Polyhedral results for a class of cardinality constrained submodular minimization problems

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1 Poyhedra resuts for a cass of cardinaity constrained submoduar minimization probems Jiain Yu and Shabbir Ahmed Georgia Institute of Technoogy, Atanta, GA August 28, 2014 Abstract Motivated by concave cost combinatoria optimization probems, we study the foowing mixed integer noninear set: P = {(w, x) R {0, 1 n : w f (a x), e x where f : R R is a concave function, n and are positive integers, a R n is a nonnegative vector, e R n is a vector of ones, and x y denotes the scaar product of vectors x and y of same dimension. A standard inearization approach for P is to expoit the fact that f (a x) is submoduar with respect to the binary vector x. We extend this approach to tae the cardinaity constraint e x into account and provide a fu description of the convex hu of P when the vector a has identica components. We aso deveop a famiy of facet-defining inequaities when the vector a has nonidentica components. Computationa resuts using the proposed inequaities in a branch-and-cut framewor to sove mean-ris napsac probems show significant decrease in both time and the number of nodes over standard methods. 1 Introduction Optimization probems in various appications invoving economies of scae or ris averse behavior can be formuated as concave cost combinatoria optimization probems of the form { L (CCO) : min f (a x) : x X {0, 1n, =1 where f : R R is concave and a R n for = 1,..., L. The set X denotes combinatoria constraints, for exampe, those modeing feasibe paths, assignments or napsac soutions. Some specific exampes of (CCO) are mentioned next. Concave cost faciity ocation: The concave cost faciity ocation probem is an extension of the usua faciity ocation probem to mode economies of scae that can be achieved by connecting mutipe customers to the same faciity. With m customers and n faciities, the probem can be formuated as min { m i=1 n c i x i + n f ( m i=1 d i x i ) : n x i = 1 i, x i {0, 1 i,, where c i is the cost of assigning customer i to faciity, d i is the demand of customer i, and f is a nondecreasing concave cost function modeing the economies of scae of serving demand from faciity. This (CCO) probem generaizes the we-nown fixed-charge faciity ocation probem and has been studied in [9, 16, 24, 25]. Corresponding Author. Emai: sahmed@isye.gatech.edu 1

2 Mean-ris combinatoria optimization: Consider a stochastic combinatoria optimization probem min{ c x : x X {0, 1 n where the cost vector has independenty distributed components c i with mean µ i and variance ν i for i = 1,..., n. A typica deterministic equivaent formuation is to optimize a weighted combination of the mean and standard deviation of the cost, i.e. { min µ x + λ ν x : x X {0, 1 n, where λ is a nonnegative weight. Such mean-ris combinatoria optimization probems in various settings have been studied in [3, 18, 21, 23]. Approximate submoduar minimization: Various combinatoria optimization probems with nondecreasing submoduar cost functions of the form min {F(x) : x X {0, 1 n have been considered (cf. [5, 12, 14, 19]). Often the cost function F is ony avaiabe through a vaue orace. It has been shown [13] that a genera nondecreasing submoduar function F can be approximated upto a factor of n og n by a function of the form f (x) = c x using ony a poynomia number of function vaue queries, i.e. f (x) F(x) O( n og n) f (x) for a x {0, 1 n. Thus we can write an expicit formuation for an approximate version of the above submoduar cost combinatoria optimization probem as the (CCO) probem: min { c x : x X {0, 1 n. With a view towards deveoping mixed integer inear programming (MILP) based approaches for (CCO) we consider the foowing reformuation { L (CCO ) : min w : x X {0, 1 n, (w, x) conv(q ) = 1,..., L, =1 where each set Q for = 1,..., L is of the form Q = {(w, x) R {0, 1 n : w f (a x), (1) and f is a concave function. Note that since Q is the union of a finite set of haf ines with a common recession direction, its convex hu, denoted by conv(q), is a poyhedron. When the vector a is nonnegative (or nonpositive), the function F(x) := f (a x) is submoduar over the binary hypercube. A function of binary variabes F : {0, 1 n R is submoduar if for a x, y {0, 1 n with x y (component-wise) and for some i {1,..., n with x i = y i = 0, we have F(x + e i ) F(x) F(y + e i ) F(y), where e i R n is the i-th unit vector, i.e. the margina vaues are diminishing [27]. By the cassica resuts of Edmonds [8] and Lovasz [20] on submoduar functions, we now an expicit inequaity description of conv(q). Thus (CCO ) provides a mixed integer inear programming formuation of the mixed integer noninear program (CCO). In this paper we improve such a formuation by incorporating information from the constraints x X in the set Q. In particuar we study the foowing mixed-integer set P = {(w, x) R {0, 1 n : w f (a x), e x, (2) where f is a concave function, a R n +, Z + and e is a vector of ones, and the cardinaity constraint e x is assumed to be impied by the constraints x X. The contributions of this paper are as foows. First, we give a compete description of conv(p) when the vector a in (2) has identica components. Simiar to the cassica wor in [8], we write a pair of prima-dua inear programs corresponding to the convex ower enveope of f (a x) and vaid inequaities of conv(p) respectivey. Then we construct a pair of optima prima and dua soutions that provide the expicit inequaity description of conv(p). Second, we give a famiy of facet-defining inequaities of conv(p) for genera nonnegative a. We obtain such inequaities through sequence dependent exact ifting and present a poynomia time agorithm to find the ifting coefficients corresponding to any given sequence. We give 2

