Download Convexity and Complexity in Polynomial Programming

Proceedings of the International Congress of Mathematicians
August 16-24, 1983, Warszawa
L. G. KHAOHIYAN
Convexity and Complexity in
Polynomial Programming
The problems to bo considered, in the talk are the problems of convex
polynomial programming: minimize
f0(x19 ...9xn)
(1)
subject to
f1(x1,...9xn)^09
(2)
fm(x19 . . . , » „ ) < 0,
where / 0 , f x , . . . , fm are convex polynomials in Rn with integer coefficients,
the polynomials being specified by blocks of their coefficients written in
binary numerical system. Two basic cases studied in mathematical programming are the following:
(i) the variables are real x = (x19..., œn) eRn9
(ii) the variables are integer x = (x19... 9 xn) e Zn9
the problems of latter type being also called diophantine. However, the
mixed case x eRk xZn~k is sometimes also considered.
The degree d of the problem is the maximum of the degrees dt of the
polynomials, the heigJit h of the problem is the maximum modulus of the
integer coefficients of the polynomials occurring in it, and the input length L
of the problem is the number of binary symbols 0 and 1 needed for its
coding. The reader should not be misled by identifying L with the sum
of binary lengths of all non-zero entries of the problem, though, in fact,
the only property of L needed below is : L > max {n9m9 log A}.
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Section 16: L. G. Khachiyan
1. Bounds on solutions
In order to consider the bounds on solutions of problems (l)-(2) with real
and/or integer variables simultaneously, we give the following
A set Jf £ JBTO is called periodic if for any integer vector
y e Z we have Jf+y = Jf.
DEFINITION.
n
We assume in the present section that the vector (x19..., xn) of unknowns in problem (l)-(2) runs over some periodic set Jf'. In particular,
the variables may be real (Jf = Rn), integer (Jf = Zn), real and integer (Jf = RkxZn~k),
rational (Jf =Qn), etc.
As usual, a system of inequalities (2) is said to be consistent in Jf if it
has a solution x* eJf. Similarly, an optimization problem (l)-(2) is said
to be feasible in Jf it if has an optimal solution œ* e Jf when the vector
of variables runs over Jf.
Note. It can be shown that a problem of convex polynomial programming (l)-(2) with integer coefficients is feasible in Rk x Zn~k if and only
if its system of constraints (2) is consistent in Rk x Zn"k and its objective
polynomial (1) is bounded from below on the set of real solutions of the
system (2).
To state the results of this section, we also need the concept of multidegree D of a system of inequalities (2). Let a = mm{n9 m} and let the
inequalities of the system be ordered by decreasing degrees dx > d&
a
> • •. > dm. Then D = fj #*•In other words, the multidegree is the maximum
of all possible products of the degrees of a distinct inequalities of the system.
In particular, D < g^i^m}^ a n ( j adjoining further linear constraints to the
system does not change its multidegree.
1 [11]. Let Jf be an arbitrary periodic set in Rn. If the system
of convex polynomial inequalities (2) of degree at most d9 d^2, of multidegree D and of height h is consistent in Jf, then it has a solution x* e Jf in
the Fuclidean ball
THEOREM
\\x\\<(hdnf**dl2ndl\
2. Let Jf be an arbitrary periodic set in Rn. If the problem
of convex polynomial programming (l)-(2) of degree at most d9d^2, and
of height h is feasible in Jf', then it has an optimal solution x* eJf in the
THEOREM
Convexity and Complexity in Polynomial Programming
1571
Fuclidean ball
WxW^lidn)»2^-1,
wJiere D is tJie multidegree of tJie system of constraints.
We conclude the section with a brief comment on Theorems 1 and 2,
To begin with, consider the problems of linear programming, which are
of the degree not exceeding 2 and of multidegree 1. By Theorems 1 and 2,
bounds on solutions of such problems in periodic sets are exponential in
the input length, i.e., \\x\\ < Anloshn < 2p(L), where J. is a constant and p
is a polynomial. As regards problems of convex polynomial programming
of an a-rbitrary but fixed degree d > 2, we see that unlike the linear case,
Theorems 1 and 2 restrict the bounds on solutions of such problems by
a. two-stage exponential function in the input length, i.e.,
|]a?]| < J.ä2min{^™} jn^iogzm < 222J{L)
p and A being a polynomial and a constant depending on d. On the other
hand, the example of the system xx^ h, a?2> ^?? •••> 0»> ^St-i? with «J
even, shows that such super-exponential growth of the solution bounds can
really be attained in any periodic set.
