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The Symmetric Rank 1 (SR1) method is a quasi-Newton method to update the second derivative (Hessian) based on the derivatives (gradients) calculated at
Symmetric_rank-one
Optimization method
are symmetric rank-one matrices, but their sum is a rank-two update matrix. BFGS and DFP updating matrix both differ from its predecessor by a rank-two
Broyden–Fletcher–Goldfarb–Shanno algorithm
Broyden–Fletcher–Goldfarb–Shanno_algorithm
Sequence of locally optimal choices
optimization problem only depends on the partial solution of solving it for one subproblem, we can solve this problem by "greedily" considering only the
Greedy_algorithm
Numerical approximation algorithm
matrix A {\displaystyle A} is symmetric positive-definite. For symmetric (and possibly indefinite) A {\displaystyle A} one works with the minimal residual
Iterative_method
Optimization algorithm
common quasi-Newton algorithms are currently the SR1 formula (for "symmetric rank-one"), the BHHH method, the widespread BFGS method (suggested independently
Quasi-Newton_method
Optimization algorithm
problem. If the system matrix A {\displaystyle \mathbf {A} } is real symmetric and positive-definite, an objective function is defined as the quadratic
Gradient_descent
Subfield of mathematical optimization
where the variables are z. Note that there are rank(A) fewer variables. This means that, in principle, one can restrict attention to convex optimization
Convex_optimization
Methods in numerical computation
Broyden–Fletcher–Goldfarb–Shanno and L-BFGS Davidon–Fletcher–Powell Symmetric rank-one (SR1) Other methods Conjugate gradient Gauss–Newton Gradient Mirror
Rosenbrock_methods
Method of solving linear programming problems
solution, if it exists. The simplex algorithm is the original and still one of the most widely used methods for solving linear maximization problems
Big_M_method
Algorithm for solving the quadratic programming problem from training SVMs
_{2}=k,} and this reduced problem can be solved analytically: one needs to find a minimum of a one-dimensional quadratic function. k {\displaystyle k} is the
Sequential minimal optimization
Sequential_minimal_optimization
Optimization algorithm
a one-dimensional function, f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } , and assume that it is unimodal, that is, contains exactly one local
Line_search
Collective behavior of decentralized, self-organized systems
requires a symmetric network and couples the two directions together; forwards reinforcement rewards a route before the outcome is known (but then one would
Swarm_intelligence
Optimization algorithm
involves a low-rank representation for the direct and/or inverse Hessian. This represents the Hessian as a sum of a diagonal matrix and a low-rank update. Such
Limited-memory_BFGS
Type of algorithm for constrained optimization
some p0>0, such that for all p>p0, the penalized objective fp has exactly one critical point in V* (denoted by x*(p)), and x*(p) approaches x* as p→∞.
Penalty_method
Broyden–Fletcher–Goldfarb–Shanno and L-BFGS Davidon–Fletcher–Powell Symmetric rank-one (SR1) Other methods Conjugate gradient Gauss–Newton Gradient Mirror
Sequential linear-quadratic programming
Sequential_linear-quadratic_programming
Sequential model-based optimization of expensive black-box functions
function values and derivatives, but because each gradient observation adds one value per input dimension, exact Gaussian process inference becomes costly
Bayesian_optimization
Term in mathematical optimization
Broyden–Fletcher–Goldfarb–Shanno and L-BFGS Davidon–Fletcher–Powell Symmetric rank-one (SR1) Other methods Conjugate gradient Gauss–Newton Gradient Mirror
Trust_region
Optimizing objective functions that have constrained variables
function is quadratic, the problem is a quadratic programming problem. It is one type of nonlinear programming. It can still be solved in polynomial time
Constrained_optimization
Optimization method
Broyden–Fletcher–Goldfarb–Shanno (BFGS) method Limited-memory BFGS method Symmetric rank-one formula Nelder–Mead method Compact quasi-Newton representation Avriel
Davidon–Fletcher–Powell formula
Davidon–Fletcher–Powell_formula
Inequalities for inexact line search
algorithm based on Armijo's condition has a better theoretical guarantee than one based on Wolfe conditions (see the sections on "Upper bound for learning
Wolfe_conditions