3 another famiy of approximatey ifted inequaities that can be weaer than the exacty ifted facet-defining inequaities but much faster to compute within a branch-and-cut procedure. Finay, we demonstrate the effectiveness of using the proposed inequaities in a branch-and-cut framewor to sove mean-ris napsac probems. Our computationa resuts show significant decrease in both time and the number of nodes over standard methods. We cose this section by a brief discussion of reated iterature. As mentioned earier, an inequaity description of the convex hu of the epigraph of a genera submoduar function foows from the cassica wors [8, 20]. In the presence of constraints, submoduar minimization is in genera NP-hard and in most cases very hard to approximate [12, 19, 26]. Even for the cardinaity constraint, the specia case of size constrained minimum cut probem is NP-hard since it can be reduced to graph partitioning probem [10]. This suggest that there is itte hope of obtaining a tractabe inequaity description of the convex hu of the epigraph of a submoduar function under a cardinaity constrained. However the submoduar function considered here is of the specia form F(x) = f (a x) where f is a concave function. Hassin and Tamir [17] sove the probem min{c x + f (a x) : e x, x {0, 1 n where f is a concave function, in poynomia time by reducing it to a two-dimension parametric inear programming probem. Onn and Rothbum [22] generaize the univariate concave function to mutivariate case. From the equivaence of separation and optimization this suggests that for such a specia submoduar function a tractabe description of the convex hu of the epigraph under a cardinaity constraint is possibe. However, to the best of our nowedge an expicit inequaity description of such a set has not been presented before. Recenty a number of wors have used vaid inequaities expoiting the submoduarity of underying functions within mixed integer inear programming approaches for various casses of mixed integer noninear programs [1 3, 5]. In particuar, the wor of Atamtür and Narayanan [3] is most reated to our wor. They use the inequaities describing the convex hu of the epigraph of a submoduar function whithout constraints, i.e. the set conv(q), to strengthen a second order cone programming (SOCP) reaxation of a mean-ris optimization probem in a branch-and-cut procedure and show that facet-defining inequaities of conv(q) significanty improves the performance. Our wor extends this approach to incude information from a cardinaity constraint. 2 Vaid Inequaities for conv(p) In this section, we study vaid inequaities of the convex hu of the set (2): P = { (w, x) R {0, 1 n : w f (a x), e x, where f is a concave function and a is a nonnegative vector. The foowing notation wi be used throughout. Let N = {1,..., n. Denote a(s) := i S a i for any S N. Given a permutation σ := ((1),..., (n)) of N, et A (i) = i a () for a (i) σ. Reca the unconstrained version of P, i.e. the set Q = {(w, x) R {0, 1 n : w f (a x). Using the fact that f (a x) is submoduar over {0, 1 n it is nown [8, 20] that every facet of conv(q), except the trivia bounds on x, is of the form f (0) + ρ (i) x (i) w. (3) (i) σ where ρ (i) = f (A (i 1) + a (i) ) f (A (i 1) ) (4) for i 1 and σ is some permutation of N. Thus a compete inequaity description of conv(q) is given by the inequaities (3) corresponding to a permutations. Athough there is an exponentia number of such inequaities, they can be separated in poynomia time. Foowing [3] we refer to the inequaities (3) as extended poymatroid inequaities (EPI). Ceary EP inequaities are vaid for conv(p). Next we strengthen the EP inequaities using the information from the cardinaity constraint in P. Given a (w, x) R [0, 1] n with x (1) x (2) x (n) we can chec if (w, x) conv(p) by soving the the foowing (dua) inear program: 3

