Variable Objective Search

Transcription

1 Variable Objective Search Sergiy Butenko, Oleksandra Yezerska, and Balabhaskar Balasundaram Abstract This paper introduces the variable objective search framework for combinatorial optimization. The method utilizes different objective functions used in alternative mathematical programming formulations of the same combinatorial optimization problem in an attempt to improve the solutions obtained using each of these formulations individually. The proposed technique is illustrated using alternative quadratic unconstrained binary formulations of the classical maximum independent set problem in graphs. Keywords. Variable objective search, Binary quadratic programming, Maximum independent set 1 Introduction Due to the inherent computational complexity of most important combinatorial optimization problems, one cannot expect to be able to solve very large-scale instances to optimality, and (meta) heuristics are usually applied in practice. The simplicity and effectiveness of metaheuristic approaches earned them considerable popularity in the optimization community [5]. In their now classical paper, Mladenović and Hansen [8] (see also [6]) proposed to take advantage of different neighborhood structures for the same combinatorial optimization problem by systematically changing neighborhoods within the search. The resulting variable neighborhood search (VNS) Industrial and Systems Engineering, Texas A&M University, College Station, TX , {butenko,yaleksa}@tamu.edu Industrial Engineering & Management, Oklahoma State University, Stillwater, OK 74078, baski@okstate.edu 1

2 is based on the observation that a local optimum with respect to one neighborhood may not be a local optimum with respect to another neighborhood, whereas a global optimum is a local optimum with respect to any neighborhood structure. Therefore, if we find a solution which is locally optimal with respect to several neighborhood structures rather than just one, intuitively such a solution stands a higher chance of being a global optimum. VNS has been successfully applied to many different combinatorial optimization problems. Inspired by this success and motivated by the observation that the properties similar to those stated for neighborhoods also hold for alternative objective functions for the same problem, in this paper we propose a new metaheuristic framework, variable objective search (VOS). It is in the spirit of VNS, however, instead of varying the neighborhood structures we propose to consider different equivalent formulations of the combinatorial optimization problem solved. Given two or more equivalent formulations of the same combinatorial optimization problem, and assuming for simplicity that all formulations share the same feasible region but have different objective functions, we can impose a neighborhood structure for our problem that will be the same for all the considered formulations. Then the following properties hold: (i) a local optimum for one formulation may not correspond to a local optimum with respect to another formulation; (ii) a global optimal solution of the considered combinatorial optimization problem should correspond to a global optimum for any formulation of this problem. Therefore, it seems natural to use the local maximum for one of the formulations as a starting point of search for another formulation. The proposed variable objective search framework is based on these properties and can be applied to almost any optimization problem allowing multiple equivalent formulations (discrete or continuous). In particular, many classical combinatorial optimization problems, such as the quadratic assignment problem, the max-cut problem, and the maximum clique problem can be expressed using unconstrained binary quadratic formulations, and therefore allow for multiple equivalent binary quadratic reformulations [7]. One way to obtain such reformulations is as follows. Assume that one has the problem in the form max x {0,1} n f(x) = xt Qx + c T x, 2

3 where Q is a real symmetric n n matrix. Given any real vector q R n, let Q = Q + diag(q), where diag(q) is the n n diagonal matrix having q as its diagonal, and let c = c q. Then, since f(x) = f(x) for any x {0, 1} n, the above 0 1 quadratic problem is equivalent to the problem: max f(x) = x {0,1} xt Qx + c T x. n The above modification does not have any impact on a local search based on any binary neighborhood, however, it does change the curvature of the quadratic function and can be useful for continuous local search. If the vector q above is selected so that the resulting matrix Q either has zero diagonal or is positive definite then we are guaranteed that the continuous problem max f(x) has a binary global maximum, thus we obtain two equivalent continuous reformulations of the original binary quadratic problem. Applying x [0,1] n classical nonlinear programming algorithms to one of these reformulations will yield a stationary point x (0) that is not necessarily a stationary point for the other reformulation and does not have to be a local maximum with respect to a binary neighborhood structure. Thus, local search procedures can be applied for the latter two cases using x (0) as initial guess. In addition, selecting the vector q above that does not result in an equivalent reformulation but provides a continuous relaxation for the binary problem (e.g., by choosing q 0) may also be of value, as it may provide good search directions. To illustrate this point, consider the following simple example: [ ] [ ] Q = ; c = There are four feasible points for the corresponding binary quadratic maximization problem, x (1) = [0, 0] T, x (2) = [0, 1] T, x (3) = [1, 0] T, x (4) = [1, 1] T with f(x (1) ) = f(x (4) ) = 0, f(x (2) ) = 4, and f(x (3) ) = 2. Hence, x (3) is a local maximum of f(x) with respect to the binary one-flip neighborhood as well as a local maximum for the corresponding continuous relaxation (see Fig. 1). Note that all feasible directions at x (3) are descent directions for f(x). However, applying the above transformation with q = [ 4, 5] T, we obtain f(x) f(x) x [0, 1] 2, with the gradient f(x (3) ) = [ 1, 1] T, meaning that [ 1, 1] T is a feasible ascent direction for f(x) at x (3). Using this direction for line search with either zero-diagonal or a positive definite continuous reformulation of the considered binary problem, we immediately obtain the global maximum given by x (2). 3

