Inexact Solution of NLP Subproblems in MINLP

Transcription

1 Ineact Solution of NLP Subproblems in MINLP M. Li L. N. Vicente April 4, 2011 Abstract In the contet of conve mied-integer nonlinear programming (MINLP, we investigate how the outer approimation method and the generalized Benders decomposition method are affected when the respective NLP subproblems are solved ineactly. We show that the cuts in the corresponding master problems can be changed to incorporate the ineact residuals, still rendering equivalence and finiteness in the limit case. Some numerical results will be presented to illustrate the behavior of the methods under NLP subproblem ineactness. Keywords: Mied integer nonlinear programming, outer approimation, generalized Benders decomposition, ineactness, conveity. 1 Introduction Recently, mied integer nonlinear programming (MINLP has become again a very active research area [1, 2, 4, 5, 6, 7, 14, 16]. Benders [3] developed in the 60 s a technique for solving linear mied-integer problems, later called Benders decomposition. Geoffrion [11] etended it to MINLP in 1972, in what become known as generalized Benders decomposition (GBD. Much later, in 1986, Duran and Grossmann [8] derived a new outer approimation (OA method to solve a particular class of MINLP problems, which become widely used in practice. Although the authors shown finiteness of the OA algorithm, their theory was restricted to problems where the discrete variables appear linearly and the functions involving the continuous variables are conve. Both OA and GBD are iterative schemes requiring at each iteration the solution of a (feasible or infeasible NLP subproblem and one mied-integer linear programming (MILP master problem. For these particular MINLP problems, Quesada and Grossmann [15] then proved that the cuts in the master problem of OA imply the cuts in the master problem of GBD, showing that the GBD algorithm provides weaker lower bounds and generally requires more maor iterations to converge. Fletcher and Leyffer [9] generalized the OA method of Duran and Grossmann [8] into a wider class of problems where nonlinearities in the discrete variables are allowed as long as the corresponding functions are conve in these variables. They also introduced a new and Department of Mathematics, University of Coimbra, Coimbra, Portugal (limin@mat.uc.pt. Support for this author was provided by FCT under the scholarship SFRH/BD/33369/2008. CMUC, Department of Mathematics, University of Coimbra, Coimbra, Portugal (lnv@mat.uc.pt. Support for this author was provided by FCT under the grant PTDC/MAT/098214/

2 simpler proof of finiteness of the OA algorithm. The relationship between OA and GBD was then addressed, again, by Grossmann [12] in this wider contet of MINLP problems, showing once more that the lower bound predicted by the relaed master problem of OA is greater than or equal to the one predicted by the relaed master problem of GBD (see also Flippo and Rinnooy Kan [10] for the relationship between the two techniques. Recently, Bonami et al. [4] suggested a different OA algorithm using linearizations of both the obective function and the constraints, independently of being taken at the feasible or infeasible NLP subproblem, to build the MILP master problem. This technique is, in fact, different from the traditional OA (see [9], where the cuts in the master MILP problems do not involve linearizations of the obective function in the infeasible case. Westerlund and Pettersson [18] generalized the cutting plane method [13] from conve NLP to conve MINLP, in what is known as the etended cutting plane (ECP method (see also [19, 20]. While OA and GBD alternate between the solution of MILP and NLP subproblems, the ECP relies only on the solution of MILP problems. In the above mentioned OA and GBD approaches, the NLP subproblems are solved eactly, at least for the derivation of the theoretical properties, such as equivalence between original and master problem and finite termination of the corresponding algorithms. In this paper we investigate the effect of NLP subproblem ineactness in these two techniques. We show how the cuts in the master problems can be changed to incorporate the ineact residuals of the first order necessary conditions of the NLP subproblems, in a way that still renders the equivalence and finiteness properties, as long as the size of these residuals allow inferring the cuts from conveity properties. In this paper, we will adopt the MINLP formulation min f(, y P s.t. g(, y 0, X Z n d, y Y, where X is a bounded polyhedral subset of R n d and Y a polyhedral subset of R nc. The functions f : X Y R and g : X Y R m are assumed continuously differentiable. We will also assume that P is conve, i.e., that f and g are conve functions. Let be any element of X Z n d. Consider, then, the (conve subproblem min f(, y NLP( s.t. g(, y 0, y Y, and suppose it is feasible. In this case, y will represent an approimated optimal solution of NLP(. For an k in X Z n d for which NLP( k is infeasible, y k is instead defined as an approimated optimal solution of the following feasibility (conve subproblem min u NLPF( k s.t. g i ( k, y u, i = 1,..., m, y Y, u R, where one minimizes the l -norm of the measure of infeasibility of subproblem NLP( k. 2

3 For a matter of simplification, and without loss of generality, we suppose that the constraints y Y are part of the constraints g(, y 0 and g i ( k, y u, i = 1,..., m, in the subproblems NLP( and NLPF( k, respectively. In addition, let us assume that the approimated optimal solutions of the NLP subproblems satisfy an ineact form of the corresponding first order necessary Karush-Kuhn-Tucker (KKT conditions. More particularly, in the case of NLP(, we assume the eistence of λ R m +, r R nc, and s R m, such that y f(, y + λ i yg i (, y = r, (1 λ i g i(, y = s i, i = 1,..., m. (2 When NLP( k is infeasible, we assume, for NLPF( k, the eistence of µ k R m +, z k R m, w k R, and v k R nc, such that µ k i y g i ( k, y k = v k, (3 1 µ k i = w k, (4 µ k i (g i ( k, y k u k = z k i, i = 1,..., m. (5 Points satisfying the ineact KKT conditions can be seen as solutions of appropriate perturbed subproblems (see the Appendi. The following two sets will then be used to inde these two sets of approimated optimal solutions: and T = { : X Z n d, NLP( is feasible and y appr. solves NLP( } S = {k : k X Z n d, NLP( k is infeasible and y k appr. solves NLPF( k }. The ineact versions of OA and GBD studied in this paper will attempt to find the best pair among all of the form (, y corresponding to T. Implicitly, we are thus redefining a perturbed version of problem P and will denote it by P: P min T f(, y. (6 This problem is well defined if T which in turn can be assumed when the original MINLP problem P has a finite optimal value. We use the superscripts l, p, and q to denote the iteration count, superscript to inde the feasible NLP subproblems defined above, and k to indicate infeasible subproblems. The following notation is adopted to distinguish between function values and functions. f l = f( l, y l denotes the value of f evaluated at the point ( l, y l, similarly, f l = f( l, y l is the value of the gradient of f at the point ( l, y l, f l = f( l, y l is the value of the gradient of f with respect to at the point ( l, y l, and y f l = y f( l, y l is the value of the gradient of f with respect to y at the point ( l, y l. Moreover, the same conventions apply for all other functions. 3

