《数字信号处理》教学参考资料（Numerical Recipes in C，The Art of Scientific Computing Second Edition）Chapter 10.5 Minimization or Maximization of Functions 10.5 Direction Set（Powell’s）Methods in Multidimensions.pdf

412 Chapter 10. Minimization or Maximization of Functions Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). if (i != ilo) { for (j=1;j<=ndim;j++) p[i][j]=psum[j]=0.5*(p[i][j]+p[ilo][j]); y[i]=(*funk)(psum); } } *nfunk += ndim; Keep track of function evaluations. GET_PSUM Recompute psum. } } else --(*nfunk); Correct the evaluation count. } Go back for the test of doneness and the next free_vector(psum,1,ndim); iteration. } #include "nrutil.h" float amotry(float **p, float y[], float psum[], int ndim, float (*funk)(float []), int ihi, float fac) Extrapolates by a factor fac through the face of the simplex across from the high point, tries it, and replaces the high point if the new point is better. { int j; float fac1,fac2,ytry,*ptry; ptry=vector(1,ndim); fac1=(1.0-fac)/ndim; fac2=fac1-fac; for (j=1;j<=ndim;j++) ptry[j]=psum[j]*fac1-p[ihi][j]*fac2; ytry=(*funk)(ptry); Evaluate the function at the trial point. if (ytry < y[ihi]) { If it’s better than the highest, then replace the highest. y[ihi]=ytry; for (j=1;j<=ndim;j++) { psum[j] += ptry[j]-p[ihi][j]; p[ihi][j]=ptry[j]; } } free_vector(ptry,1,ndim); return ytry; } CITED REFERENCES AND FURTHER READING: Nelder, J.A., and Mead, R. 1965, Computer Journal, vol. 7, pp. 308–313. [1] Yarbro, L.A., and Deming, S.N. 1974, Analytica Chimica Acta, vol. 73, pp. 391–398. Jacoby, S.L.S, Kowalik, J.S., and Pizzo, J.T. 1972, Iterative Methods for Nonlinear Optimization Problems (Englewood Cliffs, NJ: Prentice-Hall). 10.5 Direction Set (Powell’s) Methods in Multidimensions We know (§10.1–§10.3) how to minimize a function of one variable. If we start at a point P in N-dimensional space, and proceed from there in some vector

10.5 Direction Set(Powell's)Methods in Multidimensions 413 direction n,then any function of N variables f(P)can be minimized along the line n by our one-dimensional methods.One can dream up various multidimensional minimization methods that consist of sequences of such line minimizations.Different methods will differ only by how,at each stage,they choose the next direction n to try.All such methods presume the existence of a"black-box"sub-algorithm,which we might call linmin (given as an explicit routine at the end of this section).whose definition can be taken for now as linmin:Given as input the vectors P and n,and the function f,find the scalar A that minimizes f(P+An). 三 Replace P by P+An.Replace n by An.Done. g All the minimization methods in this section and in the two sections following g fall under this general schema of successive line minimizations.(The algorithm in $10.7 does not need very accurate line minimizations.Accordingly,it has its own approximate line minimization routine,Insrch.)In this section we consider a class of methods whose choice of successive directions does not involve explicit computation of the function's gradient;the next two sections do require such gradient 9 calculations.You will note that we need not specify whether linmin uses gradient information or not.That choice is up to you,and its optimization depends on your particular function.You would be crazy,however,to use gradients in linmin and not use them in the choice of directions,since in this latter role they can drastically reduce the total computational burden. But what if,in your application,calculation of the gradient is out of the question. 、a之 You might first think of this simple method:Take the unit vectors e1,e2,...en as a set of directions.Using linmin,move along the first direction to its minimum,then from there along the second direction to its minimum,and so on,cycling through the 6 whole set of directions as many times as necessary,until the function stops decreasing. This simple method is actually not too bad for many functions.Even more interesting is why it is bad,i.e.very inefficient,for some other functions.Consider a function of two dimensions whose contour map(level lines)happens to define a long,narrow valley at some angle to the coordinate basis vectors(see Figure 10.5.1). 10621 Then the only way "down the length of the valley"going along the basis vectors at each stage is by a series of many tiny steps.More generally,in N dimensions,if Numerical Recipes 43106 the function's second derivatives are much larger in magnitude in some directions than in others,then many cycles through all N basis vectors will be required in order to get anywhere.This condition is not all that unusual;according to Murphy's (outside Law,you should count on it. North Obviously what we need is a better set of directions than the ei's.All direction set methods consist of prescriptions for updating the set of directions as the method proceeds,attempting to come up with a set which either(i)includes some very good directions that will take us far along narrow valleys,or else (more subtly) (ii)includes some number of"non-interfering"directions with the special property that minimization along one is not "spoiled"by subsequent minimization along another,so that interminable cycling through the set of directions can be avoided

