Two-stage stochastic (and distributionally robust) p-order conic mixed integer programs: Tight second stage formulations

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1 Two-stage stochastic (and distributionally robust p-order conic mixed integer programs: Tight second stage formulations Manish Bansal and Yingqiu Zhang Department of Industrial and Systems Engineering Virginia Tech, Blacksburg, VA s: bansal; May 19, 2018 Abstract. In this paper, we introduce two-stage stochastic p-order conic mixed integer programs (TSS-CMIPs in which the second stage problems have p-order conic constraints along with integer variables. We present sufficient conditions under which the addition of parametric (non-linear cutting planes provides convex programming equivalent for the second stage CMIPs. In addition, we introduce TSS-CMIPs having structured p-order CMIPs in the second stage and derive classes of parametric (non-linear cuts that satisfy the foregoing conditions. These cuts allow us to relax the integrality restrictions on the second stage integer variables without effecting the integrality of the optimal solution of the TSS-CMIP. We also perform extensive computational experiments by solving randomly generated structured TSS-CMIPs with second-order CMIPs in the second stage, i.e. p = 2, and observe that there is a significant reduction in the total time taken to solve these problems after adding our parametric cuts. Finally, we introduce two-stage distributionally robust p-order conic mixed integer programs (TSDR-CMIPs and interestingly, all results developed in this paper for the TSS-CMIPs are also applicable for the TSDR-CMIPs. 1. Introduction A p-order conic mixed integer program is a nonlinear nonconvex problem whose feasible region is defined by intersection of an affine set, p-order cones, and integrality constraints. It generalizes various optimization frameworks such as mixed integer program (MIP and quadratically constrained convex quadratic MIP, and arises in applications ranging from portfolio optimization to machine learning [2, 19]. In this paper, we introduce two-stage stochastic p-order conic mixed integer programs (TSS-CMIPs with integer variables and p-order conic constraints in the second stage. More specifically, the TSS-CMIP is defined as follows: min cx + E ξp [Q ω (x] s.t. Ax b, x Z n 1, (1 where random variable ξ P follows a known probability distribution P, and for scenario ω from a finite sample space Ω: Q ω (x := min g ω y ω + j J ĝ j ωd j ω,0 (2 s.t. W ω y ω r ω T ω x, (3 E j ω y ω + Fωx j h j d j ω,0, j J, (4 y ω Z q 1 R q q 1, d j ω,0 R +, j J. (5 1

2 Here, p denotes l p -norm, i.e. y p = ( k y k p 1/p for p 1, c R n 1, A R m 1 n 1, b R m 1, and for each ω Ω and j J := 1,..., J }, g ω R q, ĝ j ω R, E j ω R m 2 q, F j ω R m 2 n 1, h j ω R m 2, recourse matrix W ω R m 3 q, technology matrix T ω R m 3 n 1, and r ω R m 3. We refer to the formulation (2-(5 and function Q ω (x as the second stage subproblem and the recourse function, respectively. For TSS-CMIPs, we make following assumptions: (A1 T ω Z m 3 n 1 and F j ω Z m 2 n 1 for all ω Ω and j J (w.l.o.g., (A2 X := x : Ax b, x Z n 1 } is non-empty, (A3 Relatively complete recourse, i.e. K ω (x := y ω : (3-(5 hold} is non-empty for all ω Ω and x X. Observe that the TSS-CMIPs generalize various classes of optimization problems studied in literature, which include: (a two-stage stochastic mixed integer program (TSS-MIP is equivalent to TSS-CMIP (1 with J =, (b TSS-MIP with quadratic objective function in the second stage, i.e. Q ω (x = miny T ω Γ ω y ω + δ T ω y ω : W ω y ω r ω T ω x, y ω Z q 1 R q q 1 } where Γ ω R q q and δ ω R q, is equivalent to TSS-CMIP (1 where Q ω (x = min d 1 ω,0 : W ω y ω r ω T ω x, Γω 1/2 y ω + Γω 1/2 δ ω 2 d 1 ω,0 Γ 1/2 ω y ω + Γ 1/2 ω δ ω 2 (δω T Γ 1 ω δ ω 1/2, y ω Z q 1 R q q 1, d 1 } ω,0 R +, and (c TSS-MIP with sum of l p -norms in the objective function of the second stage is same as TSS-CMIP (1 where g ω = 0 and ĝ j ω = 1 for j J. Likewise, many optimization problems can be reformulated as TSS-CMIP. For example, TSS-MIP with chance-constraints, defined using independent Gaussian random vectors with zero mean, in the second stage, and TSS-MIP with robust MIP in the second stage where uncertainty set is ellipsoidal [8]. We can further generalize TSS-CMIP (1 by integrating it with distributionally robust optimization (DRO framework [28] where we seek a solution that optimizes the expected value of the objective function for the worst case probability distribution within a prescribed (ambiguity set of distributions that may be followed by the uncertain parameters. In particular, we introduce twostage distributionally robust p-order conic mixed integer program (TSDR-CMIP, which is defined as follows: min cx + max E ξ P [Q ω (x] P P Ax b, x Zn 1 }, (6 where complete information about the probablity distribution P followed by the random variable ξ P is not known but it belongs to a set of distributions P (referred to as the ambiguity or uncertainty set. 1.1 Contributions of this paper As per our knowledge, the TSS-CMIPs (1 and TSDR-CMIPs (6, in their general form, have not been studied in literature, and this paper is the first attempt to tackle these problems. We 2

