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MINIMIZATION B.
T.
METHODS
WITH
CONSTRAINTS UDC 518:519
Polyak
The p r e s e n t s u r v e y deals with n u m e r i c a l methods of solving e x t r e m u m p r o b l e m s with c o n s ~ a i n t s , i.e., m a t h e m a t i c a l p r o g r a m m i n g p r o b l e m s . The s u r v e y e n c o m p a s s e s the p e r i o d beginning with the disc o v e r y of the t h e o r y of the t h e o r y of m a t h e m a t i c a l p r o g r a m m i n g (early 1950's) to the p r e s e n t day. 1.
INTRODUCTION
The g e n e r a l p r o b l e m of m a t h e m a t i c a l p r o g r a m m i n g m a y be f o r m u l a t e d in the following way: minf(x), (1)
g(x)~(af(xk), X)q-f(x k) (for all x). Such a method was f i r s t proposed by N. Z. Shor and studied by Yu. M. E r m o l ' e v and B. T. Polyak [59, 60, 64, 107, 110, 152, 153]. N. Z. Shor also proposed methods for a c c e l e r a t i n g the convergence of the generalized gradient method [154-156]. Other m i n i m i z a tion methods for a noncontinuous function use the concept of p i e c e w i s e - l i n e a r approximation and r e q u i r e the solution of auxiliary linear p r o g r a m m i n g type problems [89, 110, 204, 248]. Special methods for minimization p r o b l e m s of noncontinuous functions of the type f ( x ) = max ~ (x, y) will be discussed in Sec. 11. uEQ Let us now pass to a second important class of optimization p r o b l e m s , cnlled linear p r o g r a m m i n g problems. The f i r s t r e s u l t s in this field a r e due to L. V. Kantorovich [76] and date f r o m the late 1930ts though the t h e o r y and methods of linear p r o g r a m m i n g did not begin to be i n t e n s i v e l y developed until the 19501swith the works of Dantzig, Von Neumann, Kuhn, T u c k e r , and others. Many original and t r a n s l a t e d monographs and textbooks on linear p r o g r a m m i n g have been published in Russian including Gass [24], E. G. GolVshtein and D. B. Yudin [28], Dantzig [38], Zoutendijk [67], S. I. Zukhovitskii and L. I. Avdeeva [68], Karlin [82], and D. B. Yudin and E. G. Goltshtein [159-160]. The simplex method, also called the method of s u c c e s s i v e improvement of a plan, is a fundamental method for solving linear p r o g r a m m i n g p r o b l e m s . In idea it is close to the Gauss elimination method (more exactly, the Jordan method) in linear algebra and yields an exact solution of a problem in a finite n u m b e r of steps, a s s u m i n g the absence of computational e r r o r s . Actual p r a c t i c e has shown that this numb e r is r e m a r k a b l y small (on the o r d e r 2 m - 3 m , where m is the number of constraints). T h e r e exist diff e r e n t computational s c h e m e s for the simplex method (direct algorithm, algorithm with inverse matrix, multiplicative algorithm) and also different modifications of it (dual method, a method of simultaneous solution of the d i r e c t and dual problem). The general method is s o m e t i m e s substantially simplified for partieular p r o b l e m s of linear p r o g r a m m i n g , for example, for the transportation problem. Iterative methods of linear p r o g r a m m i n g have been developed to a significantly l e s s e r extent. A s u r vey of c e r t a i n e a r l y studies in this field can be found in [28]. New Iterative methods have r e c e n t l y appeared, some ofwhich m a y b e competitive with finite methods [7, 13, 21, 49, 57, 111, 116, 126, 127, 138, 144146, 180,226-227, 235, 256, 281]. Still another c l a s s of p r o b l e m s for which t h e r e exist finite methods of solutions a r e quadratic p r o g r a m m i n g problems. H e r e different a l g o r i t h m s belonging to the simplex-type method and which lead to a solution in a finite number of steps have been developed. A s u r v e y of this c l a s s of problems c a n b e found in Kunzi and Krelle [88] and Boot [189] (cf. also [67]) and n u m e r i c a l e x p e r i m e n t s with different quadratic p r o g r a m m i n g methods have been d e s c r i b e d in [194]. A s i m p l e r quadratic p r o g r a m m i n g probletn is often encountered in which a quadratic f o r m is defined by a unit matrix: rain Ilx- all ~, where Ax --- b. In this case the dual problem is simply to minimize a ". 1 quadratic f o r m on an octant: rmn~ llAry--a[[ 2, where y >- 0 and can be solved using special methods, such as the conjugate gradient method generalized to this c a s e [23, 112]. 3.
