L. FAYBUSOVICH in detail in ë6ë. The set of points xèæè;æ ç 0; is usually called the central trajectory for the problem For the construction

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1 Journal of Mathematical Systems, Estimation, and Control Vol. 5, No. 3, 1995, pp. 1í13 cæ 1995 Birkhíauser-Boston A Hamiltonian Formalism for Optimization Problems æ Leonid Faybusovich Abstract We consider dynamical systems that solve general convex optimization problems. We describe in detail their Hamiltonian structure and the phase portrait. In particular, it is proved that these dynamical systems are completely integrable. The Marsden-Weinstein reduction procedure plays a crucial role in our constructions. Dikintype algorithms are brieæy discussed. Key words: convex optimization, completely integrable Hamiltonian systems 1 Introduction Consider the following optimization problem: éc;xé!max; è1.1è éa i ;x é=b i ;;:::;r; f j èxè ç 0;j =1;2;:::;m;x2 R n : è1.2è è1.3è Here c; a 1 ;:::;a r 2R n ;b 1 ;:::;b r are real numbers and f j are smooth concave functions. Suppose that for any æé0 the problem æéc;xé+ j=1 ln f j èxè! max; éa i ;xé=b i ;;:::;r;x2r n ; è1.4è è1.5è has a unique solution xèæè for any æé0. It is reasonable to expect that xèæè tends to a solution of the problem when æ! +1: This approach to solving èknown as a penalty function methodè is described æ Received January 18, 1994; received in ænal form September 2, appear in Volume 5, Number 3, Summary 1

2 L. FAYBUSOVICH in detail in ë6ë. The set of points xèæè;æ ç 0; is usually called the central trajectory for the problem For the construction of practical algorithms it is necessary to choose a sequence of points æ 1 é æ 2 é ::: such that æ k! +1 and ænd the way to calculate xèæ k è with a suæcient accuracy, knowing xèæ 1 è;:::;xèæ k,1 è: Tremendous progress in the construction of eæcient algorithms of this type has been made recently for the case where f 1 ;:::;f m are linear functions ë7ë. The corresponding algorithms are known as path-following algorithms of linear programming. Regrettably, in the case of nonlinear functions f 1 ;:::;f m progress is much more modest ë8ë. The experience of the linear case suggests that in the nonlinear situation it is necessary to consider large step èi.e. æ k, æ k,1 are largeè procedures. In this situation a trajectory of the corresponding discrete algorithm does not follow closely the central trajectory. In the linear case it turns out to be possible to introduce certain dynamical systems ècalled aæne-scaling vector æeldsè which are ëinænitesimal" versions of large-step algorithms. See e.g. ë1ë. In the present paper we study the corresponding dynamical systems for the problem Observe that our construction works only for the case of linear cost function èwhich can be assumed without loss of generalityè. See ë4ë for the case of nonlinear cost function and linear constraints. As in the linear case, these dynamical systems admit a Hamiltonian structure and, moreover, turn out to be completely integrable. We study this structure in some detail. In particular, we show that the equality constraints can be handled by means of the Marsden-Weinstein reduction procedure ë9ë. The Hamiltonian properties of our dynamical systems are quite similar to ones arising in mechanics and optimal control. There are further interesting analogies like the relation between Lagrangian and Hamiltonian approaches in mechanics and between primal and dual problems in optimization èin both cases via the Legendre transformè. 2 Dynamical Systems that Solve Optimization Problems Denote by P the set in R n deæned by constraints 1.2, 1.3 and by intèp è the set fx 2 P : f j èxè é 0;j =1;:::;mg: Throughout this paper we suppose that intèp è 6= ; ènot emptyè, P is compact and the vectors a 1 ;:::;a r èsee 1.2è are linearly independent. Denote by T the smallest vector subspace in R n containing all vectors a 1 ;:::;a r : Let ç : R n! T? be the orthogonal projection of R n onto the orthogonal complement T? of T in R n relative to the standard scalar product. Let, further, æèxè = 2 ln f i èxè; è2.1è