4 We organize the paper in the following way. In Section 2, we etend OA for the ineact solution of the NLP subproblems, rederiving the corresponding background theory and main algorithm. In Section 3 we proceed similarly for GBD, also discussing the relationship between the ineact forms of OA and GBD. Section 4 describes a set of preliminary numerical eperiments, reported to better understand some of the theoretical features encountered in our study of ineactness in MINLP. 2 Ineact outer approimation 2.1 Equivalence between perturbed and master problems for OA OA relies on the fact that the original problem P is equivalent to a MILP (master problem formed by minimizing the least of the linearized forms of f for indices in T subect to the linearized forms of g for indices in S and T. When the NLP subproblems are solved ineactly, one has to consider perturbed forms of such cuts or linearized forms in order to keep an equivalence, this time to the perturbed problem P. In turn, these ineact cuts lead to a different, perturbed MILP (master problem given by P OA min s.t. where, for i = 1,..., m, and α ( f(, y y f(, y r ( + f(, y α, ( y y g(, y y y + g(, y t, T, ( g i ( k, y k ( k y g i ( k, y k 1 v k 1 w y y k k i = 1,..., m, k S, X Z n d, y Y, α R, a k i = t i = s i λ i { mz k i w k u k, if λ i > 0, 0, if λ i = 0, mµ k i, if µ k i > 0, 0, if µ k i = 0. + g i ( k, y k a k i, Note that when r, s, v, w, and z are zero, we obtain the well-known master problem in OA. Also, optionally, one could have added the cuts ( f( k, y k k y y k + f( k, y k α, k S, (9 corresponding to linearizations of the obective function in the infeasible cases, as suggested in [4]. (7 (8 4

5 From the conveity and continuous differentiability of f and g, we know that, for any ( l, y l R n d R nc, ( f(, y f( l, y l + f( l, y l l y y l, (10 g(, y g( l, y l + g( l, y l ( l y y l. (11 In addition, when y is a feasible point of NLP(, we obtain from (11 and g(, y 0 that ( 0 g( l, y l + g( l, y l l y y l. (12 The ineact OA method reported in this section as well as the GBD method of the net section require the residuals of the ineact KKT conditions to satisfy the bounds given in the net two assumptions, in order to validate the equivalence between perturbed and master problems, and to ensure finiteness of the respective algorithms. Essentially, these bounds will ensure that the above conveity properties will still imply the ineact cuts at the remaining points. We first give the bounds on the residuals r and s for the feasible case. Assumption 2.1 Given any l, T, with l, assume that ( τ[( f l l r l y y l + f l f ] y y l, for some τ [0, 1], and s l i σ i λ l i[( g l i ( l y y l + g l i], for some σ i [0, 1], i = 1,..., m. Now, we state the bounds for the residuals v, w, and z in the infeasible case. Assumption 2.2 Given any T and any k S, and for all i {1,..., m}, if µ k i 0, assume that 1 1 w k vk y y k + 1 ( µ k zi k + uk i mµ k w k β i [( gi k k y y k + gi k ], i for some β i [0, 1], otherwise, assume that 1 1 w k vk y y k η i [( g k i ( k y y k + g k i ], for some η i [0, 1]. We are now in a position to state the equivalence between the original, perturbed MINLP problem and the MILP master problem P OA. 5

6 Theorem 2.1 Let P be a conve MINLP problem and P be its perturbed problem as defined in the Introduction. Assume that P is feasible with a finite optimal value and that the residuals of the KKT conditions of the NLP subproblems satisfy Assumptions 2.1 and 2.2. Then P OA and P have the same optimal value. Proof. The proof follows closely the lines of the proof of [4, Theorem 1]. Since problem P has a finite optimal value it follows that, for every X Z n d, either problem NLP( is feasible with a finite optimal value or it is infeasible, that the sets T and S are well defined, and that the set T is nonempty. Now, given any l X Z n d with l T S, let P OA denote the l problem in α and y obtained from P OA when is fied to l. First we will prove that problem P OA is infeasible for every k S. k Part I. Establishing infeasibility of P OA for k S. k In this case, problem NLP( k is infeasible and y k is an approimated optimal solution of NLPF( k with corresponding ineact nonnegative Lagrange multipliers µ k. When we set = k, the corresponding constraints in P OA will result in ( y g i ( k, y k 1 1 w k vk (y y k + g i ( k, y k a k i, (13 for i = 1,..., m. Multiplying the inequalities in (13 by the nonnegative multipliers µ k 1,..., µk m, and summing them up, one obtains ( µ k i y g i ( k, y k v k (y y k (zi k µ k i g i ( k, y k w k u k. (14 By using (3, one can see that the left hand side of the inequality in (14 is equal to 0. On the other hand, by using equation (5, the right hand side of the inequality in (14 results in m (zk i µ k i g i( k, y k w k u k = (Σ m µk i + w k u k, which is equal to u k by (4. Since NLP( k is infeasible, u k must be strictly negative. We have thus proved that the inequality (14 has no solution y. This derivation implies that the minimum value of P OA should be found as the minimum value of P OA over all X Z n d with T. We prove in the net two separate subparts that, for every T, the optimal value ᾱ of P OA coincides with the approimated optimal value of NLP(. Part II. Establishing that P OA has the same obective value as the perturbed NLP( for T. We will show net that (y, f(, y is a feasible solution of P OA, and therefore that f(, y is an upper bound on the optimal value ᾱ of P OA. Part II A. Establishing that f(, y is an upper bound for the optimal value of P OA for T. In this case, it is easy to see that P OA contains all the constraints indeed by l T ( f( l, y l y f( l, y l r l ( l y y l g( l, y l ( l y y l + f( l, y l α, (15 + g( l, y l t l, (16 6