3 provide sufficient conditions under which the integrality constraints on the second stage integer variables of the TSS-CMIP can be relaxed (without effecting the integrality of the optimal solution of the problem by adding globally valid parametric (non-linear inequalities a priori. We refer to this approach as second stage convexification. Furthermore, we present TSS-CMIPs with structured p-order CMIPs in the second stage and demonstrate the significance of the second stage convexification by deriving classes of parametric (non-linear cuts that satisfy the foregoing conditions. As a result, after adding these cuts we obtain convex programming equivalent for the second stage CMIPs. This paper generalizes the results of Bansal et al. [5] for two-stage stochastic mixed integer program (TSS-MIP, i.e. TSS-CMIP (1 with J =, in which the authors utilize parametric linear cuts to get linear programming equivalent for some structured second stage MIPs. We also extend the so-called conic mixed integer rounding (CMIR inequalities derived by Atamtürk and Narayanan [3] to describe the convex hull of a deterministic single constrained first-order (or polyhedral conic mixed integer set with one integer variable to TSS-CMIPs. In particular, we introduce TSS-CMIPs with multi-constraint p-order conic mixed integer sets having multiple integer variables in the second stage and derive classes of parametric (non-linear cuts akin to parametric CMIR inequalities for them. We also prove that under certain conditions, our parametric cuts are sufficient to describe the convex hull of these sets. Note that these structured sets have not been studied even in the deterministic framework. More interestingly, all results presented in this paper for TSS-CMIPs are also applicable for the TSDR-CMIP. The TSDR-CMIPs generalize the two-stage distributionally robust mixed binary programs is studied by Bansal et al. [4], where both stages have linear constraints and integer variables are bounded between 0 and 1. We also perform computational experiments to evaluate the effectiveness of the second stage convexification approach by solving randomly generated aforementioned structured TSS-CMIPs where p = 2. We observe that after adding our parametric cuts, there is a significant reduction in the number of the second stage integer variables, thereby leading to reduction in the total solution time taken to solve these problems. For instance, CPLEX (with its default settings could not solve 110 out of 210 TSS-CMIP instances within a time limit of 3 hours and the allocated memory of 24 GB RAM. In contrast, after adding our parametric cuts at the root node, CPLEX could solve 107 out of these 110 (unsolvable instances within 8.2 minutes (on average. 1.2 Literature review on special cases of TSS-CMIP and TSDR-CMIP One of the most extensively studied special case of TSS-CMIP is the class of TSS-MIPs (refer to [18] for comprehensive survey. To solve TSS-MIP, many researchers have been using globally valid linear cuts in (x, y ω space, for each scenario ω Ω, to tighten the second stage problems with binary or integer variables, which are then embedded within Benders decomposition algorithm [9] to solve TSS-MIP. These cuts are of the form γ ω y ω γ ω,0 γ ω,1 x, where x is a first-stage feasible solution, and γ ω, γ ω,0, and γ ω,1 are real-vectors, and are referred to as the parametric (linear cuts. For instance, Sherali and Fraticelli [29] derive parametric linear cuts using the reformulationlinearization technique to solve TSS-MIP with Ω = 1, only binary variables in the first-stage, and mixed binary programs in the second stage. Likewise, Gade et al. [13] utilize parametric Gomory fractional cuts for solving TSS-MIPs with only binary variables in the first stage and non-negative integer variables in the second stage. Similarly, Bodur et al. [10] use parametric cuts based on split disjunctions to solve TSS-MIP with mixed integer first stage and continuous second stage variables. Lately, Bansal et al. [4] develop decomposition algorithm for TSDR-MIP with only binary variables in the first stage and mixed binary programs in the second stage. This algorithm, referred to as distributionally robust integer L-shaped method, also utilizes the parametric linear cuts derived 3

4 using lift-and-project technique and distribution separation oracle within Benders algorithm. In the aforementioned studies, the parametric cuts are developed/added sequentially in the algorithms. Recently, Kim and Mehrotra [17] formulate an integrated staffing and scheduling problem under demand uncertainty as a TSS-MIP with the second stage MIPs having a certain structure and utilize parametric mixed integer rounding inequalities (added a priori to obtain a linear programming equivalent of the second stage MIPs. Bansal et al. [5] generalize their observation to the general TSS-MIPs and show that under suitable conditions, the second stage MIPs can be convexified by adding parametric cuts a priori. As special cases, they consider structured parameterized mixed integer sets or convex objective integer program (COIP in the second stage. In particular, they extend the results of Miller and Wolsey [22] for deterministic mixed integer sets and COIP to the two-stage stochastic framework, and consider TSS-MIPs with parametrized version of two special cases of the continuous multi-mixing set [6, 7] in the second stage. In this paper, we further generalize their results for TSS-CMIPs, which is equivalent to TSS-MIPs with parametric p-order conic constraints in the second stage. Furthermore, we utilize parametric (non-linear cutting planes to obtain convex programming equivalent for the second stage CMIPs, i.e. non-linear inequalities in the original space and linear inequalities in a higher dimensional space. Another special case of TSS-CMIP is two-stage stochastic quadratic integer program (TSS-QIP. In this direction, Özaltin et al. [26] study TSS-QIPs with only stochastic right-hand sides in the second stage along with deterministic quadratic objective functions, linear constraints, and only integer variables in both stages. They reformulate this problem using value functions for quadratic integer programs, and present a global branch-and-bound algorithm and a level-set approach to solve the problem. Lately, Mijangos [20, 21] develop a Branch and Fix Coordination based algorithm to solve TSS-QIP with quadratic terms in the second stage objective function and two-stage stochastic convex problems where objective function and constraints are nonlinear; both of these problems have binary and continuous variables in the first stage and only continuous variables in the second stage. As per our knowledge, this paper is the first attempt to tackle TSS-CMIPs in its general form, i.e. integer variables in both stages and p-order conic constraints in the second stage. Remark 1. It is important to note that although TSS-CMIP has linear objective function and constraints in the first stage, the results presented in this paper can be easily extended to TSS- CMIPs with nonlinear objective function and constraints in the first stage. 1.3 Brief literature review on CMIPs In the last two decades, researchers have extended various classes of cutting planes derived for MIPs to mixed integer nonlinear programs (MINLP. It includes extensions of Gomory mixed integer cuts, Mixed Integer Rounding (MIR cuts, split cuts [12], and n-step MIR inequalities [15] for MIPs to solve MINLPs [11], second-order CMIPs [3], second-order conic mixed integer sets [23], and polyhedral CMIPs [27], respectively. Additionally, attempts have been made to derive convex hull description of (structured conic mixed integer sets (see [3, 14, 16, 24] for few examples. In this paper, for the first time, we consider TSS-CMIPs with (structured p-order CMIPs in the second stage. Among all papers on deterministic CMIPs, the work of Atamtürk and Narayanan [3] is most closely related to our work. Specifically, Atamtürk and Narayanan [3] generalize the MIR inequalities [25] by studying a conic mixed integer set defined by a single second-order conic constraint and one integer variable. They introduce conic mixed integer rounding (CMIR cuts which are non-linear in original space and linear in higher dimensional space. (See Section 3.1 for 4

5 more details. In this paper, we introduce TSS-CMIPs with multi-constraint p-order conic mixed integer sets having multiple integer variables in the second stage and describe the convex hull of these sets (under certain conditions using parametric CMIR inequalities. We would like to point that these structured sets have not been studied even in the deterministic CMIP literature. 1.4 Extensive Formulation of TSS-CMIP We write TSS-CMIP (1 as a large-scale CMIP, referred to as the extensive formulation of the TSS-CMIP: min cx + p ω g ω y ω + ĝωd j j ω,0 (7 ω Ω j J s.t. Ax b, (8 T ω x + W ω y ω r ω, (9 Eωy j ω + Fωx j h j d j ω,0, j J, ω Ω, (10 x Z n 1, y ω Z q 1 R q q 1, d j ω,0 R +, j J, ω Ω, (11 where p ω is the probability of occurence of scenario ω Ω, and we denote its feasible region by P = (x, y ω, d ω } ω Ω : (8 (11 hold}. 1.5 Notations We use the following notations and conventions in this paper: 1. Proj x,y (Z denotes the projection of Z (R n 1 R n R m onto the subspace R n 1+n defined as follows: Proj x,y (Z := (x, y R n 1 R n : (x, y, z Z for some z R m }. 2. Usage of Proj x=ˆx,y (Z implies that we first restrict Z by setting x = ˆx and then project the restricted Z onto y space, i.e. R n subspace. 3. Partial Convex Hull: Let Z := (z, d (Z n 1 R n R m 2 + : πi z π0 i, i = 1,..., m 1, i z δ i p d i }, i = 1,..., m 2 where π i, π0 i, i, and δ i are real vectors or matrices of appropriate sizes. We define a partial convex hull of Z, i.e., convex hull of Z with respect to a subset of integer variables in Z, by another mixed integer set: ( Z pch := (z, d Z n 1 n 1 R n 1+n R m 2 : π i z π0, i i = 1,..., m 1, i z δ i p d i, i = 1,..., m 2, i jz δ i j p d i, i = 1,..., m 2, j = 1,..., m 3 } such that n 1 1,..., n 1 }, m 1 m 1, m 3 Z +, and Z Z pch conv(z = conv(z pch. Note that Z pch has lesser number of integrality constraints (but possibly more linear or nonlinear inequalities than Z. Whereas in comparison to conv(z, Z pch might have lesser inequalities but more integrality constraints. 5