4 4 3 x x x 1 Figure 1: Plots of x 3 = f(x) and x 3 = f(x) considered in the example. To illustrate the proposed VOS approach on a specific problem, we will use alternative quadratic formulations of the maximum independent set problem in graphs, therefore, some definitions from graph theory are in order. Given a simple undirected graph G = (V, E), let A G denote the adjacency matrix of G. For a vertex v V let N(v) = {u : (u, v) E} denote the neighborhood of v. The complement graph Ḡ of G is given by Ḡ = (V, Ē), where Ē = {(u, v) V V \ E : u v}. A subset of vertices I V is called an independent (stable) set if (u, v) / E for any pair of vertices u, v I. The maximum independent set problem is to find an independent set of the largest size in G. The cardinality of the largest independent set is denoted by α(g) and is called the independence (stability) number. This classical combinatorial optimization problem is of great theoretical importance and has numerous practical applications of diverse origins, therefore it has been well studied in the literature. See [3] for an in-depth survey of the maximum clique problem, which is equivalent to the maximum independent set problem in the complement graph. The remainder of this paper is organized as follows. Section 2 describes the proposed methodology in detail. The approach is illustrated numerically on the maximum independent set problem in Section 3. Finally, Section 4 concludes the paper. 4

5 2 The method A combinatorial optimization problem P can be generally defined as a set of problem instances and a minimization or maximization objective, where an instance of P is a pair (S, f), with S being the set of feasible solutions and f : S R being the objective function [1]. Without loss of generality, we will assume that P has the maximization objective. The problem consists in finding a globally optimal solution, which is a feasible solution s S such that f(s ) f(s) for all s S. Let us denote by f = f(s ) = max s S f(s) the optimal objective function value, and by S = {s S : f(s) = f } = arg max s S f(s) the set of optimal solutions of the considered instance of the problem P. For simplicity, in this paper we will assume that we are dealing with a specific instance of the problem P, therefore problem P will refer to the given instance of P. Many combinatorial optimization problems can be equivalently modeled as mathematical programs of different types, e.g., as (mixed) integer programming problems or continuous nonconvex optimization problems. The equivalence is in the sense that every global maximum of a mathematical programming formulation corresponds, in a straightforward fashion, to a global maximum of the combinatorial optimization problem that this formulation models. Assume that our problem P allows the following p alternative equivalent formulations: f = max x X i f i (x), i = 1,..., p, where X i, i = 1,...,p is a feasible set that may be either discrete or continuous. If we denote by X 0 = S and by f 0 = f, then the original combinatorial formulation of the problem P can be written as f = max x X 0 f 0 (x). Consider the set of all optimal solutions for the i th formulation: Xi = arg max f i(x). Before we proceed with the formal description of the variable x X objective framework, we need to make the following important assumptions. 5

