J. V. Burke and J. J. Mor'e. On the identification of active constraints. SIAM Journal on Numerical Analysis, 25(5):1197--1211, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....3 2, 3 0.0009 3 2, 3, 4, 11 0.8554 27 6 We employed (1, 1, 1) as the initial point for all methods. We set # 0 = 0.2 in Algorithms HIA and HIS and, # = 0.2 in Algorithm HIS. 5 Conclusions Active set identification plays an important role in optimization theory [1, 2, 5, 6, 17, 26]. Accurate identification of active constraints is important from both theoretical and practical points of view. In this paper, we have proposed some new approaches for solving MPCCs, which can be regarded as modifications of the hybrid approach suggested in [17] Theoretical analysis indicates ....

J.V. Burke and J.J. Mor e, On the identification of active constraints, SIAM Journal on Numerical Analysis, 25 (1988), 1197--1211. 19

....will require a sufficiently rich set of directions in the pattern so that at any face of the feasible region, one will have directions both normal and tangent to the constraints. This we will pursue elsewhere. One issue we have not discussed is that of identifying active constraints, as in [4, 5]. One would wish to show that if the sequence fx k g converges to a nondegenerate 17 stationary point x , then in a finite number of iterations the iterates x k land on the constraints active at x and remain thereafter on those constraints. There are three difficulties in proving such a result ....

....rules out, for instance, certain of the composite designs suggested by G.E.P. Box and K.B. Wilson [2] The most serious obstacle, which remains to be overcome, is showing that ultimately the iterates will land on the active constraints and remain there. For algorithms such as those considered in [4, 5], this is not a problem because the explicit use of the gradient impels the iterates to do this in the neighborhood of a nondegenerate stationary point. However, pattern search methods do not have this information. On the other hand, the kinship of pattern search methods and gradient projection ....

J. V. Burke and J. J. Mor' e, On the identification of active constraints, SIAM Journal on Numerical Analysis, 25 (1988), pp. 1197--1211.

....of Theorem 3.1. Since the proof of Theorem 3.4 is similar to that of Theorem 3.1, it is omitted here, see [10] 8 3.2 A hybrid algorithm for MPCCs We first introduce an active set identification technique for MPCCs. Active set identification plays an important role in optimization theory [1, 2, 6, 7, 31]. Accurate identification of active constraints is important from both theoretical and practical points of view. For problem (P) by means of active set identification, the combinatorial constraints 0, G(z) H(z) 0 (3.4) may be replaced by some equality or inequality constraints that are ....

J.V. Burke and J.J. Mor e, On the identification of active constraints, SIAM Journal on Numerical Analysis, 25 (1988), 1197--1211. 25

....compact representations of limited memory matrices described by Byrd, Nocedal and Schnabel [6] the computational cost of one iteration of the algorithm can be kept to be of order n. We used the gradient projection approach [16] 17] 3] to determine the active set, because recent studies [7] [5] indicate that it possess good theoretical properties, and because it also appears to be efficient on many large problems [8] 20] However some of the main components of our algorithm could be useful in other frameworks, as long as limited memory matrices are used to approximate the Hessian of ....

J. V. Burke, and J. J. Mor'e, "On the identification of active constraints", SIAM J. Numer. Anal. Vol. 25, No 5 (1988): 1197-1211.

....of the compact representations of limited memory matrices described by Byrd, Nocedal and Schnabel [6] the computational cost of one iteration of the algorithm can be kepttobeofordern. We used the gradient projection approach[16] 17] 3] to determine the active set, because recent studies [7] [5] indicate that it possess good theoretical properties, and because it also appears to be efficientonmany large problems [8] 20] However some of the main components of our algorithm could be useful in other frameworks, as long as limited memory matrices are used to approximate the Hessian of the ....