6 1.6 Organization of this paper In Section 2, we present sufficient conditions to convexify the second stage CMIPs of TSS-CMIPs using (non-linear cuts. In Section 3, we introduce TSS-CMIPs with structured second stage CMIPs and derive globally valid parametric (non-linear cuts which satisfy the foregoing conditions. Thereafter, in Section 4, we explore the computational effectiveness of the second stage convexification approach in solving these structured TSS-CMIPs. We provide concluding remarks along with potential future extensions of this paper in Section Conditions to Convexify Second Stage CMIPs of TSS-CMIPs In this section, we present sufficient conditions under which the integrality restrictions on the second stage integer variables of the TSS-CMIPs can be relaxed (without effecting the integrality of the optimal solution by adding parametric (non-linear inequalities. First, in Lemma 1, we provide conditions to convexify second stage program for a given scenario of the TSS-CMIP using non-linear cuts in the original space. Thereafter, in Theorem 1, we derive a partial convex hull of P using non-linear parametric cuts, thereby relaxing integrality restriction on all second stage integer variables without effecting the integrality of the optimal solution. Now, for (x, ω (X, Ω, we define K ω tight(x := (y ω, d ω R q R J + :W ωy ω r ω T ω x, Eωy j ω + Fωx j h j d j ω,0, j J, E j ωy ω + F j ωx h j ω d j ω,0, j J}, p where E j ω, F j ω, h j ω are known matrices (or vectors for all j J, and Lemma 1. Let ω 1 Ω and P tight := (x, y, d X R q Ω R J Ω + : T ω x + W ω y ω r ω, E j ω y ω + Fωx j h j d j ω,0, j J, ω Ω, E j ωy ω + F j ωx h j ω d j ω,0, j J, ω Ω}. p P tight,1 := T ω x + W ω y ω r ω, E j ω y ω + Fωx j h j d j ω,0, j J, ω Ω, E j ω 1 y ω1 + F j ω 1 x h j p ω 1 d j ω 1,0, j J, x X, d R J Ω +, y ω1 R q, y ω Z q 1 R q q 1, ω Ω \ ω 1 } }. If conv ( K ω1 (x = K ω 1 tight(x for all x X, then P tight,1 is a partial convex hull of the set P, i.e., P P tight,1 conv(p = conv ( P tight,1. Proof. Refer to Appendix 6.1 Theorem 1. If conv(k ω (x = K ω tight for all x X and ω Ω, then P tight is a partial convex hull of the set P. Proof. Refer to Appendix 6.2 6

7 2.1 Reformulation of TSS-CMIP We reformulate the second stage problem, Q ω (x, of the TSS-CMIP using additional variables as follows: Q ω (x := min g ω y ω + j J where d j ω := (d j ω,1,..., dj ω,m 2 ĝ j ωd j ω,0 (12 s.t. W ω y ω r ω T ω x, (13 e j ω,i y ω + f j ω,i x hj ω,i dj ω,i, i = 1,..., m 2, j J, (14 d j d j ω,0, j J, (15 y ω Z q 1 R q q 1, d j ω,i R +, i = 0, 1,..., m 2, j J, (16 R m 2 + for j J, and ej ω,i and f j ω,i denote the ith row of matrices Eω j and Fω, j respectively. We define the feasible region of the reformulated second stage program by K ω (x := (y ω, d ω : (13 (16 hold} for x X and ω Ω. Furthermore, we can rewrite the TSS-CMIP (1 as another large-scale CMIP, referred to as the reformulated extensive formulation of the TSS-CMIP: min cx + p ω g ω y ω + ĝωd j j ω,0 (17 ω Ω j J s.t. Ax b, (18 T ω x + W ω y ω r ω, (19 e j ω,i y ω + f j ω,i x hj ω,i dj ω,i, i = 1,..., m 2, j J, (20 d j p ω d j ω,0, j J, (21 x Z n 1, y ω Z q 1 R q q 1, d j ω,i R +, i = 0, 1,..., m 2, j J. (22 We denote the feasible region of the reformulated extensive formulation by P, i.e. P := (x, y ω, d ω } ω Ω : (18 (22 hold}. Next, we describe the feasible region of the reformulated second stage problem as intersection of a polyhedral conic mixed integer set and a set of p-order cones. Let K ω (x = Kω(x 1 Kω(x 2 where Kω(x 1 := (y ω, d ω (Z q 1 R q q 1 R m 2 J + J + : (13 (14 hold} is a polyhedral conic mixed integer set and Kω(x 2 := (y ω, d ω R q R m 2 J + J + : (15 holds} is an intersection of J number of p-order cones. In Theorem 2, we show that in order to obtain the convex hull of K ω (x, it is sufficient to obtain the convex hull of Kω(x. 1 Theorem 2. For each x X and ω Ω, conv (K ω (x = conv ( K 1 ω(x K 2 ω(x. (23 Proof. For x X and ω Ω, assume that a point ˆη ω := (ŷ ω, ˆd ω conv ( K 1 ω(x K 2 ω(x, i.e. ˆη ω conv ( K 1 ω(x and ˆη ω K 2 ω(x. Since ˆη ω conv ( K 1 ω(x, ˆη ω can be written as convex combination of a finite number of points, η k ω for k 1, 2,...}, belonging to the set K 1 ω(x. Moreover 7