7 where, for i = 1,..., m, t l i = { s l i λ l i, if λ l i > 0, 0, if λ l i = 0, as well as all the constraints indeed by k S and i {1,..., m} ( g i ( k, y k y g i ( k, y k 1 1 w k v k ( k y y k + g i ( k, y k a k i, (17 where a k i is given as in (8. First take any l T and assume that y l is an approimated optimal solution of NLP( l with corresponding ineact nonnegative Lagrange multipliers λ l. If l =, it is easy to verify that (y, f(, y satisfies (15 and (16. Assume then that l. From Assumption 2.1, we know that, for some τ [0, 1], ( (r l (y y l r l y y l τ[( f l l y y l + f l f ]. Thus, [( f l ( l y y l + f l f ] (r l (y y l (1 τ[( f l ( l y y l + f l f ] 0, where the last inequality comes from 1 τ 0 and (10 with (, y = (, y. We then see that (15 is satisfied with α = f(, y and y = y. Now, from Assumption 2.1, one has for some σ i [0, 1], i = 1,..., m, λ l i[( g l i ( l y y l ( + gi] l s l i λ l i[( gi l l y y l 0, σ i λ l i[( g l i ( l y y l + g l i] (1 σ i λ l i[( g l i ( l y y l where the last inequality is ustified by (12 and σ i [0, 1]. Thus, λ l i[( g l i ( l y y l + g l i] + g l i] + g l i] s l i, i = 1,..., m. (18 If λ l i is equal to 0, so is tl i by its definition and we see that (y, f(, y satisfies the constraints (16 with y = y. If λ l i 0, then (18 can be written as: g i ( l, y l ( l y y l + g i ( l, y l sl i λ l i = t l i, 7

8 which also shows that the constraints (16 hold with y = y. Finally, we take any k S and assume that y k is an approimate optimal solution of NLP( k with corresponding ineact Lagrange multipliers µ k. For every i {1,..., m}, if µ k i 0, from Assumption 2.2, we have for some β i [0, 1], that i.e., 1 1 w k (vk (y y k 1 µ k zi k + i uk mµ k w k i β i [( g k i ( k y y k 1 ( 1 w k (vk (y y k a k i β i [( gi k k y y k + g k i ] + g k i ], by the definition of a k i. Thus, the constraints (17 are satisfied with y = y. When µ k i = 0, it results that a k i = 0 by its definition and, also by Assumption 2.2, we have that, for some η i [0, 1], ( g k i ( k y y k + gi k 1 ( 1 w k (vk (y y k (1 η i [( gi k k y y k 0. + g k i ] This also shows that the constraints (17 hold with y = y. We can therefore say that (y, f(, y is a feasible point of P OA, and thus ᾱ f(, y. Net, we will prove that f(, y is also a lower bound, i.e., ᾱ f(, y. Part II B. Establishing that f(, y is a lower bound for the optimal value of P OA for T. Recall that y is an approimated optimal solution of NLP( satisfying the ineact KKT conditions (1 and (2. By construction, any solution of P OA has to satisfy the ineact outerapproimation constraints: ( y f(, y r (y y + f(, y α, (19 y g(, y (y y + g(, y t. (20 Multiplying the inequalities (20 by the nonnegative multipliers λ 1,..., λ m and summing them together with (19, one obtains ( y f(, y r (y y + f(, y + The left hand side of the inequality (21 can be rewritten as: ( y f(, y + λ i ( yg i (, y (y y + g i (, y s i λ i yg i (, y r (y y + α. (21 (λ i g i(, y s i + f(, y. By using (1 and (2, this quantity is equal to f(, y, and it follows that inequality (21 is equivalent to f(, y α. 8

9 In conclusion, for any X Z n d with T, problems P OA and perturbed NLP( have the same optimal value. In other words, the MILP problem P OA has the same optimal value as the perturbed problem P given by (6. Since in the eact case, all the KKT residuals are zero, it results from Theorem 2.1 what is well known for OA: Corollary 2.1 Let P be a conve MINLP problem. Assume that P is feasible with a finite optimal value and the residuals of the KKT conditions of the NLP subproblems are zero. Then P OA and P have the same optimal value. In the paper [4], the feasibility NLP subproblem is stated as min m u i s.t. g( k, y u, P F k u 0, y Y, u R m. Note that one could easily rederive a result similar to [4, Theorem 1] replacing their P F by our k NLPF( k. In fact, the argument needed here is essentially Part I of the proof of Theorem 2.1 with v, w, and z set to zero. In their approach, the cuts (9 are included in the master problem, but one can also see that the proof of Theorem 2.1 remains true in this case (it would suffice to observe that (9 is satisfied trivially in the conve case when y = y and α = f(, y. 2.2 Ineact-OA algorithm One knows that the outer approimation algorithm terminates finitely in the conve case and when the optimal solutions of the NLP subproblems satisfy the first order KKT conditions (see [9]. In this section, we will etend the outer approimation algorithm to the ineact solution of the NLP subproblems by incorporating the corresponding residuals in the cuts of the master problems. As in the eact case, at each step of the ineact OA algorithm, one tries to solve a subproblem NLP( p, where p is chosen as a new discrete assignment. Two results can then occur: either NLP( p is feasible and an approimated optimal solution y p can be given, or this subproblem is found infeasible and another NLP subproblem, NLPF( p, is solved, yielding an approimated optimal solution y p. In the algorithm, the sets T and S defined in the Introduction will be replaced by: and T p = { : p, X Z n d, NLP( is feasible and y appr. solves NLP( } S p = {k : k p, k X Z n d, NLP( k is infeasible and y k appr. solves NLPF( k }. In order to prevent any, T p, from becoming the solution of the relaed master problem to be solved at the p iteration, one needs to add the constraint α < UBD p, 9