MINIMIZATION
ON S I M P L E
SETS
In this section we will c o n s i d e r the problem m i n / ( x ) , xEQcR"
(3)
where Q is a ".4imple" set (cL Sec. 1). We will limit o u r s e l v e s to situations when the s t r u c t u r e of Q is not defined in detail, and where it is only important that simple minimization-type auxiliary p r o b l e m s for a linear or quadratic function be solvable on Q. The c a s e of an explicit definition of Q by m e a n s of linear constraints and methods using this s t r u c t u r e of Q will be t r e a t e d below (Sec. 5). It turns out that many absolute minimization methods g e n e r a l i z e to p r o b l e m s with simple constraints. i00
a. Gradient Methods. The f i r s t method for introducing constraints is to r e t u r n at each step on the set Q by projection (gradient projection method): x k+~= PQ ( x * - =~V'f (x~)). H e r e PQ is the p r o j e c t i o n o p e r a t o r on Q, i.e., HPQ(Z)--z[l-----min[lx--zi[. xE~
F o r a number of c a s e s it is possi-
ble to d e s c r i b e explicitly the f o r m of the projection o p e r a t o r . F o r example, if Q = {x fi R n, x ->. 0}, we have PQ (z) = z+ (z+ is the v e c t o r with coordinates (zt)+ . . . . . (Zn)+, where (zi) + = max {0, zi)). If Q = {x E R n, a < x < b } , w e h a v e P Q ( Z ) i = b i i f z i > b i , P Q ( z ) i = z i if a i ~ z i- 0 is s o m e n u m e r i c a l p a r a m e t e r , is the s i m p l e s t e x a m p l e of such a modified function. C l e a r l y , when K = 0 the modified function coincides with the o r d i n a r y function and when y = 0, M (x, 0) is the s a m e as in the penalty function method. The function M (x, y) is (under n a t u r a l a s s u m p t i o n s and for sufficiently l a r g e K) convex with r e s p e c t to x in the neighborhood of (x*, y* ), w h e r e a s L ( x , y) l a c k s this p r o p e r t y . M o r e o v e r a m i n i m u m for M (x, y* ) with r e s p e c t to x is attained only in the solutions of the initial p r o b l e m (the stability p r o p e r t y is not inherent to L (x, y)). This m a k e s the modified L a g r a n g i a n function v e r y suitable for f o r m u l a t i n g e x t r e m u m conditions (for e x a m p l e , it is possible to f o r m u l a t e duality t h e o r e m s for nonconvex p r o b l e m s by m e a n s of it) and a l s o in developing n u m e r i c a l m e t h o d s . The function M (x, y) was f i r s t introduced in the m o n o g r a p h of A r r o w , H u r w i c z , a n d Uzawa ([158], Chap. 11). Then, roughly s i m u l t a n e o u s l y and independently, a n u m b e r of a u t h o r s (Hestenes [239], Powell [290], and H a a r h o f f and Buys [236]) p r o p o s e d a different method, b a s e d on the u s e of M(x, y): M (x ~+', yk) = min 214(x, y~), yk§ = yk + Kh (xk+l). Powell [290] and B. T. P o l y a k and N. V. T r e t ' y a k o v [117] investigated c o n v e r g e n c e conditions for the m e t h od. It turned out that they a r e substantially l e s s r e s t r i c t e d than for methods b a s e d on the m i n i m i z a t i o n of L (x, y) (strong c o n v e x i t y - t y p e conditions a r e not required) and that the method c o n v e r g e s at the r a t e of a g e o m e t r i c p r o g r e s s i o n whose r a t i o is the l e s s e r , the g r e a t e r is K. The method h a s been given different n a m e s by v a r i o u s a u t h o r s b e c a u s e of different i n t e r p r e t a t i o n s , including the method of m u l t i p l i e r s , the d i s placed penalty method, a u x i l i a r y p a r a b o l a method, and penalty e s t i m a t e method. S e v e r a l s i m i l a r methods have been c o n s i d e r e d by F l e t c h e r [219, 220] and M a r t e n s s o n [266]. Computational efforts have a l s o been d e s c r i b e d [277, 311]. 5.
