Some Interval Approximation Techniques for MINLP

MINLP problems are hard constrained optimization problems, with nonlinear constraints and mixed discrete continuous variables. They can be solved using a Branch-and-Bound scheme combining several methods, such as linear programming, interval analysis, and cutting methods. Our goal is to integrate constraint programming techniques in this framework. Firstly, global constraints can be introduced to reformulate MINLP problems thus leading to clean models and more precise computations. Secondly, interval-based approximation techniques for nonlinear constraints can be improved by taking into account the integrality of variables early. These methods have been implemented in an interval solver and we present experimental results from a set of MINLP instances. Introduction Mixed Integer Nonlinear Programming (MINLP) problems are constrained optimization problems with both discrete and continuous variables, where the objective function may be nonconvex, and the constraints may be nonlinear (Grossmann 2002; Tawarmalani & Sahinidis 2004). They can be found in many application domains such as engineering design, computational biology, and portfolio optimization. The integrality of variables and the nonconvexity of functions make MINLP problems difficult to solve. The general Branch-and-Bound scheme is a method of choice, provided bound constraints on the variables. The main principle is to divide the initial problem into a set of subproblems by enumerating discrete domains or bisecting continuous domains. Bounding techniques are used to compute lower and upper bounds on the objective function, thus allowing the rejection of subproblems proved to be non optimal. Two kinds of bounding techniques may be combined. Interval methods (Moore 1966) can derive improved bounds on domains from relaxed NLP problems, locally considering discrete variables as continuous. Upper bounds on the objective function can also be calculated, possibly using a local optimizer to find feasible Copyright c © 2009, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. points (Neumaier 2004). Linear methods can compute global optima over relaxed LP problems, requiring nonlinearities to be reformulated and linearized (Sherali & Adams 1999). This way, lower bounds on the objective function can be obtained. It turns out that these techniques complement each other. Reducing the domain bounds leads to tighter linear relaxations, and then to more precise lower bounds. Several solvers integrate these techniques, such as Couenne (Belotti et al. 2008), which is part of the COIN-OR infrastructure. We focus on improving state-of-the-art interval-based bounding techniques for MINLP problems. Our starting point is the interval constraint propagation framework (Cleary 1987; Lhomme 1993; Van Hentenryck, Michel, & Deville 1997). Constraint propagation is a general bounding algorithm where contracting operators are applied on domains until they are not refined enough. These operators are based on interval arithmetic, numerical methods for differentiable problems, and projection techniques to tackle nonlinear constraints. They require functions and constraints to be expressed as combinations of arithmetic operations and elementary functions. In this paper, we propose two kinds of improvements. Firstly, the integrality constraints can be taken into account during constraint propagation to reinforce domain contractions, as done by Apt and Zoeteweij (Apt & Zoeteweij 2007) for pure integer problems. Secondly, MINLP problems may be reformulated by means of so-called global constraints such as the famous alldifferent constraint stating that a set of integer variables must all have different values. These constraints give a better expressiveness at modeling time. Moreover, they are associated with specialized domain contraction algorithms. In fact, we investigate the well-known approach of mixing constraints and optimization techniques (Hooker 2007) for MINLP problems. The remaining of this paper is organized as follows. The following section introduces MINLP problems. Reformulations using global constraints are illustrated for two non trivial problems. Interval approximation techniques and proposed improvements are described in the third section. The next section reviews several ways of processing the alldifferent constraint. An experimental study and a conclusion follow. MINLP and Reformulations The notion of MINLP problem is introduced in this section and we show how to reformulate two real-world problems using the alldifferent global constraint. Definition Consider the MINLP problem