Binary decision rules for multistage adaptive mixed-integer optimization

Binary decision rules for multistage

adaptive mixed-integer optimization

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Bertsimas, Dimitris, and Angelos Georghiou. “Binary Decision Rules

for Multistage Adaptive Mixed-Integer Optimization.” Mathematical

Programming, vol. 167, no. 2, Feb. 2018, pp. 395–433.

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http://dx.doi.org/10.1007/s10107-017-1135-6

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Springer Berlin Heidelberg

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Author's final manuscript

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http://hdl.handle.net/1721.1/117402

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Binary Decision Rules for Multistage Adaptive Mixed-Integer

Optimization

Dimitris Bertsimas · Angelos Georghiou

Received: date / Accepted: date

Abstract Decision rules provide a flexible toolbox for solving computationally de-manding, multistage adaptive optimization problems. There is a plethora of real-valued decision rules that are highly scalable and achieve good quality solutions. On the other hand, existing binary decision rule structures tend to produce good quality solutions at the expense of limited scalability and are typically confined to worst-case optimization problems. To address these issues, we first propose a linearly parameterised binary decision rule structure and derive the exact reformulation of the decision rule problem. In the cases where the resulting optimization problem grows exponentially with respect to the problem data, we provide a systematic methodol-ogy that trades-off scalability and optimality, resulting to practical binary decision rules. We also apply the proposed binary decision rules to the class of problems with random-recourse and show that they share similar complexity as the fixed-recourse problems. Our numerical results demonstrate the effectiveness of the proposed binary decision rules and show that they are (i) highly scalable and (ii) provide high quality solutions.

Keywords Adaptive Optimization·Binary Decision Rules·Mixed-integer optimiza-tion.

1 Introduction

In this paper, we use robust optimization techniques to develop efficient solution methods for the class of multistage adaptive mixed-integer optimization problems. The one-stage variant of the problem can be described as follows: Given matrices A ∈Rm×k, B ∈Rm×q, C ∈Rn×k, D ∈Rq×k, H ∈Rm×k and a probability measure Pξξξ supported on setΞfor the uncertain vectorξξξ ∈Rk, we are interested in choosingn Dimitris Bertsimas

Sloan School of Management and Operations Research Center Massachusetts Institute of Technology

Cambridge, MA 02139, USA E-mail: dbertsim@mit.edu Angelos Georghiou

Desautels Faculty of Management McGill University

Montreal, Quebec H3A 1G5, Canada E-mail: angelos.georghiou@mcgill.ca

real-valued functionsxxx(·)∈ Rk,nandq binary functionsyyy(·)∈ Bk,q in order to solve: minimize Eξξξ  (Cξξξ)>xxx(ξξξ) + (Dξξξ)>yyy(ξξξ) subject to xxx(·)∈ Rk,n, yyy(·)∈ Bk,q, Axxx(ξξξ) +Byyy(ξξξ)≤ Hξξξ, ) ∀ξξξ ∈ Ξ. (1.1)

Here,R_{k,ndenotes the space of all real-valued functions from R}k _{to R}n, andB_k,qthe space of binary functions from Rk to {0,1}q. Problem (1.1) has a long history both from the theoretical and practical point of view, with applications in many fields such as engineering [22, 32], operations management [5, 30], and finance [13, 14]. From a theoretical point of view, Dyer and Stougie [18] have shown that computing the optimal solution for the class of Problems (1.1) involving only real-valued decisions, is #P-hard, while their multistage variants are believed to be“computationally intractable already when medium-accuracy solutions are sought” [31]. In view of these complexity results, there is a need for computationally efficient solution methods that possibly sacrifice optimality for tractability. To quote Shapiro and Nemirovski [31], “. . . in actual applications it is better to pose a modest and achievable goal rather than an ambitious goal which we do not know how to achieve.”

A drastic simplification that achieves this goal is to usedecision rules. This func-tional approximation restricts the infinite space of the adaptive decisions xxx(·) and yyy(·) to admit pre-specified structures, allowing the use of numerical solution methods. Decisions rules for real-valued functions have been used since 1974 [19]. Nevertheless, their potential was not fully exploited until recently when new advances in robust optimization provided the tools to reformulate the decision rule problem as tractable convex optimization problems [4, 7]. Robust optimization techniques were first used by Ben-Talet al.[6] to reformulate thelinear decision rule problem. In this work, real-valued functions are parameterised as linear function of the random variables, i.e., x(ξξξ) =xxx>ξξξ forxxx ∈_Rk. The simple structure of linear decision rules offers the scal-ability needed to tackle multistage adaptive problems. Even though linear decision rules are known to be optimal in a number of problem instances [1, 11, 23], their sim-ple structure generically sacrifices a significant amount of optimality in return for scalability. To gain back some degree of optimality, attention was focused on the con-struction of non-linear decision rules. Inspired by the linear decision rule structure, the real-valued adaptive decisions are parameterised asx(ξξξ) =xxx>L(ξξξ), xxx ∈_Rk0, where L: Rk _→Rk0

, k0 ≥ k, is a non-linear operator defining the structure of the decision rule. This approximation provides significant improvements in solution quality, while retaining in large parts the favourable scalability properties of the linear decision rules. Table 1 summarizes non-linear decision rules from the literature that are parameter-ized as x(ξξξ) = xxx>L(ξξξ), together with the complexity of the induced convex opti-mization problems. A notable exception of real-valued decision rules that are alterna-tively parameterized asx(ξξξ) = max{xxx₁>ξξξ, . . . , xxx>_Pξξξ} −max{xxx>₁ξξξ, . . . , xxx>_Pξξξ}, xxxp, xxxp∈Rk, is proposed by Bertsimas and Georghiou [10] in the framework of worst-case adaptive optimization. This structure offers near-optimal designs but requires the solution of mixed-integer optimization problems.

In contrast to the plethora of decision rule structures for real-valued decisions, the literature on discrete decision rules is somewhat limited. There are, however, three notable exceptions that deal with the case of worst-case adaptive optimization. The first one is the work of Bertsimas and Caramanis [8], where integer decision rules are parameterized as y(ξξξ) = yyy>dξξξe, yyy ∈ _Zk, where d·e is the component-wise ceil-ing function. In this work, the resultceil-ing semi-infinite optimization problem is further approximated and solved using a randomized algorithm [15], providing only proba-bilistic guarantees on the feasibility of the solution. The second binary decision rule

Real-Valued Decision Rule Structures

x(ξξξ) = xxx>L(ξξξ) Computational Burden Reference

Linear LOP [6, 22, 27, 32]

Piecewise linear LOP [16, 17, 20, 21]

Multilinear LOP [20]

Quadratic SOCOP [3, 20]

Power, Monomial, Inverse monomial SOCOP [20]

Polynomial SDOP [2, 12]

Table 1: The table summarizes the literature of real-valued decision rule structures for which the resulting semi-infinite optimization problem can be reformulated exactly using robust optimization techniques. The computation burden refers to the structure of the resulting optimization problems for the case where the uncertainty set is described by a polyhedral set, with Linear Optimization Problems denoted by LOP, Second-Order Cone Optimization Problems denoted by SOCOP, and Semi-Definite Optimization Problems denoted by SDOP.

structure was proposed by Bertsimas and Georghiou [10], where the decision rule is parameterized as

y(ξξξ) = (

1, max{yyy>₁ξξξ, . . . , yyy>_Pξξξ} −max{yyy> 1ξξξ, . . . , yyy

> Pξξξ} ≤0,

0, otherwise, (1.2)

withyyy_p, yyy

p∈R

k_{, p= 1, . . . , P. This binary decision rule structure offers near-optimal} designs at the expense of scalability. Finally, in the recent work by Hanasusanto et al. [24], the binary decision rule is restricted to the so called “K-adaptable struc-ture”, resulting to binary adaptable policies withK contingency plans. This approx-imation is an adaptation of the work presented in Bertsimas and Caramanis [9], and its use is limited to two-stage adaptive problems.

The goal of this paper is to develop binary decision rule structures that can be used together with the real-valued decision rules listed in Table 1 for solving multistage adaptive mixed-integer optimization problems. The proposed methodology is inspired by the work of Bertsimas and Caramanis [8], and uses the tools developed in Georghiou et al.[20] to robustly reformulate the problem into a finite dimensional mixed-integer linear optimization problem. The motivation of this work is to propose an alternative to the highly flexible but computationally demanding decision rule (1.2). The emphasis of this work is placed on the scalability properties of the solution method, allowing to solve large instances of Problem (1.1) with minimal computational effort. The main contributions of this paper can be summarized as follows.

1. We propose a linear parameterized binary decision rule structure and derive the exact reformulation of the decision rule problem that achieves the best binary decision rule. We prove that for general polyhedral uncertainty sets and arbitrary decision rule structures, the decision rule problem is computationally intractable, resulting in mixed-integer linear optimization problems whose size can grow ex-ponentially with respect to the problem data. To remedy this exponential growth, we use similar ideas as those discussed in Georghiouet al.[20], to provide a sys-tematic trade-off between scalability and optimality, resulting to practical binary decision rules.

2. We apply the proposed binary decision rules to the class of problems with random-recourse, i.e., to problem instances where the technology matrix B(ξξξ) is a given function of the uncertain vector ξξξ. We provide the exact reformulation of the binary decision rule problem, and we show that the resulting mixed-integer linear optimization problem shares similar complexity as in the fixed-recourse case. 3. We demonstrate the effectiveness of the proposed methods in the context of a

multistage inventory control problem (fixed-recourse problem), and a multistage knapsack problem (random-recourse problem). We show that for the inventory

control problem, we are able to solve problem instances with 50 time stages and 200 binary recourse decisions, while for the knapsack problem we are able to solve problem instances with 25 time stages and 1300 binary recourse decisions, using IBM ILOG CPLEX 12.5 on an Intel Core i5−3570 CPU at 3.40GHz machine with 8GB RAM.