PROBLEMS
WITH
LINEAR
CONSTRAINTS
P r o b l e m s with l i n e a r c o n s t r a i n t s , i.e., of the f o r m
(d~,
min/r (x), ~)- 0. F i r s t of all, to solve Eq. (6) it is possible to use all the methods d e s c r i b e d in Sec. 3. They r e d u c e to a sequence of l i n e a r o r quadratic p r o g r a m m i n g p r o b l e m s that can be solved under the initial c o n s t r a i n t s . H e r e we will d e s c r i b e a different type Of method g e n e r a l l y a s s o c i a t e d with a u x i l i a r y p r o b l e m s of l e s s e r dimension that can be solved on s e p a r a t e f a c e s of a polyhedron definable by the c o n s t r a i n t s . N e a r l y all t h e s e methods a r e f e a s i b l e d i r e c t i o n methods, in t e r m s of Z o u t e n d i j k ' s c l a s s i f i c a t i o n [67]. In other w o r d s , all the i t e r a t i o n s x t . . . . . x k satisfy the c o n s t r a i n t s , while the direction of motion is selected, such that it does not s i m u l t a n e o u s l y develop beyond the c o n s t r a i n t and r e m a i n s a direction along which the function to be m i n i m i z e d d e c r e a s e s . a. Method of C o o r d i n a t e - b y - C o o r d i n a t e Descent. If the c o n s t r a i n t s have the f o r m a m~xl(ft1(x*)l-~}, and e > 0 is a given p a r a m e t e r .
The solution of the a u x i l i a r y p r o b l e m , p k s e r v e s for c o n s t r u c t i n g a new a p p r o x i m a t i o n x k+l = x k + akp k, w h e r e the length of the step 0 ~o,t~11,[ tGik
IEJk
and its solution denoted by u k, v k, then
J~s~ pk_~ _ ( V f ( x k )
-t- E .u~Vgi (x~)+ E v~Vhl(xk)), the s i m i l a r i t y of
tel ~ SEJ~ the P s h e n i c h n y i method to that d e s c r i b e d in See. 5 for the simultaneous solution of the d i r e c t and dual p r o b l e m s t h e n b e i n g e v i d e n t . If the p r o b l e m is convex, the P s h e n i c h n y i method a p p r o a c h e s the feasible d i r e c tion method with n o r m a l i z a t i o n N1. Finally if no inequality-type c o n s t r a i n t s o c c u r , we have a p r o j e c t i o n type method d e s c r i b e d in Sec. 4. A second method of solution, a l s o using only f i r s t d e r i v a t i v e s of functions, is b a s e d on the concept of e l i m i n a t i n g v a r i a b l e s . It was p r e s e n t e d in [162], w h e r e the Wolfe r e d u c e d gradient method was g e n e r alized to the c a s e of g e n e r a l c o n s t r a i n t s . b. N e w t o n - T y p e Methods. If the calculation of second d e r i v a t i v e s of the functions o c c u r r i n g in the p r o b l e m i s s i m p l y a c c o m p l i s h e d , it is p o s s i b l e to use a different analog of Newton's method. All the conm
p
s t r a i n t s a r e l i n e a r i z e d at x k in this method, the L a g r a n g i a n function L (x, u, v) = f (x) + ~ u ~ (x) + ~ vih j (x) ~-i
j-1
is a p p r o x i m a t e d by a quadratic function, and an a u x i l i a r y quadratic p r o g r a m m i n g p r o b l e m is soIved, namely, ~nin [(V f
(xk), x - - x~) q--~t (V 2xxL (xk ' uk, v~)(x--x~);
x - - x~)],
g~ (x k) + (XTg~(x~), x -- x k) ~ k. It then satisfies the r e c u r r e n c e r e l a t i o n (dynamic p r o g r a m m i n g equation) Vk_,(x~.,) = rain [f~_,, (x~_,, xk) + V~ (x~)i. The application of various approximate (and, in x~EQ~
s e p a r a t e c a s e s , exact) methods for successfully solving such r e c u r r e n c e equations makes it possible to obtain a solution of the initial problem. Some new variants of the dynamic p r o g r a m m i n g method (for continuous problems) have been d e s c r i b e d by J a c o b s e n and Mayne [245]. c. R e s o u r c e Distribution P r o b l e m s . An optimization problem in which the function to be minimized and t h e c o n s t r a i n t s a r e s e p a r a b l e , n
rain ~ A (x~),
~ g~ (x,) ..< at, ~= 1. . . . , m, ~" kjk (xk) = b 1, ] ~ 1. . . . . p,
is usually called an optimal r e s o u r c e distribution problem. Some optimization methods a r e p a r t i c u l a r l y effective for these p r o b l e m s . Since the Lagrangian function in this c a s e is separable in x, its minimization r e d u c e s to one-dimensional p r o b l e m s . T h e r e f o r e methods based on minimizing the Lagrangian function a r e
113
e x t r a o r d i n a r i l y simplified.