8 in Kω(x, 1 the variables d j ω,0 have no restrictions, except non-negativity. Therefore, we can select the points η ω k Kω(x 1 such that their d ω components satisfy constraint (15, i.e. η ω k Kω(x, 2 and ˆη ω = k=1,2,... λ k η ω k where η ω k Kω 1 Kω 2 for all k and k=1,2,... λ k = 1. This also implies that ˆη ω conv ( Kω(x 1 Kω(x 2 = conv(k ω (x, and therefore, conv ( K 1 ω(x K 2 ω(x conv(k ω (x. (24 Now, suppose that a point ˆη ω conv(k ω (x = conv ( K 1 ω(x K 2 ω(x. Therefore, ˆη ω can be written as convex combination of q (m J points belonging to K 1 ω(x K 2 ω(x, i.e., ˆη ω = k λ k η k where k λ k = 1, η k ω K 1 ω, and η k ω K 2 ω for all k = 1,..., q (m J. This also implies that ˆη ω conv ( K 1 ω and ˆηω conv ( K 2 ω = K 2 ω. So, conv(k ω (x conv ( K 1 ω(x K 2 ω(x. (25 From (24 and (25, we get conv(k ω (x = conv ( K 1 ω(x K 2 ω(x. In the following results, we show that under certain conditions, the integrality restrictions on the second stage integer variables can be relaxed, without effecting the integrality of the optimal solution, by adding globally valid parametric linear inequalities in the reformulated second stage of the TSS-CMIPs. More specfically, using linear parametric cuts (unlike non-linear parametric cuts in Lemma 1 and Theorem 1, we provide conditions to convexify reformulated second stage program for a given scenario and derive a partial convex hull of P in Lemma 2 and Theorem 3, respectively. Now, for (x, ω (X, Ω, we define K ω tight (x := (y ω, d ω, u ω R q R (m 2+1 J + R q 2 : W ω y ω r ω T ω x, e j ω,i y ω + f j ω,i x hj ω,i dj ω,i, i = 1,..., m 2, j J, N 2 ωy ω + N 3 ωd ω + N 4 ωu ω s ω N 1 ωx }, where N 1 ω, N 2 ω, N 3 ω, N 4 ω, and s ω are known matrices (or vectors, and P tight := T ω x + W ω y ω r ω, Lemma 2. Let ω 1 Ω and e j ω,i y ω + f j ω,i x hj ω,i dj ω,i, i = 1,..., m 2, j J, ω Ω, d j d j ω,0, j J, ω Ω, N 1 ωx + N 2 ωy ω + N 3 ωd ω + N 4 ωu ω s ω ω Ω, x X, y ω R q, d ω R (m 2+1 J +, u ω R q 2 ω Ω }. P tight,1 := T ω x + W ω y ω r ω, e j ω,i y ω + f j ω,i x hj ω,i dj ω,i, i = 1,..., m 2, j J, ω Ω, d j d j ω,0, j J, ω Ω, N 1 ω 1 x + N 2 ω 1 y ω1 + N 3 ω 1 d ω1 + N 4 ω 1 u ω1 s ω1, x X, d ω R (m 2+1 J +, y ω1 R q, u ω1 R q 2, y ω Z q 1 R q q 1, ω Ω \ ω 1 } }. 8

9 If conv ( Kω 1 1 (x = Proj y,d (K ω 1 tight (x for all x X, then Proj x,y,d (P tight,1 is a partial convex hull of P which implies that P Proj x,y,d (P tight,1 conv(p = conv(proj x,y,d (P tight,1. Proof. Refer to Appendix 6.3 Theorem 3. If conv(k 1 ω(x = Proj y (K ω tight for all x X and ω Ω, then Proj x,y(p tight is a partial convex hull of P. Proof. Refer to Appendix Structured Two-Stage Stochastic p-order Conic Mixed Integer Programs In the previous section, we introduced second stage convexification approach for TSS-CMIPs and provided related results in an abstract setting. Our objective in this section is to introduce TSS- CMIPs with structured p-order CMIPs in the second stage and derive classes of parametric (non- linear inequalities to convexify the second stage problems (Theorems Necessary Background First we briefly review the CMIR cut generation procedure [3] which we will use in the proofs of the subsequent theorems. Atamtürk and Narayanan [3] generalize the well-known MIR procedure for MIPs to second-order CMIPs by studying a single-constraint conic mixed integer set with one integer variable, i.e., Z := (σ, v, ρ 0 Z R 2 + : } (σ β 2 + v 2 ρ 0 where β R. They reformulated set Z using additional continuous variables to get Z := (σ, v, ρ 0, ρ 1, ρ 2 Z R 4 + : σ β ρ 1, v ρ 2, ρ ρ2 2 ρ 0}, and derive a facet-defining inequality for Z 1 := (σ, ρ 1 Z R + : σ β ρ 1 }, referred to as the CMIR inequality, ( 1 2β (1 (σ β + β (1 ρ 1 (26 where β (1 = β β. Atamtürk and Narayanan [3] show that the convex hull of Z 1 can be obtained by adding (26 to the continuous relaxation of Z Tight second stage formulations for structured TSS-CMIPs We provide tight second stage formulations for TSS-CMIPs (1 with the following two structured CMIPs in the second stage, i.e. Q ω (x (or Q ω (x where (a E j ω = 1 (vector of all ones for j J and W ω is a totally unimodular (TU matrix (Theorem 4; (b E j ω for all j J, and W ω are TU matrices (Theorem 5. 9

10 It is important to note that these structured multi-parameter multi-constrained sets, which generalize Z, have not been studied even in deterministic settings (see Corollary 1 and 2. Theorem 4. In TSS-CMIP (1, let Q ω (x := min g ω y ω + j J ĝ j ωd j ω,0 (27 s.t. W ω y ω r ω T ω x, (28 1yω j + Fωx j h j d j ω,0, j J, (29 y j ω Z, d j ω,0 R +, j J, (30 where W ω is a TU matrix and r ω is integral. The convex hull of the feasible region of Q ω (x for all x X is given by (y ω, d ω,0 R J R J + : (28, (29, and (1 2µ j ωyω j + F j ωx h j ω d j ω,0 },, j J p ( where the ith row of µ j ω, F j ω, and h j ω are denoted by µ j ω,i = hj ω,i h j ω,i, f j ω,i = 1 2µ j ω,i f j ω,i (, and h j ω,i = 1 2µ j ω,i h j ω,i µ j ω,i, respectively. Furthermore, in higher dimensional space Q ω(x can be reformulated as: Q ω (x := min g ω y ω + ĝωd j j ω,0 : (28, and (31 j J yω j + f j ω,i x hj ω,i dj ω,i, i = 1,..., m 2, j J, (32 d j p ω d j ω,0, j J, (33 y j ω Z, d j ω,0 R +, d j ω R m 2 + j J }. (34 Then, for all x X, the convex hull of the feasible region of Q ω (x, denoted by K ω (x, is obtained by adding m 2 J number of the following parametric linear inequalities (in the higher dimensional space to the continuous relaxation of K ω (x: ( ( 1 2µ j ω,i y j ω h j ω,i + f j ω,i x + µ j ω,i dj ω,i, i = 1,..., m 2, j J. Proof. Let K 1 ω(x = (y ω, d ω Z J R m 2 J + : (35 W ω y ω r ω T ω x, (36 } yω j + f j ω,i x hj ω,i dj ω,i, i = 1,..., m 2, j J (37 for all x X. First we utilize the CMIR cut generation approach (discussed in Section 3.1 for each defining inequality (37 of Kω(x. 1 In other words, we substitute σ = yω, j β = h j ω,i f j ω,ix, and 10

