GloMIQO: Global mixed-integer quadratic optimizer

Abstract

This paper introduces the Global Mixed-Integer Quadratic Optimizer, GloMIQO, a numerical solver addressing mixed-integer quadratically-constrained quadratic programs (MIQCQP) to ε-global optimality. The algorithmic components are presented for: reformulating user input, detecting special structure including convexity and edge-concavity, generating tight convex relaxations, partitioning the search space, bounding the variables, and finding good feasible solutions. To demonstrate the capacity of GloMIQO, we extensively tested its performance on a test suite of 399 problems of diverse size and structure. The test cases are taken from process networks applications, computational geometry problems, GLOBALLib, MINLPLib, and the Bonmin test set. We compare the performance of GloMIQO with respect to four state-of-the-art global optimization solvers: BARON 10.1.2,

Global optimization advances in Mixed-Integer Nonlinear Programming, MINLP, and Constrained Derivative-Free Optimization, CDFO

“…The MIQCQP solver GloMIQO (Misener and Floudas, 2013) represents a major effort to elucidate special structure via reformulations and integrates reformulation techniques that can be implemented generically and applied universally. The GloMIQO reformulation uses the observation that disaggregating bilinear terms tightens the relaxation of MIQCQP and actively takes advantage of any redundant linear constraints added to the model; GloMIQO may add bilinear terms to the model formulation to create tight reformulation-linearization technique relaxations.…”

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).…”

“…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.…”