4 max π 0 + s.t. n i=1 π (i) x (i) π 0 + π (i) f (a(s)) S, S. (i) S It foows that, given any permutation σ = {(1),..., (n), an inequaity π 0 + π (i) x (i) w is vaid for conv(p) if and ony if π 0, π is a feasibe soution of (5). Aso notice that coefficients of the EP inequaity defined by (4) form a feasibe soution of (5) since the EP inequaity is vaid for conv(p). Thus we can get an inequaity description of conv(p) if we obtained a optima soutions to (5) corresponding to a vectors x. The construction is compicated by the fact that, unie the unconstrained case, an optima soution to (5) depends on the specific vaue of the vector x rather than ust the permutation impied by its components. In the foowing subsection we show that we can do this construction and obtain a description of conv(p) when a components of a are identica which we refer to as the unweighted case. In Section 2.2 we use a different approach, ifting, to derive vaid inequaities for the weighted case where the vector a has nonidentica components. 2.1 Unweighted case: Identica Components In this section we consider the case where a 1 = = a n = a. Without oss of generaity, uness otherwise specified, we consider the permutation (1,..., n). We simpify the notation by denoting f (A i ) = f (a i) as f (i) and write the coefficients (4) of the EP inequaity as ρ i = f (i) f (i 1) for i N. We aso denote f (a(s)) = f ( S ) for any S N. Given a vector x {x [0, 1] n : e x such that x 1 x 2 x n we wi next construct an optima soution to (5). First we define a critica index i 0 {0, 1,..., 1 corresponding to x as foows. Let x 0 := 1 and z i := ( i)x i 1 =i+1 x for i {0, 1,..., so that z = 0, z 1 = x 1, z 2 = 2x 2 x 1,..., z 0 = 1 x. Let y := n = x and i 0 := argmax {0 i 1 : z i+1 y z i. (6) Note that i 0 is we defined since the range [z i+1, z i ] is vaid for any i {0, 1,..., as z i z i+1 = ( i)(x i x i+1 ) 0, and y wi be in some interva [z i+1, z i ]. The atter foows from the fact that y z = 0 and y z 0 since z 0 y = i=1 n x i 0 because x satisfies e x. Given i 0 {0,..., 1, we construct the foowing soution to (5): f (0) = 0 ρ i 0 π := f () f (i 0 ) i 0 > i 0. We wi first show that the soution constructed above corresponding to any i 0 {0,..., 1 is feasibe to (5). Then we wi show that if i 0 is constructed as in (6) then it is optima. We wi need the foowing resut. Lemma Given a concave function f : R R, for positive integers t 1 > t 2, t 3 > t 4, t 2 t 4 and t 1 t 2 t 3 t 4, we have f (t 1 ) f (t 2 ) f (t 3) f (t 4 ). t 1 t 2 t 3 t 4 (5) (7) 4

5 Proof. For any integer i, we define ρ i = f (i) f (i 1). Then we have f (t 3 ) f (t 4 ) t 3 t 4 = t 3 1 t ρ 4 t 3 t 4 t 2+t 3 t 4 1 =t ρ 2 t 3 t 4 t 2+t 3 t 4 1 =t ρ 2 + t 1 1 =t 2 +t 3 t ρ 4 t 3 t 4 + t 1 t 2 t 3 + t 4 = f (t 1) f (t 2 ), t 1 t 2 where the first inequaity foows from the fact ρ ρ when (by concavity) and t 2 t 4 ; and the second inequaity foows from the same fact and t 1 t 2 t 3 t 4. Proposition The soution π in (7) corresponding to any i 0 {0,..., 1 is feasibe for probem (5). Proof. For an i 0 {0,..., 1, consider an arbitrary set S such that S. Let S 1 = S {1,..., i 0 and S 2 = S\S 1. Let i 1 = S 1, i 2 = S 2. Then from the construction (7), π satisfies π 0 + π i = π 0 + π i + π i f (i 1 ) + π i = f (i 1 ) + i 2 f () f (i 0). i i S i S 1 i S 2 i S 2 0 The first inequaity in the above chain foows from the fact that for i S 1, π i s are coefficients of EPI and hence satisfies π 0 + i S1 π i f (S 1 ) = f (i 1 ). The second equaity foows from the definition of the dua soution in (7). If i 2 = 0, then we aready now it is feasibe. Assume i 2 > 0, we need to show f (i 1 ) + i 2 f () f (i 0) i 0 f (i 1 + i 2 ), which is equivaent to We consider two cases: f () f (i 0 ) i 0 f (i 1 + i 2 ) f (i 1 ) i 2. (*) i 2 i 0 : Inequaity (*) foows from Lemma by setting t 1 =, t 2 = i 0, t 3 = i 1 + i 2 and t 4 = i 1. Note that t 2 t 4 since i 0 i 1 and t 1 t 2 t 3 t 4 since i 0 i 2. i 2 > i 0 : First by concavity, we now We aso have i 2 ( f () f (i 1 + i 2 )) i 2 ( f (i) f (i 1 + i 2 ( i 0 ))). (i 2 + i 0 )( f (i 1 + i 2 ) f (i 1 )) i 2 ( f (i 1 + i 2 ( i 0 )) f (i 1 )) from Lemma by setting t 1 = i 1 + i 2, t 2 = i 1, t 3 = i 1 + i 2 + i 0 and t 4 = i 1. To chec that the condition of Lemma is satisfied, first note t 3 = i 1 + i 2 ( i 0 ) 0, then t 2 = t 4, and finay t 1 t 2 = i 2 t 3 t 4 = i 2 + i 0 since i 0. Add these two inequaities together, and after rearrangement, we get inequaity (*). Next we show that when i 0 is constructed as in (6) then π constructed as in (7) is an optima soution to probem (5). We proceed by constructing a compementary soution to the foowing prima probem 5