3 r 2 æèxè =æèxè= Here æ is deæned on the set OPTIMIZATION PROBLEMS ræèxè =èèxè= ë r2 f i èxè f i èxè rf i èxè f i èxè ; è2.2è, rf ièxèrf i èxè T f 2 i èxè ë: è2.3è intèqè =fx2r n :f i èxèé0;;:::;mg: è2.4è We think of r 2 æèxè as n by n symmetric matrix. Observe that due to the imposed conditions the matrix æèxè is nonpositive deænite. We assume throughout this paper that the matrix æèxè is negative deænite for any x 2 intèqè: Let ~ è = ç æ è : intèp è! T? : Proposition 2.1 The map ~ è is a diæeomorphism of intèp è on T? : Proof: Consider the problem f æ èxè =æéc;xé+æèxè!max; è2.5è x 2 P: è2.6è Here c 2 R n : Observe that for any æ 2 R the problem 2.5,2.6 has a unique solution xèæè: Moreover, xèæè 2 intèp è: Indeed, let y 2 intèp è and P y = fx 2 P : f æ èxè ç f æ èyèg: It is clear that maxff æ èxè :x2pg= maxff æ èxè : x 2 P y g: Suppose that ~x belongs to the closure of P y : Let x i ;i =1;2;::: be a sequence of points of P y such that x i! ~x; i! +1: It is clear that ~x 2 P: If ~x 2 P nintèp è; then æèx i è!,1;i! +1: But then f æ èx i è!,1;i! +1 contradicts the fact that x i 2 P y for any i: Hence, ~x 2 intèp è: But f æ is continuous on intèp è: Consequently, f æ è~xè ç f æ èyè: We conclude that ~x 2 P y : Hence, P y is a closed subset of P; i.e. P y is compact.we proved that P y is a compact subset of intèp è: Since r 2 æ is strictly concave on intèp è, we conclude that the problem 2.5,2.6 has a unique solution xèæè 2 intèp è: It is clear now that or æc + èèxèæèè 2 T: Hence, rf æ èxèæèè 2 T æèçcè+ ~ èèxèæèè = 0: è2.7è By 2.7 we see that the map ~ è is surjective. Let z 1 ;z 2 2 intèp è; ~ èèz 1 è= ~èèz 2 è=v: Consider the problem f èxè =,é v;x é +æèxè!max;x 2P: è2.8è 3

4 L. FAYBUSOVICH We haverfèz i è=,v+èèz i è2t;i =1;2:Hence, both z 1 ;z 2 are solutions to 2.8. This implies z 1 = z 2 ; since f is a strictly concave function on intèp è: Given c 2 R n ; consider a vector æeld V c on intèp è: V c èxè=,æèxè,1 c+æèxè,1 A T èaæèxè,1 A T è,1 Aæèxè,1 c: è2.9è Here æèxè was deæned in 2.3. It is an invertible matrix according to imposed conditions. Further, A is r by n matrix such that A T =ëa 1 ;:::;a r ë: Proposition 2.2 Let xèæè be èa unique è solution to the problem 2.5,2.6, æ 2 R: Then dxèæè dæ = V cèxèæèè: Remark 2.3 The curves xèæè are usually called central trajectories for the problem The point xè0è is called the analytic center of the set P: Proposition 2.2 follows from the more general. Proposition 2.4 For any x 2 intèp è the integral curve yèæè of 2.9 such that yè0è = x has the form: yèæè = ~ è,1 è ~ èèxè,æçcè;æ 2R: è2.10è In particular, intèp è is an invariant manifold for V c : Proof: It is clear that yè0è = x: Further, if æèæè = èèyèæèè; ~ we haveby 2.10: dæ èæè =,çc: dæ On the other hand, Further, dæ dæ = D ~ èèyèæèè _yèæè: æèæè =D ~ èèyèæèèv c èyèæèè = çæèyèæèèv c èyèæèè = çè,c + A T æèæèè: Here æèæè = èaæèyèæèè,1 A T è,1 Aæèyèæèè,1 c: But ImA T ç T: Hence, çèa T æèæèè = 0: We conclude that æèæè =,çc: In other words,,çc = D ~ èèyèæèèv c èyèæèè = D ~ èèyèæèè _yèæè: Since D ~ èèyè is a linear bijection of T? = KerA; it follows that V c èyèæèè = _yèæè: Indeed, it is clear that Ayèæè = b: Hence, _yèæè 2 KerA and AV c èyèæèè = 0: 4