Problem optimization method
the first rank (i.e., row) and you wanted to know the shortest path (the sum of the minimum costs at each visited rank) to get to the last rank; assuming
Dynamic_programming
Subfield of mathematical optimization
problem ("MST"), and the knapsack problem. In many such problems, such as the ones previously mentioned, exhaustive search is not tractable, and so specialized
Combinatorial_optimization
Population-based search algorithm
last ns-nb flower patches with randomly generated solutions. At the end of one search cycle, the scout population is again composed of ns scouts: nr scouts
Bees_algorithm
Mathematical optimization problem restricted to integers
which unknowns are binary, and only the restrictions must be satisfied, is one of Karp's 21 NP-complete problems. If some decision variables are not discrete
Integer_programming
Algorithm used to solve non-linear least squares problems
vector β {\displaystyle {\boldsymbol {\beta }}} . In cases with only one minimum, an uninformed standard guess like β T = ( 1 , 1 , … , 1 )
Levenberg–Marquardt_algorithm
Subfield of convex optimization
\mathbb {S} ^{n}} the space of all n × n {\displaystyle n\times n} real symmetric matrices. The space is equipped with the inner product (where t r a c
Semidefinite_programming
Special case of discrete optimization
even when all the members are themselves continuous, a model containing one or more special ordered sets becomes a discrete optimization problem requiring
Special_ordered_set
Optimization technique
therefore to be understood as an example. One approach is to characterize the type of search strategy. One type of search strategy is an improvement on
Metaheuristic
Broyden–Fletcher–Goldfarb–Shanno and L-BFGS Davidon–Fletcher–Powell Symmetric rank-one (SR1) Other methods Conjugate gradient Gauss–Newton Gradient Mirror
Bat_algorithm
Solving an optimization problem with a quadratic objective function
Given: a real-valued, n-dimensional vector c, an n×n-dimensional real symmetric matrix Q, an m×n-dimensional real matrix A, and an m-dimensional real
Quadratic_programming
solution) is alternatively updating x , y {\displaystyle x,y} by fixing one of them and solving the corresponding convex optimization problem. The generalization
Biconvex_optimization
Numerical optimization algorithm
vertices in n dimensions. Examples of simplices include a line segment in one-dimensional space, a triangle in two-dimensional space, a tetrahedron in
Nelder–Mead_method
Algorithm for finding zeros of functions
generally require fewer iterations to converge if the guess is close to one of the function's roots. The method will usually converge if f ′ ( x 0
Newton's_method
Optimization algorithm
point). In this case, the Lagrangian Hessian must be regularized, for example one can add a multiple of the identity to it such that the resulting matrix is
Sequential quadratic programming
Sequential_quadratic_programming
Algorithm for computing the maximal flow of a network
in the Ford–Fulkerson algorithm, if each augmenting path is the shortest one, then the length of the augmenting paths is non-decreasing and the algorithm
Dinic's_algorithm
Linear programming algorithm
matrix A has full row rank and that the problem is feasible, i.e., there is at least one x ≥ 0 such that Ax = b. If A is rank-deficient, either there
Revised_simplex_method
Technique for finding an extremum of a function
Avriel, Mordecai; Wilde, Douglass J. (1966), "Optimality proof for the symmetric Fibonacci search technique", Fibonacci Quarterly, 4 (3): 265–269, doi:10
Golden-section_search
Algorithm for solving linear programming problems
others replaced the projective transformations that Karmarkar used by affine ones. After a few years, it was realized that the "new" affine scaling algorithms
Affine_scaling
Broyden–Fletcher–Goldfarb–Shanno and L-BFGS Davidon–Fletcher–Powell Symmetric rank-one (SR1) Other methods Conjugate gradient Gauss–Newton Gradient Mirror
Gradient_method
Solution process for some optimization problems
objective function is not a linear function. An optimization problem is one of calculation of the extrema (maxima, minima or stationary points) of an
Nonlinear_programming
Optimization by removing non-optimal solutions to subproblems
solution, and is discarded if it cannot produce a better solution than the best one found so far by the algorithm. The algorithm depends on efficient estimation
