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Inequalities for inexact line search
the unconstrained minimization problem, the Wolfe conditions (also known as the Armijo-Wolfe conditions in some books) are a set of inequalities for
Wolfe_conditions
Optimization algorithm
The Frank–Wolfe algorithm is an iterative first-order optimization algorithm for constrained convex optimization. Also known as the conditional gradient
Frank–Wolfe_algorithm
Optimization algorithm
method, or a sequence η n {\displaystyle \eta _{n}} satisfying the Wolfe conditions (which can be found by using line search). When the function f {\displaystyle
Gradient_descent
Optimization method
be enforced explicitly e.g. by finding a point xk+1 satisfying the Wolfe conditions, which entail the curvature condition, using line search. Instead of
Broyden–Fletcher–Goldfarb–Shanno algorithm
Broyden–Fletcher–Goldfarb–Shanno_algorithm
Optimization algorithm
scaled and therefore the unit step length is accepted in most iterations. A Wolfe line search is used to ensure that the curvature condition is satisfied
Limited-memory_BFGS
Sequential model-based optimization of expensive black-box functions
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Bayesian_optimization
Sequence of locally optimal choices
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Greedy_algorithm
Study of mathematical algorithms for optimization problems
similarities with Quasi-Newton methods. Conditional gradient method (Frank–Wolfe) for approximate minimization of specially structured problems with linear
Mathematical_optimization
Method of solving linear programming problems
approach for solving problems with >= constraints Karush–Kuhn–Tucker conditions, which apply to nonlinear optimization problems with inequality constraints
Big_M_method
Algorithm for linear programming
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Simplex_algorithm
Mathematical optimization problem restricted to integers
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Integer_programming
Algorithm for finding zeros of functions
non-real. In this case almost all real initial conditions lead to chaotic behavior, while some initial conditions iterate either to infinity or to repeating
Newton's_method
Algorithm used to solve non-linear least squares problems
fitting exactly. This equation is an example of very sensitive initial conditions for the Levenberg–Marquardt algorithm. One reason for this sensitivity
Levenberg–Marquardt_algorithm
Problem optimization method
and f {\displaystyle f} is a production function satisfying the Inada conditions. An initial capital stock k 0 > 0 {\displaystyle k_{0}>0} is assumed.
Dynamic_programming
Optimization algorithm
_{k}B_{k}^{-1}\nabla f(x_{k})} , with α {\displaystyle \alpha } chosen to satisfy the Wolfe conditions; x k + 1 = x k + Δ x k {\displaystyle x_{k+1}=x_{k}+\Delta x_{k}} ;
Quasi-Newton_method
Numerical optimization algorithm
converge to a non-stationary point, unless the problem satisfies stronger conditions than are necessary for modern methods. Modern improvements over the Nelder–Mead
Nelder–Mead_method
Collective behavior of decentralized, self-organized systems
connected together by real-time swarming algorithms, could diagnose medical conditions with substantially higher accuracy than individual doctors or groups of
Swarm_intelligence
Subfield of mathematical optimization
can also be solved by the following contemporary methods: Bundle methods (Wolfe, Lemaréchal, Kiwiel), and Subgradient projection methods (Polyak), Interior-point
Convex_optimization
Solution process for some optimization problems
Karush–Kuhn–Tucker (KKT) conditions are available. Under convexity, the KKT conditions are sufficient for a global optimum. Without convexity, these conditions are sufficient
Nonlinear_programming
Optimization by removing non-optimal solutions to subproblems
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Branch_and_bound
Subfield of mathematical optimization
is a combinatorial optimization problem with the following additional conditions. Note that the below referred polynomials are functions of the size of
Combinatorial_optimization
Linear programming algorithm
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Karmarkar's_algorithm
Optimization algorithm
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Hill_climbing
Numerical approximation algorithm
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Iterative_method
Solving an optimization problem with a quadratic objective function
Besides the Lagrangian duality theory, there are other duality pairings (e.g. Wolfe, etc.). For positive definite Q, the minization problem is convex. Hence
Quadratic_programming
Subfield of convex optimization
possible to attain strong duality for SDPs without additional regularity conditions by using an extended dual problem proposed by Ramana. Consider three random
Semidefinite_programming
Method to solve optimization problems
interior-point algorithms, large-scale problems, decomposition following Dantzig–Wolfe and Benders, and introducing stochastic programming.) Edmonds, Jack; Giles
Linear_programming
Type of algorithm for constrained optimization
computational mechanics, especially in the Finite element method, to enforce conditions such as e.g. contact. The advantage of the penalty method is that, once
Penalty_method
Optimization algorithm
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Ant colony optimization algorithms
Ant_colony_optimization_algorithms
Concept in mathematics