6 Assumption 1. There are mappings h ij : X i X j, i = 0,..., p (where h ii, i = 0,...,p are the identity maps) between the feasible sets of the i th and j th formulations such that any global optimal solution of the i th formulation is mapped to a global optimum of the j th formulation, i.e., h ij (X i ) X j. In some cases the mappings may be straightforward, while in others they may need to be calculated using efficient algorithms. Given x (i), one would want to have f 0 (h i0 (x (i) )) f i (x (i) ), however this may not be the case in general. Assumption 2. We assume that the same neighborhood structure N can be imposed on each feasible region X i, i = 1,...,p. This neighborhood structure will be used for local search applied to each formulation within the proposed variable objective framework. Note that if we deal with alternative continuous formulations of P, the continuous ǫ-neighborhood definition based on the Euclidean norm is the natural choice of the neighborhood structure. Assumption 3. We assume that the mappings h ij preserve the neighborhood structure N, i.e., for any x X i we have h ij (N(x)) = N(h ij (x)). This assumption will allow us to perform a local search for all problems by essentially concentrating on one of X i s. The last two assumptions are not critical and are used to simplify the presentation. They can be relaxed resulting in the generalized variable objective search or variable neighborhood/objective search, which naturally generalizes both VOS and VNS. 2.1 Basic variable objective search In the basic variable objective search (BVOS), we start with some feasible solution for the first formulation and perform a local search in the neighborhood N yielding a locally optimal solution x (1) for max f 1 (x). This solution x X 1 is then transformed into a feasible solution h 12 ( x (1) ) of the second formulation, which is then used as a starting point of a local search for the second formulation. Because of Assumption 3 above, the local search for the second formulation is performed by searching the neighborhood of x (1) in X 1 and mapping the neighbors to the corresponding points of X 2, and the resulting local maximum for the second formulation is given by h 12 ( x (2) ), where x (2) X 1. Next, h 13 ( x (2) ) is used as the starting point of local search with the third formulation, etc. To avoid cycling, local search moves for the i th 6

7 Algorithm 1 BVOS with best improvement move Require: f i, X i, i = 0,...,p; N; Ensure: a feasible solution s of P; 1: Find an initial solution x X 1 ; 2: x (i) = h 1i (x), i = 1,...,p; 3: fi = f i ( x (i) ), i = 1,...,p; 4: repeat 5: ˆx = x; 6: for i = 1,..., p do 7: for each y N(x) do 8: if f i (h 1i (y)) > f i then 9: x = y; 10: x (i) = x; 11: fi = f i (h 1i ( x (i) )); 12: end if 13: end for 14: end for 15: until ˆx = x; 16: X0 = {h 10 ( x (i) ) : i = 1,...,p}; 17: X0 = arg max{f 0 (s) : s X 0 }; 18: f0 = f 0 (s), s X 0 ; 19: return s X 0, f 0 ; formulation are performed only if there is an improvement compared to the best previously computed solution x (i). The corresponding pseudocodes are given in Algorithms 1 and 2, which describe BVOS with best improvement and first improvement local search strategies, respectively. 2.2 Uniform variable objective search The uniform variable objective search (UVOS) is similar to BVOS, however, the local search moves are performed based on analyzing the objective function values for all p formulations simultaneously rather than one at a time. Like in BVOS, we start with a feasible solution x of the first formulation. Throughout the execution of the algorithm, we store the best computed solutions with respect to each of the p formulations, where, as before, x (i) X 1 is the best current solution with respect to the i th formulation such that 7

8 Algorithm 2 BVOS with first improvement move Require: f i, X i, i = 0,...,p; N; Ensure: a feasible solution s of P; 1: Find an initial solution x X 1 ; 2: x (i) = h 1i (x), i = 1,...,p; 3: fi = f i ( x (i) ), i = 1,...,p; 4: repeat 5: ˆx = x; 6: for i = 1,..., p do 7: for each y N(x) do 8: if f i (h 1i (y)) > f i then 9: x = y; 10: x (i) = x; 11: fi = f i (h 1i ( x (i) )); 12: go to line 7; 13: end if 14: end for 15: end for 16: until ˆx = x; 17: X0 = {h 10 ( x (i) ) : i = 1,...,p}; 18: X0 = arg max{f 0 (s) : s X 0 }; 19: f0 = f 0 (s), s X 0 ; 20: return s X 0, f 0 ; h 1i ( x (i) ) X i is the corresponding local maximum. The local search proceeds as follows: for each y in the neighborhood of the current solution x, for each i = 1,..., p we check whether y improves the previously best objective value f i, and if it does, we update x (i) and f i and either continue searching the same neighborhood until all neighbors of x have been explored (the best improvement strategy), or, after updating all fi, i = 1,...,p that improved, move to y and start exploring the neighborhood of y (the first improvement strategy). The pseudocodes given in Algorithms 3 and 4 describe UVOS with best improvement and first improvement local search strategies, respectively. 8