6 corresponding to (5): min s.t. We construct a soution to (8) as foows: Note that if i 0 = 0, then P(S) f ( S ) S P(S) = x i S:i S, S P(S) = 1 S i N P(S) 0, S, S. P( ) = 1 x 1 ; P({1,... ) = x x +1 < i 0 ; and P({1,..., i 0 ) = z i 0 y i 0. (9) P( ) = z 0 y. Let S := {S : S {i 0 + 1,..., n, S = i 0. For sets S S we set P(S) according to those Lemma beow. A remaining sets S N with S are assigned P(S) = 0. Lemma Suppose i 0 is constructed as in (6) then the foowing inear system in P(S) for S S P({1,..., i 0 S) = x i, i {i 0 + 1,..., n S:i S (8) has a nonnegative soution and Proof. See Appendix. P({1,..., i 0 S) = n i=i 0 +1 x i. i S S 0 Lemma Suppose i 0 is constructed as in (6) then the soution P(S) defined by (9) and Lemma is a feasibe soution to the prima probem (8). Proof. First we chec that S:i S, S P(S) = x i for a i N. If i > i 0, this hods by Lemma For i i 0, we have (9), Lemma 2.1.3, and the definitions of z i0 and y: i 0 P({1,..., ) + P(S {1,..., i 0 ) = x i x i0 + z i 0 y + n =i 0 +1 x i =i S S 0 i 0 = x i x i0 + ( i 0)x i0 n =i 0 +1 x i 0 = x i. + n =i 0 +1 x i 0 Next we chec that the P(S) s sum up to one. For i 0 > 0, we have by the same steps as above for i = 1 and the construction of P( ): P( ) + i 0 P({1,..., ) + P(S {1,..., i 0 ) = 1 x 1 + x 1 = 1. S S For i 0 = 0, we have P( ) + P(S) = n x + n x S S = 1. 6

7 Proposition Suppose i 0 is constructed as in (6) then the soution π constructed as in (7) is an optima soution for probem (5). Proof. By Lemma and Lemma 2.1.4, we aready now we have a pair of prima and dua feasibe soutions. Now we verify that the obective vaue of the prima soution is the same as the dua soution. For i 0 > 0, we have For i 0 = 0, we have f ( )P( ) + = f (0) + i 0 i 0 =1 P({1,..., ) f () + P(S {1,..., i 0 ) f () S S ( f () f ( 1))x + n =i 0 +1 f ( )P( ) + P(S) f () S S n f () f (i 0 ) i 0 x. = n x f (0) + n x f () f () f (0) = f (0) + x Theorem When a 1 = = a n = a then conv(p) is defined by the trivia inequaities x [0, 1] n, e x and the foowing inequaities f (0) + i 0 ( f () f ( 1))x () + n =i 0 +1 f () f (i 0 ) i 0 x () w (10) corresponding to every permutation σ = {(1),..., (n) of N and i 0 {0, 1,..., 1, where f () = f (a ) for N. Moreover given a x [0, 1] n, we can decide whether x conv(p) and find a vioated inequaity in O(n og n) time. Proof. The first part foows from Propositions and Given x [0, 1] n we can first chec in O(n) if e x. If yes, then to compute the coefficients of (10), first we sort the components of x, which taes O(n og n) time. Then finding the desired i 0 taes O(n) time. Once i 0 is found, the coefficients can be computed according to (7) in O(n) time. Therefore we can chec for a vioated inequaity of the form (10) in O(n og n) time. We refer to the inequaities (10) as separation inequaities (SI) since they can be exacty separated. 2.2 Weighted case: Nonindentica Components Now we consider the more genera case where the components of the nonnegative vector a are not necessariy identica. We derive a famiy of facet-defining inequaities of conv(p) through ifting. Consider a set S N such that S =. Without oss of generaity, we consider the set S = {1,...,. The restriction of P by setting x i = 0 for a i N\S is denoted by: { P 0 = (w, x) R {0, 1 S : w f (a x). Since there is no constraint on x in P 0, we now that the extend poymatroid inequaity (3) is facet-defining for P 0 : f (0) + i=1 7 ρ i x i w (11)

