An Exact Method for Constrained Maximization of the Conditional Value-at-Risk of a Class of Stochastic Submodular Functions
aa r X i v : . [ m a t h . O C ] A p r An Exact Method for Constrained Maximization of the ConditionalValue at Risk of a Class of Stochastic Submodular Functions
Hao-Hsiang Wu
Department of Management Science, National Chiao Tung University, Hsinchu, Taiwan [email protected]
Simge K¨u¸c¨ukyavuz ∗ Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL [email protected]
April 17, 2020
Abstract:
We consider a class of risk-averse submodular maximization problems (RASM) where the objective is theconditional value-at-risk (CVaR) of a random nondecreasing submodular function at a given risk level. We proposevalid inequalities and an exact general method for solving RASM under the assumption that we have an efficientoracle that computes the CVaR of the random function. We demonstrate the proposed method on a stochastic setcovering problem that admits an efficient CVaR oracle for the random coverage function.
Keywords : conditional value-at-risk; stochastic programming; oracle; stochastic set covering; lifting; submodularmaximization
1. Introduction
We consider a class of risk-averse submodular maximization problems (RASM) re-cently formulated by Maehara [13]. Formally, let V = { , . . . , n } be a finite set, and ¯ Ω be a probabilityspace. We define an outcome mapping σ : 2 | V | × ¯ Ω → R . The random outcome σ ( X ) : ¯ Ω → R is defined by σ ( X )( ω ) = σ ( X, ω ) for all X ⊆ V and ω ∈ ¯ Ω . We assume that σ ( X )( ω ) is a nondecreasing submodular setfunction. With a slight abuse of notation, we refer to the submodular function σ ( X ) for X ⊆ V as σ ( x ) for x ∈ { , } n interchangeably, where x is the characteristic vector of X , and the usage will be clear from thecontext. We measure the risk associated with this function using conditional value-at-risk (CVaR), wherelarger function values correspond to less risky random outcomes. In this context, risk measures are referredto as acceptability functionals . CVaR was first introduced by Artzner et al. [3] and has been widely used,especially in finance, due to its desirable properties (e.g., coherence and tractability).Formally, let [ z ] + = max( z,
0) be the positive part of a number z ∈ R . Let η be a variable that measuresthe value-at-risk (VaR) at a given risk level. For a given ¯ x , the value-at-risk at a risk level α ∈ (0 ,
1] isdefined as VaR α ( σ (¯ x )) = max n η : P ( σ (¯ x ) ≥ η ) ≥ − α, η ∈ R o . (1)For a given ¯ x , the conditional value-at-risk at a risk level α ∈ (0 ,
1] is defined asCVaR α ( σ (¯ x )) = max n η − α E ([ η − σ (¯ x )] + ) : η ∈ R o , (2)and it provides the conditional expected value of σ (¯ x ) that is no larger than the value-at-risk at the risk level α [17]. In general, the risk level α is small, such as α = 0 .
05 or 0 .