10.5 Direction Set (Powell’s) Methods in Multidimensions 413 Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). direction n, then any function of N variables f(P) can be minimized along the line n by our one-dimensional methods. One can dream up various multidimensional minimization methods that consist of sequences of such line minimizations. Different methods will differ only by how, at each stage, they choose the next direction n to try. All such methods presume the existence of a “black-box” sub-algorithm, which we might call linmin (given as an explicit routine at the end of this section), whose definition can be taken for now as linmin: Given as input the vectors P and n, and the function f, find the scalar λ that minimizes f(P+λn). Replace P by P + λn. Replace n by λn. Done. All the minimization methods in this section and in the two sections following fall under this general schema of successive line minimizations. (The algorithm in §10.7 does not need very accurate line minimizations. Accordingly, it has its own approximate line minimization routine, lnsrch.) In this section we consider a class of methods whose choice of successive directions does not involve explicit computation of the function’s gradient; the next two sections do require such gradient calculations. You will note that we need not specify whether linmin uses gradient information or not. That choice is up to you, and its optimization depends on your particular function. You would be crazy, however, to use gradients in linmin and not use them in the choice of directions, since in this latter role they can drastically reduce the total computational burden. But what if, in your application, calculation of the gradient is out of the question. You might first think of this simple method: Take the unit vectors e 1, e2,... eN as a set of directions. Using linmin, move along the first direction to its minimum, then from there along the second direction to its minimum, and so on, cycling through the whole set of directions as many times as necessary, until the function stops decreasing. This simple method is actually not too bad for many functions. Even more interesting is why it is bad, i.e. very inefficient, for some other functions. Consider a function of two dimensions whose contour map (level lines) happens to define a long, narrow valley at some angle to the coordinate basis vectors (see Figure 10.5.1). Then the only way “down the length of the valley” going along the basis vectors at each stage is by a series of many tiny steps. More generally, in N dimensions, if the function’s second derivatives are much larger in magnitude in some directions than in others, then many cycles through all N basis vectors will be required in order to get anywhere. This condition is not all that unusual; according to Murphy’s Law, you should count on it. Obviously what we need is a better set of directions than the ei’s. All direction set methods consist of prescriptions for updating the set of directions as the method proceeds, attempting to come up with a set which either (i) includes some very good directions that will take us far along narrow valleys, or else (more subtly) (ii) includes some number of “non-interfering” directions with the special property that minimization along one is not “spoiled” by subsequent minimization along another, so that interminable cycling through the set of directions can be avoided

10.5 Direction Set(Powell's)Methods in Multidimensions 415 In the approximation of(10.5.1),the gradient of f is easily calculated as Vf=A·x-b (10.53) (This implies that the gradient will vanish-the function will be at an extremum- at a value of x obtained by solving A.x=b.This idea we will return to in 810.7!) How does the gradient Vf change as we move along some direction?Evidently 6(7f)=A·(6x) (10.5.4) Suppose that we have moved along some direction u to a minimum and now propose to move along some new direction v.The condition that motion along v not spoil our minimization along u is just that the gradient stay perpendicular to u,i.e., that the change in the gradient be perpendicular to u.By equation(10.5.4)this is just 0=u·6(7f)=u·A·v (10.5.5) When (10.5.5)holds for two vectors u and v,they are said to be conjugate. 2 When the relation holds pairwise for all members of a set of vectors,they are said to be a conjugate set.If you do successive line minimization of a function along a conjugate set of directions,then you don't need to redo any of those directions (unless,of course,you spoil things by minimizing along a direction that they are not conjugate to). 县气%∽ 9 A triumph for a direction set method is to come up with a set of N linearly independent,mutually conjugate directions.Then,one pass of N line minimizations OF SCIENTIFIC will put it exactly at the minimum of a quadratic form like (10.5.1).For functions f that are not exactly quadratic forms,it won't be exactly at the minimum;but 61 repeated cycles of N line minimizations will in due course converge quadratically to the minimum Powell's Quadratically Convergent Method Powell first discovered a direction set method that does produce N mutually Numerica 10621 conjugate directions.Here is how it goes:Initialize the set of directions u;to the basis vectors, 431 Recipes ui=ei i=1,...,N (10.5.6) (outside Now repeat the following sequence of steps("basic procedure")until your function North stops decreasing: Save your starting position as Po. For i=1,...,N,move Pi-1 to the minimum along direction ui and call this point Pi. ·Fori=1,,N-1,set ui←ui+1 ·Set un-Pw-Po. Move Pn to the minimum along direction un and call this point Po

