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.

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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.

In this paper we present a strategy to improve the relaxation for the global optimization of nonconvex MINLPs. The main idea consists in recognizing that each constraint or set of constraints has a meaning that comes from the physical interpretation of the problem. When these constraints are relaxed part of this meaning is lost. Adding redundant constraints that recover that physical meaning strengthens the relaxation. We propose a methodology to find such redundant constraints based on engineering knowledge and physical insight.

In a recent paper, Birgin, Floudas and Martínez introduced an augmented Lagrangian method for global optimization. In their approach, augmented Lagrangian subproblems are solved using the αBB method and convergence to global minimizers was obtained assuming feasibility of the original problem. In the present research, the algorithm mentioned above will be improved in several crucial aspects. On the one hand, feasibility of the problem will not be required. Possible infeasibility will be detected in finite time by the new algorithms and optimal infeasibility results will be proved. On the other hand, finite termination results that guarantee optimality and/or feasibility up to any required precision will be provided. An adaptive modification in which subproblem tolerances depend on current feasibility and complementarity will also be given. The adaptive algorithm allows the augmented Lagrangian subproblems to be solved without requiring unnecessary potentially high precisions in the intermediate steps of the method, which improves the overall efficiency. Experiments showing how the new algorithms and results are related to practical computations will be given. Key words: deterministic global optimization, augmented Lagrangians, nonlinear programming, algorithms, numerical experiments. 1