01. Let X be the deterministic constraintson the variables x . Given a risk level α ∈ (0 , x ∈X CVaR α ( σ ( x )) . (3)Let E [ σ ( x )] represent the expectation of σ ( x ) with respect to ω . In the next observation, we reviewproperties of CVaR that will be useful in developing valid inequalities for RASM. Observation 1.1
From the definition of CVaR in (2) , for a given ¯ x ∈ X , we have(i) CVaR ( σ (¯ x )) = E [ σ (¯ x )] , and because E [ σ (¯ x )] is submodular (see, e.g., [24]), so is CVaR ( σ (¯ x )) .(ii) CVaR α ( σ (¯ x )) ≤ CVaR α ( σ (¯ x )) for α , α ∈ (0 , and α ≤ α . ∗ Corresponding author u and K¨u¸c¨ukyavuz: CVaR Maximization of Stochastic Submodular Functions α = 1, problem (3) is equivalent to the N P -hard risk-neutral submodularmaximization problem max x ∈X E [ σ ( x )] (see, e.g., [10, 11, 20] for the associated applications).Note that for a nondecreasing submodular function σ ( · ), Proposition 2 of [14] shows that the submodularinequality σ ( x ) ≤ σ ( S ) + X j ∈ V \ S ( σ ( { j } ∪ S ) − σ ( S )) x j , ∀ S ⊆ V, (4)is valid. Ahmed and Atamt¨urk [1] and Yu and Ahmed [26] strengthen the submodular inequalities by lifting,under the assumption that the submodular utility function is strictly concave, increasing, and differentiable,assumptions we do not make in this paper. However, Maehara [13] shows that CVaR of a stochastic sub-modular function is no longer submodular for any risk level α ∈ (0 , P = N P , there is no polynomial-time approximation algorithm with a multiplicative error for RASM undersome reasonable assumptions on the given risk level. Hence it is theoretically intractable to find a solutionthat is close to optimal in polynomial time without any assumption on the feasible region or the probabilitydistribution. Along this line of work, Zhou and Tokekar [28] propose a sequential greedy algorithm for theCVaR maximization problem in [13]. The authors show that the proposed algorithm gives a solution thatis within a constant factor of optimal with an additional additive term that depends on the optimal valueand a parameter related to the curvature of the submodular set function. However, the running time of thisalgorithm is exponential as it depends quadratically on the cardinality of the feasible set.Instead of solving the generic CVaR maximization problem in [13] directly, Ohsaka and Yoshida [16]consider a specific risk-averse submodular optimization problem arising in social networks. The authorsrelax the problem by replacing the combinatorial decisions that determine a single set of a predeterminedsize with a choice of a portfolio (convex combination) of sets of the given size. They also give a polynomial-time algorithm that obtains a portfolio with a CVaR value that has a provable guarantee with a specifiedprobability. Wilder [22] generalizes these results and proposes an approximation algorithm for maximizingthe CVaR of a generic monotone continuous submodular function (or a portfolio of dicrete sets) with a worst-case guarantee within (1 − e ) factor of the optimal solution. That is, a function F : ¯ V → R is continuoussubmodular if and only if F ( a ) + F ( b ) ≥ F ( a ∨ b ) + F ( a ∧ b ) for any a, b ∈ ¯ V ⊆ R n + , where ¯ V i is a compactsubset of R , ¯ V = Q ni =1 ¯ V i ⊆ R n + , and notations ∨ and ∧ denote the coordinate-wise minimum and maximumoperations, respectively (see, e.g., [6] and [7]). In contrast to this line of work that considers a portfolio ofdiscrete sets as decision variables, we consider the discrete case, where we must choose one set of decisionsfor implementability purposes. In another line of work, Wu and K¨u¸c¨ukyavuz [25] propose an exact methodfor the minimal cost selection of x ∈ X that satisfies the chance constraint P ( σ ( x ) ≥ τ ) for a given τ , wherethe authors assume that there exists a probability oracle for evaluating the chance constraint for a givensolution. However, the exact method of [25] cannot be applied to the CVaR maximization problem (or itsVaR maximization counterpart) directly.Note that for a given ¯ x ∈ X and a finite probability space ( Ω, Ω , P ) with a set of N ∈ N realizations(scenarios) Ω = { ω , . . . , ω N } , P ( ω i ) = p i for i = 1 , . . . N , CVaR is given by [17]CVaR α ( σ (¯ x )) = max n η − α X i ∈ [ N ] p i w i : w i ≥ η − σ i (¯ x ) , ∀ i ∈ [ N ] , w ∈ R N + , η ∈ R o , (5)where [ N ] = { , . . . , N } represents the set of the first N positive integers, σ i (¯ x ) = σ ( x )( ω i ), and σ q (¯ x ) =VaR α ( σ (¯ x )) for at least one ω q ∈ Ω . Therefore, when an oracle for evaluating CVaR is not available, wecan take a sampling-based approach. To this end, in [23], we apply the stochastic submodular optimizationmethods of [24] to problem (3) based on the CVaR definition (5). The resulting solutions are then testedout-of-sample and statistical performance guarantees are provided.In this paper, our goal is to give (near-)optimal solutions for RASM without sampling. We start byconsidering high-quality (ideally optimal) feasible solutions to RASM under a true (non-trivial) distributionof the uncertain parameters assuming that there is an efficient oracle to evaluate the true value of CVaR ofthe random function at a given risk level. To solve RASM, we propose a decomposition method with various u and K¨u¸c¨ukyavuz: CVaR Maximization of Stochastic Submodular Functions