The rest of this paper is organized as follows. In Section 2, we outline our approach for binary decision rules in the context of one-stage adaptive optimization problems with fixed-recourse, and in Section 3 we discuss the extension to random-recourse problems. In Section 4, we extend the proposed approach to multistage adaptive opti-mization problems and in Section 5, we present our computational results. Throughout the paper, our work is focused on problem instances involving only binary decision rules for ease of exposition. The extension to problem instances involving both real-valued and binary recourse decisions can be easily extrapolated, and thus omitted. Notation. We denote scalar quantities by lowercase, non-bold face symbols and vector quantities by lowercase, boldface symbols, e.g., x ∈_{R and}xxx ∈_Rn, respectively. Sim-ilarly, scalar and vector valued functions will be denoted by, x(·)∈R andxxx(·)∈Rn, respectively. Matrices are denoted by uppercase symbols , e.g.,A ∈Rn×m. We model uncertainty by a probability space (Rk_{, B(R}k_),Pξ

ξ

ξ) and denote the elements of the sam-ple space Rk byξξξ. The Borelσ-algebraB_(Rk) is the set of events that are assigned probabilities by the probability measure Pξξξ. The supportΞof Pξξξrepresents the small-est closed subset of Rk _{which has probability 1, and Eξ}

ξ

ξ(·) denotes the expectation operator with respect to Pξξξ. Tr (A) denotes the trace of a square matrix A ∈Rn×n and 111(·) is the indicator function. Finally,eeek the kth canonical basis vector, whileeee denotes the vector whose components are all ones. In both cases, the dimension will be clear from the context.

2 Binary Decision Rules for Fixed-Recourse Problems

In this section, we present our approach for one-stage adaptive optimization problems with fixed recourse, involving only binary decisions. Given matrices B ∈Rm×q, D ∈ Rq×k, H ∈_Rm×k _{and a probability measure Pξξξ} supported on setΞ for the uncertain vectorξξξ ∈Rk, we are interested in choosing binary functionsyyy(·)∈ Bk,q in order to solve: minimize Eξξξ  (Dξξξ)>yyy(ξξξ) subject to yyy(·)∈ B_k,q, Byyy(ξξξ)≤ Hξξξ,  ∀ξξξ ∈ Ξ. (2.3)

Here, we assume that the uncertainty set Ξ is a non-empty, convex and compact polyhedron

Ξ=nξξξ ∈Rk:∃ζζζ ∈Rv such thatWξξξ+Uζζζ ≥ hhh, ξ1= 1 o

, (2.4)

where W ∈Rl×k, U ∈Rl×v andhhh ∈Rl. Moreover, we assume thatΞ spans Rk_{. The} parameterξ1 is set equal to 1 without loss of generality as it allows us to represent affine functions of the non-degenerate outcomes (ξ2, . . . , ξk) in a compact manner as linear functions ofξξξ= (ξ1, . . . , ξk).

If the distribution governing the uncertaintyξξξis unknown or if the decision maker is very risk-averse, then one might choose to alternatively minimize the worst-case costs with respect to all possible scenariosξξξ ∈ Ξ. This can be achieved, by replacing

the objective function in Problem (2.3) with the worst-case objective maxξξξ∈Ξ 

(Dξξξ)>yyy(ξξξ). By introducing the auxiliary variable τ ∈ R and an epigraph formulation, we can equivalently write the worst-case problem as the following adaptive robust optimiza-tion problem. minimize τ subject to τ ∈R, yyy(·)∈ Bk,q, (Dξξξ)> yyy(ξξξ)≤ τ, Byyy(ξξξ)≤ Hξξξ,      ∀ξξξ ∈ Ξ. (2.5)

Notice thatξξξappears in the left hand side of the constraint (Dξξξ)>yyy(ξξξ)≤ τ, multiplying the binary decisions yyy(ξξξ). Therefore, Problem (2.5) is an instance of the class of problems with random-recourse, which will be investigated in Section 3.

Problem (2.3) involves a continuum of decision variables and inequality con-straints. Therefore, in order to make the problem amenable to numerical solutions, there is a need for suitable functional approximations foryyy(·).

2.1 Structure of Binary Decision Rules

In this section, we present the structure of the binary decision rules. We restrict the feasible region of the binary functions yyy(·) ∈ B_k,q to admit the following piecewise constant structure:

yyy(ξξξ) =Y G(ξξξ), Y ∈Zq×g, 0≤ Y G(ξξξ)≤ eee,



∀ξξξ ∈ Ξ, (2.6a)

where G: Rk _{→ {0,1}}g _{is a piecewise constant function:} G1(ξξξ) := 1, Gi(ξξξ) := 111  α αα>i ξξξ ≥ βi  , i= 2, . . . , g, (2.6b) for givenαααi∈Rkandβi∈R,i= 2, . . . , g. Notice, that sinceG(·) is a piecewise constant function mapping to {0,1}and the entries of matrix Y can take any integer values, then yyy(ξξξ) = Y G(ξξξ) gives rise to integer decision rules. By imposing the additional constraint 0 ≤ Y G(ξξξ) ≤ eee, for all ξξξ ∈ Ξ, yyy(·) is restricted to the space of binary decision rules.

We require the pairs (αααi, βi) definingG(·) to satisfy the following conditions. First, we choose (αααi, βi) which result in unique hyperplanesααα>iξξξ−βi= 0 for alli ∈ {2, . . . , g}, i.e., we avoid creating both 111(ξ ≥0.5) and 111(ξ ≤0.5), or, both 111(ξ ≤0.5) and 111(2ξ ≤1). Second, using (αααi, βi) we partition the uncertainty setΞ intoP polyhedra, whereΞp is defined as follows: Ξp=  ξξξ ∈ Ξ :ααα>_iξξξ ≥ βi, i ∈ Gp⊆ {2, . . . , g}, ααα>_iξξξ ≤ βi, i ∈ {2, . . . , g}\Gp  , p= 1, . . . , P. (2.7)

Here, given (αααi, βi) the setsGp are chosen such that the number of non-empty parti-tions P is maximized. We require that (αααi, βi),i= 2, . . . , g are chosen such thatΞp have (i) non-empty relative interior, (ii) the setsΞp span Rk, (iii) any partition pair Ξi and Ξj, i 6= j, can overlap only on one of their facets, and (iv), Ξ = SPp=1Ξp. The number of partitionsP that satisfy these conditions is upper bounded by 2g. We remark that in condition (ii) due to the definition ofΞ in (2.4), Ξp spans Rk if and only if it has dimensionk −1.

Applying decision rules (2.6) to Problem (2.3) yields the following semi-infinite problem, which involves a finite number of decision variablesY ∈Zq×g, and an infinite

number of constraints: minimize Eξξξ  (Dξξξ)>Y G(ξξξ) subject to Y ∈Zq×g, B Y G(ξξξ)≤ Hξξξ, 0≤ Y G(ξξξ)≤ eee,      ∀ξξξ ∈ Ξ. (2.8)

Notice that all decision variables appear linearly in Problem (2.8). Nevertheless, the objective and constraints are non-linear functions ofξξξ.

The following example illustrates the use of the binary decision rule (2.8) in a simple instance of Problem (2.3), briefly discussing the choice of vectorsαααi∈Rk and βi∈R,i= 2, . . . , g, and showing the relationship between Problems (2.3) and (2.8). Example 1 Consider the following instance of Problem (2.3):

minimize Eξξξ yyy(ξξξ)
subject to y(·)∈ B2,1,

y(ξξξ)≥ ξ2, 

∀ξξξ ∈ Ξ, (2.9)

where Pξξξ is a uniform distribution supported on Ξ = 

(ξ1, ξ2) ∈R2 : ξ1 = 1, ξ2 ∈ [−1,1] . The optimal solution of Problem (2.9) is y∗(ξξξ) = 111(ξ2 ≥ 0) achieving an optimal value of 1₂. One can attain the same solution by solving the following semi-infinite problem where G(·) is defined to be G1(ξξξ) = 1, G2(ξξξ) = 111(ξ2 ≥ 0), i.e., ααα2= (0,1)>,β2= 0 andg= 2, see Figure 1.

minimize Eξξξ yyy > G(ξξξ)
subject to yyy ∈_Z2, yyy>G(ξξξ)≥ ξ2, 0≤ yyy>G(ξξξ)≤1,      ∀ξξξ ∈ Ξ. (2.10)

The optimal solution of Problem (2.10) isyyy∗= (0,1)>

achieving an optimal value of 1

2, and is equivalent to the optimal binary decision rule in Problem (2.9).

0 1

-1 0 1

Figure 1: Plot of piecewise constant function G2(ξξξ) = 111(ξ2≥ 0). This structure of G(·) will give

rise to piecewise constant decision rules y(ξξξ) = yyy>_{G(ξξξ), for some yyy ∈ R}2_{. If one imposes constraints}

yyy ∈ Z2 _{and 0 ≤ yyy}>_{G(ξξξ) ≤ 1, y(·) is restricted to the space of binary decision rules induced by G.}

In the following section, we present robust reformulations of the semi-infinite Prob-lem (2.8).

2.2 Computing the Best Binary Decision Rule

In this section, we present an exact reformulation of Problem (2.8). The solution of the reformulated problem will achieve the best binary decision rule associated with structure (2.6). The main idea used in this section is to map Problem (2.8) to an equivalent lifted adaptive optimization problem on a higher-dimensional probability space. This will allow us to represent the non-convex constraints with respect to ξξξ in Problem (2.8), as linear constraints in the lifted adaptive optimization problem. The relation between the uncertain parameters in the original and the lifted problems is determined through the piecewise constant operatorG(·). By taking advantage of the linear structure of the lifted problem, we will employ linear duality arguments to reformulate the semi-infinite structure of the lifted problem into a finite dimensional mixed-integer linear optimization problem. The work presented in this section uses the tools developed by Georghiou et al.[20, Section 3].