In p a r t i c u l a r , i n t h e s i m p l e s t p r o b l e m tl
min ~ A (xk), /t--I
~ , , x ~ = l , x k > O k = l . . . . . n,
n the solution is r e d u c e d to s e l e c t i n g a n u m b e r y such that
~ xk ( y ) = 1, w h e r e x k (y) is the point that rain"
i m i z e s f k ( X k ) + yx k with r e s p e c t to x k -> 0. It is also possible to use dynamic p r o g r a m m i n g - t y p e methods when m and p a r e of low dimension. A s u r v e y of methods of solving optimal r e s o u r c e distribution p r o b l e m s h a s been given by L. S. Gurin, Ya. S. D y m a r s k i i , and D. A. Merkulov [32]. d. G e o m e t r i c P r o g r a m m i n g .
If a m a t h e m a t i c a l p r o g r a m m i n g p r o b l e m h a s the f o r m rain go (x),
g~(x) ~ 1, i = I . . . . . m, r
w h e r e go, gt, . . . ,
gm a r e polynomials, i.e., e x p r e s s i o n s of the f o r m g ( x ) = ~ c j x ~ ; ~ . . "-nraJn' it is called a j=l
g e o m e t r i c p r o g r a m m i n g p r o b l e m . T h e r e e x i s t special methods of solution for such p r o b l e m s b a s e d on the fact that the p r o b l e m b e c o m e s convex following substitutions of the f o r m x k = e zk and the p r o b l e m dual to it p o s s e s s e s l i n e a r c o n s t r a i n t s (Daffin, P e t e r s o n , and Z e n e r [39]). e. Block P r o b l e m s . The b a s i c concept of decomposition methods is the s a m e a s for a dynamic p r o ' g r a m m i n g method, n a m e l y to r e d u c e an initial p r o b l e m with higher dimension to the solution of a sequence" of p r o b l e m s , each of l e s s e r dimension. Methods of solving block p r o b l e m s of linear p r o g r a m m i n g have b e e n h i g h l y developed [19, 28, 37, 61, 126, 127, 210, 235, 256]. Block p r o g r a m s of convex p r o g r a m m i n g a r e of the s a m e f o r m a s r e s o u r c e distribution p r o b l e m s , with the difference that the x i a r e not s c a l a r s but v e c t o r s and c o n s t r a i n t s of the type x~EQ~cRnt a r e imposed on them. Decomposition methods a r e b a s e d on the s a m e concepts a s methods of solving s e p a r a b l e p r o b l e m s [106, 241]. f. Composite Punctionals. Suppose the function to b e m i n i m i z e d has the f o r m f ( x ) = r where (z), z E R r , is a continuous function and R :R n - * R r is a nonlinear o p e r a t o r . Then in addition to g e n e r a l methods for m i n i m i z i n g f ( x ) on a given s e t Q, s p e c i a l methods e x i s t i n t e r m e d i a t e between Newton's m e t h od and the gradient method b a s e d on the quadratic a p p r o x i m a t i o n of r and a linear a p p r o x i m a t i o n of R (x). F o r e x a m p l e , at the k - t h step it is possible to solve the a u x i l i a r y p r o b l e m min [(VR (xk)r~z+ (R (xk)), X--x~) +-~1 [~zR(x~)r~'2,e(R(x~))V R(x~) (x-x~), x - x ~ ) ] x~q and to take its solution as the Xk+i. If O = R n and if ~o(z) = llz]l 2, this method coincides with the well-known G a u s s - N e w t o n method for m i n i m i z i n g a sum of s q u a r e s . Other v a r i a n t s of the method for p r o b l e m s without c o n s t r a i n t s can be found in Daniel [209] and p r o b l e m s with c o n s t r a i n t s have also been c o n s i d e r e d [115, 185, 304]. Many r e s u l t s on n u m e r i c a l e x p e r i m e n t s for m i n i m i z i n g c o m p o s i t e functions have been p r e s e n t e d by B a r d [176l. E x t r e m u m P r o b l e m s on G r a p h s . A s p e c i a l c l a s s