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The method of coordinate descent in conditional minimization problems
208
D. V. Denisov
The system was solved by the modified chord method. Suppose it is required to find a root of the equation Y(U)= 0, situated on the segment [uu, ul]. We find v. = v(uo) and v1 = I+~), then calculate ~u=(v,-v~)/(~~-u~), u~=u,,-v~/w and u,=v(u,). If (&I
1.
POLAK, E. Numerical methods of optimization (Chialennye metody optlmizataii). “Mu”, Moscow, 1974.
2.
CHERNOUS’KO, F. L. Optimal search for the extrema of unlmodal functions. Zh. vjkhisl. Mat. mat. Fiz.,
3.
AFANAS’EV, A. Yu. The search for the minimum of a function with a bounded second derivative. Zh. vjkhisl. Mat. mat. Fiz., 14, 1018-1021, 1974.
4.
PONTRYAGIN, L. S., BOLTYANSKII, V. G., GAMKRELIDZE, R. V. and MISHCHENKO,E. F. Mathematical theory of optimal processes f.Matematicheskaya teoriya optlmal’nykh protsessov). Fizmatgiz, Moscow, 196 1.
10,4,922-933,197o.
Cr.S.S.R. Comput. MathsMath. Phys. Vol. 17, pp. 208-210 0 Pergamon Press Ltd. 1978. Printed in Great Britain.
0041-5553/77/0801-0208$07.50/O
THE METHOD OF COORDINATE DESCENT IN CONDITIONAL MINIMIZATION PROBLEMS* D. V. DENISOV
Moscow (Received 15 March 1976)
THE CONVERGENCE of the method of random coordinate descent in the solution of minimization problems with constraints of a special form is studied. Suppose it is required to find a vector Z= (zi, . . . , z,)=, belonging to an n-dimensional Euclidean space E” , which is the solution of the following problem: minimize cp(x) subject to the constraints j=l,
s~X={xl--oo~aj~sj~bj~+oo,
It is assumed that
cp(x) EC’~~(X), that is, for all
2,. . . , ?Z}.
x, y=X
the inequality
Ill’--cp’(~)il~Llls-yll,
(2)
is satisfied, where L=const>O,
cp’W=M4,...,
1141= (&,2 1-t
) ‘“,
cpn’(s))’ is the gradient of cp(x) at the point x.
*Zh. vjkhisl. Mat. mat. Fiz., 17,4, 1034-1036,
1977.
(1)
209
Short communications
Let ei be the j-th coordinate vector. To solve the problem we will construct the sequence of points {z”}: Cl?k+l=Zk-~kej(k),
k=O, 1, . . . ,
(3)
where x0 is an arbitrary point of the set X, and the number of the coordinate j(k) is chosen at random from the n numbers 1,2, . . . , n. For the choice of the quantity ok we consider the following three schemes. Scheme 1. Ok is chosen by the condition (see [l] ) (4)
Uk= inf
Cp(Z’-@j(k)),
X;(k)-_Bej(k)EX.
P
Scheme 2. We can choose as flk any number p>c@, a~ (0, l] ; here -/k is the greatest of the numbers satisfying the inequality (see [ 1,2] )
Since cp(x) =C*~’(X), where can choose a -/k such that
for Scheme 3.
(6) bJ(k)-zjk{k).
1(P;(k) tzk)1 L
,
if
(Polk)
(sk)
G
0.
Let the point xk be known. For every j=~, 2, . . . , n we introduce the quantities BP, which are calculated by one of the schemes 1-3, ifj occurs as j(k) in (4)-(6). We first show that if flk and the number j(k) in (3) are chosen deterministically from the condition .
‘P(zk-gj:kpjck,)=
mincpG2k-pjkej)r
I
j=i,
2,‘a.,%
(7)
k bk=h(kh
then the sequence {z”)convergences to the set X* of stationary points, that is, of all points z*=X, such that (c$(Y), J-Y) 30 for any ZEX. It is easy to see that the set X* consists of points at which a necessary condition for a local minimum of the function cp(x) on the set X is satisfied. In the case where cp(x) is convex, X* is the set of points of minimum of the function &) (see [l I >. We write
p(x, x’) = inf //z--5*)1. X*=X.*
Let x0 be any point of X and the sequence {z+} be constructed in accordance with (3), (7) and with the aid of one of the schemes l-3. Then the following theorem holds. Theorem If cp(r) EC’*‘(X) and the set
D. V. Denisov
210
is bounded, then lim p (sk, x’) = 0.
(8)
k-rm
Proof. We denote the set of indexes of our sequence by K={k=O,
1, . . .}.
