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.

Abstract. This tutorial presents a theory of valid inequalities for mixed integer linear sets. It introduces the necessary tools from polyhedral theory and gives a geometric understanding of several classical families of valid inequalities such as lift-and-project cuts, Gomory mixed integer cuts, mixed integer rounding cuts, split cuts and intersection cuts, and it reveals the relationships between these families. The tutorial also discusses computational aspects of generating the cuts and their strength. Key words: mixed integer linear program, lift-and-project, split cut, Gomory cut, mixed integer rounding, elementary closure, polyhedra, union of polyhedra 1.

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.

A new continuous time MILP model for the short-term scheduling of multipurpose batch plants is presented. The proposed model relies on the State Task Network (STN) and addresses the general problem of batch scheduling, accounting for resource (utility) constraints, variable batch sizes and processing times, various storage policies (UIS/FIS/NIS/ZW), batch mixing/splitting, and sequence- dependent changeover times. The key features of the proposed model are the following: (a) a continuous time representation is used, common for all units, (b) assignment constraints are expressed using binary variables that are defined only for tasks, not for units, (c) start times of tasks are eliminated; thus, time matching constraints are used only for the finish times of tasks, and (d) a new class of valid inequalities that improves the LP relaxation is added to the MILP formulation. Compared to

The standard approach to formulating stochastic programs is based on the assumption that the stochastic process is independent of the optimization decisions. We address a class of problems where the optimization decisions influence the time of information discovery for a subset of the uncertain parameters. We extend the standard modeling approach by presenting a disjunctive programming formulation that accommodates stochastic programs for this class of problems. A set of theoretical properties that lead to reduction in the size of the model is identified. A Lagrangean duality based branch and bound algorithm is also presented.

Given a finite number of closed convex sets whose algebraic representation is known, we study the problem of optimizing a convex function over the closure of the convex hull of the union of these sets. We derive an algebraic characterization of the feasible region in a higher-dimensional space and propose a solution procedure akin to the interior-point approach for convex programming. Research partly supported by NSF HPCC Grant DMS 95-27124. Author's address: 417 Uris Hall, Graduate School of Business, Columbia University, New York NY 10027. E-mail sebas@cumparsita.gsb.columbia.edu. Also affiliated with Computational Optimization Research Center (CORC), Columbia University. y Supported by Subprograma Ciencia e Tecnologia do 2 o Quadro comunit'ario de Apoio grant Praxis XXI/BD/2831/94. Author's address: 804 Uris Hall, Graduate School of Business, Columbia University, New York NY 10027. E-mail jsoares@corc.ieor.columbia.edu. 1 Introduction The literature in optimality condition...

In this work, we consider the investment and operational planning of gas field developments under uncertainty in gas reserves. The resolution of uncertainty in gas reserves, and hence the shape of the scenario tree associated with the problem depends on the investment decisions. A novel stochastic programming model that incorporates the decision-dependence of the scenario tree is presented. A decomposition based approximation algorithm for the solution of this model is also proposed. We show that the proposed approach yields solutions with significantly higher Expected Net Present Value (ENPV) than that of solutions obtained using a deterministic approach. For a small sized example, the proposed approximation algorithm is shown to yield the optimal solution with more than one order of magnitude reduction in solution time, as compared to the full space method. "Good" solutions to larger problems, that require up to 165,000 binary variables in full space, are obtained in a few hours using the proposed approach.

Contents 1 Introduction 2 Disjunctive programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Two basic ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Compact Representation of the Convex Hull 3 Projection and polarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Adjacency on the higher dimensional polyhedron . . . . . . . . . . . . . . . . . . 5 3 Sequential Convexication 7 Disjunctive rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Fractionality of intermediate points . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4 Another Derivation of the Basic Results 12 5 Generating Cuts 13 Deepest cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Cut lifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Cut strengthening . . . . . . . .

A multiperiod MINLP model for offshore oilfield infrastructure planning is presented where nonlinear reservoir behavior is incorporated directly into the formulation. Discrete decisions include the selection of production platforms, well platforms and wells to be installed/drilled, as well as the drilling schedule for the wells over the planning horizon. Continuous decisions include the capacities of the platforms, as well as the production profile for each well in each time period. For the solution of this model, an iterative aggregation/disaggregation algorithm is proposed in which logic-based methods, a bilevel decomposition technique, the use of convex envelopes and aggregation of time periods are integrated. Furthermore, a novel dynamic programming sub-problem is proposed to improve the aggregation scheme at each iteration in order to obtain an aggregate problem that resembles the disaggregate problem more closely. A number of examples are presented to illustrate the performance of the proposed method. Keywords Oilfield planning, MINLP, aggregation, decomposition

Learning from ambiguous training data is highly relevant in many applications. We present a new learning algorithm for classification problems where labels are associated with sets of pattern instead of individual patterns. This encompasses multiple instance learning as a special case. Our approach is based on a generalization of linear programming boosting and uses results from disjunctive programming to generate successively stronger linear relaxations of a discrete non-convex problem. 1