Finite Dierence Smoothing Solution of Nonsmooth Constrained Optimal Control Problems Xiaojun Chen 22 April 24 Abstract The nite dierence method and smoothing approximation for a nonsmooth constrained optimal control problem are considered. Convergence of solutions of discretized smoothing optimal control problems is proved. Error estimates of nite dierence smoothing solution are given. Numerical examples are used to test a smoothing SQP method for solving the nonsmooth constrained optimal control problem. Key words. Optimal control, nondierentiability, nite dierence method, smoothing approximation AMS subject classications. 49K2, 35J25 Introduction Recently signicant progress has been made in studying elliptic optimal control problem Z minimize ; z d ) 2 (y 2 dx + Z (u ; u d ) 2 dx 2 subject to ;4y = f(x y u) in y = on ; (.) u 2U where z d u d 2 L 2 () f 2 C( R 2 ), >isa constant, is an open, bounded convex subset of R N N 3 with smooth boundary ;, and U = fu 2 L 2 () j u(x) q(x) a:e in g q 2 L (): If f is linear with respect to the second and third variables, (.) is equivalentto its rst order optimality system. Based on the equivalence, the primal-dual active set strategy [2] can solve problem (.) eciently. Moreover, Hintermuller, Ito and Department of Mathematical System Science, Hirosaki University, Hirosaki 36-856, Japan. This work was supported in part by a Grant-in-Aid for Scientic Research from the Ministry of Education, Science, Sports and Culture of Japan.
Kunisch [4] proved that the optimality system is slantly dierentiable [7] and the primal-dual active set strategy is a specic semismooth Newton method for the rst order optimality system. The nondierentiable term in the optimality system arises from the constraint u 2 U. Without this constraint, the optimality system is a linear system. For such a case, Borzi, Kunisch and Kwak [3] gave convergence properties of the nite dierence multigrid solution for the optimality system. For the case where f is nonlinear with respect to the second and third variables, the rst order optimality system and second order optimality system have been studied in order to obtain error estimates in discretization approximations and convergence theory in sequential quadratic programming algorithms. See for instance [4] and the references therein. Most of papers assume that f is of class C 2 with respect to the second and third variables. However, this assumption does not hold for the following optimal control problem Z Z minimize ; z d ) 2 (y 2 d! + (u ; u d ) 2 d! 2 subject to ;4y + max(y ) = u + g in y = on ; (.2) u 2U where > is a constant andg 2 C(). The nonsmooth elliptic equations can be found in equilibrium analysis of conned MHD(magnetohydrodynamics) plasmas [6, 7, 8], thin stretched membranes partially covered with water[5], or reactiondiusion problems []. The discretized nonsmooth constrained optimal control problems derived from a nite dierence approximation or a nite element approximation of (.2) with mass lump has the form: minimize 2 (Y ; Z d) T H(Y ; Z d )+ 2 (U ; U d) T M(U ; U d ) subject to AY + D max( Y)=NU + c (.3) U b: Here Z d c 2 R n, U d b 2 R m, H 2 R nn, M 2 R mm, A 2 R nn,d 2 R nn N 2 R nm and max( ) is understood componentwise. Moreover H M A D are symmetric positive denite matrices, and D =diag(d ::: d n ) is a diagonal matrix. It is aware that for the discretized nonsmooth constrained optimal control problem, a solution of the rst order optimality system is not necessarily a solution of the optimal control problem. Also a solution of the optimal control problem is not necessarily a solution of the rst order optimality system, see Example 2. in the 2