2. RASM with an Exact CVaR Oracle
In this section, we assume that for a given incumbentsolution ¯ x ∈ X , we have an efficient oracle that computes CVaR α ( σ (¯ x )) exactly. Under this assumption,we solve problem (3) without sampling by using a two-stage optimization model and an exact algorithmwith various valid inequalities. In the proposed method, we solve a relaxed master problem (RMP) at anyiteration in the form max { ψ : ( x, ψ ) ∈ C , x ∈ X , ψ ∈ R } , (6)where ψ is a variable that is an upper bounding approximation of CVaR α ( σ ( x )) for a solution x ∈ X , and C is a set of optimality cuts to be defined later. It is an approximation in that not all necessary optimalitycuts may have been generated at an intermediate iteration. A full master problem includes all optimalitycuts in C such that for any x ∈ X , we have ψ ≤ CVaR α ( σ ( x )). Therefore, solving the full master problem isequivalent to solving the original problem (3). In Algorithm 1, we propose a delayed constraint generationalgorithm for solving problem (3) using RMP (6). The algorithm starts with a set of optimality cuts C (couldbe empty). In the while loop, we solve RMP (6) and obtain an incumbent solution (Line 3). Based on theincumbent solution, we add an optimality cut to RMP (6) (Line 5). In this algorithm, ǫ is a user-definedoptimality tolerance. Let UB be the upper bound obtained from the optimal objective value of RMP (6) ateach iteration. Let LB be the lower bound equal to CVaR α ( σ (¯ x )), obtained from calling the CVaR oraclewith the given incumbent solution ¯ x ∈ X as input. If the optimality gap is below ǫ , then we terminate thealgorithm and return the near-optimal solution. Algorithm 1:
A Delayed Constraint Generation Algorithm with a CVaR oracle Start with an initial set of optimality cuts in C (could be empty), UB= ∞ and LB= −∞ ; while UB-LB > ǫ do Solve RMP (6) and obtain ( ¯ ψ, ¯ x ). ; UB ← the optimal objective value of RMP (6), LB ← CVaR α ( σ (¯ x )); Add an optimality cut to C ; end Output ¯ x as the optimal solution.As we mentioned earlier, CVaR α ( σ ( x )) is not submodular in x even if σ ( x ) is submodular [13]. Hence,there is no direct way use the submodular inequality (4) as a class of optimality cuts in C . Therefore, in thissection, we propose new optimality cuts that are valid for (6). Throughout, we let e j be a unit vector ofdimension | V | whose j th component is 1, and let be a | V | -dimensional vector with all entries equal to 1.For a given ¯ x ∈ X , we first propose an optimality cut given by ψ ≤ CVaR α ( σ (¯ x )) + X j ∈ V \ ¯ X (cid:0) CVaR ( σ (¯ x + e j )) − CVaR α ( σ (¯ x )) (cid:1) x j . (7)Before formally proving the validity of inequality (7), a few remarks are in order. It may be temptingto think that inequality (7) is in the form of a submodular inequality (4). However, we highlight that thecoefficients (cid:0) CVaR ( σ (¯ x + e j )) − CVaR α ( σ (¯ x )) (cid:1) for j ∈ V \ ¯ X are different from their submodular counterparts (cid:0) CVaR α ( σ (¯ x + e j )) − CVaR α ( σ (¯ x )) (cid:1) for j ∈ V \ ¯ X and because CVaR α ( σ ( · )) is not submodular, the lattercoefficients are not valid. To provide some intuition for the validity of the coefficients of the variables x j , j ∈ V \ ¯ X in inequality (7), we consider the upper bound on ψ provided by the right-hand side of theinequality (recall that ψ ≤ CVaR α ( σ ( x )) for all x ∈ X ). First, observe that for x = ¯ x , inequality (7) holdstrivially, because x j = 0 for j ∈ V \ ¯ X . Now consider the point x = ¯ x + e j for some j ∈ V \ ¯ X . Thenthe right-hand side of inequality (7) is CVaR ( σ (¯ x + e j )), which is a valid upper bound on ψ (because ψ ≤ CVaR α ( σ (¯ x + e j )) ≤ CVaR ( σ (¯ x + e j )) from Observation 1.1 (ii)). The validity for other x is provenby exploiting the properties of CVaR α ( σ ( x )) given in Observation 1.1 as we show next. u and K¨u¸c¨ukyavuz: CVaR Maximization of Stochastic Submodular Functions Proposition 2.1
Inequality (7) for a given ¯ x ∈ B n is valid for RMP (6) . Proof.