10.5 Direction Set (Powell’s) Methods in Multidimensions 415 Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). In the approximation of (10.5.1), the gradient of f is easily calculated as ∇f = A · x − b (10.5.3) (This implies that the gradient will vanish — the function will be at an extremum — at a value of x obtained by solving A · x = b. This idea we will return to in §10.7!) How does the gradient ∇f change as we move along some direction? Evidently δ(∇f) = A · (δx) (10.5.4) Suppose that we have moved along some direction u to a minimum and now propose to move along some new direction v. The condition that motion along v not spoil our minimization along u is just that the gradient stay perpendicular to u, i.e., that the change in the gradient be perpendicular to u. By equation (10.5.4) this is just 0 = u · δ(∇f) = u · A · v (10.5.5) When (10.5.5) holds for two vectors u and v, they are said to be conjugate. When the relation holds pairwise for all members of a set of vectors, they are said to be a conjugate set. If you do successive line minimization of a function along a conjugate set of directions, then you don’t need to redo any of those directions (unless, of course, you spoil things by minimizing along a direction that they are not conjugate to). A triumph for a direction set method is to come up with a set of N linearly independent, mutually conjugate directions. Then, one pass of N line minimizations will put it exactly at the minimum of a quadratic form like (10.5.1). For functions f that are not exactly quadratic forms, it won’t be exactly at the minimum; but repeated cycles of N line minimizations will in due course converge quadratically to the minimum. Powell’s Quadratically Convergent Method Powell first discovered a direction set method that does produce N mutually conjugate directions. Here is how it goes: Initialize the set of directions ui to the basis vectors, ui = ei i = 1,...,N (10.5.6) Now repeat the following sequence of steps (“basic procedure”) until your function stops decreasing: • Save your starting position as P0. • For i = 1,...,N, move Pi−1 to the minimum along direction ui and call this point Pi. • For i = 1,...,N − 1, set ui ← ui+1. • Set uN ← PN − P0. • Move PN to the minimum along direction uN and call this point P0

