?BB: A global optimization method for general constrained nonconvex problems

Abstract

Abstract.A branch and bound global optimization method, BB, for general continuous optimization problems involving nonconvexities in the objective function and/or constraints is presented. The nonconvexities are categorized as being either of special structure or generic. A convex relaxation of the original nonconvex problem is obtained by (i) replacing all nonconvex terms of special structure (i.e. bilinear, fractional, signomial) with customized tight convex lower bounding functions and (ii) by utilizing the parameter as defined in [17] to underestimate nonconvex terms of generic structure. The proposed branch and bound type algorithm attains finite -convergence to the global minimum through the successive subdivision of the original region and the subsequent solution of a series of nonlinear convex minimization problems. The global optimization method, BB, is implemented in C and tested on a variety of example problems.

GloMIQO: Global mixed-integer quadratic optimizer

“…-αBB [6,7,15,52,89] The primary contribution of the αBB code base is the implementation of a method determining univariate quadratic perturbations that convexify twice-continuously differen-tiable functions. This quadratic perturbation technique is integrated into a branch-andbound algorithm addressing MINLP to ε-global optimality.…”

“…Variable bounding algorithms relevant to this work include: interval arithmetic bounds tightening [6,7,15,18,27,127,128], optimality-based bounds tightening [89,133], reduced cost bounds tightening [117,118], and quadratic equation constraint satisfaction [44,45,77].…”

“…As we discussed in our previous work [99], we follow Androulakis et al [15], Sherali and Tuncbilek [127], and Audet et al [18] in augmenting a FBBT scheme with several additional sets of constraint equations. We perform FBBT strategies on: -The RLT equations identified in Section 3.2.1 -The convex hull of each auxiliary variable w xx (Equations 12 -13) -All facets (including the non-dominant ones) associated with the low-dimensional edge-concave aggregations elucidated in preprocessing ( §3.2.2).…”

Augmented Lagrangians with possible infeasibility and finite termination for global nonlinear programming

“…In simple constrained cases, several well-known algorithms accomplish that purpose efficiently. This is the case of the αBB algorithm [1,2,3,14], that has been used in [21] as subproblems solver in the context of an augmented Lagrangian method.…”

“…Derivative-free versions of Algencan were introduced in [31] and [49]. There exist many global optimization techniques for nonlinear programming problems, e.g., [2,3,4,14,38,40,41,42,43,44,50,52,53,58,62,63,65,66,67,68,69,70,73,75]. The main appeal of the augmented Lagrangian approach in this context is that the structure of this method makes it possible to take advantage of global optimization algorithms for simpler problems, i.e.…”