11 ρ 1 = d j ω,i in Z 1 and get the following valid parametric CMIR inequalities (26 for K 1 ω(x: (1 2µ j ω,i (y ω j h j ω,i f j ω,i x + µ j ω,i dj ω,i, i = 1,..., m 2, j J. (38 Since f j ω,i x is integral for all x X, f j ω,i x = f j ω,ix and therefore, inequality (38 is equivalent to (1 2µ j ω,i (y ω j h j ω,i + f j ω,i x + µ j ω,i dj ω,i, i = 1,..., m 2, j J. (39 Thus, conv ( K 1 ω(x K 3 ω(x := } (y ω, d ω R J R m 2 J + : (36, (37, and (39 hold (40 for all x X. Notice that we can rewrite the set Kω(x 3 as Kω(x 3 = (y ω, d ω R J R m 2 J + : W ω y ω r ω T ω x, d j ω,i yj ω (h j ω,i f j ω,i x, i = 1,..., m 2, j J, (41 ( d j ω,i h j ω,i f j ω,i x yω, j i = 1,..., m 2, j J, (42 ( } d j ω,i (1 2µ j ω,i yω j h j ω,i + f j ω,i x + µ j ω,i, i = 1,..., m 2, j J. (43 Now let K 4 ω(x be a bounded face of K 3 ω(x and has maximum possible dimension. Since d j ω,i, for i = 1,..., m 2 and j J, must be minimal on this face, we have d j ω,i y = max ω j (h j ω,i f j ω,i x, (h j ω,i f j ω,i x yω, j }. (1 2µ j ω,i (y ω j h j ω,i + f j ω,i x + µ j ω,i Moreover, if h j ω,i Z for any j J then µj ω,i = 0, and as a result inequality (39 reduces to inequality (41. However, when h j ω,i / Z for j J, then there are three possible cases: Case I. d j ω,i = yj ω (h j ω,i f j ω,i x : This case will happen if and only if yω j (h j ω,i f j ω,i x (h j ω,i f j ω,i x yω j and ( ( yω j (h j ω,i f j ω,i x 1 2µ j ω,i yω j which are equivalent to yω j h j ω,i f j ω,i x and yj ω h j ω,i h j ω,i + f j ω,i x + µ j ω,i, f j ω,ix, respectively. Note that the last inequality is stronger than the second last inequality. Therefore, we can claim that d j ω,i = yω j (h j ω,i f j ω,i x if and only if yω j h j ω,i f j ω,i x. Case II. d j ω,i = h j ω,i f j ω,i x yj ω: This case will happen if and only if h j ω,i f j ω,i x yj ω ( ( yω j (h j ω,i f j ω,i x and hj ω,i f j ω,i x yj ω 1 2µ j ω,i yω j h j ω,i + f j ω,i x + µ j ω,i, which are 11

12 equivalent to yω j h j ω,i f j ω,i x and yj ω h j ω,i f j ω,i x, respectively, as µj ω,i 1. Again, note that the last inequality is stronger than the second last inequality. Therefore, d j ω,i = hj ω,i f j ω,i x yj ω if and only if yω j h j ω,i f j ω,i x. ( Case III. d j ω,i (1 = 2µ j ω,i yω j h j ω,i + f j ω,i x + µ j ω,i : This case will happen if and only if ( ( ( ( 1 2µ j ω,i yω j h j ω,i + f j ω,i x +µ j ω,i hj ω,i f j ω,i x yj ω and 1 2µ j ω,i yω j h j ω,i + f j ω,i x + µ j ω,i yj ω (h j ω,i f j ω,i x, which are equivalent to yω j h j ω,i f j ω,i x and yj ω h j ω,i f j ω,i ( x, respectively. Therefore, d j ω,i (1 = 2µ j ω,i yω j h j ω,i + f j ω,i +µ x j ω,i if and only if h j ω,i f j ω,i x yω j h j ω,i f j ω,i x. Next, for each j J, we partition the set I := 1,..., m 2 } into the sets I j 1, Ij 2, and Ij 3, i.e. I = I j 1 Ij 2 Ij 3, such that I j 1 := i I : d j ω,i = yj ω (h j ω,i f j ω,i x}, (44 I j 2 := i I : d j ω,i = (hj ω,i f j ω,i x ω} yj, and (45 } I j 3 := i I : d j ω,i = (1 2µj ω,i (y ω j h j ω,i + f j ω,i x + µ j ω,i. (46 Therefore, in the light of the above discussed cases, we can rewrite Kω(x 4 as Kω(x 4 = (y ω, d ω R J R m 2 J + : W ω y ω r ω T ω x, yω j h j ω,i f j ω,i x, dj ω,i = yj ω (h j ω,i f j ω,i x, i I j 1, j J, yω j h j ω,i f j ω,i x, dj ω,i (h = j ω,i f j ω,i x yω, j i I j 2, j J, h j ω,i f j ω,i x yj ω h j ω,i f j ω,i x, i Ij 3, j J, ( } d j ω,i (1 = 2µ j ω,i yω j h j ω,i + f j ω,i x + µ j ω,i, i Ij 3, j J. Interestingly, in a compact form K 4 ω(x can be written as K 4 ω(x = (y ω, d ω R J R m 2 J + : W ω y ω r ω T ω x, d j ω,i = θj ω,i yj ω + σ j ω,i, lj ω y j ω u j ω, i = 1,..., m 2, j J }, where θ j ω,i, σj ω,i R and lj ω, u j ω Z, + } for j J and i = 1,..., m 2. Since W ω is a TU matrix (because of assumption, transpose of (Wω T, I, I is also a TU matrix. Therefore, each bounded face Kω(x 4 of Kω(x, 3 for x X, has extreme points with integral yω j as r ω T ω x is integral. Since Kω(x 3 is a subset of the continuous relaxation of Kω(x, 1 and all bounded faces of Kω(x 3 have extreme points with integral y ω components, Kω(x 3 conv ( Kω(x 1. Hence, conv ( Kω(x 1 = Kω(x 3 for all x X because of (40. Finally using Theorem 2, we get conv (K ω (x = conv ( Kω(x 1 Kω(x 2 = Kω(x 3 Kω(x 2 where Kω(x 2 := (d ω,0, d ω R J + Rm 2 J + : d j ω d j ω,0, j J}. In other words, we obtain the convex p 12

13 hull of K ω (x by adding m 2 J number of linear inequalities: d j ω,i (1 2µj ω,i (y ω j h j ω,i x + f j ω,i + µ j ω,i, i = 1,..., m 2, j J, (47 to the continuous relaxation of K ω (x. Moreover, Kω(x 3 Kω(x 2 when projected to (y ω, d ω,0 space gives convex hull of the feasible region of Q ω (x, i.e. (y ω, d ω,0 R J R J + : (28, (29, and (1 2µ j ωyω j + F j ωx h j ω d j ω,0 },, j J p where the ith row of µ j ω, F j ω, and h j ω are denoted by µ j ω,i = hj ω,i h j ω,i, f j ω,i = ( and h j ω,i = 1 2µ j ω,i h j ω,i µ j ω,i, respectively. This completes the proof. ( 1 2µ j ω,i f j ω,i, Corollary 1. For all x X and ω Ω, let Q ω (x := min g ω y ω + j J ĝ j ωd j ω,0 (48 s.t. W ω y ω r ω T ω x, (49 1yω j + Fωx j h j d j ω,0, j J, (50 y j ω Z, d j ω,0 R +, j J, (51 where W ω is a totally unimodular matrix and r ω is integral. Then, P tight is a partial convex hull of the feasible region of the associated extensive formulation, i.e. P, where P tight := (x, y ω, d ω,0 } ω Ω X R Ω R Ω J + : T ω x + W ω y ω r ω, 1yω j + Fωx j h j d j ω,0, j J, ω Ω, (1 2µ j ωyω j + F j ωx h j ω d j ω,0 },, j J p such that the ith row of µ j ω, F j ω, and h j ω are given by µ j ω,i = hj ω,i h j ω,i, f j ω,i = ( and h j ω,i = 1 2µ j ω,i h j ω,i µ j ω,i, respectively. Proof. Using Theorems 1 and 4, we can easily prove this. ( 1 2µ j ω,i f j ω,i, Theorem 5. In TSS-CMIP (1, let Q ω (x := min g ω y ω + ĝωd j j ω,0 (52 j J s.t. W ω y ω r ω T ω x, (53 E j ω yω j + Fωx j h j d j ω,0, j J, (54 yω j Z q, d j ω,0 R +, j J, (55 13