Branch_and_bound
Algorithm to compute the maximum flow in a flow network
found in O ( | E | ) {\displaystyle O(|E|)} time, that every time at least one of the E edges becomes saturated (an edge which has the maximum possible
Edmonds–Karp_algorithm
Primal-Dual algorithm optimization for convex problems
Broyden–Fletcher–Goldfarb–Shanno and L-BFGS Davidon–Fletcher–Powell Symmetric rank-one (SR1) Other methods Conjugate gradient Gauss–Newton Gradient Mirror
Chambolle–Pock_algorithm
Algorithm for linear programming
=\mathbf {b} ,\,\forall \ x_{i}\geq 0} It is also useful to assume that the rank of A {\displaystyle \mathbf {A} } is the number of rows. This results in
Simplex_algorithm
Optimization algorithm
programming and binary search. To attempt to avoid getting stuck in local optima, one could use restarts (i.e. repeated local search), or more complex schemes
Hill_climbing
Mathematical combinatorial optimization method
Broyden–Fletcher–Goldfarb–Shanno and L-BFGS Davidon–Fletcher–Powell Symmetric rank-one (SR1) Other methods Conjugate gradient Gauss–Newton Gradient Mirror
Branch_and_price
Method to solve optimization problems
problem as: Maximize cTx subject to Ax ≤ b, x ≥ 0; with the corresponding symmetric dual problem, Minimize bTy subject to ATy ≥ c, y ≥ 0. An alternative primal
Linear_programming
Mathematical algorithm
the simplest case of cyclic coordinate descent, one cyclically iterates through the directions, one at a time, minimizing the objective function with
Coordinate_descent
Algorithm for solving linear programs
of a problem where it is successfully used is the cutting stock problem. One particular technique in linear programming which uses this kind of approach
Column_generation
Mathematical algorithm for eliminating variables from a system of linear inequalities
all variables are eliminated from a system of linear inequalities, then one obtains a system of constant inequalities. It is then trivial to decide whether
Fourier–Motzkin_elimination
Iterative optimisation algorithm
Broyden–Fletcher–Goldfarb–Shanno and L-BFGS Davidon–Fletcher–Powell Symmetric rank-one (SR1) Other methods Conjugate gradient Gauss–Newton Gradient Mirror
Powell's_dog_leg_method
Combinatorial optimization method
(1991). "A Branch-and-Cut Algorithm for the Resolution of Large-Scale Symmetric Traveling Salesman Problems". SIAM Review. 33 (1): 60–100. doi:10.1137/1033004
Branch_and_cut
Optimization algorithm
Broyden–Fletcher–Goldfarb–Shanno and L-BFGS Davidon–Fletcher–Powell Symmetric rank-one (SR1) Other methods Conjugate gradient Gauss–Newton Gradient Mirror
Frank–Wolfe_algorithm
Concept in mathematics
_{n})_{n\geq 0}} applied to a differentiable function F {\displaystyle F} , one starts with a guess x 0 {\displaystyle \mathbf {x} _{0}} for a local minimum
Mirror_descent
Form of Newton's method used in statistics
Broyden–Fletcher–Goldfarb–Shanno and L-BFGS Davidon–Fletcher–Powell Symmetric rank-one (SR1) Other methods Conjugate gradient Gauss–Newton Gradient Mirror
Scoring_algorithm
Algorithm for finding a local minimum of a function
the search vector which contributed most to the new direction, i.e. the one which was most successful ( i d = arg max i = 1 N | α i | ‖ s i ‖ {\textstyle
Powell's_method
Algorithms for solving convex optimization problems
algorithm was the first one. Path-following methods: the algorithms of James Renegar and Clovis Gonzaga were the first ones. Primal-dual methods. Given
Interior-point_method
Optimization technique for solving (mixed) integer linear programs
an optimal solution, and if the feasible region does not contain a line), one can always find an extreme point or a corner point that is optimal. The obtained
Cutting-plane_method
Unit hypercube of variable dimension whose corners have been perturbed
Dantzig's simplex algorithm has poor worst-case performance when initialized at one corner of their "squashed cube". On the three-dimensional version, the simplex
Klee–Minty_cube
British mathematician
optimization problems. Moreover, he was among those who derived the symmetric rank-one updating formula, and his name was also attributed to Broyden's methods
Charles_George_Broyden
Concept in convex optimization mathematics
constant step-length and scaled subgradients having Euclidean norm equal to one, the subgradient method converges to an arbitrarily close approximation to
Subgradient_method
Quantum physics-based metaheuristic for optimization problems