10 where UBD p = min p, T p f(, y. Then we define the following ineact relaed MILP master problem (P OA p min α s.t. α < UBD p, ( f(, y ( y f(, y r y y + f(, y α, ( g(, y y y + g(, y t, T p, ( g i ( k, y k ( k y g i ( k, y k 1 v k 1 w y y k + g i ( k, y k a k i, k i = 1,..., m, k S p, X Z n d, y Y, α R, where t and a k i were defined in (7 and (8, respectively. The presentation of the ineact OA algorithm (given net and the proof of its finiteness in Theorem 2.2 follows the lines in [9]. Algorithm 2.1 (Ineact Outer Approimation Initialization Let 0 be given. Set p = 0, T 1 =, S 1 =, and UBD = +. REPEAT 1. Ineactly solve the subproblem NLP( p, or the feasibility subproblem NLPF( p provided NLP( p is infeasible, and let y p be an approimated optimal solution. At the same time, obtain the corresponding ineact Lagrange multipliers λ p of NLP( p (resp. µ p of NLPF( p. Evaluate the residuals r p and s p of NLP( p (resp. v p, w p, and z p of NLPF( p. 2. Linearize the obective functions and constraints at ( p, y p. Renew T p = T p 1 {p} or S p = S p 1 {p}. 3. If (NLP( p is feasible and f p < UBD, then update current best point by setting = p, ȳ = y p, and UBD = f p. 4. Solve the relaed master problem (P OA p, obtaining a new discrete assignment p+1 to be tested in the algorithm. Increment p by one unit. UNTIL ((P OA p is infeasible. If termination occurs with UBD = +, then the algorithm visited every discrete assignment X Z n d but did not obtain a feasible point for the original MINLP problem P, or perturbed version P. In this case, the MINLP is declared infeasible. Net, we will show that the ineact OA algorithm also terminates in a finite number of steps. 10

11 Theorem 2.2 Let P be a conve MINLP problem and P be its perturbed problem as defined in the Introduction. Assume that either P has a finite optimal value or is infeasible, and that the residuals of the KKT conditions of the NLP subproblems satisfy Assumptions 2.1 and 2.2. Then Algorithm 2.1 terminates in a finite number of steps at an optimal solution of P or with an indication that P is infeasible. Proof. Since the set X is bounded by assumption, finite termination of Algorithm 2.1 will be established by proving that no discrete assignment is generated twice by the algorithm. Let q p. If q S p, it has been shown in Part I of the proof of Theorem 2.1 that the corresponding constraint in P OA p, derived from the feasibility problem NLPF(q, cannot be satisfied, showing that q cannot be feasible for (P OA p. We will now show that q cannot be feasible for (P OA p when q T p. For this purpose, let us assume that q is feasible in (P OA p and try to reach a contradiction. Let y q be an approimated optimal solution of NLP( q satisfying the ineact KKT conditions, that is, there eist λ q R m +, r q R nc, and s q R m, such that y f q + λ q i yg i ( q, y q = r q, (22 λ q i g i( q, y q = s q i, i = 1,..., m. (23 If q would be feasible for (P OA p it would satisfy the following set of inequalities for some y: where, for i = 1,..., m, ( f q y f q r q ( ( g q ( t q i = α p < UBD p f q, (24 0 y y q + f q α p, (25 0 y y q { s q i λ q, if λ q i > 0, i 0, if λ q i = 0. + g q t q, (26 Multiplying the rows in (26 by the Lagrange multipliers λ q i 0, i = 1,..., m, and adding (25, we obtain that ( y f q r q (y y q + f q + λ q i yg i ( q, y q (y y q + λ q i gq i α p + λ q i tq i, which, by the definition of t q, is equivalent to ( y f q r q (y y q + f q + λ q i yg i ( q, y q (y y q + The left hand side of this inequality can be written as: [ y f q r q + λ q i yg i ( q, y q ] (y y q + 11 (λ q i gq i sq i αp. (λ q i gq i sq i + f q. =1

12 Using (22 and (23, this is equal to f q and therefore we obtain the inequality f q α p, which contradicts (24. The rest of the proof is eactly as in [9, Theorem 2] but we repeat here for completeness and possible changes in notation. Finally, we will show that Algorithm 2.1 always terminates at a solution of P or with an indication that P is infeasible (which occurs when UBD = + at the eit. If P is feasible, then let (, y be an optimal solution of P with optimal value f. Without loss of generality, we will not distinguish between (, y and any other optimal solution with the same obective value f. Note that from Theorem 2.1, (, y, f is also an optimal solution of P OA. Now assume that the algorithm terminates indicating a nonoptimal point (, y with f > f. In such a situation, the previous relaation of the master problem P OA after adding the constraints at the point (, y, f, called (P OA p, is infeasible, causing the above mentioned termination. We will get a contradiction by showing that (, y, f is feasible for (P OA p. First, by the assumption that UBD = f > f, the first constraint α = f < UBD of (P OA p holds. Secondly, since (, y, f is an optimal solution to P OA, it must be feasible for all other constraints of (P OA p. Therefore, the algorithm could not terminate at (, y with UBD = f. 3 Ineact generalized Benders decomposition 3.1 Equivalence between perturbed and master problems for GBD In the generalized Benders decomposition (GBD, the MILP master problem involves only the discrete variables. When considering the ineact case, the master problem of GBD is the following: P GBD min s.t. α f(, y + f(, y ( + m λ i g i (, y ( α, T, m µk i [g i( k, y k + g i ( k, y k ( k ] + w k u k m zk i 0, k S, X Z n d, α R. One can easily recognize the classical form of (eact GBD master problem when w k = 0 and z k = 0. Moreover, as we show in the Appendi, this MILP can also be derived in the ineact case from a perturbed duality representation of the original, perturbed problem. A proof similar to the one of eact GBD and eact and ineact OA (Theorem 2.1 allows us to establish the desired equivalence between the original, perturbed MINLP problem and the MILP master problem P GBD. Theorem 3.1 Let P be a conve MINLP problem and P be its perturbed problem as defined in the Introduction. Assume that P is feasible with a finite optimal value and that the residuals of the KKT conditions of the NLP subproblems satisfy Assumptions 2.1 and 2.2. Then P GBD and P have the same optimal value. 12

