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  2. Ellipsoid method - Wikipedia

    en.wikipedia.org/wiki/Ellipsoid_method

    Leonid Khachiyan applied the ellipsoid method to the special case of linear programming: minimize c T x s.t. Ax ≤ b, where all coefficients in A,b,c are rational numbers. 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.

  3. Constrained optimization - Wikipedia

    en.wikipedia.org/wiki/Constrained_optimization

    The constrained-optimization problem (COP) is a significant generalization of the classic constraint-satisfaction problem (CSP) model. [1] COP is a CSP that includes an objective function to be optimized.

  4. Karush–Kuhn–Tucker conditions - Wikipedia

    en.wikipedia.org/wiki/Karush–Kuhn–Tucker...

    The system of equations and inequalities corresponding to the KKT conditions is usually not solved directly, except in the few special cases where a closed-form solution can be derived analytically. In general, many optimization algorithms can be interpreted as methods for numerically solving the KKT system of equations and inequalities. [7]

  5. List of optimization software - Wikipedia

    en.wikipedia.org/wiki/List_of_optimization_software

    The use of optimization software requires that the function f is defined in a suitable programming language and connected at compilation or run time to the optimization software. The optimization software will deliver input values in A , the software module realizing f will deliver the computed value f ( x ) and, in some cases, additional ...

  6. Optimization problem - Wikipedia

    en.wikipedia.org/wiki/Optimization_problem

    g i (x) ≤ 0 are called inequality constraints; h j (x) = 0 are called equality constraints, and; m ≥ 0 and p ≥ 0. If m = p = 0, the problem is an unconstrained optimization problem. By convention, the standard form defines a minimization problem.

  7. 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 probably polynomial time (() operations on L-bit numbers, where n is the number of variables and constants), and is also very ...

  8. Second-order cone programming - Wikipedia

    en.wikipedia.org/wiki/Second-order_cone_programming

    Semidefinite programming subsumes SOCPs as the SOCP constraints can be written as linear matrix inequalities (LMI) and can be reformulated as an instance of semidefinite program. [4] The converse, however, is not valid: there are positive semidefinite cones that do not admit any second-order cone representation. [ 3 ]

  9. Linear complementarity problem - Wikipedia

    en.wikipedia.org/wiki/Linear_complementarity_problem

    In mathematical optimization theory, the linear complementarity problem (LCP) arises frequently in computational mechanics and encompasses the well-known quadratic programming as a special case. It was proposed by Cottle and Dantzig in 1968. [1] [2] [3]

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