We define the non-linear operator

L: Rk→Rk0, L(ξξξ) =  ξξξ G(ξξξ)  , (2.11)

where k0 = k+g, and define the projection matricesRξξξ ∈ Rk×k 0 , RG ∈Rg×k 0 such that R_ξξξL(ξξξ) =ξξξ, RGL(ξξξ) =G(ξξξ).

We will refer to L(·) as the lifting operator. Using L(·) and R_ξξξ, RG, we can rewrite Problem (2.8) into the following optimization problem:

minimize Eξξξ  (DR_ξξξL(ξξξ)) > Y RGL(ξξξ)  subject to Y ∈Zq×g, B Y RGL(ξξξ)≤ HRξξξL(ξξξ), 0≤ Y RGL(ξξξ)≤ eee,      ∀ξξξ ∈ Ξ. (2.12)

Notice that this simple reformulation still has infinite number of constraints. Never-theless, due to the structure ofR_ξξξ andRG, the constraints are now linear with respect toL(ξξξ). The objective function of Problem (2.12) can be further reformulated to

Eξξξ  (DRξξξL(ξξξ)) > Y RGL(ξξξ)  = TrM R>_ξξξD > Y RG  ,

where M ∈ _Rk0×k0, M _{= Eξξξ}(L(ξξξ)L(ξξξ)>) is the second order moment matrix of the uncertain vectorL(ξξξ). In some special cases where the Pξξξ has a simple structure, e.g., Pξξξ is a uniform distribution, and G(·) is not too complicated, M can be calculated analytically. If this is not the case, an arbitrarily good approximation of M can be calculated using Monte Carlo simulations. This is not computationally demanding as it does not does not involve an optimization problem, and can be done offline.

We now define the uncertain vectorξξξ0= (ξξξ0>1 , ξξξ 0> 2 ) > ∈Rk0, such thatξξξ0:=L(ξξξ) = (ξξξ>, G(ξξξ)> )>

. The random vectorξξξ0 has probability measure Pξξξ0 which is defined on the space Rk0, B Rk0

and is completely determined by the probability measure Pξξξ through the relation

Pξξξ0 B 0
:= Pξξξ  {ξξξ ∈Rk : L(ξξξ)∈ B0}  ∀B0∈ B(Rk0). (2.13) We also introduce the probability measure support

Ξ0:=L(Ξ) =nξξξ0∈_Rk0 : ∃ξξξ ∈ Ξ such that L(ξξξ) =ξξξ0o, =nξξξ0∈Rk0 : R_ξξξξξξ 0 ∈ Ξ, L(R_ξξξξξξ 0 ) =ξξξ0o, (2.14)

and the expectation operator Eξξξ0(·) with respect to the probability measure Pξξξ0. We will refer to Pξξξ0 andΞ

0

as the lifted probability measure and uncertainty set, respec-tively. Notice that although Ξ is a convex polyhedron, Ξ0 can be highly non-convex due to the non-linear nature of L(·). Using the definition ofξξξ0 we introduce the fol-lowinglifted adaptive optimization problem:

minimize TrM0R_ξ>ξξD > Y RG  subject to Y ∈_Zq×g, B Y R_Gξξξ0≤ HR_ξξξξξξ0, 0≤ Y RGξξξ0≤ eee,      ∀ξξξ0∈ Ξ0, (2.15) where M0 ∈ _Rk0×k0, M0 _{= Eξ}ξξ0(ξξξ 0

ξξξ0>) is the second order moment matrix associated

withξξξ0_{. Since there is a one-to-one mapping between Pξξξ and Pξξξ}0,M=M 0

.

Proposition 1 Problems (2.12) and (2.15) are equivalent in the following sense: both problems have the same optimal value, and there is a one-to-one mapping between feasible and optimal solutions in both problems.

Proof See [20, Proposition 3.6 (i)]. ut

The lifted uncertainty setΞ0is an open set due to the discontinuous nature ofG(·). This can be problematic in an optimization framework. We now defineΞ0 := cl(Ξ0) to be the closure of setΞ0, and introduce the following variant of Problem (2.15).

minimize TrM0R_ξ>_ξξD>Y RG  subject to Y ∈_Zq×g, B Y RGξξξ 0 ≤ HRξξξξξξ 0 , 0≤ Y RGξξξ0≤ eee,      ∀ξξξ0∈ Ξ0, (2.16)

The following proposition demonstrates that Problems (2.15) and (2.16) are in fact equivalent.

Proposition 2 Problems (2.15) and (2.16) are equivalent in the following sense: both problems have the same optimal value, and there is a one-to-one mapping between feasible and optimal solutions in both problems.

Proof To prove the assertion, it is sufficient to show that Problems (2.15) and (2.16) have the same feasible region. Notice, that since the constraints are linear inξξξ0, the semi-infinite constraints in Problems (2.15) can be equivalently written as

(HRξξξ− B Y RG)∈(cone(Ξ0)∗)m, (Y RG)∈(cone(Ξ0)∗)q,

(eeeeee>₁ − Y RG)∈(cone(Ξ0)∗)q, where cone(Ξ0)∗

is the dual cone of cone(Ξ0). Similarly, the semi-infinite constraints in Problems (2.16) can be equivalently written as

(HRξξξ− B Y RG)∈(cone(Ξ 0 )∗ )m_, (Y RG)∈(cone(Ξ0)∗)q, (eeeeee>₁ − Y R_G)∈(cone(Ξ0)∗)q. From [29, Corollary 6.21], we have that cone(Ξ0)∗

= cone(Ξ0)∗

, and therefore, we can conclude that the feasible regions of Problems (2.15) and (2.16) are equivalent. ut

Problem (2.16) is linear in both the decision variablesY and the uncertain vector ξξξ0. Despite this nice bilinear structure, in the following we demonstrate that Problems (2.12) and (2.16) are generically intractable for decision rules of type (2.6).

Theorem 1 Problems (2.12) and (2.16) defined through L(·) in (2.11) and G(·) in (2.6b), areNP-hard even when Y ∈_Zq×g is relaxed to Y ∈_Rq×g.

Proof See Appendix A. ut

Theorem 1 provides a rather disappointing result on the complexity of Problems (2.12) and (2.16). Therefore, unless P = NP, there is no algorithm that solves generic problems of type (2.12) and (2.16) in polynomial time. This difficulty stems from the generic structure of G(·) in (2.6b), combined with a generic polyhedral uncertainty set Ξ, resulting in highly non-convex sets Ξ0. Nevertheless, in the following we de-rive the exact polyhedral representation of conv(Ξ0), allowing us to use linear duality arguments from [4, 7], to reformulate Problem (2.16) into a mixed-integer linear op-timization problem. We remark that Theorem 1 also covers decision rule problems involving real-valued, piecewise constant decisions rules constructed using (2.6b), and demonstrates that these problems are also computationally intractable.

We construct the convex hull ofΞ0 by taking convex combinations of its extreme points, denoted by ext(Ξ0). These points are either the extreme points ofΞ0, or the limit points Ξ0\Ξ0. We construct these points using the definition ofG(·) and theP partitions Ξp given in (2.7). Then, for all extreme pointsξξξ ∈ ext(Ξp), we calculate the one-side limit point atL(ξξξ). The set of all one-side limit points will coincide with the set of extreme points ofΞ0, see Figure 2.

We defineV andV(p) such that

V := P [ p=1

V(p), V(p) := ext(Ξp), p= 1, . . . , P. (2.17)

Due to the discontinuity of G(·), the set of points {ξξξ0 ∈Rk0 : ξξξ0=L(ξξξ), ξξξ ∈ V }does not contain the pointsΞ0\Ξ0. To construct these points, we now define the one-sided limit points bLp(ξξξ) = (ξξξ>,Gbp(ξξξ)>)> for all ξξξ ∈ V(p) and each partition p= 1, . . . , P. Here, bGp(ξξξ) : Rk_→Rg _{is given by:}

b Gp(ξξξ) := lim u u u∈Ξp, uuu→ξξξ G(uuu), ∀ξξξ ∈ V(p), p= 1, . . . , P. (2.18)

From definition (2.6b), for each partition p,G(ξξξ) is constant for allξξξin the relative interior ofΞp, which we denote by relint(Ξp). Therefore, for each ˜ξξξ ∈ V(p), the one-side limit bGp(˜ξξξ) is equal toG(ξξξ) for allξξξ ∈relint(Ξp). Furthermore, sinceΞpspans Rk and thus has dimensionk −1, then there exists at leastkvertices inΞp, and therefore, at leastkone-sided limit points bGp(ξξξ). From the definition (2.18), we have that bLp(ξξξ) are the extreme points ofΞ0 for allξξξ ∈ V(p) andp= 1, . . . , P, see Figure 2.

The following proposition gives the polyhedral representation of the convex hull ofΞ0.

0 1

0 10

Figure 2: Convex hull representation of Ξ0 induced by lifting ξξξ0_{= (ξ}0

1, ξ02) = L(ξ) = (ξ, G(ξ))>,

where G(ξ) = 111(ξ ≥ 5) and ξ ∈ Ξ = [0, 10]. Here, V = {ext(Ξ1), ext(Ξ2)} = {0, 5, 10}. The convex

hull is constructed by taking convex combinations of the points L(0), L(5), L(10) (black dots), and point (5, bG1(5))>(white dot), where bG1(5) is the one-side limit point at ξ = 5 using partition Ξ1.

