The Branch-And-Reduce Optimization Navigator (BARON) is a computational system for facilitating the solution of nonconvex optimization problems to global optimality. We provide a brief description of the algorithms used by the software, describe the types of problems that can be currently solved and summarize our recent computational experience. BARON is available by anonymous ftp from aristotle. me. uiuc. edu.

BARON: A general purpose global optimization software package

Many industrial problems can be naturally formulated using mixed integer non-linear programming (MINLP) models and can be solved by spatial Branch&Bound (sBB) techniques. We study the impact of two important parts of sBB methods: bounds tightening (BT) and branching strategies. We extend a branching technique originally developed for MILP, reliability branching, to the MINLP case. Motivated by the demand for open-source solvers for real-world MINLP problems, we have developed an sBB software package named couenne (Convex Over-and Under-ENvelopes for Non-linear Estimation) and used it for extensive tests on several combinations of BT and branching techniques on a set of publicly available and real-world MINLP instances. We also compare the performance of couenne with a state-of-the-art MINLP solver.

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Branching and bounds tighteningtechniques for non-convex MINLP

This work addresses the development of an efficient solution strategy for obtaining global optima of continuous, integer, and mixed-integer nonlinear programs. Towards this end, we develop novel relaxation schemes, range reduction tests, and branching strategies which we incorporate into the prototypical branch-and-bound algorithm. In the theoretical/algorithmic part of the paper, we begin by developing novel strategies for constructing linear relaxations of mixed-integer nonlinear programs and prove that these relaxations enjoy quadratic convergence properties. We then use Lagrangian/linear programming duality to develop a unifying theory of domain reduction strategies as a consequence of which we derive many range reduction strategies currently used in nonlinear programming and integer linear programming. This theory leads to new range reduction schemes, including a learning heuristic that improves initial branching decisions by relaying data across siblings in a branch-and-bound tree. Finally, we incorporate these relaxation and reduction strategies in a branch-and-bound algorithm that incorporates branching strategies that guarantee finiteness for certain classes of continuous global optimization problems. In the computational part of the paper, we describe our implementation discussing, wherever appropriate, the use of suitable data structures and associated algorithms. We present computational experience with benchmark separable concave quadratic programs, fractional 0–1 programs, and mixed-integer nonlinear programs from applications in synthesis of chemical processes, engineering design, just-in-time manufacturing, and molecular design.

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Global optimization of mixed-integer nonlinear programs: A theoretical and computational study

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/pdf/complete-search-in-continuous-global-optimization-and-5ecbfk4fdi.pdf

Complete search in continuous global optimization and constraint satisfaction

This paper presents an overview of the research progress in deterministic global optimization during the last decade (1998---2008). It covers the areas of twice continuously differentiable nonlinear optimization, mixed-integer nonlinear optimization, optimization with differential-algebraic models, semi-infinite programming, optimization with grey box/nonfactorable models, and bilevel nonlinear optimization.

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A review of recent advances in global optimization

Researchers first examined the problem of separable concave programming more than thirty years ago, making it one of the earliest branches of nonlinear programming to be explored. This paper proposes a new algorithm that finds the exact global minimum of this problem in a finite number of iterations. In addition to proving that our algorithm terminates finitely, the paper extends a guarantee of finiteness to all branch-and-bound algorithms for concave programming that (1) partition exhaustively using rectangular subdivisions and (2) branch on the incumbent solution when possible. The algorithm uses domain reduction techniques to accelerate convergence; it solves problems with as many as 100 nonlinear variables, 400 linear variables and 50 constraints in about five minutes on an IBM RS/6000 Power PC. An industrial application with 152 nonlinear variables, 593 linear variables, and 417 constraints is also solved in about ten minutes.

A Finite Algorithm for Global Minimization ofSeparable Concave Programs

The problem of selecting processes and planning expansions of a chemical complex to maximize net present value has been traditionally formulated as a multiperiod, mixed-integer linear program. In this paper, the problem is approached using an entirely continuous model. Compared to previous models, the proposed formulation allows for more general objective functions. In solving the continuous model, minimizing its nonconvex objective function poses the major obstacle. We overcome this obstacle by means of a branch-and-bound global optimization algorithm that exploits the concavity and separability of the objective function and the linearity of the constraint set. The algorithm terminates with the exact global optimum in a finite number of iterations. In addition, computational results demonstrate that the proposed algorithm is very efficient as, for a number of problems from the literature, it outperforms OSL, a popular integer programming package. We also develop a procedure for generating test problems of this kind.

J. Parker Shectman

Papers

A Finite Algorithm for Global Minimization ofSeparable Concave Programs

Planning of Chemical Process Networks via Global Concave Minimization