Dynamically generated cutting planes for mixed-integer quadratically constrained quadratic programs and their incorporation into GloMIQO 2

Abstract

The Global Mixed-Integer Quadratic Optimizer, GloMIQO, addresses mixed-integer quadratically-constrained quadratic programs (MIQCQP) to ε-global optimality. This paper documents the branch-and-cut framework integrated into GloMIQO 2. Cutting planes are derived from reformulation-linearisation technique equations, convex multivariable terms, αBB convexifications, and low-and high-dimensional edge-concave aggregations. Cuts are based on both individual equations and collections of nonlinear terms in MIQCQP. Novel contributions of this paper include: development of a corollary to Crama's [35] necessary and sufficient condition for the existence of a cut dominating the termwise relaxation of a bilinear expression; algorithmic descriptions for deriving each class of cut; presentation of a branch-and-cut framework integrating the cuts. Computational results are presented along with comparison of the GloMIQO 2 performance to several state-of-the-art solvers.

A Framework for Globally Optimizing Mixed-Integer Signomial Programs

“…GloMIQO [1][2][3] and ANTIGONE [21] are the numerical optimization code bases most similar to the current work; this framework encompasses and extends GloMIQO and is publically available within ANTIGONE (beginning GAMS 24.1). GloMIQO and this framework are equivalent when there are no signomial terms in MISO (c m, s = 0 ∀ s, m).…”

“…The proposed framework expands and extends GloMIQO [1][2][3]; we therefore limit the discussion to additional algorithmic components.…”

“…Beyond identifying individual functional forms, the ReformulationLinearization Technique (RLT) is a strategy that effectively interlinks variables, nonlinear terms, and equations [30,[71][72][73][74][75][76]. The GloMIQO RLT implementation identifies all variable/inequality equation and equation/equation products that do not introduce new bilinear terms into the model formulation [1]; GloMIQO 2.0 will even add nonconvex bilinear terms to the problem formulation to increase the number of variable/equality equation products [3]. Depending on the product, GloMIQO may directly add the RLT equation to the model formulation, dynamically introduce the equation as a cutting plane, or use the equation in a bounds-updating strategy.…”

“…GloMIQO uses the special structure elucidated in the detection phase to generate tight convex relaxations, dynamically add separating cuts, partition the search space, bound the variables, and find feasible solutions [1,3].…”

“…Recent work developed a generic computational framework for addressing mixedinteger quadratically-constrained quadratic programs (MIQCQP) to ε-global optimality [1][2][3]. The Global Mixed-Integer Quadratic Optimizer, GloMIQO, a software implementation of the framework, (1) reformulates user input, (2) detects special mathematical structure, and (3) uses a branch-and-cut global optimization algorithm to deterministically solve the MIQCQP.…”