14 where Eω, j j J, and W ω are TU matrices and r ω is integral. The convex hull of the feasible region of Q ω (x for all x X is given by (y ω, d ω,0 R q J R J + : (53, (54, and E j ωyω j + F j ωx h j ω d j ω,0 },, j J p ( where the ith row of E j ω, F j ω, and h j ω are denoted by e j ω,i (1 = 2µ j ω,i e j ω,i, f j ω,i = 1 2µ j ω,i f j ω,i (, and h j ω,i = 1 2µ j ω,i h j ω,i µ j ω,i, respectively, and µj ω,i = hj ω,i h j ω,i. Furthermore, in higher dimensional space Q ω (x can be reformulated as: Q ω (x := min g ω y ω + ĝωd j j ω,0 : (53, and (56 j J e j ω,i yj ω + f j ω,i x hj ω,i dj ω,i, i = 1,..., m 2, j J, (57 d j d j ω,0, j J, (58 } yω j Z q, d j ω,0 R +, d j ω R m 2 + j J. (59 Then, for all x X, the convex hull of the feasible region of Q ω (x, denoted by K ω (x, is obtained by adding m 2 J number of the following linear inequalities (in the higher dimensional space to the continuous relaxation of K ω (x: ( ( 1 2µ j ω,i e j ω,i yj ω h j ω,i + f j ω,i x + µ j ω,i dj ω,i, i = 1,..., m 2, j J. Proof. Let K 1 ω(x = (y ω, d ω Z q J R m 2 J + : W ω y ω r ω T ω x, (60 } e j ω,i yj ω + f j ω,i x hj ω,i dj ω,i, i = 1,..., m 2, j J (61 for all x X. Since E j ω is a TU matrix, e j ω,i yj ω is integral. Similar to the proof of the previous theorem, we again utilize the CMIR cut generation approach (discussed in Section 3.1 for each defining inequality (37 of K 1 ω(x. However, in this case we substitute σ = e j ω,i yj ω Z, β = h j ω,i f j ω,i x, and ρ 1 = d j ω,i in Z 1 and get the following valid parametric CMIR inequalities (26 for Kω(x: 1 ( ( 1 2µ j ω,i e j ω,i yj ω h j ω,i f j ω,i x + µ j ω,i dj ω,i, i = 1,..., m 2, j J (62 which is equivalent to ( ( 1 2µ j ω,i e j ω,i yj ω h j ω,i + f j ω,i x + µ j ω,i dj ω,i, i = 1,..., m 2, j J, (63 14

15 as f j ω,i x is integral for all x X. Thus, conv ( K 1 ω(x K 3 ω(x for all x X, where Kω(x 3 = (y ω, d ω R q J R m 2 J + : W ω y ω r ω T ω x, d j ω,i ej ω,i yj ω (h j ω,i f j ω,i x, i = 1,..., m 2, j J, (64 ( d j ω,i h j ω,i f j ω,i x e j ω,i yj ω, i = 1,..., m 2, j J, (65 ( } d j ω,i (1 2µ j ω,i e j ω,i yj ω h j ω,i + f j ω,i x + µ j ω,i, i = 1,..., m 2, j J. (66 Notice that Kω(x 3 is a subset of the continuous relaxation of Kω(x. 1 Therefore, if we can prove that all bounded faces of Kω(x 3 have extreme points with integral y ω component, then it will imply that Kω(x 3 conv ( Kω(x 1 for all x X. So, we now consider a bounded face of Kω(x 3 with maximum possible dimension, denoted by Kω(x, 4 and on this face, d j ω,i, i = 1,..., m 2 and j J, is minimal, i.e. ( } d j ω,i = max z j ω,i hj ω,i, hj ω,i zj ω,i, (1 2µj ω,i z j ω,i h j ω,i + µ j ω,i where z j ω,i = ej ω,i yj ω + f j ω,i x Z. Now if hj ω,i Z for any j J, then µj ω,i = 0 and as a result inequality (63 reduces to inequality (64. However, when h j ω,i / Z for j J, then again there are three possible cases: Case I. d j ω,i = zj ω,i hj ω,i : This case will happen if and only if zj ω,i hj ω,i hj ω,i zj ω,i and z j ω,i hj ω,i (1 2µ j ω,i (z j ω,i h j ω,i +µ j ω,i, which are equivalent to zj ω,i hj ω,i and zj ω,i h j ω,i, respectively. Therefore, we can claim that d j ω,i = zj ω,i hj ω,i if and only if zj ω,i h j ω,i. Case II. d j ω,i = hj ω,i zj ω,i : This case will happen if and only if hj ω,i zj ω,i zj ω,i hj ω,i and h j ω,i zj ω,i (1 2µ j ω,i (z j ω,i h j ω,i +µ j ω,i, which are equivalent to zj ω,i hj ω,i and zj ω,i h j ω,i, respectively, as µ j ω,i 1. Therefore, dj ω,i = hj ω,i zj ω,i if and only if zj ω,i h j ω,i. Case III. d j ω,i (1 = 2µ j ω,i (z j ω,i h j ω,i + µ j ω,i : This case will happen if and only if ( 1 2µ j ω,i (z j ω,i h j ω,i + µ j ω,i hj ω,i zj ω,i ( and 1 2µ j ω,i (z j ω,i h j ω,i + µ j ω,i z j ω,i hj ω,i, which are equivalent to zj ω,i h j ω,i and ( z j ω,i h j ω,i, respectively. Therefore, d j ω,i = 1 2µ j ω,i (z j ω,i h j ω,i + µ j ω,i if and only if h j ω,i z j ω,i h j ω,i. For each j J, let I := 1,..., m 2 } be partitioned into the disjoint sets I j 1, Ij 2, and Ij 3, i.e. I = I j 1 Ij 2 Ij 3, such that Ij 1 := i I : d j ω,i = zj ω,i } hj, I j 2 := i I : d j ω,i = hj ω,i ω,i} zj, and I j 3 := ( i I : d j ω,i = (1 2µj ω,i z j ω,i h j ω,i + µ j ω,i}. Therefore, in the light of the above 15