Proposition 3 Let L(·)be given by (2.11) and G(·) being defined in (2.6b). Then, the exact representation of the convex hull of Ξ0 is given by the following polyhedron:

conv(Ξ0) =nξξξ0= (ξξξ0>1 , ξξξ 0> 2 ) > ∈Rk0:∃ζp(vvv)∈R+, ∀vvv ∈ V(p), p= 1, . . . , P, such that P X p=1 X v vv∈V(p) ζp(vvv) = 1, ξξξ01= P X p=1 X v v v∈V(p) ζp(vvv)vvv, ξξξ02= P X p=1 X v v v∈V(p) ζp(vvv) bGp(vvv)o. (2.19) Proof The proof is split in two parts: First, we prove that L(Ξ)⊆conv (Ξ0) holds, and then we show that conv (cl(L(Ξ)))⊇conv (Ξ0) is true as well, concluding that conv (cl(L(Ξ))) = conv (Ξ0).

To prove assertion L(Ξ)⊆conv (Ξ0), pick any ξξξ ∈ Ξ. By constructionξξξbelongs to some partition of Ξ, say Ξp, for which L(ξξξ) is an element of Ξ0. There are two possibilities: (a)ξξξ lies on a facet ofΞp, or (b)ξξξ lies in the relative interior of Ξp. If ξξξ lies on a facet, and sinceΞp spans Rk, then Carath´eodory’s Theorem implies that there are k −1 extreme points on the facet,vvv1, . . . , vvvk−1 ∈ V(p), such that for some δ(vvv)∈Rk−1+ with

Pk−1

i=1δ(vvvi) = 1, we haveξξξ=

Pk−1

i=1 δ(vvvi)vvvi. SinceG(·) is constant

on each facet, then thek −1 one-side limit points bGp(vvvi),vvv1, . . . , vvvk−1∈ V(p) attain the same value asG(ξξξ). Therefore, we have

G k−1_X i=1 δ(vvvi)vvvi ! = k−1_X i=1 δ(vvvi) bGp(vvvi).

Hence, settingζp(vvvi) =δ(vvvi), i= 1, . . . , k −1, in the definition of the convex hull with the rest of theζ(vvv) equal to zero proves part (a) of the assertion. Ifξξξlies in the relative interior ofΞp, then again by Carath´eodory’s Theorem there arek extreme points of Ξp such that for someδ(vvv)∈Rk+ andvvv1, . . . , vvvk ∈ V(p) withPki=1δ(vvviii) = 1, we have ξξξ=Pk_i=1δ(vvvi)vvvi. By construction, there also existskone-sided limit in the collection

{Gbp(vvv) :vvv ∈ V(p)}that attain the same value asG(ξξξ) for allξξξ ∈relint(Ξp). Thus, we have that G k X i=1 δ(vvvi)vvvi ! = k X i=1 δ(vvvi) bGp(vvvi).

Setting ζp(vvvi) =δ(vvvi), i= 1, . . . , k, with the rest of theζ(vvv) equal to zero proves part (b) of the assertion.

To prove the second part of assertion, conv (cl(L(Ξ)))⊇conv (Ξ0), fixξξξ0∈ Ξ0. By construction, there exists ζp(v)∈_R+, p= 1, . . . , P,vvv ∈ V, such that

P X p=1 X v v v∈V(p) ζp(vvv) = 1, ξξξ01= P X p=1 X vvv∈V(p) ζp(vvv)vvv, ξξξ02= P X p=1 X vvv∈V(p) ζp(vvv) bGp(vvv).

This implies that

 ξξξ0₁ ξξξ02  = P X p=1 X v v v∈V(p) ζp(vvv)  vvv b Gp(vvv)  ,

that is, ξξξ000 is a convex combination of either corner points or limit points ofL(Ξ). Therefore, conv (Ξ0) is equal to conv (cl(L(Ξ))) almost surely. This concludes the

proof. ut

The convex hull (2.19) is a closed and bounded polyhedral set described by a finite set of linear inequalities. Therefore, one can rewrite (2.19) as the following polyhedron

conv(Ξ0) =nξξξ0∈Rk0:∃ζζζ0∈Rv0 such thatW0ξξξ0+U0ζζζ0≥ hhh0 o

, (2.20)

where W0 ∈ _Rl0×k0, U0 ∈ _Rl0×v0 and hhh0 ∈ _Rl0 are completely determined through Propositions 3.

The following proposition captures the essence of robust optimization and provides the tools for reformulating the infinite number of constraints in Problem (2.16), see [4, 7]. The proof is repeated here to keep the paper self-contained.

Proposition 4 For any Z ∈_Rm×k0 andconv(Ξ0)given by (2.20), the following state-ments are equivalent.

(i) Zξξξ0≥0 for all ξξξ0∈conv(Ξ0),

(ii) ∃ Λ ∈_Rm×l₊ 0 with ΛW0=Z, ΛU0= 0, Λhhh0≥0.

Proof We denote byZµ>theµth row of the matrixZ. Then, statement (i) is equivalent to

Zξξξ0≥0 for allξξξ0∈conv(Ξ0),

⇐⇒ 0≤min ξξξ0 n Zµ>ξ:∃ζζζ0∈Rv 0 , W0ξξξ0+U0ζζζ0≥ hhh0o, ∀ µ= 1, . . . , m ⇐⇒ 0≤ max Λµ∈Rl0 n h h h0>Λµ:W0>Λµ=Zµ, U0>Λµ= 0, Λµ≥0 o , ∀ µ= 1, . . . , m ⇐⇒ ∃ Λµ∈Rl 0 withW0>Λµ=Zµ, U0>Λµ= 0, hhh0>Λµ≥0, Λµ≥0, ∀ µ= 1, . . . , m              (2.21) The equivalence in the third line follows from linear duality. InterpretingΛ>µ as the µth row of a new matrixΛ ∈Rm×lshows that the last line in (2.21) is equivalent to

Using Proposition 4 together with the polyhedron (2.20), we can now reformulate Problem (2.16) into the following mixed-integer linear optimization problem.

minimize TrM0R_ξ>_ξξD>Y RG  subject to Y ∈Zq×g, Λ ∈Rm×l+ 0, Γ ∈R q×l0 + , Θ ∈R q×l0 + BY RG+ΛW0=HRξξξ, ΛU 0 = 0, Λhhh0≥0, Y R_G=Γ W0, Γ U0= 0, Γhhh0≥0, eeeeee>1 − Y RG=ΘW 0 , ΘU0= 0, Θhhh0≥0. (2.22)

Here, Λ ∈ _Rm×l₊ 0, Γ ∈ _R+q×l0 and Θ ∈ _Rq×l₊ 0 are the auxiliary variables associated with constraints B Y R_Gξξξ0 ≤ HR_ξξξξξξ0, 0 ≤ Y R_Gξξξ0 and Y R_Gξξξ0 ≤ eee, respectively. We emphasize that Problem (2.22) is the exact reformulation of Problem (2.16), since (2.19) is the exact representation of the convex hull of Ξ0. Therefore, the solution of Problem (2.22) achieves the best binary decision rule associated with G(·) in (2.6b). Notice that the size of the Problem (2.22) grows quadratically with respect to the numberqof binary decisions, andl0the number of constraints of conv(Ξ0). However, l0 can be very large as it depends on the cardinality of V, which is constructed using the extreme points of theP partitionsΞp. Therefore, the size of Problem (2.22) can grow exponentially with respect to the description of Ξ and the complexity of the binary decision rule,g, defined in (2.6).

In the following, we present instances of Ξ and G(·), for which the proposed solution method will result to mixed-integer optimization problems whose size grow only polynomially with respect to the input parameters.

2.3 Scalable Binary Decision Rule Structures

In this section, we derive a scalable mixed-integer linear optimization reformulation of Problem (2.8) by considering simplified instances of the uncertainty setΞand binary decision rules yyy(·). We will show that for the structure of Ξ and yyy(·) considered in this section, the size of the resulting mixed-integer linear optimization problem grows polynomially in the size of the original Problem (2.3) as well as the description of the binary decision rules. We consider two cases: (i) uncertainty sets that can be described as hyperrectangles and (ii) general polyhedral uncertainty sets.

We now consider case (i), and define the uncertainty set of Problem (2.3) to have the following hyperrectangular structure:

Ξ=ξξξ ∈_Rk : lll ≤ ξξξ ≤ uuu, ξ1= 1 , (2.23) for givenlll, uuu ∈Rk. We restrict the feasible region of the binary functionsyyy(·)∈ Bk,q to admit the piecewise constant structure (2.6a), but nowG: Rk_{→ {0,1}}g_{is written} in the form

G(·) = (G1(·), G2,1(·), . . . , G2,r(·), . . . , Gk,r(·))>,

withg= 1+(k−1)r. Here,G1: Rk→ {0,1}andGi,j: Rk→ {0,1}, forj= 1, . . . , r, i= 2, . . . , k, are piecewise constant functions given by

G1(ξξξ) := 1, Gi,j(ξξξ) := 111 (ξi≥ βi,j), j= 1, . . . , r, i= 2, . . . , k, (2.24) for fixedβi,j∈R, j= 1, . . . , r, i= 2, . . . , k. By construction, we assume thatli< βi,1< . . . , βi,r < ui, for alli= 2, . . . , k. Structure (2.24) gives rise to binary decision rules that are discontinuous along the axis of the uncertainty set, and haverdiscontinuities in each direction. Notice that (2.24) is a special case of (2.6b), where vectorααα are replaced byeeei. One can easily modify (2.24) such that the number of discontinuities

r is different for each directionξi. We will refer toβi,1, . . . , βi,r as the breakpoints in directionξi.

Problem (2.8) still retains the same semi-infinite structure using (2.23) and (2.24). In the following, we use the same arguments as in Section 2.2, to (a) redefine Problem (2.8) into a higher dimensional space and (b) construct the convex hull of the lifted uncertainty set and use it together with Proposition 4 to reformulate the infinite number of constraints of the lifted problem.