Consider a feasible point ( ˆ ψ, ˆ x ) to the full master problem, in other words, let ˆ x ∈ X such thatˆ ψ ≤ CVaR α ( σ (ˆ x )). We show that ( ˆ ψ, ˆ x ) satisfies inequality (7) written for any ¯ x ∈ X . Let ˆ X = { i ∈ V : ˆ x i =1 } . From the definition of CVaR in (2), it follows that because σ ( x )( ω ) is nondecreasing in x for all ω ∈ ¯ Ω ,CVaR α ( σ ( x )) is also nondecreasing in x . For the case that ˆ X ⊆ ¯ X , since CVaR α ( σ ( x )) is a monotonicallynondecreasing function in x and ˆ x j = 0 for all j ∈ V \ ¯ X , we have ˆ ψ ≤ CVaR α ( σ (ˆ x )) ≤ CVaR α ( σ (¯ x )), whichshows that inequality (7) is valid for ˆ X ⊆ ¯ X . For the case that ˆ X \ ¯ X = ∅ , we select an arbitrary j ′ ∈ ˆ X \ ¯ X ,where ˆ x j ′ = 1. Thenˆ ψ ≤ CVaR α ( σ (ˆ x )) ≤ CVaR ( σ (ˆ x )) (8a) ≤ CVaR ( σ (¯ x )) + X j ∈ V \ ¯ X [CVaR ( σ (¯ x + e j )) − CVaR ( σ (¯ x ))]ˆ x j (8b)= X j ∈ V \ ¯ X CVaR ( σ (¯ x + e j ))ˆ x j − X j ∈ V \ ( ¯ X ∪{ j ′ } ) CVaR ( σ (¯ x ))ˆ x j (8c) ≤ X j ∈ V \ ¯ X CVaR ( σ (¯ x + e j ))ˆ x j − X j ∈ V \ ( ¯ X ∪{ j ′ } ) CVaR α ( σ (¯ x ))ˆ x j (8d)= (cid:18) X j ∈ V \ ¯ X CVaR ( σ (¯ x + e j ))ˆ x j − X j ∈ V \ ¯ X CVaR α ( σ (¯ x ))ˆ x j (cid:19) + CVaR α ( σ (¯ x ))ˆ x j ′ (8e)= CVaR α ( σ (¯ x )) + X j ∈ V \ ¯ X (CVaR ( σ (¯ x + e j )) − CVaR α ( σ (¯ x )))ˆ x j . (8f)Inequality (8a) follows from Observation 1.1 (ii) that CVaR α ( σ ( x )) is nondecreasing in α . Inequality (8b)follows from the submodular inequality (4) written for the monotone submodular function CVaR ( σ ( x )) (seeObservation 1.1 (i)) and for S = ¯ X evaluated at the point x = ˆ x . Arranging terms in inequality (8b) andrecalling that ˆ x j ′ = 1, we obtain equality (8c). Inequality (8d) follows from Observation 1.1 (ii), this timeobserving that − CVaR α ( σ ( x )) is nonincreasing in α . To obtain equality (8e), we add CVaR α ( σ (¯ x ))ˆ x j ′ − CVaR α ( σ (¯ x ))ˆ x j ′ to inequality (8d) and reorganize the terms. Finally, equality (8f) follows from ˆ x j ′ = 1.This completes the proof. (cid:3) Next, we introduce a class of valid inequalities obtained by a sequential lifting procedure [15, Proposition1.1 in Section II.2.1]. Given ¯ x ∈ X , which is a characteristic vector of the set ¯ X , consider a restriction with x j = 0 for j ∈ V \ ¯ X . For this restriction, we know that the base inequality ψ ≤ CVaR α ( σ (¯ x )) is valid,because for any x satisfying this restriction, we have X ⊆ ¯ X , and ψ = CVaR α ( σ ( x )) ≤ CVaR α ( σ (¯ x )) duethe property that CVaR α ( σ ( x )) is monotonically nondecreasing in x . However, this base inequality is notvalid when the restriction is lifted. To obtain a valid inequality, we