glass. In the case of annealing a purely mathematical objective function, one may consider the variables in the problem to be classical degrees of freedom
Quantum_annealing
Local search algorithm
such as frequency and impact of changes made. One example of an intermediate-term memory structure is one that prohibits or encourages solutions that contain
Tabu_search
Linear programming algorithm
patents. This left many mathematicians uneasy, such as Ronald Rivest (himself one of the holders of the patent on the RSA algorithm), who expressed the opinion
Karmarkar's_algorithm
Chinese scientist and revolutionary (born 1961)
movement's organizing body. As a result, he was sixth on a list of twenty-one activists whose arrests were ordered by the government. Liu went into hiding
Liu_Gang
Study of mathematical algorithms for optimization problems
it is also the global minimum, but a nonconvex problem may have more than one local minimum not all of which need be global minima. A large number of algorithms
Mathematical_optimization
Class of algorithms that find approximate solutions to optimization problems
for every ϵ > 0. Domination analysis considers guarantees in terms of the rank of the computed solution. PTAS - a type of approximation algorithm that takes
Approximation_algorithm
Broyden–Fletcher–Goldfarb–Shanno and L-BFGS Davidon–Fletcher–Powell Symmetric rank-one (SR1) Other methods Conjugate gradient Gauss–Newton Gradient Mirror
Lemke's_algorithm
Concept in mathematics
{\displaystyle \nabla _{x}f} indicates the direction of maximum increase. One simply starts in the opposite (steepest descent) direction: Δ x 0 = − ∇ x
Nonlinear conjugate gradient method
Nonlinear_conjugate_gradient_method
Branch of mathematical optimization
Broyden–Fletcher–Goldfarb–Shanno and L-BFGS Davidon–Fletcher–Powell Symmetric rank-one (SR1) Other methods Conjugate gradient Gauss–Newton Gradient Mirror
Discrete_optimization
Broyden–Fletcher–Goldfarb–Shanno and L-BFGS Davidon–Fletcher–Powell Symmetric rank-one (SR1) Other methods Conjugate gradient Gauss–Newton Gradient Mirror
Guided_local_search
Metaheuristic proposed by Xin-She Yang
j end for i Rank fireflies and find the current best; end while end Note that the number of objective function evaluations per loop is one evaluation per
Firefly_algorithm
Continuous function whose value increases to infinity
optimization problem: minimize f(x) subject to x ≤ b where b is some constant. If one wishes to remove the inequality constraint, the problem can be reformulated
Barrier_function
Broyden–Fletcher–Goldfarb–Shanno and L-BFGS Davidon–Fletcher–Powell Symmetric rank-one (SR1) Other methods Conjugate gradient Gauss–Newton Gradient Mirror
Generalized_iterative_scaling
Method for mathematical optimization
simplex algorithm first finds a (primal-) feasible basis by solving a "phase-one problem"; in "phase two", the simplex algorithm pivots between a sequence
Criss-cross_algorithm
Iterative method for minimizing convex functions
inequality and equality constraints). One way to do this is by combining the primal and dual linear programs together into one program, and adding the additional
Ellipsoid_method
Algorithm in computer science
bees: employed bees, onlookers and scouts. It is assumed that there is only one artificial employed bee for each food source. In other words, the number
Artificial bee colony algorithm
Artificial_bee_colony_algorithm
Class of algorithms for solving constrained optimization problems
and to encourage parsimony in the optimal solution (e.g., sparsity and low rank). ADMM's effectiveness for solving regularized problems may mean it could
Augmented_Lagrangian_method
Solving multiple machine learning tasks at the same time
R T {\displaystyle f:{\mathcal {X}}\rightarrow \mathbb {R} ^{T}} is a symmetric matrix-valued function Γ : X × X → R T × T {\displaystyle \Gamma :{\mathcal
Multi-task_learning
Optimization algorithm
satisfy the following condition: min i = 1 , … , m { max j = 1 , … , m { rank [ d j , i ( 0 ) R ( θ ) d j , i ( 0 ) ⋯ R ( θ ) 2 n − 1 d j , i
Spiral_optimization_algorithm
Finding multiple solutions of a problem
Schäpermeier, Lennart; Grimme, Christian; Kerschke, Pascal (2020). "One PLOT to Show Them All: Visualization of Efficient Sets in Multi-Objective
Evolutionary multimodal optimization
Evolutionary_multimodal_optimization