2. Finding an exact real and/or integer solution
In the present section we assume that the variables are real and/or integer
m eRk xZn~k. As the coefficients of the problem (l)-(2) are integer,
in case of its feasibility there exists an optimal solution x* *= (x*,..., x*)
e Ak x Zn"k, each real component of whioh is an algebraic number. If we
agree to code algebraic numbers x* by their (irreducible over Q) algebraic
equations Pj(x*) = 0 and, if necessary, by rational segments x* e(a$, bf)
not containing other roots of univariate polynomials pi e # [ • ] , we may
consider exact algorithms of polynomial programming. Eor a problem
(l)-(2) such an algorithm must check its feasibility and, if the problem
is feasible, print an algebraic solution x* e Ak x Zn~k. Applying Theorem 2
jointly with the decision procedure [1], we obtain
3. There exists an exact algorithm for convex polynomial
programming witli real andjor integer variables, wJwse running lime t is
bounded by a two-stage exponential function in tJie input length, t < 22 .
THEOREM
In the subsequent sections we shall consider less burdensome problems
of convex polynomial programming, allowing an essential reduction of
their computational complexity.
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Section 16: L. G-. Khachiyan
3 . Regularization of systems of convex polynomial inequalities
As we have mentioned in Section 1, the bounds on solutions of systems of
convex polynomial inequalities in periodic sets can grow as 22P{L). The
following result [11] shows that in some "computational sense" these
bounds can be effectively reduced to 2P^.
THEOREM 4. Consider systems of convex polynomial inequalities (2)
of a fixed degree d^2in
some periodic setJf. There exists a polynomial-lime
in the input length L algorithm, called the 'regularization^ algorithm, which
for any given system (2) finds a subsystem
f{(x)<0,
ieJISk
{l,2,...,m}
(3)
and a sequence of integer vectors yx, ...9yreZn
and natural scalars
ôx,..., ôr e Z+ written in binary number system, r < min {n, m} +1,
such that the following conditions hold.
(i) The system (2) is consistent in Jf if and only if the subsystem (3) is
consistent in Jf.
, (ii) If the subsystem (3) is consistent in Jf9 then it has a solution x° e Jf in
the Euclidean ball
\\x\\ < (hdn)idMl2ndl2 < 2*W.
(i)
(iii) If a solution x° eJf of subsystem (3) in the ball (ê) is known, then
a solution x* e Jf ofthe initial system (2) can easily be found from the formula
x* =yr.2dr+
. . . +y1-2âi+x°.
(5)
Moreover, the binary representation of x* can be obtained from the binary
representations of x°,y19 ...9yr9 d19..., dr in the form
^.-thplace
óVthplace
òVthplace
I
I
I
{a?*} = {yr}Q0 ... 00{#,}00 ... OOfyjOO ... 00{x0}
(6)
where { } stands for the binary representation of the coresponding vectors.
Note 1. The running time of the regularization algorithm is bounded by
an (absolute) polynomial in dnd9 m9 log ft, the algorithm being independent
OÎJf.
Note 2. If all the parameters in the r.h.s. of (5) are written in binary
numerical system, we call the representation (5) binary-exponential. Thus,
we see from (6) that the use of the binary-exponential representation of
Convexity and Complexity in Polynomial Programming
1573
solution instead of the usual binary one enables us to contract the output
information by omitting in its record a number of O's exponential in L.
Note 3. If a vector x° from the ball (4) satisfies the subsystem (3) with
an accuracy e e (0,1)
Mafl)<89
ieJt9
(3')
then the vector x* obtained by (5) satisfies the initial system (2) with the
same accuracy e.
4. Complexity of systems of convex diopliantine inequalities
Applying Theorem 4 in the diophantine case Jf = Zn9 we obtain the
following results.
THEORJBM 5 [11]. For a fixed d > 1 the problem of determining tlie consistency of systems of convex diophantine inequalities of degree at most d belongs
to tJie class NP.
Note. The problem of determining the consistency in Zn of a single
convex quadratic inequality is JTP-complete.
THEOKEM 6. If it is permitted to print integral solutions of nonlinear
systems in binary-exponential form9 then for any fixed d > 2 tJie problem of
determining tJie consistency and of finding a solution of systems of convex
diophantine inequalities of degree at most d is polynomially transformable to
tJie same problem for systems of linear diophantine inequalities. In particular9
these problems can be solved for sure in exponential time t < 2P^, the latter
assertion holding even for binary representation of the output.
All in all, to solve systems of convex diophantine inequalities is not
much harder than to solve systems of linear diophantine inequalities.
5. The ellipsoid method
Suppose that for a feasible in Rn problem of convex polynomial programming (l)-(2) a bound B is known such that the problem has some optimal real solution in the ball ||a?|| < B. Then, to solve the problem with an
accuracy e e (0,1), i.e., to find its e-solution &
fo(A)<f + *,
fi(x)^e,
ie{l92,...9m}
(1«)
(2e)
where /* is the minimum value of the objective polynomial, the ellipsoid
metJwd ([12], [8]) may be used. The following result estimates the com-
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Section 16: L. G-. Khachiyan
plexity of the ellipsoid method for convex polynomial programming,
taking into account finite precision of arithmetical operations performed
over binary numbers in a digital computer.