sequentially lift the base inequality withthe variables x j , j ∈ V \ ˆ X . Let j , . . . , j r be an ordering of the elements in V \ ¯ X , where r = | V \ ¯ X | .Sequential lifting following this order produces a valid inequality ψ ≤ CVaR α ( σ (¯ x )) + r X i =1 δ j i (¯ x ) x j i , (9)where δ j t (¯ x ) for t = 1 , . . . , r is an exact lifting function given by the t -th lifting problem δ j t (¯ x ) = − CVaR α ( σ (¯ x )) + max CVaR α ( σ ( x )) − t − X i =1 δ j i (¯ x ) x j i (10a)s.t. x j t = 1 (10b) x j i = 0 , i = t + 1 , . . . , r (10c) x ∈ X . (10d) u and K¨u¸c¨ukyavuz: CVaR Maximization of Stochastic Submodular Functions t = 1, the term P t − i =1 ( · ) = 0. Note that in the t -th lifting problem,to obtain the coefficient of the variable x j t in inequality (9), we let x j t = 1 in constraint (10b), we removethe restriction on the variables preceding this variable in the lifting sequence, while keeping the restrictionthat the variables following x j t in the lifting sequence are fixed to zero in constraint (10c). The exactlifting problem is hard to solve since it is related to the submodular maximization problem (3). Instead ofsolving problem (10) exactly, we propose to solve a relaxation that provides an upper bound for the liftingcoefficients. Then, we can obtain another valid inequality by using the upper bounds on the coefficientsinstead of the exact coefficients. We describe this approach next.Given ¯ x ∈ X , we define ¯ δ j t (¯ x ) as an upper bound of δ j t (¯ x ) for t = 1 , . . . , r , where ¯ δ j t (¯ x ) is given by solvingthe following relaxation of the exact lifting problem (10):¯ δ j t (¯ x ) = max { CVaR α ( σ ( x )) , (10b) , (10c) , x ∈ B n } − CVaR α ( σ (¯ x )); (11)here we relax constraints (10d) and remove the term − P t − i =1 δ j i (¯ x ) x j i from the objective function in problem(10). This is a valid relaxation, because δ j i ( x ) ≥ α ( σ ( x )) is monotonically nondecreasing in x . Furthermore, because x j t = 1 and x j i = 0 for i = t + 1 , . . . , r , the feasible solution with x ∗ j i = 1 for i = 1 , . . . , t has the largest CVaR α ( σ ( x ∗ )) in the objective function of (11). Thus, the optimal solution ofthe relaxed lifting problem (11) is given by ¯ x j t = − P ri = t +1 e j i , where we can use an efficient oracle forCVaR α ( σ ( x )) to obtain the optimal value of ¯ δ j t (¯ x ) efficiently. Proposition 2.2
Given ¯ x ∈ X , its support ¯ X and an ordering of V \ ¯ X given by { j , . . . , j r } , inequality ψ ≤ CVaR α ( σ (¯ x )) + r X i =1 ¯ δ j i (¯ x ) x j i , (12) where ¯ δ j i (¯ x ) = CVaR α ( σ (¯ x j i )) − CVaR α ( σ (¯ x )) , is valid for RMP (6) . Proof.