J. V. Burke,andJ.J.Mor'e, "On the identification of active constraints", SIAM J. Numer. Anal. Vol. 25, No 5( 1197-1211.

....(3.23) is needed to guarantee that every step taken is non negligible. Mor e shows that it is always possible to pick such a value of t C(k;j) using a backtracking linesearch, starting on or near to the trust region boundary. Similar methods have been proposed by Calamai and Mor e (1987) Burke and Mor e (1988), Toint (1988) and Burke et al. 1990) 2. Secondly, we pick p (k;j) so that x (k;j) p (k;j) lies within (2.6) kp (k;j) k t fi 2 Delta (k;j) and m (k;j) x (k;j) Gamma m (k;j) x (k;j) p (k;j) fi 3 [m (k;j) x (k;j) Gamma m (k;j) x (k;j) p C(k;j) ....

J. V. Burke and J. J. Mor'e. On the identification of active constraints. SIAM Journal on Numerical Analysis, 25(5):1197--1211, 1988.

....for instance, if the objective is believed to be untrustworthy in its accuracy, or if f(x) is not available as a numerical value and only comparison of objective values is possible. 7.2. Identifying active constraints. Another practical issue is that of identifying active constraints, as in [2, 3, 4]. A desirable feature of an algorithm for linearly constrained minimization is the identification of active constraints in a finite number of iterations, that is, if the sequence x k converges to a stationary point x # , then in a finite number of iterations the iterates x k land on the ....

....on those constraints. As discussed in [8] for the case of bound constraints, there are several impediments to proving such results for pattern search algorithms and showing that ultimately the iterates will land on the active constraints and remain there. For algorithms such as those considered in [2, 3, 4], this is not a problem because the explicit use of the gradient impels the iterates to do so in the neighborhood of a constrained stationary point. However, pattern search methods do not have this information, and at this point it is not clear how to avoid the possibility that these algorithms ....

J. V. Burke and J. J. Mor e, On the identification of active constraints, SIAM Journal on Numerical Analysis, 25 (1988), pp. 1197--1211.

....Indeed, as the condition P (x; r x 9(x; k) s (k) 0 (3:14) is required at optimality for such a problem, 3.5) can be viewed as an inexact stopping rule for iterative algorithms for solving it. We merely mention here that the projected gradient methods of Calamai and Mor e (1987) Burke and Mor e (1988), Conn et al. 1988a) Conn et al. 1988b) and Burke et al. 1990) and the interior point method of Nash and Sofer (1991) are all appropriate, but that methods which take special account of the nature of (1.4) may yet be prefered. 3.4 Further discussion We should also comment on the rather ....

J. V. Burke and J. J. Mor'e. On the identification of active constraints. SIAM Journal on Numerical Analysis, 25:1197--1211, 1988.

....of nonlinear, nonconvex, optimization problems. They have been studied and applied to unconstrained problems (see [6] 16] 24] 28] 29] 30] 31] 33] 34] 38] and to problems involving various classes of constraints: simple bounds ( 5] 10] 11] 27] 32] convex constraints ([1], 2] 9] 41] and also nonconvex ones ( 4] 7] 15] 35] 42] This long lasting interest is probably justified by the attractive combination of a solid convergence theory, a noted algorithmic robustness, the existence of numerically efficient implementations and an intuitively appealing ....

J. V. Burke and J. J. Mor'e. On the identification of active constraints. SIAM Journal on Numerical Analysis, 25:1197--1211, 1988.

....(3.23) is needed to guarantee that every step taken is non negligible. Mor e shows that it is always possible to pick such a value of t C(k;j) using a backtracking linesearch, starting on or near to the trust region boundary. Similar methods have been proposed by Calamai and Mor e (1987) Burke and Mor e (1988), Toint (1988) and Burke et al. 1990) 2. Secondly, we pick p (k;j) so that x (k;j) p (k;j) lies within (2.6) kp (k;j) k t fi 2 1 (k;j) and m (k;j) x (k;j) 0m (k;j) x (k;j) p (k;j) fi 3 [m (k;j) x (k;j) 0m (k;j) x (k;j) p C(k;j) 0 (3:26) for ....

J. V. Burke and J. J. Mor'e. On the identification of active constraints. SIAM Journal on Numerical Analysis, 25:1197--1211, 1988.