9 Algorithm 3 UVOS with best improvement move Require: f i, X i, i = 0,...,p; N; Ensure: a feasible solution s of P; 1: Find an initial solution x X 1 ; 2: x (i) = x, i = 1,...,p; 3: fi = f i (h 1i ( x (i) )), i = 1,...,p; 4: repeat 5: ˆx = x; 6: for each y N(x) do 7: for i = 1,..., p do 8: if f i (h 1i (y)) > f i then 9: x = y; 10: x (i) = x; 11: fi = f i (h 1i ( x (i) )); 12: end if 13: end for 14: end for 15: until ˆx = x; 16: X0 = {h 10 ( x (i) ) : i = 1,...,p}; 17: X0 = arg max{f 0 (s) : s X 0 }; 18: f0 = f 0 (s), s X 0 ; 19: return s X 0, f 0 ; 3 Experiments with the maximum independent set problem In this section, we evaluate and compare the performance of the proposed algorithms using the maximum independent set problem as the testbed. The main objective of this exercise is to explore how much the proposed approaches improve the solutions found using similar local search strategies with different formulations individually. The maximum independent set problem can be equivalently formulated using the following continuous nonconvex programs (see, e.g., [2]): α(g) = max x [0,1] n et x 1 2 xt A G x. (1) 9

10 Algorithm 4 UVOS with first improvement move Require: f i, X i, i = 0,...,p; N; Ensure: a feasible solution s of P; 1: Find an initial solution x X 1 ; 2: x (i) = x, i = 1,...,p; 3: fi = f i (h 1i ( x (i) )), i = 1,...,p; 4: repeat 5: ˆx = x; 6: for each y N(x) do 7: improvement = 0; 8: for i = 1,..., p do 9: if f i (h 1i (y)) > f i then 10: x = y; 11: x (i) = x; 12: fi = f i (h 1i ( x (i) )); 13: improvement = 1; 14: end if 15: if (improvement ==1) then 16: go to line 5; 17: end if 18: end for 19: end for 20: until ˆx = x; 21: X0 = {h 10 ( x (i) ) : i = 1,...,p}; 22: X0 = arg max{f 0 (s) : s X 0 }; 23: f0 = f 0 (s), s X 0 ; 24: return s X 0, f 0 ; 1 1 α(g) = max x S xt A Ḡ x, (2) where S = {x R V : e T x = 1, x 0} and e is the vector with all components equal to 1. In formulation (1), the continuous feasible region [0, 1] n can be replaced with the discrete one, {0, 1} n without altering the optimal objective function value. Also, the feasible region of the Motzkin-Straus [9] formulation (2) can 10

11 be discretized by considering S = {y = x/(e T x) : x {0, 1} n, x 0} in place of S, since there is always a global optimal solution x of (2) in the form of the characteristic vector of a maximum independent set I divided by the independence number: x i = { 1/ I, i I; 0, i / I. Thus, we obtain the following discrete formulations: α(g) = max x {0,1} n et x 1 2 xt A G x. (3) 1 1 α(g) = max x S xt A Ḡ x, (4) where S = {y = x/(e T x) : x {0, 1} n, x 0}. We will use these two formulations in our experiments, with f 1 (x) = e T x 1 2 xt A G x; f 2 (x) = x T A Ḡ x. Then the obvious choice for transformation h 12 (x) is { 0, x = 0; h 12 (x) = x, x 0. e T x As for the choice of the neighborhood, it should be noted that we are not interested in the most efficient neighborhood that would yield an optimal solution for most test instances. In fact, in order to be able to make a fair comparison, we are more interested in a neighborhood structure that would leave some space for improvement after finding an initial local optimum with respect to one of the formulations used. Hence, we select the standard one-flip neighborhood, which is commonly used in heuristics for unconstrained optimization of functions of binary variables. Namely, given a vector x {0, 1} n, its neighborhood consists of all y {0, 1} n that are Hamming distance 1 away from x (i.e., y differs from x in exactly one component). 3.1 Results of experiments To test the proposed algorithms, complements of selected DIMACS clique benchmark graphs [4] were used. The number of vertices of tested graphs 11

12 ranged from 64 to 496. A set of 100 runs of each algorithm was performed for every instance. The initial solution for each run was generated randomly and was the same for all algorithms. Table 1 represents the results of executing BVOS method. The columns in each row of this table contain the graph instance name ( Instance ) fol- Table 1: Results for BVOS BVOS (best improvement) f 0 1 BVOS (first improvement) Instance V f0 1 f 1 Best % f1 f 1 Best % f1 brock brock brock brock c-fat c-fat c-fat hamming hamming hamming johnson johnson johnson johnson keller mann a p hat p hat p hat san san san sanr