Consider a feasible point ( ˆ ψ, ˆ x ). We show that ( ˆ ψ, ˆ x ) satisfies inequality (12) written for ¯ x ∈ B n . ψ ≤ CVaR α ( σ (¯ x )) + r X i =1 δ j i (¯ x )ˆ x j i (13a) ≤ CVaR α ( σ (¯ x )) + r X i =1 ¯ δ j i (¯ x )ˆ x j i . (13b)Inequality (13a) follows from the valid inequality (9). Inequality (13b) follows from ¯ δ j i (¯ x ) ≥ δ j i (¯ x ) for i = 1 , . . . , r since problem (11) is a relaxation of problem (10). This completes the proof. (cid:3) While inequality (12) is derived by approximate lifting, next we provide some intuition on the validity of thecoefficients of the variables x j , j ∈ V \ ¯ X in inequality (12). Consider the upper bound on ψ ≤ CVaR α ( σ ( x ))for some x ∈ X provided by the right-hand side of the inequality. First, observe that for x = ¯ x , inequality(12) holds trivially, because x j = 0 for j ∈ V \ ¯ X . Now consider the point x = ¯ x + e j t , for some t ∈{ , . . . , r } . Then the right-hand side of inequality (12) is CVaR α ( σ (¯ x j t )), which is a valid upper bound on ψ = CVaR α ( σ (¯ x + e j t )) from Observation 1.1 (ii). The approximate lifting argument establishes that theinequality is valid for other choices of x as well, because the coefficients of the variables are no smaller thanthose obtained from an exact lifting problem.In Algorithm 2, we propose a greedy method that generates an inequality (12) given an incumbent ¯ x . Inthe for loop, Line 3 determines the entry j i by choosing the candidate index s for which ¯ δ j i = s (¯ x ) attains itssmallest value, where the candidate s has not been chosen previously.Finally, we show the correctness of Algorithm 1 based on the inequalities (7) or (12). Proposition 2.3
Algorithm 1 with optimality cuts (7) and/or (12) finitely converges to an optimal solution. u and K¨u¸c¨ukyavuz:
CVaR Maximization of Stochastic Submodular Functions Algorithm 2:
GreedyUp-lifting( α, ¯ x ) S ← V \ ¯ X ; for i = 1 to r do j i ← arg min s ∈ S ¯ δ j i = s (¯ x ) ; x ∗ ← − P ri = t +1 e j i ; ¯ δ j i (¯ x ) ← ¯ δ j i ( x ∗ ) ; S ← S \ { j i } ; end Output an inequality (12) as the solution.
Proof.
From Proposition 2.1 and 2.2, for all ¯ X ⊆ V and x ∈ X , we have ψ ≤ CVaR α ( σ (¯ x )) ≤ CVaR α ( σ ( x )) + X j ∈ V \ ¯ X (cid:0) CVaR ( σ (¯ x + e j )) − CVaR α ( σ (¯ x )) (cid:1) x j , and ψ ≤ CVaR α ( σ (¯ x )) ≤ CVaR α ( σ ( x )) + r X i =1 ¯ δ j i (¯ x ) x j i . Hence, the following two linear integer programs,max { ψ : ψ ≤ CVaR α ( σ ( x )) + X j ∈ V \ ¯ X (cid:0) CVaR ( σ (¯ x + e j )) − CVaR α ( σ (¯ x )) (cid:1) x j ∀ ¯ X ⊆ V, x ∈ X , ψ ∈ R } and max { ψ : ψ ≤ CVaR α ( σ ( x )) + r X i =1 ¯ δ j i (¯ x ) x j i ∀ ¯ X ⊆ V, x ∈ X , ψ ∈ R } , are equivalent to Problem (3). Since the number of feasible solutions and the number of inequalities (7) and(12) are finite, the result follows. (cid:3) Next we report our computational experience with the proposed method.