13 Proof. Given any l X Z n d with l T S, let P GBD denote the problem in α obtained l from P GBD when is fied to l. First we will prove that problem P GBD is infeasible for every k k S. When we set = k, in the corresponding constraint of P GBD, one obtains µ k i g i ( k, y k + w k u k zi k 0 From (4 and (5, it results that u k 0, but one knows that u k is strictly positive when NLP( k is infeasible. Net, we will prove that for each X Z n d, with T, P GBD has the same optimal value as the perturbed NLP(. First, we will prove that the following constraints of P GBD f( l, y l + f( l, y l ( l + λ l i g i ( l, y l ( l α, l T, (27 µ k i [g i ( k, y k + g i ( k, y k ( k ] + w k u k zi k 0, k S. (28 are satisfied with α = f(, y. Under Assumptions 2.1 and Assumption 2.2, we know from the proof of Theorem 2.1 (Part II A that the following hold: (15 with y = y and α = f(, y, (16 with y = y, and (17 with y = y. When l T, multiplying the inequalities (16 with y = y by the nonnegative multipliers λ 1,..., λ m and summing them together with (15 with y = y and α = f(, y, one obtains f( l, y l + f( l, y l ( l + f(, y [ y f( l, y l + λ l i g i ( l, y l ( l λ l i y g i ( l, y l r l ] (y y l λ l ig( l, y l + λ l it l i. The right hand side is equal to f(, y by the definitions of r l, s l, and t l, showing that (27 holds with α = f(, y. When k S, multiplying the inequalities in (17 with y = y by the nonnegative multipliers µ k 1,..., µk m, and summing them up, one obtains using (3 and (4 µ k i g i ( k, y k ( k + µ k i g i ( k, y k µ k i a k i, which, by the definition of a k, is the same as (28. Thus, f(, y is a feasible point of P GBD, and therefore f(, y is an upper bound on the optimal value ᾱ of P OA. To show that is also a lower bound, i.e., that ᾱ f(, y, note that from (27, when l =, P GBD contains the constraint: f(, y α. We have thus proved that for any X Z n d, with T, problems P GBD and perturbed NLP( have the same optimal value, which concludes the proof. When all the KKT residuals are zero we obtain as a corollary the known equivalence result in GBD: 13

14 Corollary 3.1 Let P be a conve MINLP problem. Assume that P is feasible with a finite optimal value and the residuals of the KKT conditions of the NLP subproblems are zero. Then P GBD and P have the same optimal value. Remark 3.1 It is well known that the constraints of the GBD master problem can be derived from the corresponding ones of the OA master problem, in the conve, eact case (see [15]. The same happens naturally in the ineact case. In fact, from the proof of Theorem 3.1 above, we can see that the constraints in P OA, for T, imply the corresponding ones in P GBD. Moreover, one can easily see that any of the constraints in P OA imply the corresponding ones in P GBD. Thus, one can also say in the ineact case that the lower bounds produced iteratively by the OA algorithm are stronger than the ones provided by the corresponding GBD algorithm (given net. 3.2 Ineact GBD algorithm As we know for eact GBD, it is possible to derive an algorithm for the ineact case, terminating finitely, by solving at each iteration a relaed MILP formed by the cuts collected so far. The definitions of UBD p, T p, and S p are the same as those in Section 2.2. The relaed MILP to be solved at each iteration is thus given by min α (P GBD p s.t. α < UBD p f(, y + f(, y ( + m λ i g i (, y ( α, T p m µk i [g i( k, y k + g i ( k, y k ( k ] + w k u k m zk i 0, k S p X Z n d, α R. The ineact GBD algorithm is given net (and follows the presentation in [9] for OA. Algorithm 3.1 (Ineact GBD Approimation Initialization Let 0 be given. Set p = 0, T 1 =, S 1 =, and UBD = +. REPEAT 1. Ineactly solve the subproblem NLP( p, or the feasibility subproblem NLPF( p provided NLP( p is infeasible, and let y p be an approimated optimal solution. At the same time, obtain the corresponding ineact Lagrange multipliers λ p of NLP( p (resp. µ p of NLPF( p. Evaluate the residuals r p and s p of NLP( p (resp. v p, w p, and z p of NLPF( p. 2. Linearize the obective functions and constraints at p. Renew T p = T p 1 {p} or S p = S p 1 {p}. 14

15 3. If (NLP( p is feasible and f p < UBD, then update current best point by setting = p, ȳ = y p, and UBD = f p. 4. Solve the relaed master problem (P GBD p, obtaining a new discrete assignment p+1 to be tested in the algorithm. Increment p by one unit. UNTIL ((P GBD p is infeasible. Similarly as in Theorem 2.2 for OA, one can establish that the above ineact GBD algorithm terminates in a finite number of steps. Theorem 3.2 Let P be a conve MINLP problem and P be its perturbed problem as defined in the Introduction. Assume that either P has a finite optimal value or is infeasible, and that the residuals of the KKT conditions of the NLP subproblems satisfy Assumptions 2.1 and 2.2. Then Algorithm 3.1 terminates in a finite number of steps at an optimal solution of P or with an indication that P is infeasible. 4 Numerical eperiments We will illustrate some of the practical features of ineact OA and GBD algorithms by reporting numerical results on three test problems: Eample 1, taken from [8, test problem no. 1], has 3 discrete variables and 3 continuous variables; Eample 2, taken from [8, test problem no. 2], has 5 discrete variables and 6 continuous variables; Eample 3, taken from [8, test problem no. 3], and has 8 discrete variables and 9 continuous variables. All the three eamples are conve and linear in the discrete variables, and consist of simplified versions of process synthesis problems. In the third eample we found a point better than the one in [8]: = (0, 2, , , 2, 0, 0, , , y = (0, 1, 0, 1, 0, 1, 0, 1 with corresponding optimal value f = The implementation and testing of Algorithms 2.1 and 3.1 was made in MATLAB (version , R2010b. We used fmincon (from MATLAB to solve the NLP subproblems and ip1 [17] to solve the MILP problems, arising in both algorithms. The linear equality constraints possibly present in the original problems were kept in the MILP master problems. For both methods, we report results for two variants, depending on the form of the cuts. In a first variant the subproblems are solved ineactly (with tolerances varying from 10 6 to 10 1 but the cuts are the eact ones. When the tolerance is set to 10 6 we are essentially running eact OA and GBD. The second variant also incorporates ineact solution of NLP subproblems (again with tolerances varying from 10 6 to 10 1 but the cuts are now the ineact ones. In the tables of results we report the number N of iterations taken by Algorithms 2.1 and 3.1. We also report, in the tables corresponding to the second variant, the number C of constraint inequalities of Assumptions 2.1 and 2.2 that were violated by more than The stopping criteria of both algorithms consisted of the corresponding master program being infeasible or the number of iterations eceeding 50 or the solution of the MILP master program coinciding with a previous one (these third cases were marked with NPC, standing for no proper convergence. We note that in all the NPC cases found, the repeated integer solution of the MILP master problem was indeed the solution of the original MINLP. 15

