Discrete OptimizationMathematical programming approaches for generating p-efficient points
Introduction
This study is devoted to the concept of p-efficiency (Prékopa, 1990) and proposes a new mathematical programming approach to generate p-efficient points. Definition 1 Let p ∈ [0, 1]. A point is called a p-efficient point of the probability distribution function F, if F(v) ⩾ p and there is no y ⩽ v, y ≠ v such that F(y) ⩾ p.
Along with mixed-integer programming (MIP) approaches (Kücükyavuz, 2009, Luedtke et al., 2010, Ruszczyński, 2002), robust optimization and approximation (Calafiore and Campi, 2005, Nemirovski and Shapiro, 2006), the concept of p-efficiency has been successfully and extensively used to solve probabilistically constrained stochastic programming problems (Charnes et al., 1958, Prékopa, 1970, Prékopa, 1973), in which the random vector has a multivariate discrete probability distribution (Dentcheva et al., 2001, Dentcheva et al., 2002, Prékopa, 1990, Sen, 1992). The generic formulation of such problems reads:where x is the m-dimensional vector of m1 continuous () and m2 integer () decision variables, , and ξ is a n-dimensional random vector having a multivariate probability distribution. The set of deterministic constraints is represented by (2) with and , while (3) is a joint probabilistic (chance) constraint that imposes that the n inequalities hi(x) ⩾ ξi (i = 1, …, n) hold jointly with a large probability at least equal to p. The formulation (3) allows us to model dependencies between random variables and it does not assume any restrictions on the type of dependencies between components of the random vector ξ. Probabilistic problems with discretely distributed random variables are non-convex in general. They have been receiving significant attention over the last few years (Kress et al., 2007, Kücükyavuz, 2009, Lejeune, 2008, Lejeune, 2009, Lejeune and Ruszczyński, 2007, Luedtke et al., 2010, Saxena et al., 2010), and have been applied in a variety of fields (see Dentcheva, 2006, Prékopa, 1995, Prékopa, 2003 for a review and a comprehensive list of references).
Existing solution methods for problem (1), (2), (3), (4) based on the concept of p-efficiency involve an enumerative phase (Avital, 2005, Beraldi and Ruszczyński, 2002, Lejeune, 2008, Prékopa, 2003, Prékopa et al., 1998) for generating the pLEPs, which are then used to derive a deterministic equivalent reformulation. Preprocessing methods have been proposed to alleviate the enumeration of candidate points (Lejeune, 2008). Cone (Dentcheva et al., 2001), primal–dual (Dentcheva et al., 2004), and column generation (Lejeune and Ruszczyński, 2007) algorithms have been successfully employed and a convexification method (Dentcheva et al., 2001) has been proposed to obtain a tight relaxation of problem (1), (2), (3), (4).
This study contributes to the literature in the following ways: (i) the p-efficiency concept is applied to a random vector whose possible values are discretized with a finite set of scenarios. Other applications of the p-efficiency concepts are generally proposed for random variables following a discrete probability distribution which has finite support; (ii) an exact mathematical programming method is proposed to generate pLEPs; (iii) a mathematical programming-based heuristic is developed for generating “quasi pLEPs”. The term “quasi pLEP” refers to a point that is very close to being a pLEP, i.e., that enforces requirements that are marginally more demanding than those defined by a pLEP; (iv) a new preprocessing method is introduced to reduce the number of scenarios and so to reduce the complexity of the proposed mathematical programming formulations.
This paper is related to a recent study of Kress et al. (2007) who consider a specific variantof (1), (2), (3), (4), in which there is no deterministic constraint besides the non-negativity and integrality restrictions (7), and the coefficients associated with the decision variables x in the objective function (5) and in the probabilistic constraint (6) are all equal to 1. Kress et al. reformulate this problem as an NP-hard, minmax multidimensional knapsack problem (MKP), and propose an enumerative algorithm to solve it. It can be seen that the optimal solution x∗ of problem (5), (6), (7) defines a pLEP of the probability distribution function of the random vector ξ, when ξ has integer-valued components.