Meta-optimization from numerical optimization is the use of one optimization method to tune another optimization method. Meta-optimization is reported
Meta-optimization
Optimization algorithm
p_{a}} ) of the worse nests are abandoned and new ones are built; Keep the best solutions/nests; Rank the solutions/nests and find the current best; Pass
Cuckoo_search
Computer compiler optimization technique
methods, and storing it into one register during its whole lifetime. Many register allocation approaches optimize for one or more specific categories of
Register_allocation
Optimization algorithm
Kaufmann, pp. 252–260, 1995 L.M. Gambardella and M. Dorigo, "Solving Symmetric and Asymmetric TSPs by Ant Colonies", Proceedings of the IEEE Conference
Ant colony optimization algorithms
Ant_colony_optimization_algorithms
Numerical optimization algorithm
Robert Hall, and Jerry Hausman. If a nonlinear model is fitted to the data one often needs to estimate coefficients through optimization. A number of optimization
Berndt–Hall–Hall–Hausman algorithm
Berndt–Hall–Hall–Hausman_algorithm
Mathematical optimization algorithms
Broyden–Fletcher–Goldfarb–Shanno and L-BFGS Davidon–Fletcher–Powell Symmetric rank-one (SR1) Other methods Conjugate gradient Gauss–Newton Gradient Mirror
Truncated_Newton_method
exists a large set of different techniques strongly or loosely based in these ones, whose behavior encompasses the multiple parallel execution of algorithm
Parallel_metaheuristic
Tensor invariant under permutations of vectors it acts on
characteristic zero, the graded vector space of all symmetric tensors can be naturally identified with the symmetric algebra on V. A related concept is that of
Symmetric_tensor
Approximation for nonlinear optimization
methods. While solving a QP subproblem takes more time than solving an LP one, the overall decrease in the number of iterations, due to improved convergence
Successive_linear_programming
successively fitting parabolas (polynomials of degree two) to a function of one variable at three unique points or, in general, a function of n variables
Successive parabolic interpolation
Successive_parabolic_interpolation
{\displaystyle \alpha } is not necessarily an injection, i.e., one agent may own more than one variables. It is also not necessarily a surjection, i.e., some
Distributed constraint optimization
Distributed_constraint_optimization
Algorithm in mathematical optimization
print(paste("Relabel =", relabCtr, "Push =", pushCtr)) # Check, flow matrix is skew-symmetric. # Check, flow exiting source (=1) equals flow entering sink (=nV). flow
Push–relabel maximum flow algorithm
Push–relabel_maximum_flow_algorithm
Statistical hypothesis test
can be assumed symmetric, then the null and alternative hypotheses are the following: Null hypothesis H0 F {\displaystyle F} is symmetric about μ = 0 {\displaystyle
Wilcoxon_signed-rank_test
Broyden–Fletcher–Goldfarb–Shanno and L-BFGS Davidon–Fletcher–Powell Symmetric rank-one (SR1) Other methods Conjugate gradient Gauss–Newton Gradient Mirror
Great_deluge_algorithm
(pseudo-)Riemannian manifold whose geodesics are reversible
curvature −1) is a locally symmetric space but not a symmetric space. Every lens space is locally symmetric but not symmetric, with the exception of the
Symmetric_space
Crucial concept of quantum information
1]} . Then for μ ∈ [ 0 , 1 / 2 ] {\displaystyle \mu \in [0,1/{\sqrt {2}}]} one can verify that the following POVM M + , + := 1 4 [ I + μ ( σ x + σ z ) ]
Incompatibility of quantum measurements
Incompatibility_of_quantum_measurements
Type of optimization heuristic
Broyden–Fletcher–Goldfarb–Shanno and L-BFGS Davidon–Fletcher–Powell Symmetric rank-one (SR1) Other methods Conjugate gradient Gauss–Newton Gradient Mirror
Extremal_optimization
Disproved conjecture in multilinear algebra on the rank of symmetric tensors
fact that the rank of a symmetric matrix can always be realized by a symmetric decomposition, as in the eigendecomposition of a real symmetric matrix. The
Comon's_conjecture
Matrix equal to its transpose
a symmetric matrix is a square matrix that is equal to its transpose. Formally, A is symmetric ⟺ A = A T . {\displaystyle A{\text{ is symmetric}}\iff
Symmetric_matrix
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SYMMETRIC RANK-ONE
SYMMETRIC RANK-ONE
SYMMETRIC RANK-ONE
SYMMETRIC RANK-ONE
SYMMETRIC RANK-ONE
SYMMETRIC RANK-ONE
SYMMETRIC RANK-ONE
SYMMETRIC RANK-ONE
SYMMETRIC RANK-ONE
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