next section. In [5], a sucient condition for the two problems to be equivalent is given. In this paper, we consider the following smoothing approximation of (.3) minimize 2 (Y ; Z d) T H(Y ; Z d )+ 2 (U ; U d) T M(U ; U d ) subject to AY + (Y )=NU + c (.4) U b where : R n! R n is dened by a smoothing function : R! R as (Y )=D B @ (Y ) (Y 2 ). (Y n ) Here is called a smoothing parameter. For >, is continuously dierentiable and its derivative satises. Moreover, it holds C A j (t) ; max( t)j with a constant > for all. We can nd many smoothing functions having such properties. In this paper, we simply choose [2] (t) = 2 (t + p t 2 + 2 ): It is not dicult to verify that for a xed >, is continuously dierentiable and Moreover, for any t and >, and (t) = 2 ( + t p t2 + 2 ) > : j (t) ; max( t)j 2 : o (t) := lim # (t) = 2 8 >< >: : 2 if t> if t = if t<: It was shown in [8] that o (t) is an element of the subgradient [9] of max( ). Following these properities of, the smoothing approximation function is continuously dierentiable for > and satises k (Y ) ; D max( Y)k p nkdk (.5) 3
for. Here kk denotes the Euclidean norm. In particular, for =, (Y )=D max( Y). Moreover, the matrix o (Y ):=Ddiag( o (Y ) o (Y 2 ) ::: o (Y n )) is an element ofthe Clarke generalized Jacobian [9] of D max( Y): In this paper we investigate the convergence of the discretized smoothing problem (.4) derived from the ve-point dierence method and smoothing approximation. In section 2, we describe the nite dierence discretization of the optimal control problem. We prove that a sequence of optimal solutions of the discretized smoothing problem (.4) converges to a solution of the discretized nonsmooth optimal control problem (.3) as!. Moreover, under certain conditions, we prove that the distance between the two solution sets of (.3) and (.4) is O(). We show that the dierence between the optimal objective value of (.3) and the optimal objective value of the continuous problem (.2) is O(h ), where < <. In section 3we use numerical examples to show that the nonsmooth optimal control problem can be solved by a nite dierence smoothing SQP method eciently. 2 Convergence of smoothing discretized problem In this section, we investigate convergence of the smoothing discretized optimal control problem (.4) derived from the smoothing approximation and the vepoint nite dierence method with = ( ) ( ). To simplify our discussion, we use the same mesh size for discretization approximation of y and u. Let be a positive integer. Set h ==( + ). Denote the set of grids by h = f(ih jh) j i j = ::: g: The Dirichlet problem in the constraints of (.2) is approximated by the ve-point nite dierence approximation with uniform mesh size h. For grid functions W and V dened on, we denote the discrete L 2 h-scalar product X (W V) L 2 = h 2 W (ih jh)v (ih jh): h i j= Let P be the restriction operator from L 2 () to L 2 h(). Let Z d = Pz d, U d = Pu d and b = Pq. Then we obtain the discretized optimal control problem (.3). Denote the objective functions of the continuous problem and the discretized problems J(y u) = 2 Z (y ; z d ) 2 dx + 2 4 Z (u ; u d ) 2 dx