16 Table 1: Application of OA (ineact solution of NLP subproblems and eact cuts to Eample 1. The table reports the number N of iterations taken. Tolerances initial point 1e-6 1e-5 1e-4 1e-3 1e-2 1e-1 (0, 0, NPC (1, 0, NPC (0, 1, NPC (1, 0, NPC (0, 1, NPC (0, 0, NPC NPC stands for no proper convergence (repeated solution of MILP master problem. Table 2: Application of OA (ineact solution of NLP subproblems and ineact cuts to Eample 1. The table reports the number N of iterations taken as well as the number C of inequalities violated in Assumptions 2.1 and 2.2. Tolerances 1e-6 1e-5 1e-4 1e-3 1e-2 1e-1 initial point N C N C N C N C N C N C (0, 0, NPC 0 (1, 0, NPC 0 (0, 1, NPC 0 (1, 0, NPC 0 (0, 1, NPC 0 (0, 0, NPC 0 The maimum number for C is 13t(t 1/2 + 12st, where t = T, s = S, and s + t = N. NPC stands for no proper convergence (repeated solution of MILP master problem. 4.1 Results for ineact OA method Tables 1 6 summarize the application of ineact OA (Algorithm 2.1 (variant ineact solution of NLP subproblems and eact cuts, and variant ineact solution of NLP subproblems and ineact cuts to Eamples 1 3. Comparing Tables 1 and 2, one can see little difference between using eact or ineact cuts for the first, smaller eample. However, looking at Tables 3 and 4 for the second eample and Tables 5 and 6 for the third eample, one can see, for larger values of the tolerances, that the ineact case with eact cuts has more tendency for unproper convergence (i.e., the MILP is incapable of either provide a new integer solution or render infeasible, while the variant incorporating the ineactness in the cuts does not. We also observe that ineact OA converged even neglecting the imposition of the inequalities of Assumptions 2.1 and

17 Table 3: Application of OA (ineact solution of NLP subproblems and eact cuts to Eample 2. The table reports the number N of iterations taken. Tolerances initial point 1e-6 1e-5 1e-4 1e-3 1e-2 1e-1 (1, 0, 0, 0, NPC NPC NPC (0, 1, 0, 0, NPC NPC NPC (1, 0, 1, 0, NPC NPC NPC (1, 0, 0, 1, NPC NPC NPC (1, 0, 0, 0, NPC NPC NPC (0, 1, 1, 0, NPC NPC NPC (0, 1, 0, 1, NPC NPC NPC (0, 1, 0, 0, NPC NPC NPC (1, 0, 1, 1, NPC NPC NPC (1, 0, 1, 0, NPC NPC NPC (0, 1, 1, 1, NPC NPC NPC (0, 1, 1, 0, NPC NPC NPC NPC stands for no proper convergence (repeated solution of MILP master problem. Table 4: Application of OA (ineact solution of NLP subproblems and ineact cuts to Eample 2. The table reports the number N of iterations taken as well as the number C of inequalities violated in Assumptions 2.1 and 2.2. Tolerances 1e-6 1e-5 1e-4 1e-3 1e-2 1e-1 initial point N C N C N C N C N C N C (1, 0, 0, 0, (0, 1, 0, 0, (1, 0, 1, 0, (1, 0, 0, 1, (1, 0, 0, 0, (0, 1, 1, 0, (0, 1, 0, 1, (0, 1, 0, 0, (1, 0, 1, 1, (1, 0, 1, 0, (0, 1, 1, 1, (0, 1, 1, 0, The maimum number for C is 27t(t 1/2 + 26st, where t = T, s = S and s + t = N. 17

18 Table 5: Application of OA (ineact solution of NLP subproblems and eact cuts to Eample 3. The table reports the number N of iterations taken. Tolerances initial point 1e-6 1e-5 1e-4 1e-3 1e-2 1e-1 (1, 0, 0, 0, 0, 0, 0, NPC NPC NPC (1, 0, 0, 0, 0, 0, 0, NPC NPC NPC (1, 0, 1, 0, 0, 0, 0, NPC NPC NPC (1, 0, 0, 0, 1, 0, 0, NPC NPC NPC (1, 0, 0, 0, 1, 0, 0, NPC NPC NPC (1, 0, 1, 0, 1, 0, 0, NPC NPC NPC (1, 0, 0, 1, 0, 1, 0, NPC NPC NPC (1, 0, 0, 1, 0, 1, 0, NPC NPC NPC (1, 0, 0, 1, 0, 0, 1, NPC NPC NPC (1, 0, 0, 1, 0, 0, 1, NPC NPC NPC (0, 1, 0, 0, 0, 0, 0, NPC NPC NPC (0, 1, 0, 0, 0, 0, 0, NPC NPC NPC (0, 1, 1, 0, 0, 0, 0, NPC NPC NPC (0, 1, 0, 0, 1, 0, 0, NPC NPC NPC (0, 1, 0, 0, 1, 0, 0, NPC NPC NPC (0, 1, 1, 0, 1, 0, 0, NPC NPC NPC (0, 1, 0, 1, 0, 1, 0, NPC NPC NPC (0, 1, 0, 1, 0, 1, 0, NPC NPC NPC (0, 1, 1, 1, 0, 1, 0, NPC NPC NPC (0, 1, 0, 1, 0, 0, 1, NPC NPC NPC (0, 1, 0, 1, 0, 0, 1, NPC NPC NPC (1, 0, 1, 1, 0, 1, 0, NPC NPC NPC NPC stands for no proper convergence (repeated solution of MILP master problem. 18