Since the
sequence {cp(sk)}, k =K, is monotonic and underbounded, it converges. Consequently, for any e>O g(q--(p(XR+*)
(9)
for sufficiently large k= K. Because of the boundedness of the set X, , we can select a convergent subsequence from the sequence {xR} To prove the theorem it is sufficient to show that any limit point y=P. Sincey is the limit of the sequence {I~}, k=K,cK,
For some jE{i, 2, . . . , n} let aj
~(5’)-o(.k+‘)rBkrpi’(Zh)-~ >[fJJj’(y)lTmin
yj-aj,
@k)”
b,-a,,
ITj’(Y) I ~ L 1
1 for some r= (0, 11, valid for any of the schemes 1-3 for all sufficiently large k= Ki. This and (9) imply that cpl( y) =0 for ajcy,c b,. Similarly, ‘p,‘(y)>0 for yj=aj and CP~(Y)GO for y,= bj. Therefore, any limit point y EX*. The theorem is proved. Remark 1. If cp(x)is convex, then obviously lim rp(zk) = min cpb).
x
k-co
Remark 2. The theorem will also be valid in the case where (7) is satisfied only for some infinite subsequence of the indexes IU, and for the remaining k the numbers of the coordinates j(k) are chosen arbitrarily, provided that conditions (4)-(6) are valid for the schemes l-3 respectively.
In the method of random coordinate descent the subscripts j(k) in (3) are chosen at random from the set of numbers 1, 2, . . . , n,where P{j(k)=i}=l/n,
i=l,
2 (...(
n.
In this case by the law “zero or unity” there exists with probability 1 an infinite subsequence of numbers (k z) for which (7) holds. Accordingly, for random coordinate descent subject to the conditions of the theorem proved above the following relation holds: P ( lim p (z?, X’) = 0) = 1. k-m
If in addition cp(x)is convex, then by the continuity of cp(x) and the remark 1, P{ limcp(zk)= k-rcc
minq(z))= x
1.
Translated by J. Berry
REFERENCES 1. 2.
KARMANOV, V. G. Mufhemutical progrumming (Matematicheskoe programmirovanie), “Nauka”, Moscow, 1975. PSHEMCHNYI, B. N. and DANILIN, Yu. M. Numerical methods in extremal problems (Chislennye metody v ekstremal’nykh zadachakh). “Nauka”, Moscow, 1975.
D. V. Denisov
The system was solved by the modified chord method. Suppose it is required to find a root of the equation Y(U)= 0, situated on the segment [uu, ul]. We find v. = v(uo) and v1 = I+~), then calculate ~u=(v,-v~)/(~~-u~), u~=u,,-v~/w and u,=v(u,). If (&I
1.
POLAK, E. Numerical methods of optimization (Chialennye metody optlmizataii). “Mu”, Moscow, 1974.
2.
CHERNOUS’KO, F. L. Optimal search for the extrema of unlmodal functions. Zh. vjkhisl. Mat. mat. Fiz.,
3.
AFANAS’EV, A. Yu. The search for the minimum of a function with a bounded second derivative. Zh. vjkhisl. Mat. mat. Fiz., 14, 1018-1021, 1974.
4.
PONTRYAGIN, L. S., BOLTYANSKII, V. G., GAMKRELIDZE, R. V. and MISHCHENKO,E. F. Mathematical theory of optimal processes f.Matematicheskaya teoriya optlmal’nykh protsessov). Fizmatgiz, Moscow, 196 1.
10,4,922-933,197o.
Cr.S.S.R. Comput. MathsMath. Phys. Vol. 17, pp. 208-210 0 Pergamon Press Ltd. 1978. Printed in Great Britain.
0041-5553/77/0801-0208$07.50/O
THE METHOD OF COORDINATE DESCENT IN CONDITIONAL MINIMIZATION PROBLEMS* D. V. DENISOV
Moscow (Received 15 March 1976)
THE CONVERGENCE of the method of random coordinate descent in the solution of minimization problems with constraints of a special form is studied. Suppose it is required to find a vector Z= (zi, . . . , z,)=, belonging to an n-dimensional Euclidean space E” , which is the solution of the following problem: minimize cp(x) subject to the constraints j=l,
s~X={xl--oo~aj~sj~bj~+oo,
It is assumed that
cp(x) EC’~~(X), that is, for all
2,. . . , ?Z}.
x, y=X
the inequality
Ill’--cp’(~)il~Llls-yll,
(2)
is satisfied, where L=const>O,
cp’W=M4,...,
1141= (&,2 1-t
) ‘“,
cpn’(s))’ is the gradient of cp(x) at the point x.