and J h (Y U) = 2 (Y ; Z d) T H(Y ; Z d )+ 2 (U ; U d) T M(U ; U d ) respectively. Let S S h and S h denote the solution sets of (.2), (.3), and (.4), respectively. 2. Problems (.4) and (.3) For a xed h, we investigate the limiting behaviour of optimal solutions of (.4) as the smoothing parameter! : Theorem 2. For any mesh size h and smoothing parameter, the solution sets S h and S h are nonempty and bounded. Moreover, there exists a constant > such that for all 2 [ ]. S h f(y U) j J h (Y U) g (2.) Proof: First we observe that for xed U b and, the system of equations AY + (Y )=NU + c is equivalent to the strongly convex unconstrained minimization problem min Y 2 Y T AY + nx i= Z Yi (t)dt ; Y T (NU + c): By the strong convexity, the problem has a unique solution Y. Hence the feasible sets of (.3) and (.4) are nonempty. Moreover, we notice that the objective function J h of (.3) and (.4) is strongly convex, which implies that the solution sets of (.3) and (.4) are nonempty and bounded. Now we prove (2.). In (.3), the constraint AY + D max( Y)=NU + c can be written as (A + DE(Y ))Y = NU + c where E(Y )is a diagonal matrix whose diagonal elements are E ii (Y )= 8 < : if Y i > if Y i : 5
Since A is an M-matrix, from the structure of DE(Y ), we have thata + DE(Y ) is also an M-matrix and ky kk(a + DE(Y )) ; (NU + c)k ka ; k(knkkuk + kck): (2.2) See Theorem 2.4. in [6]. Moreover, for >, let Y satisfy By the mean value theorem, we nd where and AY + (Y )=NU + c: = A(Y ; Y )+ (Y ) ; D max( Y) = A(Y ; Y )+ (Y ) ; (Y )+ (Y ) ; D max( Y) = (A + 2 ~ D)(Y ; Y )+ 2 ^D ~Y ~Y ~D = Ddiag( + q ::: + q n ) Y ~ 2 + 2 Y ~ n 2 + 2 ^D = Ddiag( Here ^ 2 ( ]and ~Y i lies between (Y ) i and Y i. ^ ^ q ::: q ): Y 2 +^ 2 Y 2 n +^ 2 Using that all diagonal elements of ~D are positive and < ^ q i = ::: n Yi 2 +^ 2 we obtain ky ; Y k 2 k(a + D) 2 ~ ; ^Dk 2 ka; kkdk: (2.3) Therefore, using (2.2), we nd that for 2 ( ], ky k 2 ka; kkdk + ky kka ; k( kdk + knkkuk + kck): 2 Let (Y U ) 2S h and (Y U ) 2S h. Let Y be the solution of AY + (Y )=NU + c 6
that is, (Y U )isa feasible point of (.4). By the argument above, we have J h (Y U ) J h (Y U ) 2 khkky ; Z d k 2 + 2 kmkku ; U d k 2 2 khk(ky k + kz d k) 2 + 2 kmk(ku k + ku d k) 2 2 khk(ka; k( 2 kdk + knkku k + kck)+kz d k) 2 + 2 kmk(ku k + ku d k) 2 : In the rst part of the proof, we have shown that the solution set S h is bounded, that is, ku k is smaller than a positive constant. Hence there exists a constant > such that 2 khk(ka; k( 2 kdk + knkku k + kck)+kz d k) 2 + 2 kmk(ku k + ku d k) 2 : This completes the proof. Theorem 2.2 Letting!, any accumulation point of a sequence of optimal solutions of (.4) is an optimal solution of (.3), that is, f lim (Y U)2S h # (Y U)g S h : Proof: Let (Y U ) 2S h, (Y U ) 2S h and (Y U ) satisfy Then the following inequality holds Using the Talyor expansion, we nd AY + (Y )=NU + c: J h (Y U ) J h (Y U ): (2.4) J h (Y U )=J h (Y U )+(Y ; Y ) T H(Y ; Z d )+ 2 (Y ; Y ) T H(Y ; Y ): Following the argument on(2.3), we have ky ; Y k 2 ka; kkdk: Since the solution set of (.3) is bounded, there is a constant > such that for 2=(kA ; kkdk), (Y ; Y ) T H(Y ; Z d )+ 2 (Y ; Y ) T H(Y ; Y ) (ky ; Y k + ky ; Y k 2 ) ka ; kkdk: 7