3. An Application: the Risk-Averse Set Covering Problem (RASC)
We demonstrate the pro-posed methods on a variant of a risk-averse set covering problem (RASC). We represent RASC on a bipartitegraph G = ( V ∪ V , E ). There are two groups of nodes V and V in G , where all arcs in E are from V to V . Let V := { , . . . , m } be a set of items. Let S j ⊆ V , j ∈ V := { , . . . , n } be a collection of n subsets,where S nj =1 S j = V . There exists an arc ( i, j ) ∈ E if j ∈ S i for i ∈ V representing the covering relationship.In RASC, there is uncertainty on whether an arc appears in the graph. To formulate RASC, first consider adeterministic set covering problem with a feasibility set { x ∈ B n | P j ∈ V t ij x j ≥ h i , ∀ i ∈ V } , where h i = 1for all i ∈ V , and t ij = 1 if i ∈ S j ; otherwise, t ij = 0 for all i ∈ V \ S j , j ∈ V . We say that an item i ∈ V is covered, if there exists x j = 1 for j ∈ V and t ij = 1. Now suppose that there is uncertainty onwhether a chosen subset can cover an item, where constraint t ij x j ≥ h i has random constraint coefficients t ij or random right-hand side h i for i ∈ V , j ∈ V . A related class of risk-averse problems, referred to as theprobabilistic set covering problems, consider a feasible set { x ∈ B n | P ( P j ∈ V t ij x j ≥ h i , ∀ i ∈ V ) ≥ − α } ,where α is a given risk level and t ij and/or h i are random variables for i ∈ V , j ∈ V . Here the objectiveis to select a minimum cost selection of subsets such that the probability of covering all items is at leastthe risk threshold 1 − α (see, e.g., [2, 5, 8, 19]). In contrast, in this paper, we assume that h i = 1 for all i ∈ V , and that there is uncertainty on t ij for all i ∈ V and j ∈ V and consider a different type of riskaversion, where we aim to choose at most k subsets from the collection so that the CVaR of the number ofcovered items at a risk level α is maximized. In this section, we consider an independent probability coveragemodel , where each node j has an independent probability a ij of being covered by node i ∈ V for j ∈ S i , i.e., P ( t ij = 1) = a ij . u and K¨u¸c¨ukyavuz: CVaR Maximization of Stochastic Submodular Functions σ ( x ) be a random variable representing the number of covered items in V for a given x , i.e., σ ( x ) := |{ i ∈ V : ∃ j ∈ V with x j = 1 and t ij = 1 }| . It is known that σ ( x )( ω ) is submodular for ω ∈ Ω [25]. Givenan integer k and α ∈ (0 , { CVaR α ( σ ( x )) : X i ∈ V x i ≤ k, x ∈ B n } , (14)which is in the form of (3), where X is given by X = { x ∈ B n : P i ∈ V x i ≤ k } . Next we propose an efficientCVaR oracle to evaluate the objective function in problem (14) for a given x ∈ X under the probabilitydistribution of interest. Proposition 3.1
There exists a polyniamial-time oracle that computes the function CVaR α ( σ ( x )) for x ∈ X for RASC under the independent probability coverage model. Proof.
We follow the notation described in [25] to describe the CVaR oracle. From [9, 18, 21], we knowthat function P ( σ ( x ) = b ) is equal to the probability mass function of a Poisson binomial distribution anduse a dynamic program (DP) to evaluate A ( x, i, j ), which is defined as the probability that the selection x covers j nodes among the first i nodes of V for 0 ≤ j ≤ i, i ∈ V . Barlow and Heidtmann [4], Zhang et al.[27], and Wu and K¨u¸c¨ukyavuz [25] use the DP to calculate P ( σ ( x ) ≥ b ) := P mj = b A ( x, m, j ). Next, we showthat we can use the same recursion to evaluate CVaR α ( σ ( x )). From the definition of VaR α ( σ ( x )) in (1), wehave VaR α ( σ ( x )) = min { j ∈ Z + : P ji =0 A ( x, m, i ) ≥ α } . The function CVaR α ( σ ( x )) is given byCVaR α ( σ ( x )) = VaR α ( σ ( x )) − α VaR α ( σ ( x )) − X j =0 P ( σ ( x ) = j )(VaR α ( σ ( x )) − j ) = VaR α ( σ ( x )) − α VaR α ( σ ( x )) − X j =0 A ( x, m, j )) + 1 α VaR α ( σ ( x )) − X j =0 jA ( x, m, j ) . For a given x , the running time of the DP is O ( nm + m ), because obtaining P ( x, j ) for all j ∈ V is O ( nm ),and computing the recursion is O ( m ). (cid:3) Table 1: Algorithm 1 with different inequalities