8 We then ift variabes x +1,..., x n sequentiay in that order. The intermediate set of feasibe points are: { P i = (w, x) R {0, 1 {1,...,,...,i : w f (a x), e x i = + 1,..., n. Given a facet-defining inequaity f (0) + i 1 ρ x + > ζ x w for conv(p i 1 ), the ifting probem associated with P i is: ζ i := min s.t. w f (0) ( i 1 ρ i x > ζ x ) f a x + a i w <i x 1, x {0, 1 = 1,..., i 1 <i (12) From [15], We have the foowing resut. Lemma If the ifting coefficients ζ for = + 1,..., i are computed as in (12) then the inequaity is facet-defining for conv(p i ). f (0) + ρ x + i > ζ x w Lemma The ifting coefficient ζ i in (12) can be computed in O(i 3 ) time. Proof. The optimization probem (12) is equivaent to the foowing probem of minimizing concave function over cardinaity constraint min { f ( <i a x + a i ) i 1 ρ x > ζ x, x 1, x {0, 1 = 1,..., i 1. <i This probem can be reduced to a two-dimension parametric inear programming probem and soved by enumerating a its O(i 2 ) extreme points. Atamtür and Narayanan give such an agorithm in [4] that finds an optima soution in O(i 3 ). The foowing theorem directy foows from Lemma and Lemma Theorem For a permutation ((1),..., (n)), the inequaity is facet-defining for conv(p), and can be computed in time O(n 4 ). f (0) + ρ (i) x (i) + ζ (i) x (i) w (13) i i> Next we reate the above inequaity to the separation inequaities (10) when a has identica components. Proposition If the components of a are identica, then ζ i = f () f ( 1) for i >. Moreover for a i >, the ifted coefficient ζ i = π i, where π i is given by (7) corresponding to i 0 = 1. 8

9 Proof. Denote f ( i a i x i ) as f ( i x i ) and f (a i) as f (i). We have { ( ) ζ i = min x {0,1 i min x {0,1 i f <i = f () f ( 1), x + 1 f (0) { ( ) ( f x + 1 <i f ) x <i i 1 ρ x > : x 1 <i ζ x : x 1 where the inequaity comes from the vaidity of the coefficients ρ for 1 and ζ for >, i.e., f (0) + ρ x + i 1 > ζ x f ( <i x ) and the equaity comes from the concavity of f. However ζ i f () f ( 1) since x = 1 for {1,..., 1 is a soution of (12). Finay we observe that ζ i = π i defined in (7) when i 0 = 1 and i >. The O(n 4 ) time compexity of the exact ifted inequaity (13) mae it computationay ineffective in a branch-and-cut framewor. Next, we derive another set of approximatey ifted vaid inequaities that wi be used in our computationa experiments. Proposition For i >, et T (i) = argmax {a(t) : T {(1),..., (i 1), T = 1, and γ (i) = f (a(t (i) ) + a (i) ) f (a(t (i) ). The inequaity, is vaid for conv(p) and can be computed in O(n og n) time. f (0) + ρ (i) x (i) + γ (i) x (i) w (14) i i> Proof. Without oss of generaity, we fix the sequence as (1,..., n) in the proof. Let Ti {1,..., i 1 denote the support of an optima soution of (12), i.e. x = 1 for Ti is an optima soution. We prove the statement by showing that for every i >, γ i ζ i : <i γ i = f (a(t i ) + a i ) f (a(t i )) f (a(t i ) + a i) f (a(t i )) f (a(ti ) + a i) f (0) ρ Ti, = ζ i. T i,> ζ Above the first inequaity is by concavity of f and the fact that a(t i ) a(ti ). The second inequaity comes from the vaidity of the coefficients ρ for 1 and ζ for >, i.e., f (0) + T i, ρ + T i,> ζ f (a(ti )). Finay the ast equaity is by definition of ζ i. For time compexity, the ρ i s can be computed in time O(). Then for γ i, we maintain a sorted ist of a 1,..., a i 1 and the sum of its 1 argest items. For a new a i, we insert it into the sorted ist and update the sum, which taes O(og n) time. Therefore the tota time wi be O(n og n). We refer to the approximatey ifted inequaity (14) as a ifted inequaity (LI) in the remainder of the paper. Remar If the components of a are identica then γ i = f () f ( 1), and therefore γ i = ζ i in this case. We aso have the foowing property that the (LI) inequaity is at east as strong as the (EP) inequaity. Proposition γ (i) ρ (i) 0, and is positive if f is stricty monotone. 9