Combining this with (2.4), the Talyor expansion gives J h (Y U ) J h (Y U )+ka ; kkdk: (2.5) Moreover, from Theorem 2., there is a bounded closed set L such that S h L for all 2 [ ]: Hence, without loss of generality, we may assume that (Y U )! ( Y U) 2L as! : Now, we show that ( Y U) is a feasible point of (.3). Obviously U b as U for all >. The other constraint also holds, since b ka Y + D max( Y ) ; N U ; ck = lim kay + D max( Y ) ; N U ; ck # = lim # knu ; (Y )+D max( Y ) ; N Uk lim # kd max( Y ) ; (Y )k lim # (kdkk max( Y ) ; max( Y )k + kd max( Y ) ; (Y )k) kdk(lim # kv kk Y ; Y k + p n) =: Here V is a diagonal matrix whose diagonal elements are V ii = 8 >< >: Obviously, we have V ii : Now let! in (2.5), we get Hence ( Y U) isasolution of (.3). if ( Y ; Y ) i = max( Y ) i ; max( Y ) i ( Y ; Y otherwise ) i J h ( Y U) J h (Y U ): To estimate the distance between the two solution sets S h and S h we have to consider the rst order optimality system for (.3). We say(y U) satises the rst order conditions of (.3), or(y U) is a KKT (Karush-Kuhn-Tucker) point of (.3), if it together with some (s t) 2 R n R m satises B @ H(Y ; Z d )+As + DE(Y )s M(U ; U d ) ; N T s + t AY + D max( Y) ; NU ; c min(t b ; U) 8 C A =: (2.6)
The vectors s 2 R n and t 2 R m are referred to as Lagrange multipliers. It was shown in [5] that for any U, the system of nonsmooth equations AY + D max( Y) ; NU ; c = has a unique solution, and it denes a solution function Y (U). Moreover, (2.6) is equivalent to the following system @ ((A + DE(Y (U))); N) T H(Y (U) ; Z d )+M(U ; U d )+t min(t b ; U) A =: (2.7) However, for the discretized nonsmooth constrained optimal control problem (.3), a KKT point of (.3) is not necessarily a solution of the optimal control problem (.3). Also a solution of the optimal control problem (.3) is not necessarily a KKT point of (.3). Example 2. Let n =2 m= M = = = b = H = D = I c = A = @ 2 ; ; 2 A and N = @ 3 ;. For U d = and Z d =( ;3) T,(~ Y ~U) =( ) T is a KKT point of (.3), but ( ~ Y ~ U)isnota solution of (.3). 2. For U d = and Z d = ( ), (Y U ) = ( ) T is a solution of (.3), but the set (Y U )isnota KKT point of (.3). Let A K(Y ) be the submatrix of A whose entries lie in the rows of A indexed by Lemma 2. K(Y )=fi j Y i = i = 2 ::: ng: [5] Suppose that (Y U ) is a local optimal solution of (.3), and either K(Y )= or ((A + DE(Y )) ; N) K(Y ) =, then (Y U ) is a KKT point of (.3). A : Theorem 2.3 Let (Y U ) 2 S h Lemma 2., we have and (Y U ) 2 S h : Under assumptions of ky ; Y k + ku ; U ko(): Proof: Let us set W =(A + DE(Y )) ; N: 9
By Theorem 2. in [5], the assumptions implies that in a neighborhood of U, the solution function Y () can be expressed by Y (U) =WU: Moreover, Y () is dierentiable at U and Y (U ) = W: In such a neighborhood, we dene a function F (U t) = @ W T H(Y (U) ; Z d )+M(U ; U d )+t min(t b ; U) From Lemma 2., there is t 2 R m such that F (U t )=: The Clarke generalized Jacobian @F(U t) [9] of F at (U t ) is the set of matrices that have the version @ W T HW + M T I I + T where T is a diagonal matrix whose diagonal elements are T ii = 8 >< >: ; if (b ; U ) i <t i if (b ; U ) i >t i i if (b ; U ) i = t i i 2 [; ]: This is easy to see that all matrices in @F(U t ) are nonsingular. By Proposition 3. in [7], there is a neighborhood N of (U t ) and a constant > such that for any (U t) 2N and any V 2 @F(U t), V is nonsingular and kv ; k: Now we consider a function of F dened by the rst order condition of the smoothing problem (.4) as F (U t) = @ ((A + (Y (U))) ; N) T H(Y (U) ; Z d )+M(U ; U d )+t min(t b ; U) Here Y (U) is the unique solution of the system of smoothing equations A AY + (Y ) ; NU ; c =: Since (.4) is a smoothing problem, (Y U ) 2S h implies that there is t 2 R m such that F (U t )=: A : A :