Oracle-LShape Oracle-Ineq (7) Oracle-Ineq (12) Oracle-Ineq (7) and (12) |V| α k
Time Cuts Nodes Time Cuts Nodes Time Cuts Nodes Time Cuts Nodes50 0.025 3 9 2315 2363 3 801 1915 ≤ ≤ ≥ ≤ ≤ ≥ ≥ ≥ ≥ ≥ ≥ ≥ ≥ ≥ ≥ We study the computational performance of our proposed methods on RASC. All instances were executedon a Windows 8.1 operating system with an Intel Core i5-4200U 1.60 GHz CPU, 8 GB DRAM, and x64based processor using C++ with IBM ILOG CPLEX 12.7. We set up the mixed-integer programming searchmethod as traditional branch-and-cut with the lazycallback function of CPLEX, where the presolve processis turned off. The number of threads is equal to one. All other CPLEX options are set to their defaultvalues. The time limit is set to 1800 seconds. For RASC, we generate a complete bipartite graph with arcsfrom all nodes i ∈ V to all j ∈ V . Let V = V ∪ V . (Note that V = V for RASC, because we only selectnodes from V .) We follow the data generation scheme of [25] to set n , m , and a ij for each arc ( i, j ). We let k ∈ { , } and α ∈ { . , . } . The size of the bipartite graphs is |V| ∈ { , , } . u and K¨u¸c¨ukyavuz: CVaR Maximization of Stochastic Submodular Functions x , which is a characteristic vector of the set ¯ X , we consider the L-shaped cut ψ ≤ CVaR α ( σ (¯ x )) + X j ∈ V \ ¯ X (cid:0) CVaR ( σ ( V )) − CVaR α ( σ (¯ x )) (cid:1) x j . (15)For x j = 1 for any j ∈ V \ ¯ X , inequality (15) gives a valid upper bound for ψ = CVaR α ( σ (¯ x )) (cid:1) , givenby CVaR ( σ (¯ x )) (cid:1) for any α ∈ (0 , ( σ (¯ x + e j )) ≤ CVaR ( σ ( V )), for j ∈ V \ ¯ X . Column “Oracle-LShape”denotes Algorithm 1 with the L-shaped cut (15). Column “Oracle-Ineq (7)” denotes Algorithm 1 withinequality (7). Column “Oracle-Ineq (12)” denotes Algorithm 1 with inequality (12). Column “Oracle-Ineq(7) and (12)” denotes Algorithm 1 with both inequalities (7) and (12). Column “Time” denotes the totalsolution time for each instance for RMP, in seconds. Column “Cuts” denotes the total number of user cutsadded to RMP. Column “Nodes” denotes the number of branch-and-bound nodes traced in RMP. FromTable 1, we observe that the solution time increases as k and |V| increase for all methods. We observe thatOracle-LShape cannot solve most instances within the time limit. Oracle-Ineq (7) and Oracle-Ineq (12) arefaster than Oracle-LShape for the instances that are solvable by Oracle-LShape within the time limit. Inaddition, Oracle-Ineq (7) or Oracle-Ineq (12) generates a fewer number of optimality cuts and traces a fewernumber of nodes compared to Oracle-LShape.For most of the instances with |V| ≤ |V| = 150,the solution time of Oracle-Ineq (12) is more than Oracle-Ineq (7). Recall that for a given incumbentsolution ¯ x , inequality (12) is generated by Algorithm 2. In Algorithm 2, we observe that for a given ¯ x and1 ≤ i ≤ i ′ ≤ r , we have ¯ δ j i (¯ x ) ≤ ¯ δ j ′ i (¯ x ) ≤ CVaR α ( σ ( V )) − CVaR α ( σ (¯ x )) . From the above relation, if the sizeof |V| is large, then there may exist a large number of nodes with a high value of the lifting function δ j i (¯ x ),which is close to the coefficients in the L-shaped cut (15), i.e. CVaR α ( σ ( V )) − CVaR α ( σ (¯ x )). This explainswhy Oracle-Ineq (12) does not perform well in instances with large |V| . To benefit from the complementarystrengths of inequalities (7) and (12), we add both classes of inequalities at each iteration in Algorithm 2.In Table 1, we observe that for the instances that are not solvable by either Oracle-Ineq (12) or Oracle-Ineq(7) within 1800 seconds, the solution time of Oracle-Ineq (7) and (12) is shorter. Furthermore, for a hardinstance ( | V | , α, k ) = (150 , . , Acknowledgments
We thank the AE and the reviewer for their comments that improved the presen-tation. This work is supported, in part, by the National Science Foundation Grant
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