We define the non-linear lifting operatorL_{: R}k→_Rk0to beL(·) = (L1(·)>, . . . , L_k(·)>)> such that

L1: Rk→R2, L1(ξξξ) = (ξ1, G1(ξξξ))>,

Li: Rk →Rr+1, Li(ξξξ) = (ξi, Gi(ξξξ)>)>, i= 2, . . . , k,

(2.25) withk0= 2+(k−1)(r+1). Here, with slight abuse of notation,Gi(·) = (Gi,1(·), . . . , Gi,r)> for i = 2, . . . , k. Notice that L(·) is separable in each component Li(·), since Gi(·) only involves ξi. We also introduce matrices R_ξξξ = (R_ξξξ,1, . . . , R_ξξξ,k) ∈ _Rk×k0 with

R_ξξξ,1∈Rk×2, Rξξξ,i∈Rk×(r+1), i= 2, . . . , k, such that

RξξξL(ξξξ) =ξξξ, Rξξξ,iLi(ξξξ) =eeeiξi, i= 1, . . . , k, (2.26a) and RG = (RG,1, . . . , RG,k)∈Rg×k

0

with RG,1∈Rg×2, RG,i∈Rg×(r+1), i= 2, . . . , k,

such that

RGL(ξξξ) =G(ξξξ), RG,iLi(ξξξ) = (0, . . . , Gi(ξξξ)>, . . . ,0)>, i= 1, . . . , k. (2.26b) As in Section 2.2, using the definition ofL(·) andRξξξ, RGin (2.25) and (2.26), respec-tively, we can rewrite Problem (2.8) into Problem (2.12).

We now define the uncertain vectorξξξ0= (ξξξ0>₁ , . . . , ξξξ0>_k )>∈_Rk0 such thatξξξ0=L(ξξξ) and ξξξ0_i =Li(ξξξ) for i= 1, . . . , k.ξξξ0 _{has probability measure Pξξξ}0 defined using (2.13), and corresponding probability measure support is given by

Ξ0=L(Ξ) =nξξξ0∈_Rk0: lll ≤ R_ξξξξξξ0 ≤ uuu, L(R_ξξξξξξ0) =ξξξ0o. (2.27) Notice that since both Ξ and L(·) are separable in each component ξi, Ξ0 can be written in the following equivalent form:

Ξ0=L(Ξ) =(ξξξ0>1 , . . . , ξξξ 0> k ) > ∈Rk0 : ξξξi∈ Ξ 0 i, i= 1, . . . , k , where Ξ_i0=Li(Ξ). Therefore, we can express conv(cl(Ξ0)) as

conv(cl(Ξ0)) =(ξξξ0>1 , . . . , ξξξ0>k ) >

∈Rk0 : ξξξ0_i∈conv(cl(Ξ_i0)), i= 1, . . . , k ,

=(ξξξ0>₁ , . . . , ξξξ0>_k )>∈_Rk0 : ξξξ0_i∈conv(Ξ0i), i= 1, . . . , k . (2.28)

It is thus sufficient to derive a closed-form representation for the marginal convex hulls conv(Ξ0i).

We now construct the polyhedral representation of conv(Ξ0i). As before, conv(Ξ 0

i)

will be constructed by taking convex combinations of its extreme points. Set conv(Ξ01) has the trivial representation conv(Ξ01) = {ξξξ01 ∈ R2 : ξξξ

0

1 = eee}. We define the sets Vi={li, βi,1, . . . , βi,r, ui},i= 2, . . . , k, and introduce the following partitions for each dimension ofΞ.

Ξi,1:={ξi∈R : li≤ ξi≤ βi,1},

Ξi,p:={ξi∈R : βi,p−1≤ ξi≤ βi,p}, p= 2, . . . , r,

Ξi,r+1:={ξi∈R : βi,r≤ ξi≤ ui},

 

i= 2, . . . , k. (2.29) Therefore, Vi(1) ={li, βi,1}, Vi(p) ={βi,p−1, βi,p}, p= 2, . . . , r, Vi(r+ 1) ={βi,r, ui}. It is easy to see that points L(eeeiξi), ξi ∈ Vi, are extreme points of conv(Ξ

0 i), see

Figure 3. Notice that Gi(eeeiξi) is constant for allξi in the interior of Ξi,p, i.e.,ξi ∈ int(Ξi,p), p = 1, . . . , r+ 1, but can attain different values on the boundaries of the partitions. To this end, for each partition andξi∈ Vi(p), we define the one-sided limit points bGi,p(eeeiξi)∈Rr+1 such that

b

Gi,p(eeeiξi) = lim

u∈Ξi,p, u→ξi

Gi(eeeiu), ∀ξi∈ Vi(p), p= 1, . . . , r+ 1, (2.30) for all i = 2, . . . , k. Gi(eeeiξi) is constant for all ξi ∈ int(Ξi,p), and each partition p. Therefore, for each ˜ξi∈ Vi(p), the one-side limit bGi,p(eeeiξ˜i) is equal toGi(eeeiξi) for all ξi∈int(Ξi,p). 0 1
0 10 1

Figure 3: Convex hull representation of Ξ0iinduced by lifting ξξξ0i= Li(ξ) = (ξi, Gi(ξ)>)>, where

Gi(ξ) = (111(ξi ≥ 1), 111(ξi ≥ 2))>and ξi ∈ [0, 3]. Here, Vi = {0, 1, 2, 3}. The convex hull is

con-structed by taking convex combinations of the points L(0), L(1), L(2), L(3) (black dots), and points (1, bG1,1(1))>, (2, bG1,2(2))>(white dot), where bG1,1(1) and bG1,2(2) are the one-side limit point at

points ξi= 1 and ξi= 2, respectively.

The following proposition gives the polyhedral representation of each conv(Ξ0i), i= 2, . . . , k. A visual representation of conv(Ξ0i) is depicted in Figure 3.

Proposition 5 For each i= 2. . . , k, let Li(·)be given by(2.25) and Gi(·)being defined in (2.24). Then, the exact representation of the convex hull of each Ξ0i is given by the following polyhedron: conv(Ξ0i) = n ξξξ0i= (ξ 0> i,1, ξξξ 0> i,2) >

∈Rr+1:∃ζi,p(v)∈R+, ∀v ∈ Vi(p), p= 1, . . . , r+ 1, such that

r+1_X p=1 X vvv∈Vi(p) ζi,p(v) = 1, ξi,10 = r+1 X p=1 X v vv∈Vi(p) ζi,p(v)v, ξξξ0i,2= r+1 X p=1 X v∈Vi(p) ζi,p(v) bGp(eeeiv) o . (2.31) Proof Proposition 5 can be proven using similar arguments as in Proposition 3. ut Set conv(Ξ0i) has 2r+ 2 extreme points. Using the extreme point representation (2.31), conv(Ξ0i) can be represented through 4r+ 4 constraints. Therefore, conv(Ξ

0 ) =

×k_i=1conv(Ξ0i) can be represented using a total of 2 +k(4r+ 4) constraints, i.e., its size grows quadratically in the dimension ofΞ,k, and the complexity of the binary decision ruler. One can now use the polyhedral description (2.31) together with Proposition 4, to reformulate Problem (2.16) into a mixed-integer linear optimization problem which can be expressed in the form (2.22). The size of the mixed-integer linear optimization problem will be polynomial inq, m, kandr. We emphasize that since the convex hull representation (2.31) is exact, the solution of Problem (2.22) achieves the best binary decision rule associated with G(·)defined in (2.24). For G(·) and Ξ defined in (2.24) and (2.23), respectively, the following proposition allows to instead of optimizing over integers in Problem (2.22), to restrict the integer variables to Y ∈ {−1,0,1}q×g without introducing an additional approximation. This will significantly reduce the computational time needed to solve instances of Problem (2.22).

Proposition 6 Let G(·) being defined in (2.24) and Ξ being a box uncertainty set de-fined in (2.23). Then, constraints (2.6a) imply that we can restrict Y ∈ Zq×g to Y ∈ {−1,0,1}q×g without loss of generality.

Proof First notice that by definition ofG(·) in (2.24), the break pointsβi,jare unique in each direction i ∈ {2, . . . , k}, and therefore the components of G(ξξξ) are linearly independent for allξξξ ∈ Ξ, i.e,

vvv>G(ξξξ) = 0, ∀ξξξ ∈ Ξ =⇒ vvv= 0.

First we discuss problem instances involving one uncertain parameter, i.e., Ξ = {ξξξ ∈ R2 : ξ1 = 1, l ≤ ξ2 ≤ u}, and y(ξξξ) =yyy>G(ξξξ), yyy ∈Zr+1 is one of the decision rules satisfying (2.6a). We will now prove by induction that constraint (2.6a) implies yi ∈ {−1,0,1}for all i= 1, . . . , r+ 1. We will prove the basis step by contradiction. Assume thatyr+1>1, then eithery1= 0, . . . , yr= 0, which implies that

yyy>G(ξξξ)>1, ∀ξξξ ∈ {ξξξ ∈R2 : ξ1= 1, ξ2∈[β2,r, u]}, yyy>G(ξξξ) = 0, ∀ξξξ ∈ {ξξξ ∈_R2 : ξ1= 1, ξ2∈[l, β2,r)},

or Pr_j=1yj ≤1− yr+1 <0 which implies that there exists a subset Ξr ⊆ {ξξξ ∈R2 : ξ1= 1, ξ2∈[l, β2,r)}such that

yyy>G(ξξξ)≤1, ∀ξξξ ∈ {ξξξ ∈R2 : ξ1= 1, ξ2∈[β2,r, u]},

yyy>G(ξξξ)<0, ∀ξξξ ∈ Ξr. (2.32)

In both cases, the constraint 0≤ yyy>G(ξξξ)≤1 for allξξξ ∈ Ξ, is violated, and therefore it must be the case that yr+1 ≤1. Demonstrating the case that yr+1 < −1 is also infeasible can be shown in a similar way, thus concluding thatyr+1∈ {−1,0,1}.