have Applying the mean value theorem for Lipschitz continuous functions in [9], we F (U t ) ; F (U t )=co@f(u U t t ) @ U ; U t ; t where co@f(u U t t ) denotes the convex hull of all matrices V 2 @F(Z) for Z in the line segment bewteen (U t ) and (U t ): Therefore, we can claim that Note that @ U ; U t ; t A A 2kF (U t ) ; F (U t )k: (2.8) lim # (Y )= o (Y ) and o (Y ) 2 @D max( Y ): By Lemma 2.2 in [5] (A + o (Y )) ; N = W: From the nonsingularity of @F(Y U ), (Y U ) is the unique solution of (.3). From Theorem 2.2, Y! Y as!. Hence there are constants >, 2 > and > such that for all 2 ( ], and for i 62 K(Y ), i 62 K(Y ) and kh(y j o (Y ) ; (Y )j i = 2 ; Z d )k Therefore, from F (U t ) = and F (U t )=,we nd kf (U t ) ; F (U t )k = kf (U t ) ; F (U t )k q (Y ) 2 i + 2 ;jy j i q 2 : (2.9) (Y ) 2 i + 2 = k(((a + o (Y )) ; ; (A + (Y )) ; )N) T H(Y ; Z d )k k((a + o (Y )) ; ; (A + (Y )) ; )Nk k(a + (Y )) ; ( (Y ) ; o (Y ))(A + o (Y )) ; Nk ka ; kk( (Y ) ; o (Y ))(A + o (Y )) ; Nk 2 p nka ; k 2 knk: The last inequality uses k(a + o (Y )) ; kka ; k (2.9) and ((A + o (Y )) ; N) K(Y ) =:
This, together with (2.8), gives ku ; U ko(): Furthermore, from the convergence of Y suciently small, and the assumptions, we have that for This completes the proof. 2.2 Problems (.3) and (.2) ky ; Y k = kw (U ; U )k O(): Note that L 2 h()-scalar product (Y Y ) L 2 associated with Y = Py can be considered as the Riemann sum for the multidimensional integral y 2 dx: By the h Z error bound (5.5.5) in [], we have where Z j y 2 dx ; (Y Y ) L 2 h V (y 2 )= max x z2 h x6=z j 2V (y2 ) p n jy 2 (x) ; y 2 (z)j : kx ; zk If y is Lipschitz continuous in with a Lipschitz constant K, then there is such that max x2 jy(x)j and jy 2 (x) ; y 2 (z)j = jy(x)+y(z)jjy(x) ; y(z)j 2Kkx ; zk: Hence the Lipschitzan continuity of f yields an error bound for the Riemann sum Z j y 2 dx ; (Y Y ) L 2 j 4K p = O(h): (2.) h n For a given function u, error bounds for the ve-point nite dierence method to solve the nonsmooth Dirichlet problem can be found in [6]. Lemma 2.2 ;4y + max( y)=u + g in y =on ;: (2.) [6] Let y 2 C 2 () be a solution of (2.), and let Y be the nite dierence solution of (2.). Then we have A(Py)+D max( Py)=Nu+ c + O(h ) and kpy; Y ko(h ): 2
Here stands for the exponent of Holder-continuity, and <<. Theorem 2.4 Suppose that (.2) has a Lipschitz continuous solution (y u ) and y 2 C 2. Let (Y U ) be a solution of (.3). Then we have J h (Y U ) J h (Py Pu )+O(h ): (2.2) Moreover, if there exists u 2 C, together with y 2 C 2, satises the constraints of (.2) and kp u ; U ko(h ) then we have J h (Y U ) J h (Py Pu ) ; O(h ): (2.3) Proof: By Lemma 2.2, the truncation error of the nite dierence method yields A(Py )+D max( Py )=N(Pu )+c + O(h ): (2.4) We enlarge the feasible set of (.3) and consider a relaxing problem minimize 2 (Y ; Z d) T H(Y ; Z d )+ 2 (U ; U d) T M(U ; U d ) subject to AY + D max( Y)=NU + c (2.5) U b + h e where e =( ::: ) T 2 R n, and is a positive constant such that (Py Pu )is a feasible point of(2.5). Let ( ~Y ~U) be a solution of (2.5). Then it holds J h ( ~Y ~U) J h (Py Pu ): (2.6) Moreover, since the feasible set of (.3) is contained in that of (2.5), we have J h ( ~Y ~U) J h (Y U ): Take a point ^U = min(b ~U), together with ^Y satisfying A ^Y + D max( ^Y )=N ^U + c: Then ( ^Y ^U) is a feasible point of (.3). Moreover, from ~U b + h e, we have k ^U ; ~Uk O(h ) and k ^Y ; ~Y kka ; kknkk ^U ; ~Uk O(h ): 3
Therefore, we nd J h (Y U ) J h ( ^Y ^U) J h ( ~Y ~U)+O(h ): This, together with (2.6), implies J h (Y U ) J h (Py Pu )+O(h ): To prove (2.3), we let Y be the nite dierence solution of the Diriclet problem ;4y + max( y)=u + g in y =on ;: (2.7) By Lemma 2.2, we have kp y ; Y ko(h ): (2.8) Moreover, from and A Y + D max( Y )=N(P u)+c AY + D max( Y )=NU + c we nd k Y ; Y k = k(a + V ) ; N(P u ; U )kka ; kknkkp u ; U ko(h ): Here V is a nonnegative diagonal matrix. (See the proof of Theorem 2.2.) This, together with (2.8), implies that kp y ; Y ko(h ): Therefore, we obtain J h (P y P u) J h (Y U )+O(h ): (2.9) From the assumption that y u y and u are Lipschitz continuous functions, we can estimate the errors of the integrals in J and get J h (Py Pu ) ; O(h) J(y u ) (2.2) and J(y u) J h (P y P u)+o(h): (2.2) Finally, using the optimality of(y u ), that is, J(y u ) J(y u) we obtain (2.3) from (2.9), (2.2),(2.2). From Theorem 2.3 and Theorem 2.4, we nd a nice relation between the solution (y u )ofthenonsmooth optimal control problem (.2) and the solution (Y U ) of the nite dierence smoothing approximation (.4) as follows: kj h (Y U ) ; J h (Py Pu )ko(h )+O(): 4
3 Numerical Examples Convergence analysis and error estimates in Section 2 suggest that the discretized smoothing constrained optimal control problem (.4) is a good approximation of the nonsmooth optimal control problem (.2). In this section, we propose a smoothing SQP (sequential quadratic programming ) method for solving (.4) and report numerical results. Examples are generated by adding the nonsmooth term max( y) to examples in [2]. Several tests for dierentvalues of were performed. The tests were carried out on a IBM workstation using Matlab. Smoothing SQP method(ssqp) Choose parameters >, > and a feasible point (Y U ) of (.3). For k we solve the quadratic program minimize subject to 2 (Y ; Z d) T H(Y ; Z d )+ 2 (U ; U d) T M(U ; U d ) AY + (Y k )+ (Y k )(Y ; Y k )=NU + c U b and let the optimal solution be (Y k+ U k+ ): We stop the iteration when jj h (Y k+ U k+ ) ; J h (Y k U k )j: The SSQP method is a standard SQP method for solving the smoothing optimization problem (.4). Convergence analysis can be found in []. Furthermore, the quadratic program at each step can be solved by an optimization toolbox, for example, quadprog in MATLAB. In the numerical test, we chose = ( ) ( ), n = m = 2 = ;6, = ;8, g =, and (Y U ) = ( ::: ) T. In Tables -3, k is the number of iterations, L(Y k U k )=k min(;((a+de(y k )) ; N) T H(Y k ;Z d );M(U k ;U d ) b;u k )k jj k h ; J k; h j = jj h (Y k U k ) ; J h (Y k; U k; )j and r = kay k + D max( Y k ) ; NU k ; ck : Example 3. Let q(x) and z d (x) = 6 exp(2x ) sin(2x ) sin(2x 2 ): 5
Table : Example 3.(a) u d, = ;2 k L(Y k U k ) J h (Y k U k ) jj k h ; J k; h j r.2 4 8.4e-9.49.e-2 8.9e-3.8 4 2.5e-7.49 3.e- 3.6e-2.6 4 2.e-7.49 2.6e- 7.e-2 3.2 4 6.e-8.49 5.3e-2.4e- 6.4 4.6e-7.49 2.7e- 2.6e- 2.8 4 2.9e-7.49 2.8e- 3.2e- Table 2: Example 3.(b) u d, = ;6 k L(Y k U k ) J h (Y k U k ) jj k h ; J k; h j r.2 4 8.e-9.32 8.7e-3 6.6e-4.8 4.e-6.32 7.6e- 2.7e-3.6 4 7.e-.33 5.4e-5 5.3e-3 3.2 4 6.e-7.32 8.6e-.e-2 6.4 4 7.8e-9.32 5.4e-5 2.e-2 2.8 4.e-7.32 7.6e-2 4.e-2 Table 3: Example 3.2 = ;6 k L(Y k U k ) J h (Y k U k ) jj k h ; J k; h j r.2 4.5e-5.584 2.4e- 3.4e-.8 4.5e-4.584 5.6e-6 5.8e-2.6 4 3.e-4.584 8.3e-6 9.9e-2 3.2 4 6.3e-4.584 2.e-6.5e- 6.4 4.3e-3.584.2e-7.3e- 2.8 5 3.e-.584 2.e-6.2e- 6
Example 3.2 Let q(x), u d, =: ;6 and z d (x) = 8 < : 2x x 2 (x ; 2 )2 ( ; x 2 ) if <x =2 2x 2 (x ; )(x ; 2 )2 ( ; x 2 ) if =2 <x Acknowledgements. The author is very grateful to Prof. Tetsuro Yamamoto of Waseda University for his helpful comments on the nite dierence method. References [] A.K. Aziz, A.B. Stephens and Manil Suri, Numerical methods for reactiondiusion problems with non-dierentiable kinetics, Numer. Math., 5 (988) -. [2] M. Bergounioux, K. Ito, K. Kunisch, Primal-dual strategy for constrained optimal control problems, SIAM J. Control Optim., 37 (999) 76-94. [3] A. Borzi, K. Kunisch and D.Y. Kwak, Accuracy and convergence properties of the nite dierence multigrid solution of an optimal control optimality system, SIAM J. Control Optim. 4(23) 477-497. [4] E. Casas and M. Mateos, Second order optimality conditions for semilinear elliptic control problems with nitely many state constraints, SIAM J. Control Optim., 4 (22) 43-454. [5] X. Chen, First order conditions for discretized nonsmooth constrained optimal control problems, SIAM J. Control Optim., 42(24) 24-25. [6] X. Chen, N. Matsunaga and T. Yamamoto, Smoothing Newton methods for nonsmooth Dirichlet problems, Reformulation - Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, M. Fukushima and L. Qi, eds., (Kluwer Academic Publisher, Dordrecht, The Netherlands, 999) 65-79. [7] X. Chen, Z. Nashed and L. Qi, Smoothing methods and semismooth methods for nondierentiable operator equations, SIAM J. Numer. Anal., 38 (2) 2-26. [8] X. Chen, L. Qi and D. Sun, Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities, Math. Comp., 67 (998), 59-54. 7
[9] F.H.Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 983. [] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Academic Press, Inc., 975. [] R. Fletcher, Practical Methods of Optimization, Second Edition, John Wiley & Sons Ltd, 987. [2] C. Kanzow, Some noninterior continuation methods for linear complementarity problems, SIAM J. Matrix Anal. Appl. 7(996) 85-868. [3] H. Niederreiter: Random Number Generation and Quasi-Monte Carlo Methods, SIAM, Philadelphia, 992. [4] M. Hintermuller, K. Ito and K. Kunisch, The primal-dual active set strategy as a semismooth Newton method, SIAM J. Optim., 3 (23), 865-888. [5] F. Kikuchi, K. Nakazato and T. Ushijima, Finite element approximation of a nonlinear eigenvalue problem related to MHD equilibria, Japan J.Appl. Math., (984) 369-43. [6] J.M. Ortega and W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 97. [7] L.Qi and J.Sun, A nonsmooth version of Newton's method, Math. Programming, 58(993) 353-367. [8] J. Rappaz, Approximation of a nondierentiable nonlinear problem related to MHD equilibria, Numer. Math., 45 (984) 7-33. Appendix: Proof of Example 2. The solution function Y () can be given explicitly as Y (U) = 8 >< >: @ @ 5=3 =3 A U if U A U if U < :. Since ~Y = Y ( ~U) =( ) and E(Y ( ~U)) is a zero matrix, we have (A ; N) T H( ~Y ; Z d )+M( ~U ; U d )+t = 3 (5 ) @ 3 A ; +t = 8
with t =, and min(t b ; ~U) =: Hence ( ~U ~U) is a KKT point of(.3). However, J h ( ~Y ~U) = 2 ( 3) @ 3 A + 2 =5 and J h ( Y U)= 2 ( 2 3) @ 2 3 A + 2 ( 2 ; )2 = 39 8 <J h( ~ Y ~ U) where U ==2 and Y =(=2 ) T. Hence ( Y ~ U)isnota ~ solution of (.3). 2. For U, J h (Y (U) U)= 2 (U 2 +)+ 2 U 2 : For U <, J h (Y U) = 2 (25 9 U 2 +( U 3 ; )2 )+ 2 U 2 : Hence (Y U )=isthe solution of (.3). However (A ; N) T (Y ; Z d )+t = 3 (5 ) @ ; A + t = implies t ==3 and min(t b ; U ) = min(=3 ) = =3 that is, (Y U ) is not a KKT point. 9