of p r o b l e m s a r e those r e l a t e d to optimizing n e t w o r k s , for e x a m p l e , the t r a n s p o r t a t i o n p r o b l e m in a network formulation. It is 0sually unsuitable to r e d u c e them to g e n e r a l m a t h e m a t i c a l p r o g r a m m i n g p r o b l e m s , since their s t r u c t u r e would not then be used. The r e a d e r who w i s h e s to b e c o m e acquainted with e x t r e m u m p r o b l e m s on graphs and methods for solving them is r e f e r r e d to the monograph of Yu. M. E r m o l ' e v and I. M. Mel'nik [62]. We mention still other special types of m a t h e m a t i c a l p r o g r a m m i n g p r o b l e m s . T h e s e include f r a c t i o n a l linear p r o g r a m m i n g and p a r a m e t r i c p r o g r a m m i n g [24, 28], the g e n e r a l i z e d linear p r o g r a m m i n g p r o b l e m [38, 210], linear dynamic p r o g r a m m i n g [121], optimization p r o b l e m s for multiply connected s y s t e m s [97], and oth or s.
114
11.
RELATED
PROBLEMS
H e r e we will t r e a t a n u m b e r of p r o b l e m s which, though not f o r m u l a t e d d i r e c t l y a s e x t r e m u m p r o b l e m s , a r e c l o s e l y r e l a t e d to t h e m . a) Solution of Inequalities and Finding the C o m m o n Point of Convex Sets. The p r o b l e m of finding the solution of a s y s t e m of convex inequalities gi (x) ~ 0, w h e r e i = 1, . . . . m, can be c o n s i d e r e d as a p a r t i c u l a r c a s e of a convex p r o g r a m m i n g p r o b l e m in which the function to be m i n i m i z e d is a constant. On the other rn
hand, this p r o b l e m r e d u c e s to absolute m i n i m i z a t i o n o f a discontinuous function, for e x a m p l e , f { x ) ~ ~ g~(x)+. H o w e v e r , t h e r e a l s o e x i s t specific methods f o r solving inequalities which we now c o n s i d e r . We begin with the g e n e r a l p r o b l e m of finding a c o m m o n point of a s y s t e m of convex s e t s Q~cR", w h e r e i = 1 . . . . . m. The s u c c e s s i v e p r o j e c t i o n method c o n s i s t s in p r o j e c t i n g in turn on each of the s e t s , i.e., a sequence ~' = PQ,(X~ x2=Pr .. :, x ~§ =PQm(x ~) . . . . is c o n s t r u c t e d . In a second v a r i a n t of the method p r o j e c t i o n is p e r f o r m e d a~
to the m o s t distant set. It is p o s s i b l e to p r o v e that x k - ~ x * E Q = QQ~.
T h i s m e t h o d w a s p r o p o s e d in 1937 by
K a c z m a r z for the c a s e of Qi a h y p e r p l a n e (i.e., for solving a s y s t e m of linear equations), and by Agmon [165] for l i n e a r inequalities, while Motzkin and Shoenberg [279] extended it in 1954 to the c a s e of a r b i t r a r y s e t s . D i f f e r e n t modifications and g e n e r a l i z a t i o n s of the method as well as the r a t e of c o n v e r g e n c e have been i n v e s t i g a t e d by L. M. B r e g m a n [9], L, G. Gurin, B. T. Polyak, and ]~. V. Raik, [31], I. I. E r e m i n [51], A. I. L o b y r e v [93], V. A. Yakubovich [161]; and T o m p k i n s [308]. In p a r t i c u l a r , it h a s been p r o v e d [31, 107, 161] that if Q h a s inner points and a step is m a d e in the d i r e c t i o n f r o m x k to P i ( x k ) , but of the length 7k, where "[4-~ 0, ~ Tk= c~,
the method l e a d s to a solution in a finite n u m b e r of steps. Other finite v a r i a n t s of the
k=l
method have a l s o been p r e s e n t e d [279, 161]. If all the Qi a r e s e m i s p a c e s , i.e., Qi = {x : gi (x)