13 Table 2: Results for UVOS UVOS (best improvement) UVOS (first improvement) Instance V f 1 f1 f 2 f2 f 1 f1 f 2 f2 brock brock brock brock c-fat c-fat c-fat hamming hamming hamming johnson johnson johnson johnson keller mann a p hat p hat p hat san san san sanr lowed by its number of vertices ( V ). The next four columns contain the following information: the mean (over the total number of runs) of the values of the first found local optimal solutions for the formulation (3) ( f 1 ), 0 the mean of the maximum values of the objective function (3) found ( f 1 ), the maximum (over the 100 trials) percentage improvement from the first found 13

14 local optimal solution to the best found solution ( Best % ), and the best overall value found for the objective function (3) ( f1 ) obtained under the BVOS method with best improvement move. The last four columns contain the same information for the first improvement move strategy. The results of applying UVOS algorithm to the same graph instances are shown in Table 2. Similar to Table 1, the first two columns in this table represent the graph instance name and its number of vertices, while the other two groups of four columns contain the mean ( f 1 ) and the maximum ( f1 ) of objective values for formulation (3), and the mean ( f 2 ) and the maximum ( f2 ) of objective values for formulation (4) obtained using best and first improvement approach, respectively. The results of the experiments indicate that the overall performance of the VOS on the given set of instances was rather encouraging. As can be seen from Table 1 in many cases it yielded a considerable (more than 50%) improvement in the value of the objective function under either the BVOS with best improvement or with first improvement, or both. We obtained only one result with no improvement in both BVOS strategies. However, there were three instances with a very large percentage of improvement indicating that the first obtained value of objective function was far from the optimal and it considerably improved with the help of BVOS with first improvement move. Although, in general the better results of the maximum value of objective function were generated by BVOS with best improvement move. In general we observed a tendency (about 70% of all cases) of getting the same maximum values of objective functions under both BVOS and UVOS methods. The majority of the rest 30% of the problems was better solved using the UVOS algorithm. We observed no significant difference in performance of UVOS with either best improvement or first improvement approach. In most cases both strategies showed similar results. 4 Conclusion We introduced the general framework of the variable objective search for combinatorial optimization, which is based on a simple observation that systematically combining different mathematical programming formulations of the same combinatorial optimization problem may lead to better heuristic solutions than those obtained by solving any of the considered formulations individually. We proposed two different variations of the method, the basic 14

15 variable objective search (BVOS), in which local search performed sequentially with respect to the considered formulations, and the uniform variable objective search (UVOS), which runs one local search that tries to look for better solutions with respect to each formulation simultaneously. In the latter part of the paper, we demonstrate the merit of BVOS and UVOS by applying them to two different formulations of the maximum independent set problem in graphs. While in this paper we restricted the discussion to two versions of VOS, many other variations of the method can be developed, similarly to how numerous versions of the VNS have been proposed in the literature [6]. Since the main objective of this paper is to introduce the general idea of this new framework, we leave the exploration of its various potentially useful variations and evaluation of their practical performance on large-scale instances of different combinatorial optimization problems for future research. In particular, the case of alternative continuous nonconvex formulations of the same problem, where the neighborhood of a feasible point is naturally defined as the intersection of the feasible region with an ǫ-ball centered at this point, is an interesting direction to investigate. References [1] E. Aarts and J. K. Lenstra, editors. Local Search in Combinatorial Optimization. John Wiley & Sons, Chichester, [2] J. Abello, S. Butenko, P. Pardalos, and M. Resende. Finding independent sets in a graph using continuous multivariable polynomial formulations. Journal of Global Optimization, 21: , [3] I. M. Bomze, M. Budinich, P. M. Pardalos, and M. Pelillo. The maximum clique problem. In D.-Z. Du and P. M. Pardalos, editors, Handbook of Combinatorial Optimization, pages Kluwer Academic Publishers, Dordrecht, The Netherlands, [4] DIMACS. Cliques, Coloring, and Satisfiability: Second DIMACS Implementation Challenge Accessed April [5] F. Glover and G. Kochenberger, editors. Handbook Of Metaheuristics. Springer, London,

16 [6] P. Hansen and N. Mladenović. Variable neighborhood search: Principles and applications. European Journal of Operational Research, 130: , [7] R. Horst, P. M. Pardalos, and N. V. Thoai. Introduction to Global Optimization. Kluwer Academic Publishers, Dordrecht, The Netherlands, 2 edition, [8] N. Mladenović and P. Hansen. Variable neighborhood search. Computers and Operations Research, 24: , [9] T. S. Motzkin and E. G. Straus. Maxima for graphs and a new proof of a theorem of Turán. Canad. J. Math., 17: ,