5 OPTIMIZATION PROBLEMS We now introduce a Riemannian metric g on intèqè : gèx;ç; çè =,ç T æèxèç: è2.11è Recall that æèxèis negative deænite on intèqè due to imposed conditions. It is clear that intèp è ç intèqè can be thought of as a Riemannian submanifold. Proposition 2.5 The vector æeld V c is a gradient æow of the function fèxè =éc;xéon the manifold intèp è: Proof: Since the vector æeld V c is tangent to the manifold intèp è, it is suæcient to verify that éc;çé=gèx;v c èxè;çè for any ç 2 R n such that Aç = 0 and x 2 intèp è: But gèx; V c èxè;çè=éç;æèxèæèxè,1 c,æèxèæèxè,1 A T æèxè é=é ç;xé; since éç;a T æèxè é=é Aç;æèxè é= 0:Here æèxè =èaæèxè,1 A T è,1 Aæèxèè,1 c: Theorem 2.6 Let yèæè;æ 2 R; be an integral curve of the vector æeld V c such that yè0è 2 intèp è: Then lim éc;yèæèé=éc;x æ é; æ! +1; Here x æ èresp. minè, x 2 P: lim éc;yèæèé=éc;x æ é; æ!,1: x æ è is a solution to the problem é c; x é! max èresp. Corollary 2.7 If x æ èresp. x æ è is a unique solution to the problem é c; x é! max èresp. minè, x 2 P; then lim yèæè = x æ ;æ! +1 èresp. lim yèæè =x æ ;æ!,1è. Proof: Observe that yèæè is the solution to the optimization problem: éæc+d; x é +æèxè! max;x2p; è2.12è where d =, ~ èèyè0èè: Indeed, this easily follows by 2.7,2.10. Fix æé0:set u i èæè = 1 æf i èyèæèè ;;:::;m;ç æèxè=éc+d=æ; x é : 5

6 We have: ç æ èyèæèè + L. FAYBUSOVICH ç æ èx æ è ç ç æ èx æ è+ Here we used a standard inequality for the concave function u i èæèf i èx æ è ç u i èæèf i èyèæèè+ éx æ,yèæè;c+d=æ + èèyèæèè=æ é : çèxè, çèyè çé rçèyè;x,yé çèxè =ç æ èxè+ u i èæèf i èxè for x = x æ ;y =yèæè: Observe now that x æ, yèæè 2 T? and çèc + d=æ + èèyèæèè è=0: æ Hence, Taking into account we arrive at the inequality éx æ,yèæè;c+d=æ + èèyèæèè æ u i èæèf i èyèæèè = m æ ; é= 0: éc+d=æ; xæ éçé c+d=æ; yèæè é +m=æ; æ é 0: è2.13è Similarly, éc+d=æ; x æ éçé c+d=æ; yèæè é +m=æ; æ é 0: è2.14è Observe now that yèæè belongs to the compact set P: This implies that lim éd;yèæèé æ = 0;jæj!+1:By Proposition 2.5 the function éc;yèæèé is monotonically nondecreasing. We conclude from 2.13, 2.14 that lim éc;yèæèéçéc;x æéçéc;x æ éç lim éc;yèæèé: æ!,1 æ!1 It is obvious, however, that é c; x æ éç lim é c; yèæè é; æ! +1; é c; x æ éç lim éc;yèæèé; æ!,1: 6

