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  2. Dual linear program - Wikipedia

    en.wikipedia.org/wiki/Dual_linear_program

    This linear combination gives us an upper bound on the objective. The variables y of the dual LP are the coefficients of this linear combination. The dual LP tries to find such coefficients that minimize the resulting upper bound. This gives the following LP: [1]: 81–83 Minimize b T y subject to A T y ≥ c, y ≥ 0

  3. Duality (optimization) - Wikipedia

    en.wikipedia.org/wiki/Duality_(optimization)

    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.

  4. Interior-point method - Wikipedia

    en.wikipedia.org/wiki/Interior-point_method

    An interior point method was discovered by Soviet mathematician I. I. Dikin in 1967. [1] The method was reinvented in the U.S. in the mid-1980s. In 1984, Narendra Karmarkar developed a method for linear programming called Karmarkar's algorithm, [2] which runs in provably polynomial time (() operations on L-bit numbers, where n is the number of variables and constants), and is also very ...

  5. Linear programming - Wikipedia

    en.wikipedia.org/wiki/Linear_programming

    More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope , which is a set defined as the intersection of finitely many half spaces , each of which is defined by a linear inequality.

  6. Basic feasible solution - Wikipedia

    en.wikipedia.org/wiki/Basic_feasible_solution

    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 ...

  7. Ellipsoid method - Wikipedia

    en.wikipedia.org/wiki/Ellipsoid_method

    He showed that linear programs can be solved in polynomial time. Here is a sketch of Khachiyan's theorem. [7]: Sec.8.4.2 Step 1: reducing optimization to search. The theorem of linear programming duality says that we can reduce the above minimization problem to the search problem: find x,y s.t. Ax ≤ b ; A T y = c ; y ≤ 0 ; c T x=b T y.

  8. Duality (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Duality_(mathematics)

    A linear program may be specified by a system of real variables (the coordinates for a point in Euclidean space ), a system of linear constraints (specifying that the point lie in a halfspace; the intersection of these halfspaces is a convex polytope, the feasible region of the program), and a linear function (what to optimize).

  9. Duality gap - Wikipedia

    en.wikipedia.org/wiki/Duality_gap

    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 ...