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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}. [1569] 1570 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. 1572 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- 1574 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 47 — Proceedings.,., t. II 1576 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]. References [1] Collins G. E., Quantifier elimination for real-closed fields by cylindrical alge. braic decomposition. In: Automata Theory and Formal Languages, Lecture Note^ in Comput. Sci 33, Springer, Berlin, 1975. Convexity and Complexity in Polynomial Programming 1577 [2] Xa'iHHH JI. T., nojiHHOMHajiLHHfi anropnTM B JiHiieöHOM nporpaMMHpoBaiiHH, ffAH CCCP 244, Xs 5 (1979); translated in: Soviet Math. Bold. 20, No. 1 (1979). [3] XaiHHH JI. T., üojiHHOMHajiBiiHe ajiropiiTMH B jiHHeftHOM nporpaMMnpoBairHH, 2KB M u M®, T. 20, J\fe 1 (1980), translated i n : USSR Clomp. Math, and Math. Phys. 20, Ko. 1 (1980). [4] Xa^HHH JI. P., 0 TOHHOM pemeHHH CHCTeni jiHiieaiiLix iiepaBencTB H aa^a i i jiHHefiHoro nporpaMMHpoBairHH, SKBM u M® 22, Ks 6 (1982). [6] Xa^iHHH JI. T., BtmyKJiocTB H ajiropuTMiwecKaH cjiomHocTB pemeiinn Basan noJiHHOMnajiLiioro nporpaMMiipoBaniiH, lise. AH CCCP, TexHwiecnan miôepnemuKa 6 (1982). [6] KOBJIOB M. K., TapacoB C. II., X&mwn JI. F., IIojiHHOMiiajihiiaH paspeimiMOCTt BHnyKJioro KBap;paTHliHoro nporpaaiMiipoBannH, ff AH CCCP 248; translated i n : Soviet Math. Dolci. 20, No. 5 (1979). [7] Lenstra H. W. Jr., Integer programming with a fixed number of variables, Dept. of Math., Univ. of Amsterdam, Rept. No. 81-95 (1981). [8] IIIop H. 3., MeToji; OTceneHHH c pacTHJKeiraeM npocTpancTBa EJIH pemeHHH sa^an BBinyKJioro nporpaMMHpoBaHHH, Kuóepnemuna 13, N° 1 (1977), translated i n : Cybernetics 13 (1977). [9] TapacoB C. H., Ajieeopaunecnuü nodxod n nenomopuM 3adanaM eunynjiozo npoepaMMupooanun, flnc. KaHR. $HB.-MaT. nayK, BHHHCJiHTejiBHHÄ imnTp All CCCP, M., 1979. [10] TapacoB C. II., XaiiHHH JI. T., Onpedejienue coeMecmnocmu cucmeM eunynjiux KeadpamuHJtbix uepaeencme B Rn, TesncHflOKJiaftOBcoBeTCKo-nojiLCKoro nayraoro ceMHnapa no MaTeMaT. MeTop;aM B nnannpoBaiiHH H ynp. DKOHOMHKOH, IJ3MM, M., 1979. [11] TapacoB C. n . , Xa*niHH JI. T., FpaHHHH pemenafi H ajiropiiTMimecKafl cjiomHOCTB cncTeM BunyKJiBix RHO$anTOBLix iiepaBeHCTB, JIAH CCCP 255, Ne 2, translated in: Soviet Math. Dolci. 22, No. 3 (1980). [12] IOftHH fl. B., HeMiipoBCKHfi A. C , HH<J)opMau;HOHHaH cjiomnocTB II B $ $ e K T H B H t i e MOTORM pemeHHH BtinyKJiLix BKCTpeMajiBHHx sa^an, SKOHOMUKU U MameM. Memodu, XII, X° 2 (1976), translated in: Matelcon 13, 3 (1977). COMPUTING OENTBE OF THE AOADEMY OP SOIENOES OF THE USSR