7 OPTIMIZATION PROBLEMS 3 A Hamiltonian Structure Our next goal is to introduce a Hamiltonian structure for the totality of vector æelds V c : Namely, we consider the dynamical systems _x = V c èxè; _c =0 è3.1è on the manifold M = intèp è æ T? : We introduce a symplectic structure! on M and prove that 3.1 is a completely integrable Hamiltonian system. We explicitly construct action-angle variables for 3.1 using the Legendre transform ~ è: Here we follow the pattern of ë3ë, where the linear programming problem in the canonical form was considered. We begin by deæning a symplectic structure on intèqèær n : Recall that intèqè =fx2r n :f i èxèé0;;:::;mg and we deæned a Riemannian metric gèx; ç; çè =,éç;æèxèçéèsee 2.11, 2.3è on intèqè: Given èx; cè 2 intèqè æ R n ; set æèx; c;èç 1 ;ç 1 è;èç 2 ;ç 2 èè = gèx; ç 1 ;ç 2 è,gèx;ç 2 ;ç 1 è; è3.2è èç i ;ç i è 2 R n ær n ;i =1;2:Recall èsee e.g. ë9ëè that there is a canonical symplectic structure on R n æ R n :! can èx; y;èç 1 ;ç 1 è;èç 2 ;ç 2 èè =é ç 2 ;ç 1 é,éç 1 ;ç 2 é; x; y; ç i ;ç i 2R n ;;2:By Proposition 2.1 èunder the assumption that Q is compactè the map èæid R n : intèqèær n! R n ær n is a diæeomorphism èsee 2.2è. We claim that æ is a pullback of! can under this map. Proposition 3.1 We have: èè æ Id R nè æ è! can è=æ: è3.3è In particular, æ is a symplectic two-form. Proof: Let, = èè æ Id R nè æ è! can è: We have:,èx; c;èç 1 ;ç 1 è;èç 2 ;ç 2 èè =! can èèèxè;c;dèèxèç 1 ;ç 1 è;èdèèxèç 2 ;ç 2 èè = éæèxèç 2 ;ç 1 é,éæèxèç 1 ;ç 2 é=æèx; c;èç 1 ;ç 1 è;èç 2 ;ç 2 èè: Hence,, = æ: Here we used Dèèxè =æèxè: Let H be a smooth function on the manifold N = intèqè æ R n : The corresponding Hamiltonian vector æeld X H is determined by the condition ë9ë: æèx; c; X H èx; cè; èç; çèè = D x Hç + D c Hç è3.4è 7

8 L. FAYBUSOVICH for any èç; çè 2 R n æ R n : From 3.4 we easily obtain for X H : _x =,æèxè,1 D c H; _c = æèxè,1 D x H: è3.5è In particular, for the case Hèx; cè =éc;cé=2;we arrive at the Hamiltonian system: _x =,æèxè,1 c; _c =0; è3.6è which coincides with 3.1 for the situation considered èno equality constraintsè. Let H 1 ;H 2 be two smooth functions on N: To compute their Poisson bracket observe that ë9ë: Using 3.5, we easily obtain: fh 1 ;H 2 gèx; cè =æèx H1 ;X H2 è: fh 1 ;H 2 gèx; cè = éd x H 2 èx; cè;æèxè,1 D c H 1 èx; cè é, éd x H 1 èx; cè;æèxè,1 D c H 2 èx; cè é: è3.7è Consider now the Hamiltonians h i : N! R; h i èx; cè =é a i ;c é; i = 1;:::;r: Here a i ;i = 1;:::;r are linearly independent vectors in R n : It is clear by 3.7 that the corresponding Hamiltonians are in involution èi.e. fh i ;h j g = 0;i;j = 1;:::;rè. This enable us to deæne a Hamiltonian action ë9ë of the abelian group R r on N: More precisely, according to 3.5 the Hamiltonian vector æelds X i corresponding to Hamiltonians h i have the form: _x = æèxè,1 a i ; _c =0;;:::;r: è3.8è Denote,æèxè,1 a i by W i èxè;i = 1;:::;r: Introduce a simultaneous æow G : intèqè æ R r! intèqè in the following way: Gèx; t 1 ;:::;t r è=è,1 èèèxè,a 1 t 1,:::,a m t m è: è3.9è It is clear that Gèx;~0è = x for any x 2 intèqè: By i èx;~tè=w i ègèx;~tèè;;:::r: è3.10è The above mentioned Hamiltonian action of R r on N has the form: ~t æ èx; cè =ègèx;~tè;cè;~t2r r ;x2 intèqè;c2 R n : è3.11è The Hamiltonians h 1 ;:::;h r deæne the so-called moment map æ : N! R r : æèx; cè =èh 1 èx; cè;:::;h r èx; cèè = èé a 1 ;cé;:::;éa r ;céè: è3.12è 8