We will also prove the inductive step by contradiction. For somei ∈ {1, . . . , r}, assume that yi+1 ∈ {−1,0,1}, . . . , yr+1 ∈ {−1,0,1}, and assume that yi >1. Notice thatyyy>G(ξξξ) can influence its value onξξξ ∈[β2,i, β2,i+1), by only taking combinations of basis functions G1(·), . . . , Gi(·). Then, either y1 = 0, . . . , yi−1 = 0, which implies that

yyy>G(ξξξ)>1, ∀ξξξ ∈ {ξξξ ∈R2 : ξ1= 1, ξ2∈[β2,i, β2,i+1)}, yyy>G(ξξξ) = 0, ∀ξξξ ∈ {ξξξ ∈R2 : ξ1= 1, ξ2∈[l, β2,i)},

orPi−1_j=1yj≤1− yi<0 which implies that there exists a subsetΞi⊆ {ξξξ ∈R2 : ξ1= 1, ξ2∈[l, β2,i)}such that

yyy>G(ξξξ)≤1, ∀ξξξ ∈ {ξξξ ∈R2 : ξ1= 1, ξ2∈[β2,i, β2,i+1)}, yyy>G(ξξξ)<0, ∀ξξξ ∈ Ξi.

(2.33) In both cases, the constraint 0≤ yyy>G(ξξξ)≤1,for allξξξ ∈ Ξ is violated, and therefore it must be the case that yi ≤1. Again, demonstrating the case that yi < −1 is also infeasible can be shown in a similar way, thus concluding thatyyy ∈ {−1,0,1}r+1.

If the uncertainty set involves more than one uncertain parameter, since the com-ponents of G(ξξξ) are linearly independent for allξξξ ∈ Ξ, it is easy to see that if for somep ∈ {2, . . . , k} and j ∈ {1, . . . , r},yp,j∈ {−1,1}, i.e., non-zero, then yi,j= 0 for all i ∈ {2. . . , k},i 6=pandj ∈ {1, . . . , r}. Therefore, we conclude that ifG(·) defined in (2.24) andΞ in (2.23), we can restrictY ∈Zq×g toY ∈ {−1,0,1}q×g without loss

of generality. ut

We now consider case (ii), and we assume thatΞis a generic set of the type (2.4) andG(·) is given by (2.24). From Section 2, we know that the number of constraints needed to describe the polyhedral representation of conv(Ξ0) grows exponentially with the description of Ξ and complexity of the decision rule. In the following, we present a systematic way to construct a tractable outer approximation for conv(Ξ0). Using this outer approximation to reformulate the lifted problem (2.12), will not yield the best binary decision rule structure associated with functionG(·) but rather a conservative approximation. Nevertheless, the size of the resulting mixed-integer linear optimization problem will only grow polynomially with respect to the constrains ofΞ and complexity of the binary decision rule.

To this end, let{ξξξ ∈_Rk :lll ≤ ξξξ ≤ uuu}be the smallest hyperrectangle containingΞ. We have

Ξ =nξξξ ∈Rk:∃ζζζ ∈Rv such thatWξξξ+Uζζζ ≥ hhh, ξ1= 1 o

, =nξξξ ∈_Rk:∃ζζζ ∈_Rv such thatWξξξ+Uζζζ ≥ hhh, ξ1= 1, lll ≤ ξξξ ≤ uuu

o

, (2.34) which implies that the lifted uncertainty set Ξ0 = L(Ξ) can be expressed as Ξ0 = Ξ10 ∩ Ξ20, where

Ξ₁0 :=nξξξ0∈_Rk0:∃ζζζ ∈_Rv such thatW R_ξξξξξξ0+Uζζζ ≥ hhh, ξ₁0 = 1o Ξ20 :=

n

ξξξ0∈Rk0:lll ≤ Rξξξξξξ0≤ uuu, L(Rξξξξξξ0) =ξξξ0 o

.

Notice thatΞ₂0 has exactly the same structure as (2.27). We thus conclude that b

Ξ0:={Ξ10 ∩conv(Ξ 0

2)} ⊇ conv(Ξ0), (2.35) and therefore, bΞ0 can be used as an outer approximation of conv(Ξ0). Since conv(Ξ02) can be written as the polyhedron induced by Proposition 5, set bΞ0 has a total of (l+1)+(2+k(4r+4)) constraints. As before, one can now use the polyhedral description of bΞ0 together with Proposition 4, to reformulate Problem (2.16) into a mixed-integer linear optimization problem which can be expressed in the form (2.22). The size of the mixed-integer linear optimization problem will be polynomial inq, m, k, landr. Since bΞ0 ⊇conv(Ξ0), the solution of Problem (2.22) might not achieve the best binary decision rule induced by (2.24), but rather a conservative approximation. Nevertheless, using this systematic decomposition of the uncertainty set, we can efficiently apply binary decision rules to problems where the uncertainty set has arbitrary convex polyhedral structure.

One can useG(·) given by (2.24) together with this systematic decomposition of Ξ to achieve the same binary decision rule structures as those offered byG(·) defined in (2.6b). We will illustrate this through the following example. We assume that the problem in hand requires a genericΞ of type (2.4), and we want to parameterize the binary decision rule to be a linear function ofG(ξξξ) = 111ααα>ξξξ ≥ βfor someααα ∈Rk andβ ∈R. We can introduce an additional random variable ˜ξinΞsuch that ˜ξ=ααα>ξξξ.

Here ˜ξis completely determined by the random variables already presented inΞ. The modified support can now be written as follows,

Ξ=n(ξξξ,ξ˜)∈Rk+1:∃ζζζ ∈Rv, Wξξξ+Uζζζ ≥ hhh, ξ1= 1, ξ˜=ααα>ξξξ o , =n(ξξξ,ξ˜)∈Rk+1:∃ζζζ ∈Rv, Wξξξ+Uζζζ ≥ hhh, ξ1= 1, ξ˜=ααα>ξξξ, lll ≤ ξξξ ≤ uuu,˜l ≤ξ ≤˜ ˜u o , (2.36) where {(ξξξ,ξ˜)∈Rk+1:lll ≤ ξξξ ≤ uuu,˜l ≤ξ ≤˜ u}˜ is the smallest hyperrectangle containing Ξ. We can now use

G(ξξξ,ξ˜) = 111 ˜ξ ≥ β
, (2.37) which is an instance of (2.24), to achieve the same structure asG(ξξξ) = 111ααα>ξξξ ≥ β. Defining appropriateL(·) andR_ξξξ, RG, we can use the following decomposition of the lifted uncertainty setΞ0=Ξ₁0 ∩ Ξ₂0 such that

Ξ10 := n (ξξξ0,ξ˜0)∈Rk0+1:∃ζζζ ∈Rv such thatW Rξξξξξξ 0 +Uζζζ ≥ hhh, ξ10 = 1, ξ˜ 0 =ααα>Rξξξξξξ 0o , Ξ₂0 :=n(ξξξ0,ξ˜0)∈_Rk0+1:lll ≤ R_ξξξξξξ 0 ≤ uuu,˜l ≤ξ˜0≤˜u, L(R_ξξξ(ξξξ 0 ,ξ˜0)) = (ξξξ0,ξ˜0)o.

By defining, bΞ0 := {Ξ₁0 ∩conv(Ξ₂0)} ⊇ conv(Ξ0) we can reformulate Problem (2.16) into a mixed-integer linear optimization problem with polynomial number of decision variable and constraints with respect to the the input data. Once more, we emphasize that this systematic decomposition might not achieve the best binary decision rule induced by G(·) but rather a conservative approximation. Nevertheless, we can now achieve the same flexibility for the binary decision rules as those offered by (2.6b) without the exponential growth in the size of the resulting problem.

3 Binary Decision Rules for Random-Recourse Problems

In this section, we present our approach for one-stage adaptive optimization problems with random recourse. The solution method of this class of problems is an adaptation of the solution method presented in Section 2.2. Given functionB: Rk_→

Rm×q and matricesD ∈Rq×k, H ∈Rm×k and a probability measure Pξξξ supported on setΞ for the uncertain vectorξξξ ∈Rk, we are interested in choosing binary functionsyyy(·)∈ Bk,q in order to solve: minimize Eξξξ  (Dξξξ)>yyy(ξξξ) subject to yyy(·)∈ B_k,q, B(ξξξ)yyy(ξξξ)≤ Hξξξ,  ∀ξξξ ∈ Ξ, (3.38)

where Ξ is a generic polyhedron of type (2.4). We assume that the recourse matrix B(ξξξ) depends linearly on the uncertain parameters. Therefore, theµth row ofB(ξξξ) is representable asξξξ>Bµfor matricesBµ∈Rk×q withµ= 1, . . . , m. Problem (3.38) can therefore be written as the following problem:

minimize Eξξξ  (Dξξξ)>yyy(ξξξ) subject toyyy(·)∈ Bk,q, ξξξ>Bµyyy(ξξξ)≤ Hµ>ξξξ, µ= 1, . . . , m ) ∀ξξξ ∈ Ξ, (3.39)

where Hµ> denotes theµth row of matrixH. Problem (3.39) involves a continuum of decision variables and inequality constraints. Therefore, in order to make the problem amenable to numerical solutions, we restrict yyy(·) to admit structure (2.6). Applying the binary decision rules (2.6) to Problem (3.39) yields the following semi-infinite

problem, which involves a finite number of decision variablesY ∈Zq×g, and an infinite number of constraints: minimize Eξξξ  ξξξ>D>Y G(ξξξ) subject to Y ∈Zq×g, ξξξ>BµY G(ξξξ)≤ Hµ>ξξξ, µ= 1, . . . , m, 0≤ Y G(ξξξ)≤ eee,      ∀ξξξ ∈ Ξ. (3.40)

Notice that all decision variables appear linearly in Problem (2.8). Nevertheless, the objective and constraints are non-linear functions ofξξξ.

