This survey covers the state of the art of techniques for solving general purpose constrained global optimization problems and continuous constraint satisfaction problems, with emphasis on complete techniques that provably nd all solutions (if there are nitely many). The core of the material is presented in sucient detail that the survey may serve as a text for teaching constrained global optimization.

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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Abstract. The paper describes a software package called LaGO for solving nonconvex mixed integer nonlinear programs (MINLPs). The main component of LaGO is a convex relaxation which is used for generating solution candidates and computing lower bounds of the optimal value. The relaxation is generated by reformulating the given MINLP as a block-separable problem, and replacing nonconvex functions by convex underestimators. Results on medium size MINLPs are presented.

A method is presented for deterministic global optimization in the estimation of parameters in models of dynamic systems. The method can be implemented as an ɛ-global algorithm, or, by use of the interval-Newton method, as an exact algorithm. In the latter case, the method provides a mathematically guaranteed and computationally validated global optimum in the goodness of fit function. A key feature of the method is the use of a new validated solver for parametric ODEs, which is used to produce guaranteed bounds on the solutions of dynamic systems with intervalvalued parameters, as well as on the first- and second-order sensitivities of the state variables with respect to the parameters. The computational efficiency of the method is demonstrated using several benchmark problems.

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