....to the current estimated active set many constraints at each iteration and yet find an active set in a finite number of steps. This work has motivated a lot of further studies on projection techniques, both for the general linearly constrained case and for the box constrained case (see , e.g. [6, 7, 8, 15] and [33, 34] Trust region type algorithms for unconstrained optimization have been successfully extended to handle the presence of bounds on the variables. The global convergence theory thus developed is very robust [10, 22] and, under appropriate assumptions, it is possible to establish a ....

J. Burke and J. Mor e: On the identification of active constraints. SIAM Journal on Numerical Analysis 25, 1988, pp. 1197--1211.

....will require a sufficiently rich set of directions in the pattern so that at any face of the feasible region, one will have directions both normal and tangent to the constraints. This we will pursue elsewhere. One issue we have not discussed is that of identifying active constraints, as in [4, 5]. One would wish to show that if the sequence fx k g converges to a nondegenerate stationary point x , then in a finite number of iterations the iterates x k land on the constraints active at x and remain thereafter on those constraints. There are three difficulties in proving such a result for ....

....rules out, for instance, certain of the composite designs suggested by G.E.P. Box and K.B. Wilson [2] The most serious obstacle, which remains to be overcome, is showing that ultimately the iterates will land on the active constraints and remain there. For algorithms such as those considered in [4, 5], this is not a problem because the explicit use of the gradient impels the iterates to do this in the neighborhood of a nondegenerate stationary point. However, pattern search methods do not have this information. On the other hand, the kinship of pattern search methods and gradient projection ....

J. V. Burke and J. J. Mor' e, On the identification of active constraints, SIAM Journal on Numerical Analysis, 25 (1988), pp. 1197--1211.

....l 2 L: As in [10] we can also introduce the concept of nondegeneracy using the relative interior operator. Definition 2 A stationary point (x ; y ) of problem (1. 1) is nondegenerate if Gammarf (x ; y ) 2 ri(T 0 l (x ; y ) l 2 L: If (x ; y ) is nondegenerate then (see [6]) the multiplier inequalities in the expression defining T 0 l (x ; y ) l 2 L, can be made strict (i.e. strict complementarity holds) The following theorem extends the second order sufficient conditions of smooth linearly constrained problems to problem (1.1) Theorem 3 Let f : Omega ....

J. Burke and J. Mor'e, On the identification of active constraints, SIAM Journal on Numerical Analysis 25 (1988) 1197--1211.

....and considered its role in parametric nonlinear programming. In recent years, the strict complementarity property was given a renewed emphasis in the analysis of many iterative algorithms for solving linear and nonlinear programs and complementarity problems. Dunn [10] and BurkeMor e [6] used a geometric definition of a strictly complementary solution to a nonlinear program and showed how such a solution was essential for the successful identification of active constraints in a broad class of gradient based methods for solving constrained optimization problems. Guler and Ye [19] ....

....is said to be strictly complementary, or, nondegenerate, if x F (x) 0. For an optimization problem of the form minimize f(x) subject to x 2 C; 1) where f : R n R is continuous and C R n is convex, different forms of nondegeneracy abound in the literature. Dunn [10] Burke and Mor e [6] use the relative interior condition: Gamma rf(x) 2 ri NC (x) 2) to define an optimal solution x of (1) as being nondegenerate. Here ri S denotes the relative interior of the convex set S and NC (x) denotes the normal cone to the convex set C at the point x 2 R n which is defined by NC ....

[Article contains additional citation context not shown here]

J.V. Burke and J.J. Mor'e, "On the identification of active constraints", SIAM Journal on Numerical Analysis 15 (1988), pp.1197-1211.

....for instance, if the objective is believed to be untrustworthy in its accuracy, or if f(x) is not available as a numerical value and only comparison of objective values is possible. 7.2. Identifying active constraints. Another practical issue is that of identifying active constraints, as in [2, 3, 4]. A desirable feature of an algorithm for linearly constrained minimization is the identification of active constraints in a finite number of iterations, that is, if the sequence x k converges to a stationary point x # , then in a finite number of iterations the iterates x k land on the ....

....on those constraints. As discussed in [8] for the case of bound constraints, there are several impediments to proving such results for pattern search algorithms and showing that ultimately the iterates will land on the active constraints and remain there. For algorithms such as those considered in [2, 3, 4], this is not a problem because the explicit use of the gradient impels the iterates to do so in the neighborhood of a constrained stationary point. However, pattern search methods do not have this information, and at this point it is not clear how to avoid the possibility that these algorithms ....

