Collection method


The group is an important equivalent relationship. Set S, T is the two non-air sets of the group G, H is a subgroup of g, if there is H Element g makes T = GSG = S, referred to as S and T about H conjugate, where t = gsg = {GSG | s ∈s} is called S is deformed by g. If S is a g of g, T is called S about H conjugate group; if S = {S} is a set of elements, T = GSG is called the conjugate element of H. When h = g, it usually does not add "about g" this The conjugate relationship is an equivalent relationship. Setting S is a subset of group G, H is a subgroup of g, and the set of all subsets of the H conjugate is called S about H Conjugate. When S = {S} is a collection of an element, S about G is a collection of elements, and a conjugate of g (elements).

A special relationship between the two-year direction

two - vector. Set a N × N symmetry matrix, vector P 1 , p 2 ∈R If condition (P 1 ) AP 2 = 0, P 1 and p 2 about A It is a conjugate direction, or called P 1 and p 2 about A conjugate. Generally, for non-zero vector group P 1 , p 2 , ..., p n R, if condition: (P i ) AP j = 0 ( I ≠ j, i, j = 1, 2, ..., n), referred to as the vector group is conjugated with respect.

set A is N × N symmetrical positive matrix, if there are two n-dimensional ports p and q, satisfy

paq = 0

, the vector P And Q is a conjugated, or called P, Q is a conjugate direction.


to solve a type of decrease in unconstrained nonlinear planning problem with a set of conjugated directions as a search direction. It is based on the N-Dimed secondary function of the symmetric positive matrix Q

f (x) = 1 / 2xq x + bx + c

optimal solution A type of gradient type algorithm comprising a conjugate gradient method and a variable scale method. According to the nature of the conjugate direction, it is sequentially searched in a set of directions of the Q conjugate, and the minimum dot of the secondary function can be obtained in the up to N step. The conjugated direction method is also quite effective in handling the non-secondary target function, with superlinear convergence speed, overcomes the servoidal phenomenon of the most flexible drop, while avoiding the chase involved in Newton (HESSE) The calculation and requirements of the matrix. For non-quadratic functions, N step search does not get extremely small points, need to use heavy start policies, that is, after each of the N-time search, if no extreme small point is not obtained, the negative gradient direction is refreshed in the initial direction. Construct the conjugate direction and continue to search.

Mathematical expression

For n-dimensional secondary function f, select the vector group P 0 , P of its coefficient matrix is ​​a conjugate 1 , ..., p n-1 , from any point x 0 ∈R N , successively in P 0 , p 1 , ..., p n-1 is the search direction, the iterative formula is:

Search by N times If you can find x n to f (x) minimum point. The conjugate direction is Powell, MJD) first proposed in 1964.

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