9 OPTIMIZATION PROBLEMS It is obvious that the set intèqè æ T? =æ,1 è0è is an invariant manifold under the action of R r on N: The Marsden-Weinstein reduction procedure ë9ë enables one to deæne a canonical symplectic structure on the factormanifold æ,1 è0è=r r : Our immediate goal is to identify this factor-manifold with intèp è æ T? : Lemma 3.2 Given x 2 intèqè; suppose that Gèx;~tè 2 intèp è;gèx;~uè 2 intèp è for some ~t; ~u 2 R r : Then ~t = ~u: Proof: Recall that ç : R n! T? is the orthogonal projection and ~ è = çè is a diæeomorphism of intèp è onto T? èsee Proposition 2.1è. Let ç = Gèx; ~uè 2 intèp è;ç =Gèx;~tè2 intèp è: By 3.9 èèçè =èèxè,u 1 a 1,:::, u r a r ; èèçè = èèxè,t 1 a 1,:::,t r a r : Hence, çèèçè = çèèçè = çèèxè: This implies by Proposition 2.1 that ç = ç: Hence, èèçè = èèçè and consequently, u 1 a 1 + :::+u r a r = t 1 a 1 +:::t r a r or t i = u i ;i = 1;:::;r; since the vectors a 1 ;:::a r are linearly independent. Lemma 3.3 For any x 2 intèqè there exists a unique ~tèxè 2 R r such that Gèx;~tèxèè 2 intèp è: Moreover, the map x! ~tèxè is smooth. Proof: Set yèxè = ~ è,1 èçèèxèè: It is clear that yèxè is a smooth function from intèqè into intèp è: We have: çèèxè = ~ èèyèxèè = çèèyèxèè: In other words, èèxè, èèyèxèè 2 T: Consequently, èèyèxèè = èèxè, t 1 èxèa 1, :::,t r èxèa r for some smooth functions t i èxè: We claim that Gèx;~tèxèè = yèxè 2 intèp è: Indeed, Gèx;~tèxèè = è,1 èèèxè, t 1 èxèa 1, :::t r èxèa r è: Hence, ègèx; ~tèxèè = èèyèxèè or Gèx; ~tèxèè = yèxè: The uniqueness of ~tèxè follows by Lemma 3.2. Given x 2 intèqè; denote by 'èxè the point Gèx;~tèxèè: In this way, we deæne a smooth map ' : intèqè! intèp è; which enable us to identify æ,1 è0è=r r with intèp è æ T? : The map ' æ Id T? : intèqè æ T?! intèp è æ T? è3.13è plays the role of the canonical projection under this identiæcation. The manifold æ,1 è0è=r r is endowed with a natural symplectic structure! ë9ë which is under our identiæcations has the form:!è'èxè;c;èd'èxèç 1 ;ç 1 è;èd'èxèç 2 ;ç 2 èè=æèx; c;èç 1 ;ç 1 è;èç 2 ;ç 2 èè; è3.14è x 2 intèqè;ç i 2 R n ;ç i 2 T? ;c2 T? : Consider the Hamiltonian Hèx; cè = é c; c é =2 on intèqè æ R n : It is clear by 3.5 that the corresponding Hamiltonian vector æeld has the form: _x = W c èxè; _c =0: Since fh; h i g = 0;i = 1;:::r;H can be lifted to the Hamiltonian H ~ on æ,1 è0è=r r : Under our identiæcations, Hèx; ~ cè = éc;cé 2 2 ;x 2 intèp è;c 2 T? : 9