19 Table 6: Application of OA (ineact solution of NLP subproblems and ineact cuts to Eample 3. The table reports the number N of iterations taken as well as the number C of inequalities violated in Assumptions 2.1 and 2.2. Tolerances 1e-6 1e-5 1e-4 1e-3 1e-2 1e-1 initial point N C N C N C N C N C N C (1, 0, 0, 0, 0, 0, 0, (1, 0, 0, 0, 0, 0, 0, (1, 0, 1, 0, 0, 0, 0, (1, 0, 0, 0, 1, 0, 0, (1, 0, 0, 0, 1, 0, 0, (1, 0, 1, 0, 1, 0, 0, (1, 0, 0, 1, 0, 1, 0, (1, 0, 0, 1, 0, 1, 0, (1, 0, 0, 1, 0, 0, 1, (1, 0, 0, 1, 0, 0, 1, (0, 1, 0, 0, 0, 0, 0, (0, 1, 0, 0, 0, 0, 0, (0, 1, 1, 0, 0, 0, 0, (0, 1, 0, 0, 1, 0, 0, (0, 1, 0, 0, 1, 0, 0, (0, 1, 1, 0, 1, 0, 0, (0, 1, 0, 1, 0, 1, 0, (0, 1, 0, 1, 0, 1, 0, (0, 1, 1, 1, 0, 1, 0, (0, 1, 0, 1, 0, 0, 1, (0, 1, 0, 1, 0, 0, 1, (1, 0, 1, 1, 0, 1, 0, The maimum number for C is 21t(t st, where t = T, s = S, and s + t = N. 19

20 Table 7: Application of GBD (ineact solution of NLP subproblems and eact cuts to Eample 1. The table reports the number N of iterations taken. Tolerances initial point 1e-6 1e-5 1e-4 1e-3 1e-2 1e-1 (0, 0, (1, 0, (0, 1, (1, 0, (0, 1, (0, 0, Table 8: Application of GBD (ineact solution of NLP subproblems and ineact cuts to Eample 1. The table reports the number N of iterations taken as well as the number C of inequalities violated in Assumptions 2.1 and 2.2. Tolerances 1e-6 1e-5 1e-4 1e-3 1e-2 1e-1 initial point N C N C N C N C N C N C (0, 0, (1, 0, (0, 1, (1, 0, (0, 1, (0, 0, The maimum number for C is as in Table Results for ineact GBD method Tables 7 12 summarize the application of ineact GBD (Algorithm 3.1 (variant ineact solution of NLP subproblems and eact cuts, and variant ineact solution of NLP subproblems and ineact cuts to Eamples 1 3. One can observe that there is little difference between the two variants since the eact cuts in the first variant already incorporate ineact information coming from the ineact Lagrange multipliers. One observes that ineact GBD takes more iterations than ineact OA in these eamples, which, according to Remark 3.1, is epected since ineact GBD yields weaker lower bounds and hence generally requires more maor iterations to converge than ineact OA. The number of inequalities of Assumptions 2.1 and 2.2 violated in ineact GBD is also higher than the one in ineact OA. 5 Conclusions and final remarks In this paper we have attempted to gain a better understanding of the effect of ineactness when solving NLP subproblems in two well known decomposition techniques for Mied Integer 20

21 Table 9: Application of GBD (ineact solution of NLP subproblems and eact cuts to Eample 2. The table reports the number N of iterations taken. Tolerances initial point 1e-6 1e-5 1e-4 1e-3 1e-2 1e-1 (1, 0, 0, 0, (0, 1, 0, 0, (1, 0, 1, 0, (1, 0, 0, 1, (1, 0, 0, 0, (0, 1, 1, 0, (0, 1, 0, 1, (0, 1, 0, 0, (1, 0, 1, 1, (1, 0, 1, 0, (0, 1, 1, 1, (0, 1, 1, 0, Table 10: Application of GBD (ineact solution of NLP subproblems and ineact cuts to Eample 2. The table reports the number N of iterations taken as well as the number C of inequalities violated in Assumptions 2.1 and 2.2. Tolerances 1e-6 1e-5 1e-4 1e-3 1e-2 1e-1 initial point N C N C N C N C N C N C (1, 0, 0, 0, (0, 1, 0, 0, (1, 0, 1, 0, (1, 0, 0, 1, (1, 0, 0, 0, (0, 1, 1, 0, (0, 1, 0, 1, (0, 1, 0, 0, (1, 0, 1, 1, (1, 0, 1, 0, (0, 1, 1, 1, (0, 1, 1, 0, The maimum number for C is as in Table 4. 21

22 Table 11: Application of GBD (ineact solution of NLP subproblems and eact cuts to Eample 3. The table reports the number N of iterations taken. Tolerances initial point 1e-6 1e-5 1e-4 1e-3 1e-2 1e-1 (1, 0, 0, 0, 0, 0, 0, (1, 0, 0, 0, 0, 0, 0, (1, 0, 1, 0, 0, 0, 0, (1, 0, 0, 0, 1, 0, 0, (1, 0, 0, 0, 1, 0, 0, (1, 0, 1, 0, 1, 0, 0, (1, 0, 0, 1, 0, 1, 0, (1, 0, 0, 1, 0, 1, 0, (1, 0, 0, 1, 0, 0, 1, (1, 0, 0, 1, 0, 0, 1, (0, 1, 0, 0, 0, 0, 0, (0, 1, 0, 0, 0, 0, 0, (0, 1, 1, 0, 0, 0, 0, (0, 1, 0, 0, 1, 0, 0, (0, 1, 0, 0, 1, 0, 0, (0, 1, 1, 0, 1, 0, 0, (0, 1, 0, 1, 0, 1, 0, (0, 1, 0, 1, 0, 1, 0, (0, 1, 1, 1, 0, 1, 0, (0, 1, 0, 1, 0, 0, 1, (0, 1, 0, 1, 0, 0, 1, (1, 0, 1, 1, 0, 1, 0,

