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  2. Halley's method - Wikipedia

    en.wikipedia.org/wiki/Halley's_method

    Halley's method is a numerical algorithm for solving the nonlinear equation f(x) = 0.In this case, the function f has to be a function of one real variable. The method consists of a sequence of iterations:

  3. Nonlinear programming - Wikipedia

    en.wikipedia.org/wiki/Nonlinear_programming

    SciPy (de facto standard for scientific Python) has scipy.optimize solver, which includes several nonlinear programming algorithms (zero-order, first order and second order ones). IPOPT (C++ implementation, with numerous interfaces including C, Fortran, Java, AMPL, R, Python, etc.) is an interior point method solver (zero-order, and optionally ...

  4. Relaxation (iterative method) - Wikipedia

    en.wikipedia.org/wiki/Relaxation_(iterative_method)

    Relaxation methods were developed for solving large sparse linear systems, which arose as finite-difference discretizations of differential equations. [2] [3] They are also used for the solution of linear equations for linear least-squares problems [4] and also for systems of linear inequalities, such as those arising in linear programming.

  5. Newton–Krylov method - Wikipedia

    en.wikipedia.org/wiki/Newton–Krylov_method

    Newton–Krylov methods are numerical methods for solving non-linear problems using Krylov subspace linear solvers. [1] [2] Generalising the Newton method to systems of multiple variables, the iteration formula includes a Jacobian matrix. Solving this directly would involve calculation of the Jacobian's inverse, when the Jacobian matrix itself ...

  6. 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]

  7. Levenberg–Marquardt algorithm - Wikipedia

    en.wikipedia.org/wiki/Levenberg–Marquardt...

    The primary application of the Levenberg–Marquardt algorithm is in the least-squares curve fitting problem: given a set of empirical pairs (,) of independent and dependent variables, find the parameters ⁠ ⁠ of the model curve (,) so that the sum of the squares of the deviations () is minimized:

  8. Wolfe conditions - Wikipedia

    en.wikipedia.org/wiki/Wolfe_conditions

    Each step often involves approximately solving the subproblem (+) where is the current best guess, is a search direction, and is the step length. The inexact line searches provide an efficient way of computing an acceptable step length α {\displaystyle \alpha } that reduces the objective function 'sufficiently', rather than minimizing the ...

  9. Broyden–Fletcher–Goldfarb–Shanno algorithm - Wikipedia

    en.wikipedia.org/wiki/Broyden–Fletcher...

    In numerical optimization, the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm is an iterative method for solving unconstrained nonlinear optimization problems. [1] Like the related Davidon–Fletcher–Powell method, BFGS determines the descent direction by preconditioning the gradient with curvature information.