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Quadratic programming (QP) is the process of solving certain mathematical optimization problems involving quadratic functions. Specifically, one seeks to optimize a multivariate quadratic function subject to linear constraints on the variables. Quadratic programming is a type of nonlinear programming.
Mathematical optimization or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries.
In mathematics, gradient descent is a first-order iterative optimization algorithm for finding a local minimum of a differentiable function. The idea is to take repeated steps in the opposite direction of the gradient of the function at the current point, because this is the direction of steepest descent. Conversely, stepping in the direction of the gradient will lead to a local maximum of that function; the procedure is then known as gradient ascent. It is particularly useful in machine learning for minimizing the cost or loss function. Gradient descent should not be confused with local search algorithms, although both are iterative methods for optimization.
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Pattern search is a family of numerical optimization methods that does not require a gradient. As a result, it can be used on functions that are not continuous or differentiable. One such pattern search method is "convergence", which is based on the theory of positive bases. Optimization attempts to find the best match in a multidimensional analysis space of possibilities.
Augmented Lagrangian methods are a certain class of algorithms for solving constrained optimization problems. They have similarities to penalty methods in that they replace a constrained optimization problem by a series of unconstrained problems and add a penalty term to the objective; the difference is that the augmented Lagrangian method adds yet another term designed to mimic a Lagrange multiplier. The augmented Lagrangian is related to, but not identical with the method of Lagrange multipliers.
Yurii Nesterov is a Russian mathematician, an internationally recognized expert in convex optimization, especially in the development of efficient algorithms and numerical optimization analysis. He is currently a professor at the University of Louvain (UCLouvain).
MINOS is a Fortran software package for solving linear and nonlinear mathematical optimization problems. MINOS may be used for linear programming, quadratic programming, and more general objective functions and constraints, and for finding a feasible point for a set of linear or nonlinear equalities and inequalities.
References
- ↑ Dembo, Ron S.; Steihaug, Trond (1983). "Truncated-Newton algorithms for large-scale unconstrained optimization". Mathematical Programming. Springer. 26 (2): 190–212. doi:10.1007/BF02592055. S2CID 40537623.. Convergence results for this algorithm can be found in Dembo, Ron S.; Eisenstat, Stanley C.; Steihaug, Trond (1982). "Inexact newton methods". SIAM Journal on Numerical Analysis. 19 (2): 400–408. Bibcode:1982SJNA...19..400D. doi:10.1137/0719025. JSTOR 2156954..
- 1 2 Martens, James (2010). Deep learning via Hessian-free optimization (PDF). Proc. International Conference on Machine Learning.
- ↑ Nash, Stephen G. (2000). "A survey of truncated-Newton methods". Journal of Computational and Applied Mathematics. 124 (1–2): 45–59. Bibcode:2000JCoAM.124...45N. doi: 10.1016/S0377-0427(00)00426-X .
- ↑ Nash, Stephen G. (1985). "Preconditioning of truncated-Newton methods" (PDF). SIAM J. Sci. Stat. Comput. 6 (3): 599–616. doi:10.1137/0906042.