10 L. FAYBUSOVICH Theorem 3.4 The Hamiltonian vector æeld corresponding to the Hamiltonian ~ H relative to the symplectic form! has the form 3.1 èsee 2.9è. Proof: Suppose that çètè isanintegral curve ofw c :Consider the curve çètè ='èçètèè: By Proposition 2.4 çètè =è,1 èèèçè0èè, ctè: On the other hand, by the very deænition of ' we have: è'èxè =èèxè+t 1 èxèa 1 +:::t r èxèa r ; è3.15è x 2 intèqè and some t 1 èxè;:::t r èxè: In particular, è'èçètèè = èèçè0èè, ct + t 1 èçètèèa 1 + :::t r èçètèèa r : Hence, ~ è'èçètèè = ~ èèçè0èè,çct: On the other hand, ~ èèçè0èè = ~ èè'èçè0èè: We see by Proposition 2.4 that 'èçètèè is the integral curve ofv c with the initial condition 'èçè0èè: The result follows by general considerations ë9ë. Theorem 3.5 The map è ~ æ Id T?is a symplectic diæeomorphism of èintèp è æ T? ;!è onto èt? æ T? ;! can è: Under this diæeomorphism the Hamiltonian system _x = V c èxè; _c = 0 goes to the dynamical system _x =,çc; _c =0: Proof: Set æ = è ~ è æ Id T?è æ! can : We have æè'èxè;c;èd'èxèç 1 ;ç 1 è;èd'èxèç 2 ;ç 2 è=! can è ~ è'èxè;c;èdè ~ èæ'èèxèç 1 ;ç 1 è;èdè ~ èæ'èèxèç 2 ;ç 2 è: Here ç i 2 R n ;ç i 2 T? : But ~ èè'èxèè = çèèxè by Hence, æè'èxè;c;èd'èxèç 1 ;ç 1 è;èd'èxèç 2 ;ç 2 è=! can èèdèèxèç 1 ;ç 1 è;èdèèxèç 2 ;ç 2 èè: Here we used the fact that ç i 2 T? : In other words, èè æ Id R nè æ! can èèx; cè; èç 1 ;ç 1 è;èç 2 ;ç 2 èè = æèx; c;èç 1 ;ç 1 è;èç 2 ;ç 2 èè by Proposition 3.1. The result follows by Remark 3.6 Theorem 3.5 shows that our Hamiltonian system _x = V c èxè; _c = 0 is completely integrable and provides a construction of action-angle variables. Theorem 3.7 The pair èintèp è æ T? ;!è is a symplectic submanifold of èintèqè æ R n ; æè: In particular,! =æj intèp èæt?: 10

11 OPTIMIZATION PROBLEMS Proof: By 3.14 it is suæcient toprove that æ 1 = æèx; c;èç 1 :ç 1 è; èç 2 ;ç 2 èè = æ 2 = æè'èxè;c;èd'èxèç 1 ;ç 1 è;d'èxèç 2 ;ç 2 èè è3.16è for any x 2 intèqè;c 2 T? ;ç i 2 R n ;ç i 2 T? : By Proposition 3.1 æ = èè æ Id R nè æ : Hence, æ 2 =! can èèdèè æ 'èèxèç 1 ;ç 1 è;dèèæ'èèxèç 2 ;ç 2 èè =! can èèdèèxèç 1 ;ç 1 è;èdèèxèç 2 ;ç 2 èè = èè æ Id R nè æ! can èx; c;èç 1 ;ç 1 è;èç 2 ;ç 2 èè=æ 1 : Here we used 3.14 and the condition ç i 2 T? ;;2: Remark 3.8 Thus in the situation considered the reduced manifold can be realized as a symplectic submanifold of the initial manifold. 4 Concluding Remarks In the present paper we have considered dynamical systems that solve general èsmoothè convex programming problems. We have analyzed in detail their Hamiltonian structure. The crucial part of our analysis is based on the notion of the Legendre transform. In particular, the underlying symplectic form is obtained as a pullback of the canonical symplectic structure via the Legendre transform. Dynamical systems corresponding to the problems with equality constraints are obtained by applying the Marsden-Weinstein reduction procedure. In particular, their solutions are projections of solutions of dynamical systems corresponding to the problems without inequality constraints. The reduced manifold turned out to be the symplectic submanifold of the initial manifold. The Legendre transform also provides a linearization of our dynamical systems and the explicit construction of action-angle variables. At present time it is not quite clear how to generalize the interior-point methodology for the case of nonlinear equality constraints. Our Hamiltonian framework may be helpful in this situation. It seems that it is natural to use the universal Hamiltonian éc;cé=2 and the introduced symplectic structure to construct constrained Hamiltonian systems in this more general situation. The essential part of the classical book ë6ë isdevoted to the analysis of one particular type of trajectories of our dynamical systems-central trajectories. While this is a very important class of trajectories, the consideration of large-step algorithms requires the analysis of all trajectories. Observe that we considered here only logarithmic barrier functions. More general classes of barrier functions can be analyzed using the same technique èsee ë5ë for the case of entropy barrier functionsè. 11