10 Proof. Since ( ( ) ( )) γ (i) ρ (i) = f (a(t (i) + a (i) ) f (a(t (i) ) f a () + a (i) f a (), <i <i and we now from the definition in (14) that a(t (i) ) < <i a () if i >, therefore by concavity γ (i) ρ (i). It is then positive if f is stricty monotone. 3 Computationa Resuts In this section we demonstrate the effectiveness of the proposed separation inequaities (10) and the ifted inequaities (14) for soving the foowing cass of mean-ris napsac probems [3]: { min λ x + Ω(ɛ) ν x : b x B, x {0, 1 n. (15) As discussed in Section 1, probem (15) invoves a weighted combination of the mean and standard deviation of the tota vaue of a napsac, where individua item vaues are independent stochastic with mean µ i and variance ν i for i = 1,..., n. Foowing [3, 6, 11] the tradeoff between the mean and standard deviation 1 ɛ of the obective is set as Ω(ɛ) = ɛ where ɛ (0, 1) represents a ris aversion parameter. The vector of item weights and napsac capacity are denoted by b and B, respectivey. Probem (15) can be formuated as a mixed-integer second order cone program (SOCP): { min λ x + Ω(ɛ)w : w ν i xi 2, b x B, x {0, 1 n. (16) i We incorporate the proposed inequaities in a branch-and-cut framewor for the above formuation. Note that inequaities derived in the previous section are for the cardinaity constraint e x, whie the constraint in (16) is a napsac. We derive a cardinaity constraint from the napsac constraint as a simpe cover inequaity as foows. Sort b i such that b 1 b n, and find an index such that b 1 + b B and b b +1 > B. Then the constraint e x is vaid for b x B. The impementation, written in python, is based on the mixed integer second order cone programming sover of Gurobi 5.6. The continuous reaxation of (16) at each node is soved using the Barrier method. We restrict the number of submoduar inequaities added to at most five. Gurobi s interna cut parameters are in defaut setting. We disabe mutithreading, heuristics, and the concurrent MIP sover; and set the reative MIP optimaity gap as 0.01% and the time imit of the computation to 30 minutes. A computations are on an Inte Xeon server with 16 cores and 7GB memory restriction. Our computationa experiments use randomy generated instances of (16). Our parameters setting generay foow the ones in [3]. For the weighted case, each component of λ and b is generated from a uniform distribution with range [0, 100]. The ris aversion parameter ɛ is from {0.01, 0.02, It is varied to observe the reationship between the weight on the noninear term in the obective and running time. For each component ν i of the variance vector, its square root is uniformy generated in the range [0, αλ i ] where α is from {0.5, 0.75, 1. Such generation ensure that ν i /λ i α. Finay we set B = i b i /r where r is aso a varied parameter. For a fixed b 1,..., b n, the arger r is, the smaer B is, thus the smaer is. Therefore the parameter r contros the right-hand-side of the cardinaity constraint. For the unweighted case, the uniform distribution of components in λ is changed to [1, 5] and every component of ν i is identica and equa to a number whose square root is generated according to the uniform distribution with range [1, α min i {λ i ]. The reasons for the changes are the foowing. First we need the upper bound min i {λ i to ensure ν i αλ i for every i. Second if we used a ower bound of 0 for λ, it sometimes generates very sma λ i which then maes the non-inear part of the probem negigibe. Thirdy, if we used the origina upper bound of 100 for λ, instances for the unweighted case coud be soved in ess than one second maing them too trivia. The experiments are performed over a combinations of the parameters n, α, ɛ and r; and for each combination we generate 20 instances. For each instance, we sove the probem using three different approaches: the SOCP formuation, SOCP formuation with the extended poymatroid inequaities (3) denoted by EPI, 10

11 SOCP EPI SI SOCP EPI SI (a) Weighed case (b) Unweighted Figure 1: Performance Profie and the SOCP formuation with ifted inequaities (14) denoted by LI (or separation inequaities (7) if unweighted, denoted by SI). Figure 1 presents the performance profies of time for the weighted and unweighted cases. Foowing Doan and Moré in [7], the performance profies are constructed as foows. We have a set S of three sovers: SOCP, EPI, LI/SI, and a set of probem instances P. For a sover s S and a probem instance t p P, we cacuate its performance ratio r p,s = p,s min{t p,s : s S where t p,s is the time required by sover s on instance p. Figure 1 pots the cumuative distribution function of the performance ratio defined as g s (τ) = 1 { p P : r P p,s τ. Observe that in both cases the proposed inequaities are uniformy better than SOCP formuation and SOCP with EPI. The improvement in the unweighted cases is very significant since in this case we have a compete description of the convex hu. In Tabes 1 and 2, we provide more detais of the performance improvement. These tabes report, for both the weighted and unweighted cases, the node counts, the soution times, and the corresponding percentage of improvements over EPI across various parameter settings. We note that with LI or SI, performance is better in amost every parameter group. We aso observe that the weighted case taes much more time and nodes than the unweighted case. When the parameter α, that contro the range of the variance ν i, gets smaer, the time and node needed decrease most significanty, compared with other parameters. This is because when α is sma, the non-inear part of the constraint is ess import. For both weighted and unweighted case, smaer α means that LI/SI cuts ead to better improvements. For ɛ, a smaer vaue puts more weight on the difficut noninear part of the obective and therefore the running time is onger. For the parameter r, we see that in the weighted case, arger r resuts in better improvement in case of the LI cuts. This observation matches our theoretica resuts since arger r indicates smaer which impies that there wi be more different coefficients in the LI cuts compared with EPI cuts. 11

12 SOCP EPI LI Nodes Time Nodes Time Nodes Improvement Time Improvement n = % % n = % % α = % % α = % % α = % % ɛ = % % ɛ = % % ɛ = % % r = % % r = % % r = % % Tabe 1: Performance Comparison for Weighted Case SOCP EPI SI Nodes Time Nodes Time Nodes Improvement Time Improvement n = % % n = % % α = % % α = % % α = % % ɛ = % % ɛ = % % ɛ = % % r = % % r = % % r = % % Tabe 2: Performance Comparison for Unweighted Case 12

