A novel global optimization method based on an Augmented Lagrangian framework is introduced for continuous constrained nonlinear optimization problems. At each outer iteration k the method requires the εk-global minimization of the Augmented Lagrangian with simple constraints, where εk → ε. Global convergence to an ε-global minimizer of the original problem is proved. The subproblems are solved using the αBB method. Numerical experiments are presented.

Summary. Mathematical programming is a language for describing optimization problems; it is based on parameters, decision variables, objective function(s) subject to various types of constraints. The present treatment is concerned with the case when objective(s) and constraints are algebraic mathematical expressions of the parameters and decision variables, and therefore excludes optimization of black-box functions. A reformulation of a mathematical program P is a mathematical program Q obtained from P via symbolic transformations applied to the sets of variables, objectives and constraints. We present a survey of existing reformulations interpreted along these lines, some example applications, and describe the implementation of a software framework for reformulation and optimization. 1

A reformulation of a mathematical program is a formulation which shares some properties with, but is in some sense better than, the original program. Reformulations are important with respect to the choice and efficiency of the solution algorithms; furthermore, it is desirable that reformulations can be carried out automatically. Reformulation techniques are very common in mathematical programming but interestingly they have never been studied under a common framework. This paper attempts to move some steps in this direction. We define a framework for storing and manipulating mathematical programming formulations, give several fundamental definitions categorizing reformulations in essentially four types (opt-reformulations, narrowings, relaxations and approximations). We establish some theoretical results and give reformulation examples for each type.

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In this paper we compare two different approaches to nonconvex global optimization. The first one is a deterministic spatial Branch-and-Bound algorithm (sBB), whereas the second approach is a quasi Monte Carlo (QMC) variant of a stochastic multi level single linkage (MLSL) algorithm. Both algorithms apply to problems in a very general form and are not dependent on problem structure. The test suite we chose is fairly extensive in scope, in that it includes constrained and unconstrained problems, continuous and mixed-integer problems. The conclusion of the tests is that in general the QMC variant of the MLSL algorithm is more efficient, although in some instances the Branch-and-Bound algorithm is capable of locating the global optimum of hard problems in just one iteration.

Keywords: Reformulation-Linearization Technique, lift-and-project, tight relaxations, valid inequalities, model reformulation, convex hull, convex envelopes, mixed-integer 0-1 program, polynomial programs, nonconvex programs, factorable programs, reduced relaxations. Discrete and continuous nonconvex programming problems arise in a host of practical applications in the context of production planning and control, location-allocation, distribution, economics and game theory, quantum chemistry, and process and engineering design situations. Several recent advances have been made in the development of branch-and-cut type algorithms for mixed-integer linear and nonlinear programming problems, as well as polyhedral outer-approximation methods for continuous nonconvex programming problems. At the heart of these approaches is a sequence of linear (or convex) programming relaxations that drive the solution process, and the success of such algorithms is strongly tied in with the strength or tightness of these relaxations. The Reformulation-Linearization-Technique (RLT) is a method that generates such tight linear programming relaxations for not only constructing exact solution algorithms, but also to design powerful heuristic procedures for large classes of discrete combinatorial and continuous nonconvex programming problems. Its development originated in [4, 5, 6], initially focusing on 0-1 and mixed 0-1 linear and

We present a MILP mathematical programming formulation for static scheduling of dependent tasks onto homogeneous multiprocessor system of an arbitrary architecture with communication delays. We reduce the number of constraints by applying a Reduction Constraint reformulation to the model. We solve several small-scale instances of the reformulated problem by using CPLEX 8.1. Upper bounds are computed with the Variable Neighborhood Search meta-heuristic applied directly to the graph-based formulation of the problem, whereas lower bounds are obtained by solving linear relaxations of the MILP formulation, further tightened by using load balancing and critical path method arguments. Résumé Nous présentons une formulation sous la forme d’un modèle de programmation linéaire en nombres entiers (PLNE) pour le problème d’ordonnancement avec des tâches dépendantes dans un système de multiprocesseurs homogènes associés à une architecture arbitraire en présence de délais de communications. Pour diminuer le nombre de contraintes nous utilisons une reformulation du modèle à l’aide d’une procédure de Réduction de Contraintes. Nous avons résolu plusieurs exemples de petite taille du modèle reformulé avec CPLEX 8.1. Les bornes supérieures ont été calculées par des techniques VNS appliquées directement à une formulation basée sur la théorie des graphes. Les bornes inférieures ont été obtenues en résolvant des relaxations linéaires de la formulation PLNE et améliorées par des méthodes du chemin critique et par des techniques de balancemment de charges.

This paper discusses how to build a solver for mixed integer quadratically constrained programs (MIQCPs) by extending a framework for constraint integer programming (CIP). The advantage of this approach is that we can utilize the full power of advanced MILP and CP technologies, in particular for the linear relaxation and the discrete components of the problem. We use an outer approximation generated by linearization of convex constraints and linear underestimation of nonconvex constraints to relax the problem. Further, we give an overview of the reformulation, separation, and propagation techniques that are used to handle the quadratic constraints efficiently. We implemented these methods in the branch-cut-and-price framework SCIP. Computational experiments indicating the potential of the approach and evaluating the impact of the algorithmic components are provided.

Interval-gradient cuts are (nonlinear) valid inequalities for nonconvex NLPs defined for constraints g(x) ≤ 0 with g being continuously differentiable in a box [x, ¯x]. In this paper we define intervalsubgradient cuts, a generalization to the case of nondifferentiable g, and show that no-good cuts (which have the form ‖x−ˆx ‖ ≥ ε for some norm and positive constant ε) are a special case of interval-subgradient cuts whenever the 1-norm is used. We then briefly discuss what happens if other norms are used.

Abstract In this paper we present a methodology for finding tight convex relaxations for a special set of quadratic constraints given by bilinear and linear terms that frequently arise in the optimization of process networks. The basic idea lies on exploiting the interaction between the vector spaces where the different set of variables are defined in order to generate cuts that will tighten the relaxation of traditional approaches. These cuts are not dominated by the McCormick convex envelopes and can be effectively used in conjunction with them. The performance of the method is tested in several case studies by implementing the resulting relaxation within a spatial branch and bound framework.