We now define the non-linear lifting operator L: Rk_→

Rk0 such that L: Rk→Rk0, L(ξξξ) =        ξξξ G(ξξξ) G(ξξξ)ξ1 .. . G(ξξξ)ξk        , (3.41)

k0=k+g+gk. Note that including both componentsG(ξξξ) andG(ξξξ)ξ1inL(ξξξ) is redun-dant as by construction ξ1= 1. Nevertheless, in the following we adopt formulation (3.41) for ease of exposition. We also define matrices R_ξξξ ∈Rk×k

0 , and RG ∈Rq×k 0 such that R_ξξξL(ξξξ) =ξξξ, R_GL(ξξξ) =G(ξξξ). (3.42) In addition, we define F : Rk×g _{→ R}k0

that expresses the quadratic polynomials ξξξ>BµY G(ξξξ), as linear functions ofL(ξξξ) through the following constraints:

y y y0µ=F(BµY), yyy0>µ L(ξξξ) =ξξξ > BµY G(ξξξ), yyy0µ∈Rk 0 , µ= 1, . . . , m, ∀ξξξ ∈ Ξ. (3.43) Constraints yyy0µ=F(BµY), µ= 1, . . . , m, are linear, sinceF(·) effectively reorders the entries of matrix BµY into the vector yyy0µ. Using (3.41), (3.42) and (3.43), Problem (3.40) can now be rewritten into the following optimization problem.

minimize TrM R_ξ>ξξD > Y RG  subject to Y ∈_Zq×g, y yy0>µ L(ξξξ)≤ Hµ>RξξξL(ξξξ), µ= 1, . . . , m, y yy0 µ=F(BµY),yyy0µ∈Rk 0 , µ= 1, . . . , m, 0≤ Y RGL(ξξξ)≤ eee,          ∀ξξξ ∈ Ξ. (3.44)

Problem (3.44) has similar structure as Problem (2.12), i.e., bothY andL(ξξξ) appear linearly in the constraints. Therefore, in the following we redefine Problem (3.44) in a higher dimensional space and apply Proposition 4 to reformulate the problem into a mixed-integer optimization problem.

We define the uncertain vector ξξξ0 = (ξξξ0>_1,1, ξξξ0>_1,2, ξξξ0>_2,1, . . . , ξξξ0>_2,k)> ∈ _Rk0 such that ξξξ0=L(ξξξ) andξξξ_1,10 =ξξξ,ξξξ0_1,2=G(ξξξ), andξξξ0_2,i=G(ξξξ)ξifori= 1, . . . , k.ξξξ0has probability measure Pξξξ0 defined as in (2.13) and supportΞ

0

=L(Ξ). Problem (3.44) can now be written as the equivalent semi-infinite problem defined on the lifted spaceξξξ0.

minimize TrM0R_ξ>ξξD > Y RG  subject to Y ∈_Zq×g, y y y0>µ ξξξ0≤ Hµ>Rξξξξξξ 0 , µ= 1, . . . , m, y y y0 µ=F(BµY),yyy0µ∈Rk 0 , µ= 1, . . . , m, 0≤ Y RGξξξ0≤ eee,          ∀ξξξ0∈ Ξ0, (3.45)

We will construct the convex hull ofΞ0 in the same way as in Section 2.2. Notice that conv(Ξ0) will have the same number of extreme points as the convex hull defined through the liftingL(ξξξ) = (ξξξ>, G(ξξξ)>)>, see Figure 4. By defining the partitionsΞpof

0
 0  0 0
 0  0 1

Figure 4: Visualization of L(ξξξ) = (ξ1, ξ2, G(ξξξ)) (left) and L(ξξξ)e = (ξ1, ξ2, G(ξξξ)ξ2)

(right) where G(ξξξ) = 111(ξ1 ≥ 2), for ξ1 ∈ [0, 4] and ξ2 ∈ [−2, 2]. Here, V =

{(0, −2), (0, 2), (2, −2), (2, 2), (4, −2), (4, 2)}. The convex hull of L(ξξξ) is constructed by taken convex combinations of L(ξξξ) for all ξξξ ∈ V and (2, −2, bG(2, −2)))>_{, (2, 2, bG((2, 2)))}>_{, and the}

convex hull of eL(ξξξ) is constructed by taken convex combinations of eL(ξξξ) for all ξξξ ∈ V and

(2, −2, bG(2, −2)(−2))>_{, (2, 2, bG(2, 2)2)}>_.

Ξ as in (2.7), and bGp(ξξξ) as in (2.18), it is easy to see that conv(Ξ0) can be described as a convex combination of the following points.

       ξξξ b Gp(ξξξ) b Gp(ξξξ)ξ1 .. . b Gp(ξξξ)ξk        , ξξξ ∈ V(p), p= 1, . . . , P.

The following proposition gives the polyhedral representation of the convex hull ofΞ0.

Proposition 7 Let L(·)being defined in (3.41) and G(·)being defined in (2.6b). Then, the exact representation of the convex hull of Ξ0 is given by the following polyhedron:

conv(Ξ0) =nξξξ0= (ξξξ0>1,1, ξξξ 0> 1,2, ξξξ 0> 2,1, . . . , ξξξ 0> 2,k) > ∈Rk0 :∃ζp(vvv)∈R+, ∀vvv ∈ V(p), p= 1, . . . , P, P X p=1 X vvv∈V(p) ζp(vvv) = 1, ξξξ01,1= P X p=1 X vvv∈V(p) ζp(vvv)vvv, ξξξ01,2= P X p=1 X vvv∈V(p) ζp(vvv) bGp(vvv), ξξξ0_2,i= P X p=1 X v v v∈V(p) ζp(vvv) bGp(vvv)vi, i= 1, . . . , k o . (3.46) Proof Proposition 7 can be proven using similar arguments as in Proposition 3. ut

The description of polyhedron (3.46) inherits the same exponential complexity as (2.19), since the number of constraints in (3.46) depends on the cardinality of V which is constructed using the extreme points of the partitionsΞp. Nevertheless, the description (3.46) provides the exact representation of the convex hull for conv (Ξ0).

Using Proposition 4 together with the polyhedron (3.46), we can now reformulate Problem (3.45) into the following mixed-integer linear optimization problem.

minimize TrM0R>_ξξξD>Y RG  subject to Y ∈Zq×g, Γ ∈Rq×l+ 0, Θ ∈Rq×l 0 + , y yy0µ∈Rk 0 , λλλµ∈Rm×l 0 + , y yy0µ+λλλµW0=Rξ>ξξHµ, λλλµU0= 0, λλλµhhh0≥0 y yy0µ=F(BµY), yyy0µ∈Rk 0 ,      µ= 1, . . . , m, Y RG=Γ W0, Γ U0= 0, Γhhh0≥0, eeeeee>₁ − Y RG=ΘW0, ΘU0= 0, Θhhh0≥0. (3.47)

Here, the auxiliary variables λλλµ ∈ Rm×l 0

+ are associated with constraints yyy 0>

µ ξξξ0 ≤

Hµ>Rξξξξξξ 0

, forµ= 1, . . . , m, andΓ ∈_Rq×l₊ 0 and Θ ∈_Rq×l₊ 0 with constraints 0≤ Y R_Gξξξ0 and Y RGξξξ

0

≤ eee, respectively. Problem (3.47) is the exact reformulation of Problem (3.45), since (3.46) is the exact representation of the convex hull ofΞ0, and thusthe solution of Problem (3.47) achieves the best binary decision rule associated with G(·)in (2.6b).Nevertheless, the size of Problem (3.47) is affected by the exponential growth of the constraints in conv (Ξ0). One can mitigate this exponential growth by consid-ering simplified structures of Ξ and G(·), following similar guidelines as those those discussed in Section 2.3.

4 Binary Decision Rules for Multistage Problems

In this section, we extend the methodology presented in Section 2 to cover multistage adaptive optimization problems with fixed-recourse. The mathematical formulations presented can easily be adapted to random-recourse problems by following the guide-lines of Section 3.

The dynamic decision process considered can be described as follows: A decision maker first observes an uncertain parameterξξξ1∈Rk1and then takes a binary decision yyy1(ξξξ1)∈ {0,1}q1_{. Subsequently, a second uncertain parameterξξξ}

2 ∈ Rk2 is revealed, in response to which the decision maker takes a second decisionyyy2(ξξξ1, ξξξ2)∈ {0,1}q2. This sequence of alternating observations and decisions extends overTtime stages. To enforce the non-anticipative structure of the decisions, a decision taken at stagetcan only depend on the observed parameters up to and including staget, i.e.,yyyt(ξξξt) where ξξξt = (ξξξ>₁, . . . , ξξξ>_t)> ∈ _Rkt, with kt = Pt_s=1ks. For consistency with the previous sections, and with slight abuse of notation, we assume that k1 = 1 and ξξξ1 = 1. As before, settingξξξ1 = 1 is a non-restrictive assumption which allows to represent affine functions of the non-degenerate outcomes (ξξξ>2, . . . , ξξξ>t)> in a compact manner as linear functions of (ξξξ>1, . . . , ξξξ>t)

>

. We denote byξξξ= (ξξξ>1, . . . , ξξξ>T) >

∈Rk the vector of all uncertain parameters, where k=kT. Finally, we denote by B_kt_,qt the space of binary functions from Rkt to{0,1}qt_.