6. To find an s-solution â of a feasible in Rn problem (1) —(2)
in the ball \\x\\ < B it suffices to perform
THEOREM
)
(7)
elementary operations +, —, x, /, \/~9 max over numbers having in binary
form
I ^llog2(d2hNBdn2/e) +30
(8)
places, the operations being carried out approximately with the same number
of digits as in the binary representation of the numbers. Here n, h, d are the
number of unknowns, the height, and the degree of the problem, respectively,
and N, M are the maximum and the total number of non-zero coefficients
(monomials) of the polynomials /o>/i> •••i/m- ^n addition to the input information the storage of n* + 6n such l-place numbers is also needed.
Note 1. For linear programming problems (d = 19N< n+l) the estimates (7)-(8) can be improved:
k < n*[3M +10.5n*]log2(8hnBle),
ï<îlog a (ftV 3 J3 s /s 5 )+30.
Note 2. For problems of convex quadratic programming with convex
quadratic and/or linear constraints d = 2, JV< (n+2)2/2 the estimates
(7)-(g) yield
Jc < n2[4:.5M +13.5(n+2f]log2(32hn2B2ls)9
Kllog2(hn*B2le) + 37.
COROLLARY. For problems of convex polynomial programming of an
arbitrary fixed degree d the ellipsoid method runs 'in polynomial time with
respect to L and log (Bis).
6. Finding an approximate real solution of systems of convex polynomial
inequalities
From Theorem 4, Ebte 3 in Section 3 and the last corollary follows
THEOREM 7. There exists a polynomial-time with respect to input L
and log(l/e) algorithm for finding an s-solution of systems of convex polyno-
Convexity and Complexity in Polynomial Programming
1575
miai inequalities consistent in Rn9 of an arbitrary fixed degree d, provided
tJiat the output is printed in binary-exponential form.
Note 1. If one insists on binary representation of the output, it is
impossible to design such an algorithm allowing an enormous length of
the output information. Thus, for d > 2, the binary-exponential representation of the output is essential for the validity of Theorem 7.
Note 2. The existence of a similar algorithm for systems of linear
inequalities aggravated by a single non-convex quadratic constraint would
imply P = NP.
We now turn attention to some problems of convex polynomial programming which are exactly solvable in polynomial time.
7. Polynomial solvability of linear programming
It is clear that any feasible in Rn problem of linear programming has
a rational optimal solution x* e Qn — recall that only problems with integer
coefficients are considered. Thus, in accordance with the definition from
Section 2, an exact algorithm for linear programming must check the
feasibility of an Lp, problem and find its optimal rational solution. An
exact algorithm for linear programming, polynomial in L, was announced
in [2] and described in [3]. Later on it was improved in [4].
THEOREM
8 [4]. TJiere exists an exact algorithm for linear programming
requiring
k <> a^ßlogAa <, a4ßlogJia
operations +, —, x , /, max over
I <, log/da £ alog7&a
place binary numbers and additional to the input information storage of
<, a* sucJv numbers. Here a =min{w,m} and ß = m a x { n , m} are the
minimum and the maximum dimensionalities of tJie problem, Ji is the JieigJit
of the problem, and A stands for the maximum modulus of the determinants
of the extended matrix of coefficients of tlie problem.
Note. Theorem 8 holds for linear fractional programming.
8. Polynomial solvability of convex quadratic programming
For problems of convex quadratic programming, consisting in minimization of a convex quadratic polynomial (1) under linear constraints (2), we
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Section 16: L. Gr. Khachiyan
notice that again their feasibility in Rn implies the existence of a rational
optimal solution. An exact algorithm of convex quadratic programming
was described in [6] and improved in [9]. The latter result can also be
improved as follows:
THEOREM 9. TJiere exists an exact algorithm of convex quadratic programming which requires <, n41 (n+m)loghn elementary operations over
<; nloghn-place numbers and additional to the input information storage
of <>ri*such numbers, where n,m,h are of their usual meaning.
Note. From Theorem 9 it follows that the problem of determining the
consistency in real variables of systems of linear inequalities aggravated
by a single convex quadratic constraint is polynomially solvable. This
result can be extended to any fixed number of convex quadratic constraints
[10, 9], The problem of whether there exists a polynomial algorithm for
checking the consistency in real variables of general systems of convex
quadratic inequalities is open.
9. Polynomial solvability of convex polynomial programming with a fixed
number of real and/or integer variables
In [7] a polynomial algorithm was described for solving linear programming
problems with a fixed number of integer variables. Using the technique of
[7], one can immediately derive from [1], p. 135 and Theorem 2 the following result (see also [7], p. 13).
THEOREM 10. There exists an exact algorithm, polynomial in m and
log A, for convex polynomial programming with real and jor integer variables,
provided that the degree d of problems and the number n of unknowns are fixed.
Note. From Ebte in Section 4 it follows that unlike the linear case [7],
there does not exist an algorithm, polynomial in n and log ft, for convex
quadratic integer' programming with a fixed number of constraints unless
P = NP.
In conclusion, let us mention that this talk is an abridged version of
the survey [5].
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Convexity and Complexity in Polynomial Programming
1577
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COMPUTING OENTBE OF THE AOADEMY OP SOIENOES OF THE USSR