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

Abstract

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.

Optimization of the Design and Partial-Load Operation of Power Plants Using Mixed-Integer Nonlinear Programming

“…Such methods subdivide the feasible set into smaller subregions (branching) to allow for tighter convex underestimators on the corresponding subproblems. Comparing lower bounds given by evaluating the relaxation of a subregion with upper bounds calculated from feasible points of the original problem then allows coordinating the search for a global optimum [52]. The open source software package LaGO (Lagrangian Global Optimizer) [42][43][44][45] is an implementation of such a method and is used for the plant design optimization discussed in this paper (cf.…”

“…BARON [52,53] implements a branch and reduce algorithm that is related to LaGOs methodology. Here, exact convex underestimators are constructed for nonconvex functions using a factorable reformulation of the model.…”

Identification of optimal measurement sets for complete flux elucidation in metabolic flux analysis experiments

“…TestUniq and TestOpt were solved using GAMS/CONOPT 3 (Drud, 1994). GAMS/BARON 7.5 (Tawarmalani and Sahinidis, 2004) and the EMU representation were also used whenever possible to ensure global optimality during the verification process.…”

Convexification of Learning from Constraints

“…In this case, (4)- (7) is resistant to standard optimization techniques. See Tawarmalani and Sahinidis (2004); Lee and Leyffer (2011);Belotti et al (2013) for an overview of non-convex mixed-integer non-linear programming. (1) for Y = {0, 1}, Θ = R, the Hinge loss (Tab.…”