13 Acnowedgement This research has been supported in part by the Nationa Science Foundation (Grant ) and the Air Force Office of Scientific Research (Grant FA ). References [1] S. AHMED AND A. ATAMTÜRK, Maximizing a cass of submoduar utiity functions, Mathematica programming, 128 (2011), pp [2] S. AHMED AND D. J. PAPAGEORGIOU, Probabiistic set covering with correations, Operations Research, 61 (2013), pp [3] A. ATAMTÜRK AND V. NARAYANAN, Poymatroids and mean-ris minimization in discrete optimization, Operations Research Letters, 36 (2008), pp [4] A. ATAMTÜRK AND V. NARAYANAN, The submoduar napsac poytope, Discrete Optimization, 6 (2009), pp [5] F. BAUMANN, S. BERCKEY, AND C. BUCHHEIM, Exact agorithms for combinatoria optimization probems with submoduar obective functions, in Facets of Combinatoria Optimization, M. Jünger and G. Reinet, eds., Springer Berin Heideberg, 2013, pp [6] D. BERTSIMAS AND I. POPESCU, Optima inequaities in probabiity theory: A convex optimization approach, SIAM Journa on Optimization, 15 (2005), pp [7] E. D. DOLAN AND J. J. MORÉ, Benchmaring optimization software with performance profies, Mathematica Programming, 91 (2002), pp [8] J. EDMONDS, Submoduar functions, matroids, and certain poyhedra, in Combinatoria Structures and Their Appications, Gordon and Breach, 1971, pp [9] E. FELDMAN, F. LEHRER, AND T. RAY, Warehouse ocation under continuous economies of scae, Management Science, 12 (1966), pp [10] M. R. GAREY AND D. S. JOHNSON, Computers and Intractabiity; A Guide to the Theory of NP- Competeness, W. H. Freeman & Co., New Yor, NY, USA, [11] L. E. GHAOUI, M. OKS, AND F. OUSTRY, Worst-case vaue-at-ris and robust portfoio optimization: A conic programming approach, Operations Research, 51 (2003), pp [12] G. GOEL, C. KARANDE, P. TRIPATHI, AND L. WANG, Approximabiity of combinatoria probems with muti-agent submoduar cost functions, in Proceedings of the 50th Annua IEEE Symposium on Foundations of Computer Science, FOCS, IEEE, 2009, pp [13] M. X. GOEMANS, N. J. A. HARVEY, S. IWATA, AND V. MIRROKNI, Approximating submoduar functions everywhere, in SODA 09: Proceedings of the twentieth Annua ACM-SIAM Symposium on Discrete Agorithms, Phiadephia, PA, USA, 2009, Society for Industria and Appied Mathematics, pp [14] M. X. GOEMANS AND J. A. SOTO, Symmetric submoduar function minimization under hereditary famiy constraints, CoRR, abs/ (2010). [15] Z. GU, G. L. NEMHAUSER, AND M. W. SAVELSBERGH, Sequence independent ifting in mixed integer programming, Journa of Combinatoria Optimization, 4 (2000), pp [16] M. HAJIAGHAYI, M. MAHDIAN, AND V. MIRROKNI, The faciity ocation probem with genera cost functions, Networs, 42 (2003), pp

14 [17] R. HASSIN AND A. TAMIR, Maximizing casses of two-parameter obectives over matroids, Mathematics of Operations Research, 14 (1989), pp [18] H. ISHII, S. SHIODE, T. NISHIDA, AND Y. NAMASUYA, Stochastic spanning tree probem, Discrete Appied Mathematics, 3 (1981), pp Specia Copy. [19] S. IWATA AND K. NAGANO, Submoduar function minimization under covering constraints, in Proceedings of the 50th Annua IEEE Symposium on Foundations of Computer Science, FOCS, IEEE, 2009, pp [20] L. LOVÁSZ, Submoduar functions and convexity, in Mathematica Programming The State of the Art, Springer, 1983, pp [21] E. NIKOLOVA, Approximation agorithms for reiabe stochastic combinatoria optimization, in Proceedings of the 13th Internationa Worshop on Approximation Agorithms for Combinatoria Optimization Probems, APPROX, Springer, 2010, pp [22] S. ONN AND U. G. ROTHBLUM, Convex combinatoria optimization, Discrete & Computationa Geometry, 32 (2004), pp [23] L. OZSEN, C. R. COULLARD, AND M. S. DASKIN, Capacitated warehouse ocation mode with ris pooing, Nava Research Logistics (NRL), 55 (2008), pp [24] H. E. ROMEIJN, T. C. SHARKEY, Z.-J. M. SHEN, AND J. ZHANG, Integrating faciity ocation and production panning decisions, Networs, 55 (2010), pp [25] D. STRATILA, Combinatoria Optimization Probems with Concave Costs, PhD thesis, MIT, [26] Z. SVITKINA AND L. FLEISCHER, Submoduar approximation: Samping-based agorithms and ower bounds, SIAM Journa on Computing, 40 (2011), pp [27] D. M. TOPKIS, Supermoduarity and compementarity, Princeton University Press, Appendix: Proof of Lemma Lemma Suppose i 0 is constructed as in (6) then the foowing inear system in P(S) for S S has a nonnegative soution and P({1,..., i 0 S) = x i, i {i 0 + 1,..., n S:i S P({1,..., i 0 S) = n i=i 0 +1 x i. i S S 0 Proof. To avoid confusion with the origina notation, we change the notation as foows. Set m = n i 0, = i 0, v 1 = x i0 +1, v 2 = x i0 +2,..., v m = x n. We wi prove for m, Z +, m + 1, v 1 v 2 v m 0, and S = {S : S {1,..., m, S =, the foowing inear system in q(s), S S has a nonnegative soution and S S q(s) = m v. q(s) = v, {1,..., m, S: S We wi use Agorithm 1 to construct a feasibe soution. The main idea is for any set S, the associated variabe q t (S) wi remain nondecreasing in the iteration count t. For any, v t the right-hand-side vaue that has not been satisfied yet, and it wi remain nonincreasing in t. We wi eep v t+1 + S, S,S S q t+1 (S) = v aong the way. At the end of the procedure, every item wi have its v t = 0, and we find the soution as desired. We now expoit some properties of v t and qt (S). Initiay, we have 14