16 discussed cases, we can define Kω(x 4 as Kω(x 4 = (y ω, d ω R q J R m 2 J + : W ω y ω r ω T ω x, e j ω,i yj ω h j ω,i f j ω,i x, dj ω,i = ej ω,i yj ω (h j ω,i f j ω,i x, i I j 1, j J, e j ω,i yj ω h j ω,i f j ω,i x, dj ω,i (h = j ω,i f j ω,i x e j ω,i yj ω, i I j 2, j J, h j ω,i f j ω,i x ej ω,i yj ω h j ω,i f j ω,i x, i Ij 3, j J, ( } d j ω,i (1 = 2µ j ω,i e j ω,i yj ω h j ω,i + f j ω,i x + µ j ω,i, i Ij 3, j J. Let (ŷ ω, ˆd ω be an extreme point of K 4 ω(x for a given x X. Then this point can be obtained by binding J (q + m 2 constraints of K 4 ω(x such that the matrix defining these constraints is nonsingular. Since the polyhedron K 4 ω(x already has m 2 J equality constraints, we need a combination of at least q J defining inequalities of K 4 ω(x to be binding at the point (ŷ ω, ˆd ω. Let the constraints w ω,i y ω r ω,i t ω,i x, i τ 0, e j ω,i yj ω h j ω,i f j ω,i x, i τ j 1, j J 1, e j ω,i yj ω h j ω,i f j ω,i x, i τ j 2, j J 2, be binding at the point (ŷ ω, ˆd ω where w ω,i, r ω,i, and t ω,i denote ith row of matrix/vector W ω, r ω, and T ω, respectively, and τ 0, τ j 1, j J 1 J, and τ j 2, j J 1 J, are subsets of 1,..., m 2 } such that τ 0 + j J 1 τ j 1 + j J 2 τ j 2 = q J. Then, ŷ ω satisfies the following system of equations: w ω,i ŷ ω = r ω,i t ω,i x, i τ 0, (67 e j ω,iŷj ω = h j ω,i f j ω,i x, i τ j 1, j J 1, (68 e j ω,iŷj ω = h j ω,i f j ω,i x, i τ j 2, j J 2, (69 which in compact form is written as Π ω ŷ ω = π ω (x. Observe that Π ω is a totally unimodular matrix of size q J q J as W ω, Eω j for j J are totally unimodular matrices. Also, since each component of the vector π ω (x, i.e., r ω,i t ω,i x, h j ω,i f j ω,i x, or h j ω,i f j ω,i x, is integral, ŷ ω is integral. Hence, all bounded faces of K 3 ω(x have extreme point (ŷ ω, ˆd ω with integral ŷ ω and since K 3 ω(x is a subset of the continuous relaxation of K 1 ω(x, K 3 ω(x conv ( K 1 ω(x for all x X. Therefore, we have K 3 ω(x = conv ( K 1 ω(x. Finally using Theorem 2, we get conv (K ω (x = conv ( Kω(x 1 Kω(x 2 = Kω(x 3 Kω(x 2 where Kω(x 2 := (d ω,0, d ω R J (m : d j ω d j ω,0, j J}. In other words, we obtain the convex hull p of K ω (x by adding m 2 J number of linear inequalities: d j ω,i (1 2µj ω,i (e j ω,i yj ω h j ω,i + f j ω,i + µ j ω,i, i = 1,..., m 2, j J, (70 to the continuous relaxation of K ω (x. Moreover, Kω(x 3 Kω(x 2 when projected to (y ω, d ω,0 space gives the convex hull of the feasible region of Q ω (x, i.e. } (y ω, d ω,0 R q J R J E + : (53, (54, and j ωyω j + F j ωx h j ω d j ω,0, j J p 16

17 where the ith row of E j ω, F j ω, and h j ω are denoted by e j ω,i (1 = 2µ j ω,i e j ω,i, f j ω,i = ( and h j ω,i = 1 2µ j ω,i h j ω,i µ j ω,i, respectively. This completes the proof. ( 1 2µ j ω,i f j ω,i, Corollary 2. For all x X and ω Ω, let Q ω (x := min g ω y ω + j J ĝ j ωd j ω,0 s.t. W ω y ω r ω T ω x, E j ω yω j + Fωx j h j d j ω,0, j J, y j ω Z q, d j ω,0 R +, j J, where Eω, j j J, W ω are TU matrices and r ω is integral. Then, P tight is a partial convex hull of the feasible region of the associated extensive formulation, i.e. P, where P tight := (x, y ω, d ω,0 } ω Ω X R q Ω R Ω J + : T ω x + W ω y ω r ω, Eωy j ω j + Fωx j h j d j ω,0, j J, ω Ω, E j ωyω j + F j ωx h j ω d j ω,0 },, j J p ( such that the ith row of µ j ω, E j ω, F j ω, and h j ω are given by µ j ω,i = hj ω,i h j ω,i, e j ω,i = 1 2µ j ω,i e j ω,i ( (, f j ω,i = 1 2µ j ω,i f j ω,i, and hj ω,i = 1 2µ j ω,i h j ω,i µ j ω,i, respectively. Proof. Using Theorems 1 and 5, we can easily prove this. 4. Computational Experiments We perform computational experiments to evaluate the effectiveness of adding our parametric cuts (a priori in the second stage convexification approach for solving TSS-CMIP test instances. We first describe generation of these test problems in Section 4.1 and then present our computational results in Section Generation of TSS-CMIP test instances For our computational experiments, we consider the reformulated extensive formulation of TSS- CMIP, i.e., (17-(22, with p = 2 and structured CMIPs in the second stage (as discussed in previous section, i.e. either E j ω = I (an identity matrix or E j ω is a randomly generated deterministic TU matrix for j J, and W ω is a deterministic TU matrix for all ω Ω. For each structured TSS-CMIP with only integer variables in the second stage, we generate two sets of random instances: instances from the first problem set are motivated from the Stochastic Integer Programming Library (SIPLIB TSS-MIP instances [1], in particular stochastic server location problem (SSLP and stochastic multiple binary knapsack problem (SMBKP instances, whereas for the second problem set, we consider instances with larger number of scenarios (up to 10,000. More specifically, in the first 17

18 problem set, we generate random instances with similar problem size as of SSLP and SMBKP instances but with uncertain cost-coefficients, technology matrix, F j ω, h j ω and right-hand-side. In Tables 1 and 2, we provide details of problem categories for two types of structured second-stage CMIPs, i.e., E j ω is an identity matrix or any TU matrix, used for our experiments. We denote the number of linear constraints and number of integer variables by #LCon and #IVar, respectively, in each stage. Also, we use #PCCon and #SOCcon to denote the number of polyhedral conic constraints (or number of rows in E j ω and number of second-order conic constraints, respectively, in the second stage. Table 1: Details of TSS-CMIP Instances with E j ω = I (identity matrix Problem Instance Stage I Stage II Set Category #LCon #IVar #LCon #IVar #PCCon #SOCCon I II SCMIP SCMIP SCMIP SCMIP SCMIP SCMIP SCMIP Table 2: Details of TSS-CMIP Instances where Eω j is TU for all j J Problem Instance Stage I Stage II Set Category #LCon #IVar #LCon #IVar #PCCon #SOCCon SCMIP I SCMIP SCMIP SCMIP SCMIP SCMIP II SCMIP SCMIP SCMIP We use SCMIP.α.β.λ to denote our instance category in Tables 1 and 2, where α is the number of integer variables in the first stage, β is the number of integer variables in the second stage, and λ is the number of second-order conic constraints in the second stage. Note that in Table 1, #PCCon is the same as #IVar in the second stage as Eω j is an identity matrix. For each instance category, we generate instances as follow: we draw c from uniform[0, 10], A, b from uniform[0, 100], r ω, T ω from integer uniform [ 100, 100], Fω j from integer uniform[ 10, 10], h j ω from uniform[ 10, 10]. Moreover, for instance categories in Table 1 where J = 1, we let g ω = 0 and ĝω j = 1 for ω Ω, j J, while for instance categories in Table 2 where J = 3, g ω are randomly generated from uniform[ 1, 1] and ĝω j are randomly generated from uniform[ 10, 10]. Since yω j is an unrestricted variable and g ω [ 1, 1] for j J and ω Ω, we observed that many randomly generated TSS-CMIP instances have unbounded solution values. Therefore, in order to obtain finite optimal solution values, we impose an upper bound of 100 and a lower bound of 100 on each yω j variable. 18

