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Sequential quadratic programming (SQP) is an iterative method for constrained nonlinear optimization, also known as Lagrange-Newton method.SQP methods are used on mathematical problems for which the objective function and the constraints are twice continuously differentiable, but not necessarily convex.
Quadratic programming is particularly simple when Q is positive definite and there are only equality constraints; specifically, the solution process is linear. By using Lagrange multipliers and seeking the extremum of the Lagrangian, it may be readily shown that the solution to the equality constrained problem
Limited-memory BFGS (L-BFGS or LM-BFGS) is an optimization algorithm in the family of quasi-Newton methods that approximates the Broyden–Fletcher–Goldfarb–Shanno algorithm (BFGS) using a limited amount of computer memory. [1] It is a popular algorithm for parameter estimation in machine learning.
In the EQP phase of SLQP, the search direction of the step is obtained by solving the following equality-constrained quadratic program: + + (,,).. + = + =Note that the term () in the objective functions above may be left out for the minimization problems, since it is constant.
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. Many algorithms are used to handle the optimization part.
Examples of simplices include a line segment in one-dimensional space, a triangle in two-dimensional space, a tetrahedron in three-dimensional space, and so forth. The method approximates a local optimum of a problem with n variables when the objective function varies smoothly and is unimodal .
In the SciPy extension to Python, the scipy.optimize.minimize function includes, among other methods, a BFGS implementation. [8] Notable proprietary implementations include: Mathematica includes quasi-Newton solvers. [9] The NAG Library contains several routines [10] for minimizing or maximizing a function [11] which use quasi-Newton algorithms.
The geometric interpretation of Newton's method is that at each iteration, it amounts to the fitting of a parabola to the graph of () at the trial value , having the same slope and curvature as the graph at that point, and then proceeding to the maximum or minimum of that parabola (in higher dimensions, this may also be a saddle point), see below.