J. V. Burke and J. J. Mor e, On the identification of active constraints, SIAM Journal on Numerical Analysis, 25 (1988), pp. 1197--1211.

....to the current estimated active set many constraints at each iteration and yet find an active set in a finite number of steps. This work has motivated a lot of further studies on projection techniques, both for the general linearly constrained case and for the box constrained case (see , e.g. [3], 4] 5] and [10] and it is safe to say that algorithms in this class are among the most efficient ones for the solution of large scale, convex, quadratic problems, 29] 30] More recently trust region type algorithms for unconstrained optimization have been succesfully extended to handle ....

J. Burke and J. Mor' e: On the identification of active constraints. SIAM Journal on Numerical Analysis 25, 1988, pp. 1197--1211.

....of its affine hull, that is the affine subspace with lowest dimensionality that contains Y (see [26, p. 44] for further details) Using this concept, we now express our condition as follows. AS.10 For every limit point x 2 L, one has that Gamma rf(x ) 2 ri[N(x ) 5:15) As discussed in [3], this last condition can be viewed as the generalization of the strict complementarity assumption used in [9] and [18] It was also used in [2] and in [3] in a similar context. As in [2] and [3] we note that (AS.9) AS.10) and (5.14) together imply the existence of a unique set of strictly ....

....we now express our condition as follows. AS.10 For every limit point x 2 L, one has that Gamma rf(x ) 2 ri[N(x ) 5:15) As discussed in [3] this last condition can be viewed as the generalization of the strict complementarity assumption used in [9] and [18] It was also used in [2] and in [3] in a similar context. As in [2] and [3] we note that (AS.9) AS.10) and (5.14) together imply the existence of a unique set of strictly positive multipliers. Thus, for every x 2 L, rf(x ) X i2A(x ) i rh i (x ) 5:16) for some uniquely defined i 0. We finally assume that the ....

[Article contains additional citation context not shown here]

J.V. Burke and J.J. Mor'e, "On the identification of active constraints", SIAM Journal on Numerical Analysis, vol. 25, pp. 1197--1211, 1988.

....imposes few additional requirements and as we have seen in 6, the classical pattern search methods for unconstrained minimization or straightforward variants thereof carry over to the bound constrained case. One issue we have not discussed is that of identifying active constraints, as in [4, 5]. One would wish to show that if the sequence x k converges to a nondegenerate stationary point x # , then in a finite number of iterations the iterates x k land on the constraints active at x # and remain thereafter on those constraints. There are three di#culties in proving such a ....

....to a point on the boundary. This rules out, for instance, certain of the composite designs suggested by G.E.P. Box and K.B. Wilson [2] The most serious obstacle is showing that ultimately the iterates will land on the active constraints and remain there. For algorithms such as those considered in [4, 5], this is not a problem because the explicit use of the gradient impels the iterates to do this in the neighborhood of a nondegenerate stationary point. However, pattern search methods do not have this information. On the other hand, the kinship of pattern search methods and gradient projection ....

J. V. Burke and J. J. Mor e, On the identification of active constraints, SIAM Journal on Numerical Analysis, 25 (1988), pp. 1197--1211.

....sequence generated by Algorithm 2.1, say lim k 1 x k = x : If x is a nondegenerate stationary point then there exists k 2 IN such that x k 2 F I , i.e. I(x k ) I(x ) I for k k. Proof. Follows directly from Theorem 2.2 and Corollary 3. 6 in Burke and Mor e [4]. 2 Finally, the next theorem generalizes Corollary 2.2. Theorem 2.3 If all the limit points of a sequence fx k g generated by Algorithm 2.1 are non degenerate, then there exists k 2 IN and a face F I such that x k 2 F I for all k k. Proof. Suppose, by contradiction, that the thesis is ....

J. V. Burke and J. J. Mor'e, "On the identification of active constraints ", SIAM Journal on Numerical Analysis 25, (1988) 1197-1211.

