Discrete Optimization
Beam search heuristic to solve stochastic integer problems under probabilistic constraints

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Abstract

This paper proposes a Beam Search heuristic strategy to solve stochastic integer programming problems under probabilistic constraints. Beam Search is an adaptation of the classical Branch and Bound method in which at any level of the search tree only the most promising nodes are kept for further exploration, whereas the remaining are pruned out permanently. The proposed algorithm has been compared with the Branch and Bound method. The numerical results collected on the probabilistic set covering problem show that the Beam Search technique is very efficient and appears to be a promising tool to solve difficult stochastic integer problems under probabilistic constraints.

Introduction

This paper addresses stochastic programming problems under probabilistic constraints where both the decision variables and the random variables are restricted to be integer. In particular, the class of problems considered throughout can be mathematically formulated as follows:(SIPC)mincTxAxbP{Txξ}px0integer.Here T is a m × n integer matrix, A is a q × n matrix, c, x  Rn, b  Rq, ξ a m-dimensional random vector integer-valued and P denotes probability. Constraints (2) and (3) represent the deterministic and stochastic side of the problem, respectively. In particular, (3) are probabilistic constraints which ensure the satisfaction of the stochastic constraints Txξ with a prescribed probability level p  (0,1). In the following, we shall use Z and ∣·∣1 to denote the set of integers and the ℓ1 norm, respectively. Furthermore, the inequality “⩾” for vectors will be understood coordinate-wise.

SIPC problems arise in a broad spectrum of applications in which integer and combinatorial optimization problems are formulated in the presence of uncertainty and a “reliable solution” is required. Routing and location are two classical examples of applications where probabilistically constrained integer formulations have been applied to define robust strategic plans. The interested readers are referred to the survey papers of [9], [13] and to the references therein.

As one might expect, the difficulty in addressing SIPC problems is twofold. First, SIPC problems have a combinatorial nature. Second, (3) are joint probabilistic constraints and no assumption on the independence on the components ξi, i = 1, …, m, of the random vector ξ, is imposed. To the best of our knowledge, all the probabilistic models proposed in the papers referred above either include individual chance constraints (i.e. probabilistic constraints individually imposed on all the inequalities) or assume independence of the components of the random vector. In both cases, equivalent deterministic formulations of the probabilistic constraints can be easily derived. In particular, the chance constraintsP{Tixξi}pi,i=1,,mcan be replaced by the deterministic constraints:TixFi-(pi),i=1,,m,where Ti denotes the ith row of the matrix T and Fi- is the pi-quantile of the distribution function of ξi (see [17]).

In the case of joint probabilistic constraints with independent random components, Eq. (3) can be replaced byi=1mln(Fi(Tix))lnp,and specific reformulation can be then derived for the case of integer random variables as shown in [7].

The previous assumptions are rather restrictive and often not adequate to represent real-life applications. Nevertheless, the derivation of deterministic equivalent formulations for joint probabilistic constraints requires the generation of the set of p-efficient points of the probability distribution function. The cardinality of this set is typically high even for moderate size of the random vector, amplifying the complexity associated with the solution of a single deterministic integer optimization problem.

The inherent complexity of the SIPC problems poses the crucial problem to design efficient solution approaches. In this respect the literature is rather scarce. Dentcheva et al. proposed in [7] a cone generation method based on the convexification of a deterministic formulation of the problem. A Branch and Bound method embedding efficient strategies for the determination of lower and upper bound values was proposed in [3].

Both the approaches mentioned above belong to the class of exact solution methods. Because of the complexity of the SIPC problems, the applicability of these methods is typically limited to problems of small and medium sizes. As in deterministic setting, practical-sized instances may be solved rather effectively by resorting to heuristic solution approaches. This paper represents a contribution in this direction. In particular, we propose a heuristic solution approach based on a Beam Search strategy. This is an adaptation of the Branch and Bound method proposed in [3] in which at any level of the search tree only the promising nodes are kept for further exploration, whereas the remaining are pruned off permanently.

The rest of the paper is organized as follows. In Section 2 we briefly recall the Branch and Bound method on which the heuristic approach is based on. Section 3 is devoted to the description of the Beam Search heuristic strategy for the solution of SIPC problems. The efficiency of the approach is tested on the probabilistic set covering problem formulated in [2]. The presentation and discussion of the numerical results is reported in Section 4. The paper ends with some concluding remarks.

Section snippets

The Branch and Bound method

The Branch and Bound method for SIPC problems operates on a deterministic equivalent formulation of the original problem which is obtained by means of the so called p-efficient points of the probability distribution function. In particular, given a probability level p  (0,1), a point s  Zm is called efficient point of the probability distribution function F, if F(s)⩾p. A point s is said to be p-efficient if it is efficient and satisfies the additional condition that there is no other point ys, y 

The beam search strategy

Beam search is a heuristic strategy closely related to the Branch and Bound method. It was first used in the field of artificial intelligence where it was applied to solve the speech recognition problem [10]. Successful applications of the Beam Search techniques can be also found in the area of scheduling problems (see for example [19] and the references therein).

As the Branch and Bound scheme outlined in Section 2, Beam Search implements a breadth-first search. However, unlike the Branch and

Computational experiments

In this section we report on the computational experiments of the Beam Search algorithm. In order to test the heuristic method we have considered the probabilistic set covering problem proposed in [2]. The motivation in the choice of this problem is twofold. First, set covering is one of the most fundamental models of integer programming whose probabilistic counterpart can be used to model interesting applications in presence of uncertainty. Secondly, it is one of the few problems for which

Conclusion and future research directions

In this paper we have proposed a heuristic strategy to solve stochastic integer problems under probabilistic constraints. The heuristic method implements a beam search strategy which is based on the classical Branch and Bound scheme. At any level of the search tree, only the promising nodes are kept for further exploration, whereas the remaining are pruned out permanently. The computational experiments show that the solution time of the Beam Search is far less than the time required by the

Uncited references

[4], [5], [8], [11], [12], [14], [15], [16], [18], [20]

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