Some of the key features that differentiate the present study from Kress et al. (2007) are that (i) we propose a new formulation and a new mathematical programming based framework (instead of an enumerative one) for generating exact and quasi pLEPs. Both the formulation and the solution approach are applicable to the general probabilistically constrained optimization problem (1), (2), (3), (4); (ii) our approach can be used to generate one as well as a series of pLEPs. The rationale for deriving a new solution framework comes from the observation of Kress et al. that their enumerative algorithm is outperformed by the state-of-the-art MIP solver CPLEX for problems of moderate to large size. The key feature of the proposed solution framework is that it is based on an outer approximation (Duran and Grossmann, 1986, Fletcher and Leyffer, 1994) algorithm, which is enhanced by a new family of valid inequalities, a fixing strategy, and a new preprocessing method that groups possible realizations of the random vector in bundles defining identical requirements. The proposed methods constitute an efficient alternative to the sometimes computationally intensive enumerative methods for generating pLEPs, the cardinality of which is finite yet unknown. An extended computational study analyzes the contribution of three specific algorithmic techniques (bundle preprocessing, strengthening valid inequalities and fixing strategy) integrated within the outer approximation method, and investigates the efficiency and effectiveness of the proposed heuristic algorithm. The computational results show that the heuristic approach allows the generation of quasi pLEPs in very short CPU times with significantly small optimality gaps, even when the number of scenarios used to describe the random variables is large.
The paper is organized as follows. Section 2 defines the optimization model to generate a single p-efficient point. Section 3 introduces a preprocessing method that reduces the complexity of the described mathematical programming formulation. Section 4 is devoted to the outer approximation solution framework. Section 5 describes the iterative procedure to generate a series of p-efficient points. Section 6 presents the computational results, while concluding remarks are given in Section 7.
Section snippets
Mathematical programming problem for generating a single pLEP
We denote by S the finite set of scenarios characterizing the probability distribution of the random vector ξ = (ξ1, …, ξn)T. Let denote the realization of the random variable ξi under scenario s, i = 1, …, n, s ∈ S, i.e., is the n-dimensional deterministic vector representing the joint realizations of the components ξi, i = 1, …, n, under scenario s. The probabilities associated with scenarios are denoted by πs, s ∈ S, where and . Without loss of generality, we assume
Preprocessing
The difficulty of solving MKP increases with the dimension of the random vector and, in particular, with the number of scenarios used to represent the random vector. In this section, we present a new preprocessing method that can be used to reduce the number of scenarios to which attention must be paid. The idea is to construct bundles of scenarios, such that all the scenarios included in a bundle define similar requirements. A preprocessing technique based on the quantile of the marginal
An outer approximation solution framework
In this section, we develop (exact and heuristic) mathematical programming methods to solve SMKP. These methods are based on the iterative generation of outer approximation problems obtained by relaxing the integrality restrictions on a subset of the binary variables γs, s ∈ S1. The notation OAt denotes the outer approximation problem solved at iteration t.
We shall first describe the initialization phase which involves the solution of the continuous relaxation of SMKP, the generation of an
Iterative generation of a set of p-efficient points
The satisfaction of all (n) requirements imposed by any pLEP ϑ guarantees to attain the prescribed probability level p: (v ⩾ ϑ) implies that P(ξ ⩽ v) ⩾ p. However, the cost triggered by satisfying the requirements imposed by different pLEPs can fluctuate very much. An industrial supply chain management problem described in Lejeune and Ruszczyński (2007) illustrates the differences in the costs associated with the multiple p-efficient points. This observation motivates our iterative solution approach
Computational results
In Section 6.1, we present the problem instances used in our computational study. In Section 6.2, we first assess the individual contribution of the three specific algorithmic techniques (preprocessing, valid inequalities, fixing strategy) integrated within the outer approximation method. We then analyze the efficiency and effectiveness of the outer approximation method proposed for generating quasi pLEPs.
The optimization problems are modeled with the AMPL mathematical programming language (
Concluding remarks
Probabilistically constrained problems, in which the random variables are finitely distributed, are generally not convex and thus very challenging to solve. The methods based on the p-efficiency concept have been widely used to solve such problems and typically use an enumeration algorithm to find the p-efficient points. In this study, we propose an alternative approach to the existing enumeration algorithms. First, we formulate a MIP problem whose optimal solution defines a pLEP and whose
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The author is supported by Grant # W911NF-09-1-0497 from the Army Research Office.