....compact representations of limited memory matrices described by Byrd, Nocedal and Schnabel [6] the computational cost of one iteration of the algorithm can be kept to be of order n. We used the gradient projection approach [16] 17] 3] to determine the active set, because recent studies [7] [5] indicate that it possess good theoretical properties, and because it also appears to be efficient on many large problems [8] 20] However some of the main components of our algorithm could be useful in other frameworks, as long as limited memory matrices are used to approximate the Hessian of ....

J. V. Burke, and J. J. Mor'e, "On the identification of active constraints", SIAM J. Numer. Anal. Vol. 25, No 5 (1988): 1197-1211.

....z k 1 (ff) B (z k Gamma ffrf(z k ) satisfies f(z k ) Gamma f(z k 1 (ff) oerf(z k ) z k Gamma z k 1 (ff) 8) The scalars ff max 0, and oe 2 (0; 0:5) are fixed. Conditions under which this method identifies the active set after a finite number of iterations are given in [10, 11]. Since there is potential for a large change in the current active set as a result of a single step CRASH TECHNIQUES 5 in this method, these methods are used in large scale codes to determine an active set quickly, after which a Newton method is applied on the identified face [13] Note that ....

J. V. Burke and J. J. Mor' e, On the identification of active constraints, SIAM Journal on Numerical Analysis, 25 (1988), pp. 1197--1211.

.... relate nondegeneracy to the geometry of E [ #f(x # ) by proving that x # is nondegenerate if and only if x # lies in the relative interior of the face E [ #f(x # ) These two definitions rely only on the geometry of ## If# is expressed in terms of constraints, then nondegeneracy can be shown [5] to be equivalent to the existence of a set of nonzero Lagrange multipliers. Thus, a stationary point x # is nondegenerate as defined by Dunn [17] if and only if strict complementarity holds at x # . We can also show [6, Theorem 5.3] that x # E [ #f(x # ) ## A(x # ) # A(x) 3.2) whenever ....

J. V. Burke and J. J. Mor e, On the identification of active constraints, SIAM J. Numer. Anal., 25 (1988), pp. 1197--1211.

.... geometry of E [ Gammarf (x ) by proving that x is nondegenerate if and only if x lies in the relative interior of the face E [ Gammarf (x ) These two definitions rely only on the geometry of Omega Gamma If Omega is expressed in terms of constraints, then nondegeneracy can be shown [5] to be equivalent to the existence of a set of nonzero Lagrange multipliers. Thus, a stationary point x is nondegenerate as defined by Dunn [17] if and only if strict complementarity holds at x . We can also show [6, Theorem 5.3] that x 2 E [ Gammarf (x ) A(x ) ae A(x) 3.2) ....

J. V. Burke and J. J. Mor' e, On the identification of active constraints, SIAM J. Numer. Anal., 25 (1988), pp. 1197--1211.

....degenerate case is that there may be no face F (x ) such that x k 2 ri fF (x )g for all k sufficiently large. However, we show that there is a face F (x ) such that x k 2 F (x ) for all k sufficiently large. The approach in this paper is reminiscent of the approach of Burke and Mor e [4]. In particular, the approach depends on the concept of an exposed face, and on the facial geometry of a convex set Omega Gamma These results are developed in Sections 2 and 3. The definition of an exposed face and several important properties of exposed faces are presented in Section 2, while ....

....multipliers, and that x 2 E[ Gammarf (x ) B s ae A(x) x 2 Omega : This leads, in particular, to a version of the results of Section 4 in terms of B s . Section 4 also contains a discussion of the connection of these result to the identification results of Burke and Mor e [4], and to the convergence results for the class of trust region methods for bound constrained optimization problems proposed by Conn, Gould, and Toint [7, 8] The connection between these results and the class of trust region methods for general linearly constrained methods analyzed by Mor e [18] ....

[Article contains additional citation context not shown here]

J. V. Burke and J. J. Mor' e, On the identification of active constraints, SIAM J. Numer. Anal., 25 (1988), pp. 1197--1211.

No context found.

J. V. Burke and J. J. Mor'e. On the identification of active constraints. SIAM Journal on Numerical Analysis, 25(5):1197--1211, 1988.

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC