A wide range of optimization problems arising from engineering applications can be formulated as Mixed Integer NonLinear Programmming problems (MINLPs). Duran and Grossmann (1986) suggest an outer approximation scheme for solving a class of MINLPs that are linear in the integer variables by a finite sequence of relaxed MILP master programs and NLP subproblems. Their idea is generalized by treating nonlinearities in the integer variables directly, which allows a much wider class of problem to be tackled, including the case of pure INLPs. A new and more simple proof of finite termination is given and a rigorous treatment of infeasible NLP subproblems is presented which includes all the common methods for resolving infeasibility in Phase I. The worst case performance of the outer approximation algorithm is investigated and an example is given for which it visits all integer assignments. This behaviour leads us to include curvature information into the relaxed MILP master problem, giving r...

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.

This paper has as a major objective to present a unified overview and derivation of mixedinteger nonlinear programming (MINLP) techniques, Branch and Bound, Outer-Approximation, Generalized Benders and Extended Cutting Plane methods, as applied to nonlinear discrete optimization problems that are expressed in algebraic form. The solution of MINLP problems with convex functions is presented first, followed by a brief discussion on extensions for the nonconvex case. The solution of logic based representations, known as generalized disjunctive programs, is also described. Theoretical properties are presented, and numerical comparisons on a small process network problem.

The solution of convex Mixed Integer Quadratic Programming (MIQP) problems with a general branch--and--bound framework is considered. It is shown how lower bounds can be computed efficiently during the branch--and--bound process. Improved lower bounds such as the ones derived in this paper can reduce the number of QP problems that have to be solved. The branch--and--bound approach is also shown to be superior to other approaches to solving MIQP problems. Numerical experience is presented which supports these conclusions. Key words : Integer Programming, Mixed Integer Quadratic Programming, Branch--and--Bound AMS subject classification: 90C10, 90C11, 90C20 1 Introduction One of the most successful methods for solving mixed--integer nonlinear problems is branch--and--bound. Land and Doig [16] first introduced a branch--and--bound algorithm for the travelling salesman problem. Dakin [3] introduced the now common branching dichotomy and was the first to realize that it is possible to so...

We study here a problem of scheduling n job types on m parallel machines, when setups are required and the demands for the products are correlated random variables. We model this problem as a chance constrained integer program. Methods of solution currently available -- in integer programming and stochastic programming -- are not sufficient to solve this model exactly. We develop and introduce here a new approach, based on a geometric interpretation of some recent results in Grobner basis theory, to provide a solution method applicable to a general class of chance constrained integer programming problems. Our algorithm is conceptually simple and easy to implement. Starting from a (possibly) infeasible solution, we move from one lattice point to another in a monotone manner regularly querying a membership oracle for feasibility until the optimal solution is found. We illustrate this methodology by solving a problem based on a real system. Key Words: Scheduling, Chance Constrained Progr...

Generalized Disjunctive Programming (GDP) has been introduced recently as an alternative model to MINLP for representing discrete/continuous optimization problems. The basic idea of GDP consists of representing discrete decisions in the continuous space with disjunctions, and constraints in the discrete space with logic propositions. In this paper, we describe a new convex nonlinear relaxation of the nonlinear GDP problem that relies on the use of the convex hull of each of the disjunctions involving nonlinear inequalities. The proposed nonlinear relaxation is used to reformulate the GDP problem as a tight MINLP problem, and for deriving a branch and bound method. Properties of these methods are given, and the relation of this method with the Logic Based Outer-Approximation method is established. Numerical results are presented for problems in jobshop scheduling, synthesis of process networks, optimal positioning of new products and batch process design.

This paper considers the solution of Mixed Integer Nonlinear Programming (MINLP) problems. Classical methods for the solution of MINLP problems decompose the problem by separating the nonlinear part from the integer part. This approach is largely due to the existence of packaged software for solving Nonlinear Programming (NLP) and Mixed Integer Linear Programming problems. In contrast, an integrated approach to solving MINLP problems is considered here. This new algorithm is based on branch-and-bound, but does not require the NLP problem at each node to be solved to optimality. Instead, branching is allowed after each iteration of the NLP solver. In this way, the nonlinear part of the MINLP problem is solved whilst searching the tree. The nonlinear solver that is considered in this paper is a Sequential Quadratic Programming solver. A numerical comparison of the new method with nonlinear branch-and-bound is presented and a factor of about 3 improvement over branch-and-bound is observed...

: Two new methodologies for the global optimization of MINLP models, the Special structure Mixed Integer Nonlinear ffBB, SMIN--ffBB, and the General structure Mixed Integer Nonlinear ffBB, GMIN--ffBB, are presented. Their theoretical foundations provide guarantees that the global optimum solution of MINLPs involving twice--differentiable nonconvex functions in the continuous variables can be identified. The conditions imposed on the functionality of the binary variables differ for each method : linear and mixed bilinear terms can be treated with the SMIN--ffBB; mixed nonlinear terms whose continuous relaxation is twice--differentiable are handled by the GMIN--ffBB. While both algorithms use the concept of a branch & bound tree, they rely on fundamentally different bounding and branching strategies. In the GMIN--ffBB algorithm, lower (upper) bounds at each node result from the solution of convex (nonconvex) MINLPs derived from the original problem. The construction of convex lower bound...

Abstract In this paper, we study the partitioning of constraints in temporal planning problems formu-lated as mixed-integer nonlinear programming (MINLP) problems. Constraint partitioning is

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