23 Table 12: Application of GBD (ineact solution of NLP subproblems and ineact cuts to Eample 3. The table reports the number N of iterations taken as well as the number C of inequalities violated in Assumptions 2.1 and 2.2. Tolerances 1e-6 1e-5 1e-4 1e-3 1e-2 1e-1 initial point N C N C N C N C N C N C (1, 0, 0, 0, 0, 0, 0, (1, 0, 0, 0, 0, 0, 0, (1, 0, 1, 0, 0, 0, 0, (1, 0, 0, 0, 1, 0, 0, (1, 0, 0, 0, 1, 0, 0, (1, 0, 1, 0, 1, 0, 0, (1, 0, 0, 1, 0, 1, 0, (1, 0, 0, 1, 0, 1, 0, (1, 0, 0, 1, 0, 0, 1, (1, 0, 0, 1, 0, 0, 1, (0, 1, 0, 0, 0, 0, 0, (0, 1, 0, 0, 0, 0, 0, (0, 1, 1, 0, 0, 0, 0, (0, 1, 0, 0, 1, 0, 0, (0, 1, 0, 0, 1, 0, 0, (0, 1, 1, 0, 1, 0, 0, (0, 1, 0, 1, 0, 1, 0, (0, 1, 0, 1, 0, 1, 0, (0, 1, 1, 1, 0, 1, 0, (0, 1, 0, 1, 0, 0, 1, (0, 1, 0, 1, 0, 0, 1, (1, 0, 1, 1, 0, 1, 0, The maimum number for C is as in Table 6. 23

24 Nonlinear Programming (MINLP, the outer approimation (OA and the generalized Benders decomposition (GBD. As pointed out to us by I. E. Grossmann, solving the NLP subproblems ineactly in OA positions this approach somewhere in between eact OA and the etended cutting plane method [18]. Although all the conditions required on the residuals of the ineact KKT conditions can be imposed, one can see from Assumptions 2.1 and 2.2 that the complete satisfaction of all those inequalities would ask for repeated NLP subproblem solution for all the previous addressed discrete assignments. Such requirement would then undermine the practical purpose of saving computational effort aimed by the NLP subproblem ineactness. In our preliminary numerical tests we disregarded the conditions of Assumptions 2.1 and 2.2 and verified, after terminating each run of ineact OA or GBD, how many of them were violated. The results indicated that proper convergence can be achieved without imposing Assumptions 2.1 and 2.2, and that the number of violated inequalities was relatively low. The results also seem to indicate that the cuts in OA and GBD must be changed accordingly when the corresponding NLP subproblems are solved ineactly. Testing these ineact approaches in a wider test set of larger problems is out of the scope of this paper, although it seems a necessary step to further validate these indications. Our study was performed under the assumption of conveity of the functions involved. Moreover, we also assumed that the approimated optimal solutions of the NLP subproblems were feasible in these subproblems, and that the corresponding ineact Lagrange multipliers were nonnegative. Relaing these assumptions introduces another layer of difficulty but certainly deserves proper attention in the future. A Appendi A.1 Ineact KKT conditions and perturbed problems As we said in the Introduction of the paper, the point y satisfying the ineact KKT conditions (1 (2 of the subproblem NLP( can be interpreted as a solution of a perturbed NLP subproblem, which has the form min f(, y (r (y y perturbed NLP( s.t. g(, y t 0, y Y, where t is given by (7. The data of this perturbed subproblem depends, however, on the approimated optimal solution y and ineact Lagrange multipliers λ. Similarly, the point y k satisfying the ineact KKT conditions (3 (5 of the subproblem NLPF( k can be interpreted as a solution of the following perturbed NLP subproblem min u w k (u u k (v k (y y k perturbed NLPF( k s.t. g i ( k, y u c k i 0, i = 1,..., m, y Y, u R, where, for i = 1,..., m, c k i = { z k i, if µ k µ k i > 0, i 0, if µ k i = 0. 24

25 A.2 Derivation of the master problem for ineact GBD As in the eact case, the MILP master problem P GBD can be derived from a more general master problem closer to the original duality motivation of GBD: min α P GBD 1 s.t. inf y Y {f(, y + (λ g(, y (r (y y } m s i α, T, inf y Y {(µ k g(, y (v k (y y k } + w k u k m zk i 0, k S, X Z n d, α R. In fact, we will show net that the constraints in problem P GBD 1 imply those of P GBD. When l T, one knows that NLP( l has an approimated optimal solution y l, satisfying the corresponding ineact KKT conditions with ineact Lagrange multipliers λ l. By the conveity of f and g (see (10 and (11, f(, y + (λ l g(, y (r l (y y l f( l, y l + f( l, y l ( l + y f( l, y l (y y l Thus, using the ineact KKT conditions (1, + λ l i[g i ( l, y l + g i ( l, y l ( l α inf y Y {f(, y + (λl g(, y (r l (y y l } + y g i ( l, y l (y y l ] (r l (y y l. inf y Y {f(l, y l + f( l, y l ( l + λ l i g i ( l, y l ( l + λ l ig i ( l, y l } = f( l, y l + f( l, y l ( l + λ l i g i ( l, y l ( l + λ l ig i ( l, y l = f( l, y l + f( l, y l ( l + s l i λ l i g i ( l, y l ( l. The last equality holds due to (2. When l S, we know that NLPF( l has an approimated optimal solution y l satisfying the corresponding ineact KKT conditions with ineact Lagrange multipliers µ l. Also by the conveity of g (see (11, we have that (µ l g(, y (v l (y y l (µ l [g( l, y l + g( l, y l ( l ] + ( µ l i y g i ( l, y l v l (y y l. Then, using the ineact KKT conditions (4, 0 inf y Y {(µl g(, y (v l (y y l } + w l u l = inf { m µ l i[g i ( l, y l + g i ( l, y l ( l ]} + w l u l y Y In summary we have the following property. z l i µ l i[g i ( l, y l + g i ( l, y l ( l ] + w l u l zi. l z l i s l i s l i 25

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