Conjugate gradient method L-BFGS (limited memory BFGS) Nelder–Mead method Wolfe conditions Fletcher, R.; Reeves, C. M. (1964). "Function minimization by conjugate
Nonlinear conjugate gradient method
Nonlinear_conjugate_gradient_method
Optimizing objective functions that have constrained variables
to be maximized. Constraints can be either hard constraints, which set conditions for the variables that are required to be satisfied, or soft constraints
Constrained_optimization
Optimization algorithm
applying Newton's method to the first-order optimality conditions, or Karush–Kuhn–Tucker conditions, of the problem. Consider a nonlinear programming problem
Sequential quadratic programming
Sequential_quadratic_programming
Concept in convex optimization mathematics
Many methods for minimizing differentiable functions satisfy Wolfe's sufficient conditions for convergence, where step-sizes typically depend on the current
Subgradient_method
Algorithm to compute the maximum flow in a flow network
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Edmonds–Karp_algorithm
Algorithms for solving convex optimization problems
complementarity" condition, for its resemblance to "complementary slackness" in KKT conditions. We try to find those ( x μ , λ μ ) {\displaystyle (x_{\mu },\lambda _{\mu
Interior-point_method
Optimization algorithm
a number of ways, such as a backtracking line search or using the Wolfe conditions. Like other optimization methods, line search may be combined with
Line_search
Mathematical algorithm for eliminating variables from a system of linear inequalities
achievability proofs result in conditions under which the existence of a well-performing coding scheme is guaranteed. These conditions are often described by
Fourier–Motzkin_elimination
Optimization technique
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Metaheuristic
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Sequential linear-quadratic programming
Sequential_linear-quadratic_programming
Quantum physics-based metaheuristic for optimization problems
that quantum annealing outperforms simulated annealing under certain conditions (see Heim et al and see Yan and Sinitsyn for a fully solvable model of
Quantum_annealing
Gradient descent Stochastic gradient descent Coordinate descent Frank–Wolfe algorithm Landweber iteration Random coordinate descent Conjugate gradient
Gradient_method
Algorithm for solving linear programs
technique in linear programming which uses this kind of approach is the Dantzig–Wolfe decomposition algorithm. Additionally, column generation has been applied
Column_generation
Solving multiple machine learning tasks at the same time
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Multi-task_learning
Term in mathematical optimization
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Trust_region
Mathematical algorithm
the optimum, it is possible to show formal convergence under reasonable conditions. The other problem is difficulty in parallelism. Since the nature of coordinate
Coordinate_descent
Form of Newton's method used in statistics
{J}}^{-1}(\theta _{m})V(\theta _{m}),\,} and under certain regularity conditions, it can be shown that θ m → θ ∗ {\displaystyle \theta _{m}\rightarrow
Scoring_algorithm
Mathematical algorithm
\alpha } should be chosen such that it satisfies the Wolfe conditions or the Goldstein conditions. In cases where the direction of the shift vector is
Gauss–Newton_algorithm
Algorithm for finding a local minimum of a function
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Powell's_method
Methods in numerical computation
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Rosenbrock_methods
Algorithm in computer science
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Artificial bee colony algorithm
Artificial_bee_colony_algorithm
Unit hypercube of variable dimension whose corners have been perturbed
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Klee–Minty_cube
Continuous function whose value increases to infinity
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Barrier_function
Class of algorithms for solving constrained optimization problems
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Augmented_Lagrangian_method
Linear programming algorithm
programming, the Karush–Kuhn–Tucker conditions are both necessary and sufficient for optimality. The KKT conditions of a linear programming problem in
Revised_simplex_method
Optimization technique for solving (mixed) integer linear programs
dual functions. Another common situation is the application of the Dantzig–Wolfe decomposition to a structured optimization problem in which formulations
Cutting-plane_method
Algorithm for computing the maximal flow of a network
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Dinic's_algorithm
Concept in mathematics
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Mirror_descent
Algorithm for solving the quadratic programming problem from training SVMs
violator of the Karush–Kuhn–Tucker (KKT) conditions is guaranteed to converge. The chunking algorithm obeys the conditions of the theorem, and hence will converge
Sequential minimal optimization
Sequential_minimal_optimization
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Bat_algorithm
Mathematical optimization method
limit point (if exists) can make convergence faster. For example, in Wolfe conditions, there is no mention of α 0 {\displaystyle \alpha _{0}} but another
Backtracking_line_search
Mathematical combinatorial optimization method
The algorithm typically begins by using a reformulation, such as Dantzig–Wolfe decomposition, to form what is known as the Master Problem. The decomposition
Branch_and_price
Computer compiler optimization technique
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Register_allocation
Combinatorial optimization method
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Branch_and_cut
Local search algorithm
until a user-specified stopping condition is met (two examples of such conditions are a simple time limit or a threshold on the fitness score). The neighboring
Tabu_search
Optimization method
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Davidon–Fletcher–Powell formula
Davidon–Fletcher–Powell_formula
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Evolutionary multimodal optimization
Evolutionary_multimodal_optimization
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Rider_optimization_algorithm
Technique for finding an extremum of a function
is how this search algorithm gets its name. Any number of termination conditions may be applied, depending upon the application. The interval ΔX = X4 −
Golden-section_search
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Biconvex_optimization
Optimization algorithm
convergence of CS-based algorithms Providing the sufficient and necessary conditions for the control parameter settings Employing non-homogeneous search rules
Cuckoo_search
Iterative optimisation algorithm
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Powell's_dog_leg_method
Primal-Dual algorithm optimization for convex problems
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Chambolle–Pock_algorithm
Branch of mathematical optimization
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Discrete_optimization
Metaheuristic proposed by Xin-She Yang
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Firefly_algorithm
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Great_deluge_algorithm
Class of algorithms that find approximate solutions to optimization problems
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Approximation_algorithm
Approximation for nonlinear optimization
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Successive_linear_programming
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Meta-optimization
Iterative method for minimizing convex functions
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Ellipsoid_method
Method for mathematical optimization
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Criss-cross_algorithm
Chinese scientist and revolutionary (born 1961)
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Liu_Gang
Population-based search algorithm
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Bees_algorithm
Special case of discrete optimization
ordered set of variables used as an additional way to specify integrality conditions in an optimization model. Special order sets are basically a device or
Special_ordered_set
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Distributed constraint optimization
Distributed_constraint_optimization
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Parallel_metaheuristic
Optimization algorithm
problems under the maximum iteration k max {\displaystyle k_{\max }} . The conditions on R ( θ ) {\displaystyle R(\theta )} and x i ( 0 ) ( i = 1 , … , m
Spiral_optimization_algorithm
Algorithm for solving linear programming problems
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Affine_scaling
American mountaineer who died on K2 in 1939
Dudley Francis Cecil Wolfe (February 6, 1896 – July 30, 1939) was an American socialite. As a racing yacht owner and captain, he was the first person
Dudley_Wolfe
Method for finding stationary points of a function
[f''(x_{k})]^{-1}f'(x_{k}).} This is often done to ensure that the Wolfe conditions, or much simpler and efficient Armijo's condition, are satisfied at
Newton's method in optimization
Newton's_method_in_optimization
Irish revolutionary figure (1763–1798)
Theobald Wolfe Tone (Irish: Bhulbh Teón; 20 June 1763 – 19 November 1798), posthumously known as Wolfe Tone, was a revolutionary exponent of Irish independence
Wolfe_Tone
Mathematical optimization algorithms
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Truncated_Newton_method
have other forms. The BHHH algorithm has the advantage that, if certain conditions apply, convergence of the iterative procedure is guaranteed.[citation
Berndt–Hall–Hall–Hausman algorithm
Berndt–Hall–Hall–Hausman_algorithm
generated by the SR1 method converges to the true Hessian under mild conditions, in theory; in practice, the approximate Hessians generated by the SR1
Symmetric_rank-one
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Minimum_Population_Search
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Lemke's_algorithm
Type of optimization heuristic
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Extremal_optimization
Algorithm in mathematical optimization
function is denoted by 𝓁 : V → ℕ. This function must satisfy the following conditions in order to be considered valid: Valid labeling: 𝓁(u) ≤ 𝓁(v) + 1 for
Push–relabel maximum flow algorithm
Push–relabel_maximum_flow_algorithm
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Humanoid_ant_algorithm
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Guided_local_search
Successive parabolic interpolation Gradients Convergence Trust region Wolfe conditions Quasi–Newton Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno
Successive parabolic interpolation
Successive_parabolic_interpolation
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