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  2. Linear programming - Wikipedia

    en.wikipedia.org/wiki/Linear_programming

    For example, the LP relaxations of the set packing problem, the independent set problem, and the matching problem are packing LPs. The LP relaxations of the set cover problem, the vertex cover problem, and the dominating set problem are also covering LPs. Finding a fractional coloring of a graph is another example of a

  3. Dual linear program - Wikipedia

    en.wikipedia.org/wiki/Dual_linear_program

    Suppose we have the linear program: Maximize c T x subject to Ax ≤ b, x ≥ 0.. We would like to construct an upper bound on the solution. So we create a linear combination of the constraints, with positive coefficients, such that the coefficients of x in the constraints are at least c T.

  4. Lagrange multiplier - Wikipedia

    en.wikipedia.org/wiki/Lagrange_multiplier

    In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfied exactly by the chosen values of the variables). [1]

  5. Linear programming relaxation - Wikipedia

    en.wikipedia.org/wiki/Linear_programming_relaxation

    The linear programming relaxation of an integer program may be solved using any standard linear programming technique. If it happens that, in the optimal solution, all variables have integer values, then it will also be an optimal solution to the original integer program.

  6. HiGHS optimization solver - Wikipedia

    en.wikipedia.org/wiki/HiGHS_optimization_solver

    HiGHS has an interior point method implementation for solving LP problems, based on techniques described by Schork and Gondzio (2020). [10] It is notable for solving the Newton system iteratively by a preconditioned conjugate gradient method, rather than directly, via an LDL* decomposition. The interior point solver's performance relative to ...

  7. Farkas' lemma - Wikipedia

    en.wikipedia.org/wiki/Farkas'_lemma

    Farkas' lemma implies that the decision problem "Given a system of linear equations, does it have a non-negative solution?" is in the intersection of NP and co-NP . This is because, according to the lemma, both a "yes" answer and a "no" answer have a proof that can be verified in polynomial time.

  8. 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 ...

  9. Basic feasible solution - Wikipedia

    en.wikipedia.org/wiki/Basic_feasible_solution

    For example, if is non-basic and its coefficient in is positive, then increasing it above 0 may make larger. If it is possible to do so without violating other constraints, then the increased variable becomes basic (it "enters the basis"), while some basic variable is decreased to 0 to keep the equality constraints and thus becomes non-basic ...

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