15 Agorithm 1: Recursive procedure t 0, m t m; v t v, 1 m t q t (S) 0, S S; whie m t > + 1 do C1 if v t mt vt vt m t then q t+1 ( { 1,..., 1, m t ) q( { 1,..., 1, m t ) + v t m t ; v t+1 v t vt m t, {1,..., 1; m t+1 m t 1; ese C2 = mt vt / vt +1 ; q t+1 ({1,..., ) q t ({1,..., ) + ; v t+1 v t, {1,..., ; m t+1 m t q t+1 (S) q t (S) for a the other S; SRT sort v1 t+1,..., v t+1 and index them so that v t+1 m t 1 v t+1 t t + 1; m t+1 ; q t ({1,..., + 1 \ {) q t ({1,..., + 1 \ {) + +1 τ=1 vt τ v t, {1,..., v 0 0 for every 2. q 0 (S) = 0 and v 0 + S, S,S S q 0 (S) = v 3. Since y z i0 +1, we have n = x ( i 0 1)x i =i 0 +2 x. Then m v = n =i 0 +1 x ( i 0 )x i0 +1 = v 1. We wi show that v s satisfy the foowing invariance after finishing ine SRT. 1. v t+1 0 for { 1,..., m t v t+1 + S, S,S S q t+1 (S) = v. 3. v t+1 1 mt+1 vt+1. First we prove v t+1 0. Case C1 is easy since v t vt m t for a. For Case C2 where v t > mt vt vt m t, the smaest v t+1 among the ones updated is v t = vt mt vt + v t +1 mt vt v t mt mt v + v t +1 = vt +1 vt m t 0. Then we show v t+1 + S, S,S S q t+1 (S) = v. Since in the oop each iteration we ony consider one set S, and for any item S, v t vt+1 = q t+1 (S) q t (S), the caim is true. Then we prove that v1 t+1 mt+1 vt+1 after ine SRT. In Case C1, v1 t+1 wi be either v1 t vt m t. Then mt+1 vt+1 = mt vt vt m t v1 t vt m t since the caim hods for the previous iteration, or v t. Then again mt+1 vt+1 = mt vt vt m t v t because we are at Case C1. 15

16 In the second case C2, first notice that after ine SRT, v1 t+1 = v t +1 since the other choice vt 1 = vt 1 mt vt + v t +1 vt +1 because vt 1 mt vt. Then m t+1 v t+1 = m t v t = m t v t mt vt v t +1 = v t +1. If Agorithm 1 terminates, we say that q t (S), S S is the soution desired. We consider two cases after the whie oop. For any > + 1, we have v t = 0 which is equivaent to S, S,S S q t (S) = v. For any + 1, before the ast ine, we have S, S,S S q t (S) = v v t, and after ast ine, we have ( q t (S) = v v t + S, S,S S = ) +1 τ=1 vt τ v t = v v t +1 + v t τ v t τ=1 = = v. If it does not terminate, we caim that q(s) = im t q t (S) for a S S is the soution desired. To prove that, we show that im t v t = 0. If this is true, then we get im t S, S,S S q t (S) = v. Notice that each time after we finish Case (C2), mt+1 vt+1 +1 mt vt because m t v t +1 v t m t+1 ( + 1)vt +1, and v t+1 = m t v t = v +1. Since the agorithm does not terminate, Case C2 happens infinitey many times. m0 v m, the vaue m t im t v t im t ( + 1 )t m = 0. Notice that initiay Since v t is aways nonnegative, im t mt vt = 0 and im t v t = 0. Now we cacuate the sum of a variabes. Given that we have q(s) = v, {1,..., m, S: S m S: S q(s) = Since each q(s) appears on the eft side exacty times, m v. m q(s) = v. S S 16