19 Note that when E j ω = I, the number of integer variables in the second stage, i.e., q, is same as number of polyhedral conic constraints, i.e., m 2. Whereas for instance categories in Table 2, E j ω is not a square matrix. 4.2 Computational Framework In this section, we evaluate the effectiveness of convexifying the second stage CMIPs by performing computational experiments on instances belonging to the aforementioned instance categories with different number of scenarios. The results of our experiments are presented in Tables 3 and 4. Each row in Table 3 reports the average over five randomly generated instances corresponding to the instance category, while each row in Table 4 reports the average over three randomly generated (relatively harder instances. For each instance, we perform two experiments: TSS-CMIP-NO-PCUT and TSS-CMIP-WITH-PCUTS. In TSS-CMIP-NO-PCUT, we solve the reformulated extensive formulation of the problem using CPLEX with its default settings, without adding our parametric cuts. Whereas in TSS-CMIP-WITH-PCUTS, we add our parametric linear cuts (provided in Theorem 5, a priori to the reformulated extensive formulation of the problem instance, relax the integrality constraints of second stage integer variables, and use CPLEX with its default settings to solve it. All experiments are performed on a 8-core Xeon 2.4GHz machine with 24 GB RAM running with Windows 10. In Tables 3 and 4, we report following statistic: number of integer variables in the extensive formulation with(out parametric cuts (#IVar, number of linear constraints in the extensive formulation without our parametric cuts (#LCon, and the number of linear parametric cuts added in the second stage (#LCuts. Also, we denote the total time taken to solve TSS-CMIP instances without and with our parametric cuts by T-EF and T-EFC, respectively, and the percentage of integrality gap closed by a priori addition of parametric cuts by G% = 100 (V pcut V cp /(V mip V cp, where V pcut, V cp, and V mip denote the optimal objective value of continuous relaxation of extensive formulation with our parametric cuts, continuous relaxation of extensive formulation without our parametric cuts, and extensive CMIP formulation, respectively. We use TL to notify that CPLEX cannot solve the corresponding instance within 3 hours time limit and OM is to notify that our system ran out of 24 GB memory when solving this instance. Instances for which we could not obtain the optimal value due to TL or OM, we put - in column G% Computational results for TSS-CMIPs with E j ω = I In Table 3, comparing T-EF and T-EFC, we observe that adding our parametric linear cuts (a priori significantly reduces the time taken to solve the reformulated extensive formulation of the TSS-CMIP instances, using CPLEX with its default settings. This is because after adding our cuts, the number of integer variables (#IVar reduced drastically. More specifically, after adding our parametric cuts, CPLEX solved 132 out of 135 randomly generated instances within the time limit, except three instances of SCMIP with 500 scenarios. Whereas, without our cuts, CPLEX could not solve 35 out of 135 TSS-CMIP instances within 3 hours time limit (32 instances and allocated memory (3 instances, and took 148 seconds (on average to solve the remaining 100 instances in comparison to 32 seconds (on average after adding our cuts. Additionally, TSS-CMIP- WITH-PCUTS took 22 minutes (on average for 32 out of 35 (unsolvable instances. Overall, for instances in Table 3, our parametric cuts closed the integrality gap by 25.53% (on average and reduced the time taken to solve the TSS-CMIP instances by 5 times (on average. 19

20 Table 3: Results of Computational Experiments for TSS-CMIPs with Eω j = I Problem Instance TSS-CMIP-NO-PCUTS TSS-CMIP-WITH-PCUTS Ω Set Category #IVar #LCon T-EF #IVar #PCuts T-EFC G% SCMIP SCMIP SCMIP TL SCMIP TL SCMIP TL TL - SCMIP I SCMIP SCMIP SCMIP TL SCMIP TL SCMIP TL SCMIP SCMIP SCMIP SCMIP SCMIP SCMIP SCMIP SCMIP II SCMIP TL SCMIP SCMIP SCMIP SCMIP SCMIP SCMIP SCMIP In particular, for problem set I, TSS-CMIP-WITH-PCUTS were performed at least 2 times and up to 19 times faster than TSS-CMIP-NO-PCUTS. Whereas for problem set II, TSS-CMIP- WITH-PCUTS is 5 times (on average and up to 39 times faster than TSS-CMIP-NO-PCUTS. It is interesting to observe that while using our second stage convexification approach, there is a trade-off between the decrease in the number of integer variables and increase in the number of constraints. For instances such as SCMIP where Ω 50, 100, 500, 1000}, TSS-CMIP- WITH-PCUTS took marginally longer time that TSS-CMIP-NO-PCUTS. However, for instances with larger number of scenarios, TSS-CMIP-WITH-PCUTS are notably faster than TSS-CMIP- NO-PCUTS. 20

21 Table 4: Results of Computational Experiments for TSS-CMIPs where Eω j is TU for j J, ω Ω Problem Instance TSS-CMIP-NO-PCUTS TSS-CMIP-WITH-PCUTS Ω Set Category #IVar #LCon T-EF #IVar #PCuts T-EFC G% SCMIP TL SCMIP TL SCMIP TL SCMIP TL SCMIP I SCMIP TL SCMIP TL SCMIP TL SCMIP TL SCMIP SCMIP SCMIP SCMIP TL SCMIP OM SCMIP TL SCMIP OM SCMIP OM SCMIP TL II SCMIP OM SCMIP OM SCMIP OM SCMIP OM SCMIP SCMIP OM SCMIP OM Computational results for TSS-CMIPs where E j ω is TU Notice that the problem instances considered in Table 4 are much harder than problem instances considered in Table 3, primarily because of multiple conic constraints (21 corresponding to each scenario (as J = 3 and multiple integer variables in each constraint (20. It is evident from column T-EF in Table 4 which shows that CPLEX with its default settings could not solve reformulated extensive formulation of 50 out of 75 TSS-CMIP instances (without our parametric cuts within a time limit of 3 hours and the allocated 24 GB RAM. In contrast, after adding our parametric cuts, we solved all 75 instances in seconds (on average. For instances solved to optimality using TSS-CMIP-NO-PCUTS, our parametric cuts closed the integrality gap by 47.29% (on average and reduced the time taken to solve the TSS-CMIP instances by times (on average. It is worth to note that even though G% is small for some instances (i.e., SCMIP , SCMIP , and SCMIP for Ω 50, 200}, CPLEX with its 21