12 L. FAYBUSOVICH We now brieæy outline the idea behind one particular class of large-step algorithms. Given ç 2 R n ; denote ëgèx; ç; çèë 1=2 by k ç k x : Suppose that there exists æé0 such that for any x 2 intèqè If 4.1 holds and x 2 intèp è; then x + ç 2 intèqè; k ç k x éæ: çèxè =x+ç V cèxè 2intèP è; jçj éæ: kv c èxèk x è4.1è Given x 2 intèp è; consider iterations x; çèxè;ç 2 èxè;:::;: This procedure is known as Dikin's algorithm ë2ë for the case where all constraints are linear. It is known to be an eæcient tool for solving linear programming problems. To the best of our knowledge, no convergence results have been reported for the case of nonlinear constraints. It is clear, however, that convergence to the optimal solution should take place èat least for small çè due to the global convergence of all trajectories of the vector æeld V c to the optimal solution. Observe that conditions 4.1 are satisæed provided the functions ln f i ;;:::m are self-concordant functions èthis is always the case when f i are linear or quadratic ë10ë è.the properties of this type provide a special status for logarithmic barrier functions at least for now. Our analysis can be generalized to certain inænite-dimensional situations with interesting applications to control problems like linear-quadratic regulators with quadratic constraints ë11ë. We plan to consider these applications separately. References ë1ë A.M. Bloch. Steepest descent, linear programming and Hamiltonian æows, Contemporary Math. 114 è1990è, ë2ë I. Dikin. Iterative solution of problems of linear and quadratic programming, Soviet. Math. Dokl. 8 è1967è, ë3ë L. Faybusovich. Hamiltonian structure of dynamical systems which solve linear programming problems, Physica D53 è1991è, ë4ë L. Faybusovich. Dynamical systems which solve optimization problems with linear constraints, IMA Journ. of Mathematical Control and Information 8 è1991è, ë5ë L. Faybusovich. Gibbs variational principal,interior-point methods and gradient æows, èto appearè ë6ë A.V. Fiacco and G.P. McCormick. Nonlinear Programming:Sequential Unconstrained Minimization Techniques. New York: John Wiley,

13 OPTIMIZATION PROBLEMS ë7ë C. Gonzaga. Path-following algorithms for linear programming, SIAM Review 34 è1992è, ë8ë F. Jarre. Interior-point methods for convex programming, Applied Mathematics and Optimization 26 è1992è, ë9ë J. Marsden. Lectures on Mechanics. Cambridge: Cambridge University Press, ë10ë Yu.E. Nesterov and A.S. Nemirovsky. Self-concordant functions and polynomial-time methods in convex programming. Technical report, Centr. Econ. Inst.,Moscow, USSR,1989. ë11ë V.A. Yakubovich. Nonconvex optimization problem: The inænitehorizon linear-quadratic control problem with quadratic constraints, System and Control Letters 19 è1992è, Department of Mathematics, University of Notre Dame, Notre Dame, Indiana, USA Communicated by Clyde F. Martin 13