*Zh. vjkhisl. Mat. mat. Fiz., 17,4, 1034-1036,
1977.
(1)
209
Short communications
Let ei be the j-th coordinate vector. To solve the problem we will construct the sequence of points {z”}: Cl?k+l=Zk-~kej(k),
k=O, 1, . . . ,
(3)
where x0 is an arbitrary point of the set X, and the number of the coordinate j(k) is chosen at random from the n numbers 1,2, . . . , n. For the choice of the quantity ok we consider the following three schemes. Scheme 1. Ok is chosen by the condition (see [l] ) (4)
Uk= inf
Cp(Z’-@j(k)),
X;(k)-_Bej(k)EX.
P
Scheme 2. We can choose as flk any number p>c@, a~ (0, l] ; here -/k is the greatest of the numbers satisfying the inequality (see [ 1,2] )
Since cp(x) =C*~’(X), where can choose a -/k such that
for Scheme 3.
(6) bJ(k)-zjk{k).
1(P;(k) tzk)1 L
,
if
(Polk)
(sk)
G
0.
Let the point xk be known. For every j=~, 2, . . . , n we introduce the quantities BP, which are calculated by one of the schemes 1-3, ifj occurs as j(k) in (4)-(6). We first show that if flk and the number j(k) in (3) are chosen deterministically from the condition .
‘P(zk-gj:kpjck,)=
mincpG2k-pjkej)r
I
j=i,
2,‘a.,%
(7)
k bk=h(kh
then the sequence {z”)convergences to the set X* of stationary points, that is, of all points z*=X, such that (c$(Y), J-Y) 30 for any ZEX. It is easy to see that the set X* consists of points at which a necessary condition for a local minimum of the function cp(x) on the set X is satisfied. In the case where cp(x) is convex, X* is the set of points of minimum of the function &) (see [l I >. We write
p(x, x’) = inf //z--5*)1. X*=X.*
Let x0 be any point of X and the sequence {z+} be constructed in accordance with (3), (7) and with the aid of one of the schemes l-3. Then the following theorem holds. Theorem If cp(r) EC’*‘(X) and the set
D. V. Denisov
210
is bounded, then lim p (sk, x’) = 0.
(8)
k-rm
Proof. We denote the set of indexes of our sequence by K={k=O,
1, . . .}.
Since the
sequence {cp(sk)}, k =K, is monotonic and underbounded, it converges. Consequently, for any e>O g(q--(p(XR+*)
(9)
for sufficiently large k= K. Because of the boundedness of the set X, , we can select a convergent subsequence from the sequence {xR} To prove the theorem it is sufficient to show that any limit point y=P. Sincey is the limit of the sequence {I~}, k=K,cK,
For some jE{i, 2, . . . , n} let aj
~(5’)-o(.k+‘)rBkrpi’(Zh)-~ >[fJJj’(y)lTmin
yj-aj,
@k)”
b,-a,,
ITj’(Y) I ~ L 1
1 for some r= (0, 11, valid for any of the schemes 1-3 for all sufficiently large k= Ki. This and (9) imply that cpl( y) =0 for ajcy,c b,. Similarly, ‘p,‘(y)>0 for yj=aj and CP~(Y)GO for y,= bj. Therefore, any limit point y EX*. The theorem is proved. Remark 1. If cp(x)is convex, then obviously lim rp(zk) = min cpb).
x
k-co
Remark 2. The theorem will also be valid in the case where (7) is satisfied only for some infinite subsequence of the indexes IU, and for the remaining k the numbers of the coordinates j(k) are chosen arbitrarily, provided that conditions (4)-(6) are valid for the schemes l-3 respectively.
In the method of random coordinate descent the subscripts j(k) in (3) are chosen at random from the set of numbers 1, 2, . . . , n,where P{j(k)=i}=l/n,
i=l,
2 (...(
n.
In this case by the law “zero or unity” there exists with probability 1 an infinite subsequence of numbers (k z) for which (7) holds. Accordingly, for random coordinate descent subject to the conditions of the theorem proved above the following relation holds: P ( lim p (z?, X’) = 0) = 1. k-m
If in addition cp(x)is convex, then by the continuity of cp(x) and the remark 1, P{ limcp(zk)= k-rcc
minq(z))= x
1.
Translated by J. Berry
REFERENCES 1. 2.
KARMANOV, V. G. Mufhemutical progrumming (Matematicheskoe programmirovanie), “Nauka”, Moscow, 1975. PSHEMCHNYI, B. N. and DANILIN, Yu. M. Numerical methods in extremal problems (Chislennye metody v ekstremal’nykh zadachakh). “Nauka”, Moscow, 1975.