416 Chapter 10.Minimization or Maximization of Functions Powell,in 1964,showed that,for a quadratic form like (10.5.1).iterations of the above basic procedure produce a set of directions ui whose last k members are mutually conjugate.Therefore,N iterations of the basic procedure,amounting to N(N+1)line minimizations in all,will exactly minimize a quadratic form. Brent [1]gives proofs of these statements in accessible form. Unfortunately,there is a problem with Powell's quadratically convergent al- gorithm.The procedure of throwing away,at each stage,u in favor of P-Po tends to produce sets of directions that"fold up on each other"and become linearly dependent.Once this happens,then the procedure finds the minimum of the function f only over a subspace of the full N-dimensional case;in other words,it gives the 81 wrong answer.Therefore,the algorithm must not be used in the form given above. There are a number of ways to fix up the problem of linear dependence in Powell's algorithm,among them: 1.You can reinitialize the set of directions u;to the basis vectors e;after every N or N+1 iterations of the basic procedure.This produces a serviceable method, which we commend to you if quadratic convergence is important for your application (i.e.,if your functions are close to quadratic forms and if you desire high accuracy). 2.Brent points out that the set of directions can equally well be reset to the columns of any orthogonal matrix.Rather than throw away the information 9 on conjugate directions already built up,he resets the direction set to calculated principal directions of the matrix A (which he gives a procedure for determining) The calculation is essentially a singular value decomposition algorithm (see 82.6) Brent has a number of other cute tricks up his sleeve,and his modification of Powell's method is probably the best presently known.Consult [1]for a detailed 苔2ps description and listing of the program.Unfortunately it is rather too elaborate for OF SCIENTIFIC us to include here. 3.You can give up the property of quadratic convergence in favor of a more 6 heuristic scheme (due to Powell)which tries to find a few good directions along narrow valleys instead of N necessarily conjugate directions.This is the method that we now implement.(It is also the version of Powell's method given in Acton [21, from which parts of the following discussion are drawn.) Numerica 10621 Discarding the Direction of Largest Decrease 431 The fox and the grapes:Now that we are going to give up the property of Recipes quadratic convergence,was it so important after all?That depends on the function (outside that you are minimizing.Some applications produce functions with long,twisty valleys.Quadratic convergence is of no particular advantage to a program which North must slalom down the length of a valley floor that twists one way and another (and another,and another,...-there are N dimensions!).Along the long direction, a quadratically convergent method is trying to extrapolate to the minimum of a parabola which just isn't(yet)there;while the conjugacy of the N-1 transverse directions keeps getting spoiled by the twists. Sooner or later,however,we do arrive at an approximately ellipsoidal minimum (cf.equation 10.5.1 when b,the gradient,is zero).Then,depending on how much accuracy we require,a method with quadratic convergence can save us several times N2 extra line minimizations,since quadratic convergence doubles the number of significant figures at each iteration

416 Chapter 10. Minimization or Maximization of Functions Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). Powell, in 1964, showed that, for a quadratic form like (10.5.1), k iterations of the above basic procedure produce a set of directions ui whose last k members are mutually conjugate. Therefore, N iterations of the basic procedure, amounting to N(N + 1) line minimizations in all, will exactly minimize a quadratic form. Brent [1] gives proofs of these statements in accessible form. Unfortunately, there is a problem with Powell’s quadratically convergent algorithm. The procedure of throwing away, at each stage, u 1 in favor of PN − P0 tends to produce sets of directions that “fold up on each other” and become linearly dependent. Once this happens, then the procedure finds the minimum of the function f only over a subspace of the full N-dimensional case; in other words, it gives the wrong answer. Therefore, the algorithm must not be used in the form given above. There are a number of ways to fix up the problem of linear dependence in Powell’s algorithm, among them: 1. You can reinitialize the set of directions ui to the basis vectors ei after every N or N + 1 iterations of the basic procedure. This produces a serviceable method, which we commend to you if quadratic convergence is important for your application (i.e., if your functions are close to quadratic forms and if you desire high accuracy). 2. Brent points out that the set of directions can equally well be reset to the columns of any orthogonal matrix. Rather than throw away the information on conjugate directions already built up, he resets the direction set to calculated principal directions of the matrix A (which he gives a procedure for determining). The calculation is essentially a singular value decomposition algorithm (see §2.6). Brent has a number of other cute tricks up his sleeve, and his modification of Powell’s method is probably the best presently known. Consult [1] for a detailed description and listing of the program. Unfortunately it is rather too elaborate for us to include here. 3. You can give up the property of quadratic convergence in favor of a more heuristic scheme (due to Powell) which tries to find a few good directions along narrow valleys instead of N necessarily conjugate directions. This is the method that we now implement. (It is also the version of Powell’s method given in Acton [2], from which parts of the following discussion are drawn.) Discarding the Direction of Largest Decrease The fox and the grapes: Now that we are going to give up the property of quadratic convergence, was it so important after all? That depends on the function that you are minimizing. Some applications produce functions with long, twisty valleys. Quadratic convergence is of no particular advantage to a program which must slalom down the length of a valley floor that twists one way and another (and another, and another, ... – there are N dimensions!). Along the long direction, a quadratically convergent method is trying to extrapolate to the minimum of a parabola which just isn’t (yet) there; while the conjugacy of the N − 1 transverse directions keeps getting spoiled by the twists. Sooner or later, however, we do arrive at an approximately ellipsoidal minimum (cf. equation 10.5.1 when b, the gradient, is zero). Then, depending on how much accuracy we require, a method with quadratic convergence can save us several times N2 extra line minimizations, since quadratic convergence doubles the number of significant figures at each iteration