Given matrices Bts ∈Rmt×qs, Dt ∈Rqt×k t

and Ht∈Rmt×k t

, and a probability measure Pξξξ supported on setΞgiven by (2.4), for the uncertain vectorξξξ ∈Rk, we are

interested in choosing binary functionsyyyt(·)∈ B_kt_,qt in order to solve: minimize Eξξξ T X t=1 (Dtξξξt)>yyyt(ξξξt) ! subject to yyyt(·)∈ Bkt_,qt, t X s=1 Btsyyys(ξξξs)≤ Htξξξt,      t= 1, . . . , T, ∀ξξξ ∈ Ξ. (4.48)

Problem (4.48) involves a continuum of decision variables and inequality constraints. Therefore, in order to make the problem amenable to numerical solutions, we re-strict the feasible region of the binary functionsyyyt(·)∈ Bkt_,qt to admit the following piecewise constant structure:

yyyt(ξξξ) =YtGt(ξξξ), Y ∈Zqt×g t , 0≤ YtGt(ξξξ)≤ eee, ) t= 1, . . . , T, ∀ξξξ ∈ Ξ, (4.49a)

where Gt _{: R}k → {0,1}gt can be expressed by Gt(·) = (G1(·)>, . . . , Gt(·)>), and Gt: Rk

t

→ {0,1}gt _{are the piecewise constant functions:} G1(ξξξ1) := 1, Gt,i(ξξξ) := 111  α α α>t,iξξξt≥ βt,i  , i= 1, . . . , gt, t= 2, . . . , T, (4.49b) for givenαααt,i ∈Rk

t

and βt,i∈R,i= 1, . . . , g, t= 2, . . . , T. The dimension ofGt(·) is gt, withgt=Pt_s=1gs. The non-anticipativity ofyyyt(·) is ensured by restrictingGt(·) to depend only on random parameters up to and including staget.

The pairs (αααt,i, βt,i) defining the G(·) need to satisfy the same requirements as those presented in Section (2.6), namely, (αααt,i, βt,i) are chosen such that they result to unique hyperplanes ααα>_t,iξξξt− βt,i = 0 for all i ∈ {2, . . . , g} and t ∈ {2, . . . , T }, and the partitionsΞpgiven by (2.7) that result from the choices of (αααt,i, βt,i), must have (i) non-empty relative interior, (ii)Ξp spans Rk, (iii) any partition pairΞi and Ξj, i 6=j, can overlap only on one of their facets, and (iv),Ξ=SP_p=1Ξp.

Applying decision rules (4.49) to Problem (4.48) yields the following semi-infinite problem. minimize Eξξξ T X t=1 (Dtξξξt) > YtGt(ξξξ) ! subject to Yt∈Zqt×g t , t X s=1 BtsYsGs(ξξξ)≤ Htξξξt, 0≤ YtGt(ξξξ)≤ eee          t= 1, . . . , T, ∀ξξξ ∈ Ξ. (4.50)

We can now use the lifting techniques to first express Problem (4.50) in a higher di-mensional space, compute the convex hull associated with the non-convex uncertainty set of the lifted problem, and finally reformulate the semi-infinite structure using Proposition 4. We define liftingLt: Rk_→

Rk0t, such thatLt(·) = (L1(·)>, . . . , Lt(·)>)>, where Lt: Rkt→Rk 0 t_{, Lt(ξξξ) = (ξξξ}> t , Gt(ξξξ) > )>, t= 1, . . . , T, (4.51a) Here, kt0 = kt+gt. By convention L(·) = (L1(·)>, . . . , LT(·)>)> and k0t = Pts=1k

0 s with k0 =k0T. In addition, we define matrices,Rξξξ,t∈Rk

t ×k0 andRG,t∈Rg t ×k0 such that R_ξξξ,tL(ξξξ) =ξξξt, RG,tL(ξξξ) =G(ξξξ)t, t= 1, . . . , T. (4.51b)

Using (4.51), Problem (4.50) can now be rewritten in the following form: minimize Eξξξ T X t=1 (DtRξξξ,tL(ξξξ)) > YtRG,tL(ξξξ) ! subject to Yt∈Zqt×g t , t X s=1 BtsYsRG,sL(ξξξ)≤ HtRξξξ,tL(ξξξ), 0≤ YtRG,tL(ξξξ)≤ eee          t= 1, . . . , T, ∀ξξξ ∈ Ξ, (4.52)

Problem (4.52) has similar structure as Problem (2.12), i.e., bothYtandL(ξξξ) appear linearly in the constraints. Therefore, in the following we redefine Problem (4.52) in a higher dimensional space and apply Proposition 4 to reformulate the problem into a mixed-integer optimization problem.

We now define the uncertain vectorξξξ0= (ξξξ0>₁ , . . . , ξξξ0>_T )>∈_Rk0 such thatξξξ0=L(ξξξ) andξξξ0t=Lt(ξξξ) for t= 1, . . . , T.ξξξ

0

has probability measure Pξξξ0 defined using (2.13), and probability measure support Ξ0 = L(Ξ). Using the definition of ξξξ0, the lifted semi-infinite problem is given as follows:

minimize Eξξξ0 T X t=1 (DtRξξξ,tξξξ 0 )>YtRG,tξξξ 0 ! subject to Yt∈Zqt×g t , t X s=1 BtsYsRG,sξξξ 0 ≤ HtRξξξ,tξξξ 0 , 0≤ YtRG,tξξξ0≤ eee          t= 1, . . . , T, ∀ξξξ0∈ Ξ0. (4.53)

The definition ofL(·) in (4.51a) share almost identical structure as L(·) in (2.11). Therefore, the convex hull of Ξ0 can be expressed as a slight variant of the poly-hedron given in (2.19), and will constitute the exact representation of the convex hull of conv(Ξ0). Using Proposition 4 together with the polyhedral representation of conv(Ξ0), we can now reformulate Problem (3.45) into the following mixed-integer linear optimization problem.

minimize T X t=1 TrM0R>_ξξξ,tDt>YtRG,t  subject to Yt∈Zqt×g t , Λ ∈Rmt×l0 + , Γ ∈R qt_×l0 + , Θ ∈R qt_×l0 + t X s=1 BtsY RG,s+ΛtW0=HtRξξ,tξ , ΛtU0= 0, Λthhh0≥0, YtRG,t=ΓtW0, ΓtU0= 0, Γthhh0≥0, eeeeee>₁ − YtRG,t=ΘtW0, ΘtU0= 0, Θthhh0≥0.                t= 1, . . . , T, (4.54) where M0 ∈Rk0×k0, M0= Eξξξ0(ξξξ 0

ξξξ0>). Once more, we emphasize that Problem (4.54)

is the exact reformulation of Problem (4.53). Therefore,the solution of Problem (4.54) achieves the best binary decision rule associated with G(·) in (2.6b) at the cost of the exponential growth of its size with respect to the description ofΞand the complexity of the binary decision rules (4.49). The exponential growth in the time horizon T is implicit through the description ofΞ and the structure of the binary decision rules. One can mitigate this exponential growth by considering simplified structures of Ξ andG(·), following similar guidelines as those discussed in Section 2.3.

5 Computational Results

In this section, we apply the proposed binary decision rules to a multistage inventory control problem and to a multistage knapsack problem. In both problems, the struc-ture of the binary decision rules is constructed usingG(·) defined in (2.24), where the breakpointsβi,1, . . . , βi,rare placed equidistantly within the marginal support of each random parameter ξi. To improve the scalability of the solution method, we further restrict the structure of the binary decisions rules to be

yyy(ξξξ) =Y G(ξξξ), Y ∈ {−1,0,1}q×g, 0≤ Y G(ξξξ)≤ eee,



∀ξξξ ∈ Ξ. (5.55)

As shown in Proposition 6, for problem instances where G(·) and Ξ are defined in (2.24) and (2.23), respectively, this restriction does not introduce an additional conser-vatism. All of our numerical results are carried out using the IBM ILOG CPLEX 12.5 optimization package on a Intel Core i5−3570 CPU at 3.40GHz machine with 8GB RAM [26].

5.1 Multistage Inventory Control

In this case study, we consider a single item inventory control problem where the ordering decisions are discrete. This problem can be formulated as an instance of the multistage adaptive optimization problem with fixed-recourse, and can be described as follows. At the beginning of each time periodt ∈ T :={2, . . . , T }, the decision maker observes the product demand ξt that needs to be satisfied robustly. This demand can be served in two ways: (i) by pre-ordering at stage 1 a maximum of N lots, each delivering a fixed quantityqz at the beginning of periodt, for a unit cost ofcz; (ii) or by placing an order for a maximum ofN lots, each delivering immediately a fixed quantity qy, for a unit cost ofcy, with cz < cy. For each lotn = 1, . . . , N, the pre-ordering binary decisions delivered at stage t are denoted by zn,t ∈ {0,1}, and the recourse binary ordering decisions are denoted by yn,t(·)∈ B_kt_,1. If the ordered quantity is greater than the demand, the excess units are stored in a warehouse, incurring a unit holding costch, and can be used to serve future demand. The level of available inventory at each period is given byIt(·)∈ R_kt_,1. In addition, the cumulative volume of pre-ordersPt_s=1PN_n=1zn,smust not exceed the ordering budgetBtot,t. The decision maker wishes to determine the orderszn,t andyn,t(·) that minimize the total ordering and holding costs associated with the worst-case demand realization over the planning horizon T. The problem can be formulated as the following multistage adaptive optimization problem.

minimize max ξ ξ ξ∈Ξ X t∈T N X n=1 czqzzn,t+cyqyyn,t(ξξξt) +chIt(ξξξttt) ! subject to It(·)∈ R_kt_,1, zn,t∈ {0,1}, yn,t(·)∈ Bkt_,1, ∀n= 1, . . . , N, It(ξξξttt) =It−1(ξξξt−1) + N X n=1 qzzn,t+qyyn,t(ξξξt)− ξt, It(ξξξttt)≥0, t X s=1 N X n=1 qzzn,s≤ Btot,t,                          ∀t ∈ T , ∀ξξξ ∈ Ξ. (5.56)