2] MINUIT (now MINUIT2) LGPL